Microeconomics , 6th Edition Besanko solution manuals

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 1 Analyzing Economic Problems Solutions to Review Questions 1.

What is the difference between microeconomics and macroeconomics?

Microeconomics studies the economic behavior of individual economic decision makers, such as a consumer, a worker, a firm, or a manager. Macroeconomics studies how an entire national economy performs, examining such topics as the aggregate levels of income and employment, the levels of interest rates and prices, the rate of inflation, and the nature of business cycles.

2.

Why is economics often described as the science of constrained choice?

While our wants for goods and services are unlimited, the resources necessary to produce those goods and services, such as labor, managerial talent, capital, and raw materials, are “scarce” because their supply is limited. This scarcity implies that we are constrained in the choices we can make about which goods and services to produce. Thus, economics is often described as the science of constrained choice.

3.

How does the tool of constrained optimization help decision makers make choices?

What roles do the objective function and constraints play in a model of constrained optimization?

Constrained optimization allows the decision maker to select the best (optimal) alternative while accounting for any possible limitations or restrictions on the choices. The objective function represents the relationship to be maximized or minimized. For example, a firm’s profit might be the objective function and all choices will be evaluated in the profit function to determine which yields the highest profit. The constraints place limitations on the choice the decision maker can select and defines the set of alternatives from which the best will be chosen.

4.

Suppose the market for wheat is competitive, with an upward-sloping supply curve, a downward-sloping demand curve, and an equilibrium price of $4.00 per bushel. Why would a higher price (e.g., $5.00 per bushel) not be an equilibrium price? Why would a lower price (e.g., $2.50 per bushel) not be an equilibrium price?

If the price in the market was above the equilibrium price, consumers would be willing to purchase fewer units than suppliers would be willing to sell, creating an excess supply. As suppliers realize they are not selling the units they have made available, sellers will bid down the price to entice more consumers to purchase their goods or services. By definition, equilibrium is a state that will remain unchanged as long as exogenous factors remain unchanged. Since in this case suppliers will lower their price, this high price cannot be an equilibrium.

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Chapter 1-1

Besanko & Braeutigam – Microeconomics, 6th edition

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When the price is below the equilibrium price, consumers will demand more units than suppliers have made available. This excess demand will entice consumers to bid up the prices to purchase the limited units available. Since the price will change, it cannot be an equilibrium.

5.

What is the difference between an exogenous variable and an endogenous variable in an economic model? Would it ever be useful to construct a model that contained only exogenous variables (and no endogenous variables)?

Exogenous variables are taken as given in an economic model, i.e., they are determined by some process outside the model, while endogenous variables are determined within the economic model being studied.

An economic model that contained no endogenous variables would not be very interesting. With no endogenous variables, nothing would be determined by the model so it would not serve much purpose.

6.

Why do economists do comparative statics analysis? What role do endogenous variables and exogenous variables play in comparative statics analysis?

Comparative statics analyses are performed to determine how the levels of endogenous variables change as some exogenous variable is changed. This type of analysis is very important since in the real world the exogenous variables, such as weather, policy tools, etc. are always changing and it is useful to know how changes in these variables affect the levels of other, endogenous, variables. An example of comparative statics analysis would be asking the question: If extraordinarily low rainfall (an exogenous variable) causes a 30 percent reduction in corn supply, by how much will the market price for corn (an endogenous variable) increase?

7.

What is the difference between positive and normative analysis? Which of the following questions would entail positive analysis, and which normative analysis? a) What effect will Internet auction companies have on the profits of local automobile dealerships? b) Should the government impose special taxes on sales of merchandise made over the Internet?

Positive analysis attempts to explain how an economic system works or to predict how it will change over time by asking explanatory or predictive questions. Normative analysis focuses on what should be done by asking prescriptive questions. a)

b)

Because this question asks whether dealership profits will go up or down (and by how much) – but refrains from inquiring as to whether this would be a good thing – it is an example of positive analysis.

On the other hand, this question asks whether it is desirable to impose taxes on Internet sales, so it is normative analysis. Notably, this question does not ask what the effect of such taxes would be.

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Chapter 1-2

Besanko & Braeutigam – Microeconomics, 6th edition

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Solutions to Problems 1.1 Discuss the following statement: “Since supply and demand curves are always shifting, markets never actually reach an equilibrium. Therefore, the concept of equilibrium is useless.”

While the claim that markets never reach an equilibrium is probably debatable, even if markets do not ever reach equilibrium, the concept is still of central importance. The concept of equilibrium is important because it provides a simple way to predict how market prices and quantities will change as exogenous variables change. Thus, while we may never reach a particular equilibrium price, say because a supply or demand schedule shifts as the market moves toward equilibrium, we can predict with relative ease, for example, whether prices will be rising or falling when exogenous market factors change as we move toward equilibrium. As exogenous variables continue to change, we can continue to predict the direction of change for the endogenous variables, and this is not “useless.”

1.2 In an article entitled, “Corn Prices Surge on Export Demand, Crop Data,” the Wall Street Journal identified several exogenous shocks that pushed U.S. corn prices sharply higher. (See the article by Aaron Lucchetti, August 22, 1997, p. C17. on national income.)

Suppose the U.S. market for corn is competitive, with an upward-sloping supply curve and a downward-sloping demand curve. For each of the following scenarios, illustrate graphically how the exogenous event described will contribute to a higher price of corn in the U.S. market. a) The U.S. Department of Agriculture announces that exports of corn to Taiwan and Japan were “surprisingly bullish,” around 30 percent higher than had been expected. b) Some analysts project that the size of the U.S. corn crop will hit a six-year low because of dry weather. c) The strengthening of El Niño, the meteorological trend that brings warmer weather to the western coast of South America, reduces corn production outside the United States, thereby increasing foreign countries’ dependence on the U.S. corn crop.

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a) Surprisingly high export sales mean that the demand for corn was higher than expected, at D2 rather than D1.

b) Dry weather would reduce the supply of corn, to S2 rather than S1.

c) Assuming the U.S. does not import corn, reduced production outside the U.S. would not impact U.S. corn market supply. El Nino would, however, cause demand for U.S. corn to shift out, the figure being the same as in part (a) above.

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1.3 In early 2008, the price of oil on the world market increased, hitting a peak of about $140 per barrel in July, 2008. In the second half of 2008, the price of oil declined, ending the year at just over $40 per barrel. Suppose that the global market for oil can be described by an upward-sloping supply curve and a downward-sloping demand curve. For each of the following scenarios, illustrate graphically how the exogenous event contributed to a rise or a decline in the price of oil in 2008: a) A booming economy in China raised the global demand for oil to record levels in 2008. b) As a result of the financial crisis of 2008, the U.S. and other developed economies plunged into a severe recession in the latter half of 2008. c) Reduced sectarian violence in Iraq in 2008 enabled Iraq to increase its oil production capacity. a) Booming economy in China shifts the demand curve for oil rightward (from D0 to D1 below), contributing to an increase in the price of oil.

b) Recession in the U.S. and other developed economies shifts the demand curve for oil leftward (from D0 to D1 below), contributing to a decrease in the price of oil.

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c) Increase in oil production capacity in Iraq shifts the supply for oil rightward (from S0 to S1 below), contributing to a decrease in the price of oil.

1.4 A firm produces cellular telephone service using equipment and labor. When it uses E machine-hours of equipment and hires L person-hours of labor, it can provide up to Q units of telephone service. The relationship between Q, E, and L is as follows: 𝑸 = √𝑬𝑳.

The firm must always pay PE for each machine-hour of equipment it uses and PL for each person-hour of labor it hires. Suppose the production manager is told to produce Q = 200 units of telephone service and that she wants to choose E and L to minimize costs while achieving that production target. a) What is the objective function for this problem? b) What is the constraint? c) Which of the variables (Q, E, L, PE, and PL) are exogenous? Which are endogenous?

Explain. d) Write a statement of the constrained optimization problem. a) The production manager wants to minimize total costs TC = PE*E + PL*L. b) The constraint is to produce Q = 200 units, so the manager must choose E and L so that √𝐸𝐿 = 200. c) The endogenous variables are E and L, because those are the variables over which the production manager has control. By contrast, the exogenous variables are Q, PE, and PL because the production manager has no control over their values and must take them as given. d) Student answers will vary.

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Chapter 1-6

Besanko & Braeutigam – Microeconomics, 6th edition

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1.5 The supply of aluminum in the United States depends on the price of aluminum and the average price of electricity (a critical input in the production of aluminum). Assume that an increase in the price of electricity shifts the supply curve for aluminum to the left (i.e., a higher average price of electricity decreases the supply of aluminum). The demand for aluminum in the United States depends on the price of aluminum and income shifts the demand curve for aluminum to the right (i.e., higher income increases the demand for aluminum). In 2004, national income in the United States increased, while the price of electricity fell, as compared to 2003. How would the equilibrium price of aluminum in 2004 compare to the equilibrium price in 2003? How would the equilibrium quantity in 2004 compare to the equilibrium quantity in 2003?

In 2003, the initial equilibrium is at price P1 and quantity Q1. As national income increased, demand for aluminum shifted to the right. The fall in the price of electricity shifted the supply curve to the right, from S1 to S2. Both shifts have the effect of increasing the equilibrium quantity, from Q1 to Q2. However, it is unclear whether price will rise or fall – if the demand shift dominates, price would rise; if the supply shift dominates, price would fall.

1.6 Ethanol (i.e., ethyl alcohol) is a colorless, flammable liquid that, when blended with gasoline, creates a motor fuel that can serve as an alternative to gasoline. The quantity of ethanol motor fuel that is demanded depends on the price of ethanol and the price of gasoline. Because ethanol fuel is a substitute for gasoline, an increase in the price of gasoline shifts the demand curve for ethanol rightward. The quantity of ethanol supplied depends on the price of ethanol and the price of corn (since the primary input used to produce ethanol in the U.S. is corn). An increase in the price of corn shifts the supply curve of ethanol leftward. In the first half of 2008, the price of gasoline in the U.S. increased significantly as compared to 2007, and the price of corn increased as well. How would the equilibrium price of ethanol motor fuel in the first half of 2008 compare to the price in 2007?

The increase in the price of gasoline shifted the demand curve for ethanol rightward (from D0 to D1), while the increase in the price of corn shifted the supply curve for ethanol leftward (from S0 to S1 below). Both changes had the impact of increasing the price of ethanol, moving the equilibrium from E0 in 2007 to E1 in 2008. (The impact of these changes on quantity is, in principle, ambiguous; the equilibrium quantity could either go up or down depending on the magnitude of the shifts in the demand and supply curves. The picture below shows the case in which there is a positive change in the equilibrium quantity.)

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Chapter 1-7

Besanko & Braeutigam – Microeconomics, 6th edition

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1.7 The price of gasoline in the United States depends on the supply of gasoline and the demand for gasoline. Gasoline is supplied by oil companies that sell it on several markets.

Hence the supply of gasoline in the United States depends on the price of gasoline in the United States and its price on other markets. When the price of gasoline outside the United States increases, the U.S. supply decreases because firms prefer to sell the gasoline elsewhere. How would an increase in the price of gasoline abroad affect the equilibrium price of gasoline in the United States?

When the price of gasoline abroad goes up, the supply on the domestic market decreases. Firms are willing to supply less gasoline for the same price as before. At that price the domestic demand exceeds the supply and therefore the equilibrium price in the US has to increase. When this is followed by increase in the demand – consumers are willing to buy more gasoline then before – supply would again be smaller than the demand. Hence the equilibrium price of the gasoline would increase even more.

1.8 The demand for computer monitors is given by the equation Qd = 700 - P, while the supply is given by the equation Qs = 100 + P. In both equations P denotes the market price.

Fill in the following table. For what price is the market in equilibrium—supply equals to the demand?

P Qd Qs

200

250

300

350

400

P Qd Qs

200 500 300

250 450 350

300 400 400

350 350 450

400 300 500

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Besanko & Braeutigam – Microeconomics, 6th edition

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1.9 The demand for computer memory chips is given by the equation Qd = 500 – 2P, while the supply is given by the equation Qs = 50 + P. In both equations P denotes the market price. For what price is the market in equilibrium – supply equals demand? What is the equilibrium quantity?

P Qd Qs

50

100

150

200

250

As shown in the table below, the equilibrium price is 150, and the equilibrium quantity is 200.

P Qd Qs

50 400 100

100 300 150

150 200 200

200 100 250

250 0 300

1.10 The demand for sunglasses is given by equation Qd = 1000 - 4P, where P denotes the market price. The supply of sunglasses is given by equation Qs = 100 + 6P. Fill in the following table and find the equilibrium price.

P Qd Qs

80

90

100

110

120

P 80 90 100 110 120 Qd 680 640 600 560 520 Qs 580 640 700 760 820 1.11 This year’s summer is expected to be very sunny. Hence the demand for sunglasses increased and now is given by equation Qd = 1200 - 4P. How is the equilibrium price going to change compared with the scenario described in problem 1.7? Explain and then fill in the following table to verify your explanation.

P Qd Qs

80

90

100

110

120

When the demand increases, more people are willing to buy sunglasses at the equilibrium price.

Hence, the supply is insufficient to satisfy the demand and the equilibrium price has to go up.

The table below confirms this.

P 80 90 100 110 120 d Q 880 840 800 760 720 Qs 580 640 700 760 820

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1.12 Suppose the supply curve for wool is given by Qs = P, where Qs is the quantity offered for sale when the price is P. Also suppose the demand curve for wool is given by Qd = 10 − P + I , where Qd is the quantity of wool demanded when the price is P and the level of income is I. Assume I is an exogenous variable. a) Suppose the level of income is I = 20. Graph the supply and demand relationships, and indicate the equilibrium levels of price and quantity on your graph. b) Explain why the market for wool would not be in equilibrium if the price of wool were 18. c) Explain why the market for wool would not be in equilibrium if the price of wool were 14. a) Assuming I = 20 we have Q s = P and Q d = 30 − P . Graphing these yields:

b) At a price of 18, Q s  Q d implying an excess supply of wool. Because sellers will not be able to sell all of their wool at this price, they will need to reduce price to attract buyers. At the lower price, the suppliers will offer a lower quantity of output for sale, and consumers will want to purchase more.

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c) At a price of 14, Q d  Q s , implying an excess demand for wool. Buyers will begin to bid up the price of wool until the new equilibrium is reached. At the higher price, the suppliers will offer a higher quantity of output for sale, and consumers will want to purchase less.

1.13 Consider the market for wool described by the supply and demand equations in Problem 1.12. Suppose income rises from I1 = 20 to I2 = 24. a) Using comparative statics analysis, find the impact of the change in income on the equilibrium price of wool. b) Using comparative statics analysis, find the impact of the change in income on the equilibrium quantity of wool.

a) With I1 = 20 , we had Q s = P and Q d = 30 − P , which implied an equilibrium price of 15.

With I 2 = 24 , we have Q s = P and Q d = 34 − P . Finding the point where Q s = Q d yields:

Qs = Qd P = 34 − P 2 P = 34 P = 17 Thus, a change in income of I = 4 yields a change in price of P = 2 . b) Plugging the result from part a) into the equation for Q s reveals the new equilibrium quantity is Q = 17 . Thus, a change in income of I = 4 yields a change in quantity of Q = 2 .

1.14 As commissioner of a league of recreational tennis players, you are responsible for purchasing court time from the local tennis facility. The members of the league will tell you how many hours of court time they would like for you to purchase each month. Your job is to find the least expensive way of buying the required amount of court time. After researching the options, you have found that the tennis facility offers three plans from which you can choose: Plan A: Pay $15 per hour for court time at the facility, with no additional fees.

Plan B: Buy a Basic Membership to use the facility. Here you pay a membership fee of $100 per month, with an additional charge of $10 per hour of court time.

Plan C: Buy a Preferred Membership. This requires your league to pay a monthly fee of $250, with an additional charge of $5 for every hour of court time.

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Chapter 1-11

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a) Which plan would you select if you are instructed to purchase 25 hours of court time at the lowest possible cost? b) Which plan would you select if you are instructed to purchase 50 hours of court time at the lowest possible cost? c) In this exercise is the number of hours of court time you purchase endogenous or exogenous? Explain. d) In this exercise is the type of plan you purchase (A, B, or C) endogenous or exogenous?

Explain. e) Are your league’s total expenditures endogenous or exogenous? Explain. a) Which plan would you select if you are instructed to purchase 25 hours of court time at the lowest possible cost?

Under Plan A, total expenditure = ($15/hour) x (25 hours/month) = $375 per month Under Plan B, total expenditure = $100 + ($10/hour) x (25 hours/month) = $350 per month Under Plan C, total expenditure = $250 + ($5/hour) x (25 hours/month) = $375 per month Choose Plan B because monthly expenditures are lowest. b) Which plan would you select if you are instructed to purchase 50 hours of court time at the lowest possible cost?

Under Plan A, total expenditure = ($15) x (50 hours/month) = $750 per month Under Plan B, total expenditure = $100 + ($10/hour) x (50 hours/month) = $600 per month Under Plan C, total expenditure = $250 + ($5/hour) x (50 hours/month) = $500 per month Choose Plan C because monthly expenditures are lowest. c) Since you are told how many hours to purchase, the number of hours is exogenous. d) Since you get to choose the plan, the choice of plan is endogenous. e) Since the expenditures are determined by the plan, and you get to choose the plan, the expenditures are endogenous.

1.15 Reconsider the problem of purchasing time on the tennis court in Problem 1.14.

Suppose the members of your league give you a specified amount of money to spend on court time. They then want you to maximize the number of hours the league purchases with that budget. You can choose from the same three plans (A, B and C) available in Problem 1.14. As commissioner you are responsible for purchasing court time from the local tennis club. The members of the league will tell you how many hours of court time they would like for you to purchase each month. Your job is to find the least expensive plan to buy the required amount of court time. a) Which plan would you select if you are instructed to purchase the largest number of hours while spending $300 per month? b) Which plan would you select if you are instructed to purchase the largest number of hours while spending $900 per month?

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c) In this exercise is the type of plan you purchase (A, B, or C) endogenous or exogenous?

Explain. d) In this exercise is the number of hours of court time you purchase endogenous or exogenous? Explain. e) Are your league’s total expenditures endogenous or exogenous? Explain. a) Which plan would you select if you are instructed to purchase the largest number of hours while spending $300 per month?

Let T be the number of hours you can purchase.

Under Plan A, total expenditure = $300 / month = ($15 $/hour) x (T hours) => T = 20 hours / month Under Plan B, total expenditure = $300 / month = $100 + ($10 $/hour) x (T hours) => T = 20 hours / month Under Plan C, total expenditure = $300 / month = $250 + ($5 $/hour) x (T hours) => T = 10 hours / month Choose either Plan A or Plan B because you can purchase 20 hours with either, and you can purchase only 10 hours with Plan C. b) Which plan would you select if you are instructed to purchase the largest number of hours while spending $900 per month?

Under Plan A, total expenditure = $900 / month = ($15 $/hour) x (T hours) => T = 60 hours / month Under Plan B, total expenditure = $900 / month = $100 + ($10 $/hour) x (T hours) => T = 80 hours / month Under Plan C, total expenditure = $900 / month = $250 + ($5 $/hour) x (T hours) => T = 130 hours / month Choose either Plan C because you can purchase more hours than you can with either Plan A or Plan B. c) Since you get to choose the plan, the choice of plan is endogenous. d) Since the number of hours you can purchase depends on the plan, and the choice of plan is endogenous, the number of hours is endogenous. e) Since the league tells you how much to spend, total expenditures are exogenous.

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1.16 A major automobile manufacturer is considering how to allocate a $2 million advertising budget between two types of television programs: NFL football games and PGA tour professional golf tournaments. The following table shows the new sports utility vehicles (SUVs) that are sold when a given amount of money is spent on advertising during an NFL football game and a PGA tour golf event.

The manufacturer’s goal is to allocate its $2 million advertising budget to maximize the number of SUVs sold. Let F be the amount of money devoted to advertising on NFL football games, G the amount of money spent on advertising on PGA tour golf events, and C(F,G) the number of new vehicles sold. a) What is the objective function for this problem? b) What is the constraint? c) Write a statement of the constrained optimization problem. d) In light of the information in the table, how should the manufacturer allocate its advertising budget? a) The objective function is the number of new SUVs sold, which we can denote by Q(F, G). b) The constraint is that total spending must be less than or equal to $2million, or TS  $2 million. c) The constrained optimization problem is max 𝑄(𝐹, 𝐺) subject to TS(F,G)  $2 million (𝐹,𝐺)

d) The following table shows all possible combinations of spending on football games and golf events:

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(F, G)

New sales from F

New sales from G

Total new sales

(0, 2)

0

9

9

(0.5, 1.5)

10

8

18

(1, 1)

15

6

21

(1.5, 0.5)

19

8

27

(2, 0)

20

0

20

The table indicates that new SUV sales are maximized when (F, G) = (1.5, 0.5), that is, when the manufacturer spends $1.5 million on football and $0.5 million on golf.

1.17 An electricity producer has two power plants, each of which emits carbon dioxide (CO2), a greenhouse gas. Each plant is currently emitting 1,000,000 metric tons of CO2 per year. However, new emissions rules restrict the firm’s emissions to 1,000,000 metric tons of CO2 per year from both plants combined. The cost of operating a power plant goes up as it curtails its emissions. The table below shows the cost of operating each plant for different emissions levels:

The firm’s goal is to choose emissions levels at each plant that minimize its total cost of operating its plants, subject to meeting its emissions target of 1,000,000 metric tons of CO2 per year from both plants combined. Let X denote the quantity of emissions from plant 1 and Y denote the quantity of emissions from plant 2. Let TC(X,Y) denote the total operating cost of the firm when the quantity of emissions from plant 1 is X and the quantity of emissions from plant 2 is Y. a) What is the objective function for this problem? b) What is the constraint? c) Write a statement of the constrained optimization problem.

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d) In light of the information in the table, what emissions levels from each plant should the firm choose? a) The objective function is TC(X,Y). b) The constraint is X + Y = 1,000,000 c) The statement of the problem is: min(X,Y) TC(X,Y) Subject to: X + Y = 1,000,000 d) The solution to the problem is: X = 750,000 and Y = 250,000. We can see this by creating a table that shows various combinations of X and Y that add up to 1,000,000 and computing the level of total operating cost associated with each feasible combination of emissions.

Emissions in plant 1

Total operating cost, plant 1

Emissions in plant 2

Total operating cost, plant 2

Total emissions

0 250,000 500,000 750,000 1,000,000

$490 $360 $250 $160 $90

1,000,000 750,000 500,000 250,000 0

$10 $40 $90 $160 $250

1,000,000 1,000,000 1,000,000 1,000,000 1,000,000

Total operating cost, plants 1 and 2 combined $500 $400 $340 $320 $340

1.18 The demand curve for peaches is given by the equation Qd = 100 − 4P, where P is the price of peaches expressed in cents per pound and Qd is the quantity of peaches demanded (expressed in thousands of bushels per year). The supply curve for peaches is given by Qs = RP, where R is the amount of rainfall (inches per month during the growing season) and Qs is the quantity of peaches supplied (expressed in thousands of bushels per year). Let P* denote the market equilibrium price and Q* denote the market equilibrium quantity.

Complete the following table showing how the equilibrium quantity and price vary with the amount of rainfall. Verify that when R = 1, the equilibrium price is 20 cents per pound and the equilibrium quantity is 20,000 bushels per year.

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Besanko & Braeutigam – Microeconomics, 6th edition

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When R = 1, the equilibrium occurs where Qd = Qs, or 100 – 4P* = P*, or P* = 20. The equilibrium quantity can be found from either supply or demand; using the latter we have Q* = 100 – 4(20) = 20. When R = 2, Qd = Qs implies 100 – 4P* = 2P* or P* = 16.67 and Q* = 33.33. Similarly, we can fill out the rest of the table: R Q* P*

1 20 20

2 33.33 16.67

4 50 12.5

8 66.67 8.33

16 80 5

1.19 The world-wide demand curve for pistachios is given by Qd = 10 – P, where P is the price of pistachios in U.S. dollars, and Qd is the quantity in millions of kilograms per year. 𝟗𝑷 The world supply curve for pistachios is given by 𝑸𝒔 = 𝟏+.𝟎𝟓(𝑻−𝟕𝟎)𝟐 , where T is the average temperature (measured in degrees Fahrenheit) in pistachio-growing regions such as Iran.

The supply curve implies that as the temperature deviates from the ideal growing temperature of 70o, the quantity of pistachios supplied goes down. Let P* denote the equilibrium price and Q* denote the equilibrium quantity. Complete the following table showing how the equilibrium quantity and price vary with the average temperature. Verify that when T = 70, the equilibrium price is $1 per kilogram and the equilibrium quantity is 9 million kilograms per year.

To fill in the table, one could create a set of tables, each corresponding to a particular level of T, that shows Qd and Qs for various prices, where Qd and Qs are computed using the formulas in the problem. Those tables are summarized below and then shown in detail.

T Q*(millions of kilograms per year) P* ($ per kilogram)

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30 1 9

50 3 7

65 8 2

70 9 1

80 6 4

Chapter 1-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

For each table, the equilibrium point is highlighted.

P 1 2 3 4 5 6 7 8 9 10

T = 30 Qd 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

Qs 0.11 0.22 0.33 0.44 0.56 0.67 0.78 0.89 1.00 1.11

P 1 2 3 4 5 6 7 8 9 10

T = 50 Qd 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

Qs 0.43 0.86 1.29 1.71 2.14 2.57 3.00 3.43 3.86 4.29

P 1 2 3 4 5 6 7 8 9 10

T = 65 Qd 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

Qs 4.00 8.00 12.00 16.00 20.00 24.00 28.00 32.00 36.00 40.00

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Besanko & Braeutigam – Microeconomics, 6th edition

P 1 2 3 4 5 6 7 8 9 10

T = 70 Qd 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

Qs 9.00 18.00 27.00 36.00 45.00 54.00 63.00 72.00 81.00 90.00

P 1 2 3 4 5 6 7 8 9 10

T = 80 Qd 9.00 8.00 7.00 6.00 5.00 4.00 3.00 2.00 1.00 0.00

Qs 1.50 3.00 4.50 6.00 7.50 9.00 10.50 12.00 13.50 15.00

Solutions Manual

1.20 Consider the comparative statics of the farmer’s fencing problem in Learning-ByDoing Exercise 1.4, where L is the length of the pen, W is the width, and A = LW is the area. a) Suppose the number of feet of fence given to the farmer was initially F1 = 200. Complete the following table. Verify that the optimal design of the fence (the one yielding the largest area with a perimeter of 200 feet) would be a square.

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Solutions Manual

b) Now suppose the farmer is instead given 240 feet of fence (F2 = 240). Complete the following table. By how much would the length L of the optimally designed pen increase?

c) When the amount of fence is increased from 200 to 240 (ΔF = 40), what is the change in the optimal length (ΔL)? d) When the amount of fence is increased from 200 to 240 (ΔF = 40), what is the change in the optimal area (ΔA)? Is the area A endogenous or exogenous in this example? Explain. a) L W A

10 20 30 40 50 60 70 80 90 90 80 70 60 50 40 30 20 10 900 1600 2100 2400 2500 2400 2100 1600 900

b) L 20 30 40 50 60 70 80 90 100 W 100 90 80 70 60 50 40 30 20 A 2000 2700 3200 3500 3600 3500 3200 2700 2000 The length L of the optimally designed fence increases by 10 ( F / 4 ). c) As in b), the length L of the optimally designed fence increases by 10 ( F / 4 ). d) When F = 40 , A = 1100. The area in this problem is an endogenous variable. The farmer may choose values for L and W and choices for these variables imply a value for A. So, implicitly, the farmer is choosing the area of the pen.

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Solutions Manual

1.21 Which of the following statements suggest a positive analysis and which a normative analysis? a) If the United States lifts the prohibition on imports of Cuban cigars, the price of cigars will fall. b) A freeze in Florida will lead to an increase in the price of orange juice. c) To provide revenues for public schools, taxes on alcohol, tobacco, and gambling casinos should be raised instead of increasing income taxes. d) Telephone companies should be allowed to offer cable TV service as well as telephone service. e) If telephone companies are allowed to offer cable TV service, the price of both types of service will fall. f) Government subsidies to farmers are too high and should be phased out over the next decade. g) If the tax on cigarettes is increased by 50 cents per pack, the equilibrium price of cigarettes will rise by 30 cents per pack. a) Positive analysis – this statement indicates what the consequences of the U.S. action will be, ignoring any value judgment when making the claim. b) Positive analysis – again this statement simply indicates the consequences of a change in an exogenous variable on the market, ignoring any value judgments. c) Normative analysis – here the author implies that there are two possible solutions to providing additional revenues for public schools and suggests, based on a value judgment, which of the alternatives is better. d) Normative analysis – again the author makes a claim based upon his own value judgment, namely that telephone companies offering cable TV service would be a good thing. e) Positive analysis – The author is making a positive statement. The author is predicting the effect of a policy change on the price in a market. f) Normative analysis – here the author is making a prescriptive statement about what should be done. This is a value judgment about the policy to subsidize farmers. g) Positive analysis – the author is making a prediction about what will happen if the tax on cigarettes is increased. While the claim may not be accurate, the statement is predictive and made without the author imposing any value judgments on the prediction.

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 2 Supply and Demand Analysis Solutions to Review Questions 1.

Explain why a situation of excess demand will result in an increase in the market price. Why will a situation of excess supply result in a decrease in the market price?

Excess demand occurs when price falls below the equilibrium price. In this situation, consumers are demanding a higher quantity than is being made available by suppliers. This creates pressure for the price to increase – sellers can ask for higher prices and still find buyers, and buyers offer higher prices to secure the units they want. As the price increases, quantity demanded will fall as quantity supplied increases returning the market to equilibrium.

Excess supply occurs when price is above the equilibrium price. Suppliers have made available more units than consumers are willing to purchase at the high price. This creates pressure for the price to decrease – buyers can get away with paying less because sellers are happy to find a buyer at all, and sellers are willing to sell for less wanting to make sure they find a buyer. As the price decreases, the quantity demanded will go up while at the same time the quantity supplied will decrease, returning the market to equilibrium.

2.

Use supply and demand curves to illustrate the impact of the following events on the market for coffee: a) The price of tea goes up by 100 percent. b) A study is released that links consumption of caffeine to the incidence of cancer. c) A frost kills half of the Colombian coffee bean crop. d) The price of styrofoam coffee cups goes up by 300 percent. a) An increase in the price of a substitute, such as tea, will increase demand for coffee, raising the market equilibrium price and quantity. How much demand for coffee increases, depends on how sensitive coffee demand is to the price of tea (cross-price elasticity).

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Chapter 2 - 1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

P S P’ P

D’ D

Q

Q’

Q

b) This study will reduce demand for caffeine drinks as people drink less to reduce the risk of cancer, lowering the market equilibrium price and quantity.

P S P P’ D D’ Q’ Q

Q

c) The frost will reduce supply raising the equilibrium price while lowering the equilibrium quantity.

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Chapter 2 - 2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

d) Increasing the price of an input for a cup of coffee will reduce supply, increasing market price and reducing market quantity. This will result in the same figure as that for part c).

3.

Suppose we observe that the price of soybeans goes up, while the quantity of soybeans sold goes up as well. Use supply and demand curves to illustrate two possible explanations for this pattern of price and quantity changes.

Any factor increasing demand and leaving the remainder of the market unchanged will increase both market price and quantity sold. If demand were to increase at the same time as supply changed, both market price and quantity sold could increase if the change in demand is large relative to the change in supply (in either direction).

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Chapter 2 - 3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.

A 10 percent increase in the price of automobiles reduces the quantity of automobiles demanded by 8 percent. What is the price elasticity of demand for automobiles?

 Q,P = 5.

%Q − 8 = = −0.80 %P 10

A linear demand curve has the equation Q = 50 − 100P. What is the choke price?

The choke price is the price where Q = 0 . Using the given demand curve we have Q = 50 − 100P 0 = 50 − 100P 100P = 50 P = $0.50

6.

Explain why we might expect the price elasticity of demand for speedboats to be more negative than the price elasticity of demand for light bulbs.

Speedboats could probably be categorized as a luxury item whereas light bulbs are more likely categorized as a necessity. For the necessity, the change in quantity demanded will be relatively small for any percent change in price. The change in quantity demanded may be quite large, however, for a luxury item. Since the percent change in quantity demanded is likely higher for the luxury item for any given percent change in price, the elasticity of demand would be less (more negative).

7.

Many business travelers receive reimbursement from their companies when they travel by air, whereas vacation travelers typically pay for their trips out of their own pockets. How would this affect the comparison between the price elasticity of demand for air travel for business travelers versus vacation travelers?

Because business travelers receive reimbursement for expenses, they will probably be less sensitive to price changes than the vacation traveler who pays out of her own pocket. This implies the price elasticity for vacationers would be less (more negative/smaller number) than for business travelers.

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Chapter 2 - 4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

8.

Explain why the price elasticity of demand for an entire product category (such as yogurt) is likely to be less negative than the price elasticity of demand for a typical brand (such as Dannon) within that product category.

If the prices for a particular product, such as Dannon, within a product category changes (say it increases) then it is easy for a consumer to switch to another brand, implying a relatively high percent change in quantity demanded for the product. On the other hand, if prices for the entire product category change, substitutes are not as easily found and the percent change in quantity demanded for the category will be relatively lower. This implies the elasticity for the entire product category will be higher (less negative) than the elasticity for a single product.

9.

What does the sign of the cross-price elasticity of demand between two goods tell us about the nature of the relationship between those goods?

When the cross-price elasticity is positive we have % Q A 0 % PB

Either a) both QA and PB increased or b) they both decreased. Since they are moving in the same direction, the product must be substitutes. Take coffee and tea for example; if the price of tea increases, the quantity of coffee demanded will increase.

When the cross-price elasticity is negative, QA and PB are moving in the opposite direction, implying the products are complements. Take coffee and cream for example; if the price of cream increases, the quantity of coffee demanded will decrease.

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Chapter 2 - 5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.

Explain why a shift in the demand curve identifies the supply curve and not the demand curve.

P S

D’’ D’

These points trace out the market supply curve

D Q

As the demand curve shifts, the market will reach a new equilibrium. Each new equilibrium occurs at a new price and quantity. These price/quantity combinations trace out the market supply curve. Thus, in order to identify the market supply curve one needs to observe shifts in the demand curve.

Solutions to Problems 2.1. The demand for beer in Japan is given by the following equation: Qd = 700 − 2P − PN + 0.1I, where P is the price of beer, PN is the price of nuts, and I is average consumer income. a) What happens to the demand for beer when the price of nuts goes up? Are beer and nuts demand substitutes or demand complements? b) What happens to the demand for beer when average consumer income rises? c) Graph the demand curve for beer when PN = 100 and I = 10, 000. a) When the price of nuts goes up, the beer quantity demanded falls for all levels of price (demand shifts left). Beer and nuts are demand complements. b) When income rises, quantity demanded increases for all levels of price (demand shifts rightward).

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Chapter 2 - 6

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

c) Now: Qd = 700 − 2P − 100 + 0.1*10,000 = 1,600 – 2P  P = 800 – 0.5 Qd P

800

1600

2.2.

Q

Suppose the demand curve in a particular market is given by Q = 5 − 0.5P.

a) Plot this curve in a graph. b) At what price will demand be unitary elastic? a) The inverse demand function is P = 10 – 2Q

P 10 Demand: Slope = - 2

Q 5 b) We know that the value of the price elasticity of demand is given by εQ,P. For a linear demand Q Q P P = −b and  Q , P = = −b . function: Q = a – bP, then P P Q Q  P  = −1 which implies Here, –b = –1/2. For demand to be unitary elastic it must be that − 12  P 5 − 2  that P = 5.

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Chapter 2 - 7

Besanko & Braeutigam – Microeconomics, 6th edition

2.3.

4P.

Solutions Manual

The demand and supply curves for coffee are given by Qd = 600 − 2P and Qs = 300 +

a) Plot the supply and demand curves on a graph and show where the equilibrium occurs. b) Using algebra, determine the market equilibrium price and quantity of coffee. a) P 300 S 50

D 300

500

600

Q

600 − 2 P = 300 + 4 P

b)

300 = 6 P 50 = P

Plugging P = 50 back into either the supply or demand equation yields Q=500.

2.4. Suppose that demand for bagels in the local store is given by equation Qd = 300 100P. In this equation, P denotes the price of one bagel in dollars. a) Fill in the following table:

b) At what price is demand inelastic? c) At what price is demand elastic? d) At what price is demand inelastic? e) At what price is demand elastic?

P 0.10 0.45 0.50 0.55 2.50 d Q 290 255 250 245 50 εQ,P –0.035 –0.176 –0.2 –0.225 –5 We can find elasticities of demand using the following formula Copyright © 2020 John Wiley & Sons, Inc.

Chapter 2 - 8

Besanko & Braeutigam – Microeconomics, 6th edition

 Q,P =

Solutions Manual

Q d P P P = −100  = . d P Q 300 − 100  P P − 3

This demand curve is linear. The inverse demand function is P = 3 – 1/100 Qd P $3

300 Q d

Observe that for price $1.50 the elasticity of demand is equal to 1.5  Q ,P = = −1 .

1.5 − 3 For all prices below $1.50, the demand is inelastic, while for all prices above $1.50, the demand is elastic.

2.5. The demand curve for ice cream in a small town has been stable for the past few years. In most months, when the equilibrium price is $3 per serving for the most popular ice cream, customers buy 300 servings per month. For one month the price of materials used to make ice cream increased, shifting the supply curve to the left. The equilibrium price in that month increased to $4, and customers bought only 200 portions in the month.

With these data draw a graph of a linear demand curve for ice cream in the town. Find price elasticity of demand for prices equal to $3 and $4. At what price would the demand be unitary elastic?

Using the data from the problem we can graph the demand curve. The slope of the demand curve is equal to P / Q = −1/100

So the equation of the demand curve is P = A – 0.01Q. We can find the vertical intercept A substituting P = 3 and Q = 300.

3 = A – 0.01(300), so A = 6. The vertical intercept (choke price) is P = $6.

The equation of the demand curve is then P = 6 – 0.01Q.

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Chapter 2 - 9

Besanko & Braeutigam – Microeconomics, 6th edition

P $6

 = −2

Solutions Manual

 = −1

$4 $3

200

300

600

Q

Elasticity of demand can be computed using formula EQ,P =

Q P P Q

= −100

P Q

= −100

6 − 0.01Q Q

When the elasticity is -1, Q = 300 and P = $3.

Thus demand is unitary elastic at a price P = $3.

Based on the data from the problem the graph of the demand curve is

P 3− 4 1 = =− = −0.01 Q 300 − 200 100 Find the vertical intercept in the graph or by substituting into P (Q ) = m − 0.01Q one of the two points. The inverse demand function is P(Q) = 6 − 0.01Q .

1 P 1 6 − 0.01Q 600 − Q  =  = The elasticity is  i = . P Q Q −0.01 Q −Q 600 − Q = −1  600 − Q = Q  Q = 300, P = 3 .

The function is unit-elastic at −Q 600 − Q 600 − 200 = = −2 At P = 4 and Q = 200 , the elasticity is  i = −Q −200

The slope of the demand curve is equal to

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

2.6. Granny’s Restaurant sells apple pies. Granny knows that the demand curve for her pies does not shift over time, but she wants to learn more about that demand. She has tested the market for her pies by charging different prices. When she charges $4 per pie, she sells 30 pies per week. When she charges $5, she sells 24 pies per week. If she charges $4.50, she sells 27 apple pies per week. a) With this data draw a graph of the linear demand curve for Granny’s apple pies. b) Find the price elasticity of demand at each of the three prices.

The demand for apple pies is Qd = 54 – 6P. P

$9

 = −1.25

 = −1

$5 $4.50

 = −0.8

$4

24 27 30

Q

54

To find the equation of the demand curve, observe that when she drops the price by $0.50, she sells 3 more pies. So, movement along the demand occurs so that Q / P = −3 / 0.5 = −6

The demand curve then has the form Qd = A – 6P, where A is a constant. We can determine the value of A using any one of the three data points on the demand curve. For example, if we use the point P = 5 and Q = 24, we see that 24 = A – 6(5), so that A = 54. So the demand curve can be described by the equation Qd = 54 – 6P. To find elasticity of demand at any point on the demand curve, we use formula EQ,P =

Copyright © 2020 John Wiley & Sons, Inc.

Q P P Q

= −6

P Q

Chapter 2 - 11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

2.7. Every year there is a shortage of Super Bowl tickets at the official prices P0. Generally, a black market (known as scalping) develops in which tickets are sold for much more than the official price. Use supply and demand analysis to answer these questions: a) What does the existence of scalping imply about the relationship between the official price P0 and the equilibrium price? b) If stiff penalties were imposed for scalping, how would the average black market price be affected? a) Since the price is being bid up above the official price, quantity demanded must exceed quantity supplied at the official price. This is a situation of excess demand and the official price must be below the equilibrium price. b) Most likely there were be fewer tickets offered for sale in the black market. The effect on the market price of scalped tickets depends on who is at risk of paying the penalty. If sellers are at risk, the price would most likely go up because the supply curve of scalped tickets would shift to the left. If buyers are at risk, the demand for scalped tickets would fall, shifting the demand curve to the left, leading to a lower equilibrium price for scalped tickets.

2.8 You have decided to study the market for fresh picked cherries. You learn that over the last 10 years, cherry prices have risen, while the quantity of cherries purchased has also risen. This seems puzzling because you learned in microeconomics that an increase in price usually decreases the quantity demanded. What might explain this seemingly strange pattern of prices and consumption levels?

This could occur as a result of the demand curve shifting to the right, increasing both equilibrium price and quantity. This would not contradict what was learned regarding downward sloping demand curves.

2.9 Suppose that, over a period of six months, the price of corn increased. Yet, the quantity of corn sold by producers decreased. Does this contradict the law of supply? If not, why not?

This does not contradict the law of supply. For example, farmers may have experienced something that shifted the supply curve for corn leftward (such as a flooding or a drought). This would have the effect of increasing the equilibrium price of corn, while decreasing the quantity of corn sold by producers. This is shown in the figure below. Another possibility is that, alternatively, the supply curve for corn could have shifted leftward, and the demand curves for could have also shifted, but in such a way that the overall effect is to increase the equilibrium price and decrease the equilibrium quantity. These cases are also shown in the figure below.

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Chapter 2 - 12

Besanko & Braeutigam – Microeconomics, 6th edition

Price (dollars per bushel)

Solutions Manual

S2 S1

D1 Quantity (bushels per year) Supply curve shifts leftward, demand curve remains stationary

Price (dollars per bushel)

S2 S1

D2

D1 Quantity (bushels per year)

Supply curve shifts leftward, demand curve also shifts leftward

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Chapter 2 - 13

Besanko & Braeutigam – Microeconomics, 6th edition

Price (dollars per bushel)

Solutions Manual

S2 S1

D2

D1

Quantity (bushels per year)

Supply curve shifts leftward, demand curve shifts rightward

2.10 Explain why a good with a positive price elasticity of demand must violate the law of demand.

The law of demand states that, holding other factors fixed, there is an inverse relationship between price and quantity demanded, i.e. that an increase in price decreases quantity and vice versa. If a good has a positive price elasticity of demand, it must be that an increase in the price of that good leads to an increase in the quantity demanded. Therefore, such a good violates the law of demand.

2.11 Suppose that the quantity of corn supplied depends on the price of corn (P) and the amount of rainfall (R). The demand for corn depends on the price of corn and the level of disposable income (I). The equations describing the supply and demand relationships are Qs = 20R + 100P and Qd = 4000 − 100P + 10I. a) Sketch a graph of demand and supply curves that shows the effect of an increase in rainfall on the equilibrium price and quantity of corn. b) Sketch a graph of demand and supply curves that shows the effect of a decrease in disposable income on the equilibrium price and quantity of corn.

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Chapter 2 - 14

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) S

P

S’

P* P’

D Q

*

Q Q’

An increase in rainfall will increase supply, lowering the equilibrium price and increasing the equilibrium quantity. b)

P

S

P* P’ D D’ Q’

Q*

Q

A decrease in disposable income will reduce demand, shifting the demand schedule left, reducing both the equilibrium price and quantity.

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Chapter 2 - 15

Besanko & Braeutigam – Microeconomics, 6th edition

2.12

Solutions Manual

Recall that when demand is perfectly inelastic, εQ, P = 0.

a) Sketch a graph of a perfectly inelastic demand curve. b) Suppose the supply of 1961 Roger Maris baseball cards is perfectly inelastic. Suppose, too, that renewed interest in Maris’s career caused by Mark McGwire and Sammy Sosa’s quest to break his home run record in 1998 caused the demand for 1961 Maris cards to go up. What will happen to the equilibrium price? What will happen to the equilibrium quantity of Maris baseball cards bought and sold? a) A perfectly inelastic demand curve will be vertical.

P D

Q

b) The renewed interest will shift demand to the right, raising the equilibrium price. Since supply is perfectly inelastic (and therefore vertical) there will be no change in the quantity supplied; the quantity is fixed.

P S P’ P

D’ D Q

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Chapter 2 - 16

Besanko & Braeutigam – Microeconomics, 6th edition

2.13

Solutions Manual

Consider a linear demand curve, Q = 350 − 7P.

a) Derive the inverse demand curve corresponding to this demand curve. b) What is the choke price? c) What is the price elasticity of demand at P = 50?

Q = 350 − 7 P

a) 7 P = 350 − Q P = 50 − 17 Q

b) The choke price occurs at the point where Q = 0 . Setting Q = 0 in the inverse demand equation above yields P = 50 . c) At P = 50 , the choke price, the elasticity will approach negative infinity.

2.14 Suppose that the quantity of steel demanded in France is given by Qs = 100 – 2Ps + 0.5Y + 0.2PA, where Qs is the quantity of steel demanded per year, Ps is the market price of steel, Y is real GDP in France, and PA is the market price of aluminum. In 2011, Ps = 10, Y = 40, and PA = 100. How much steel will be demanded in 2011? What is the price elasticity of demand, given market conditions in 2011?

We are given that Y = 40, and PA = 100, and so substituting these values into the equation that determines the quantity demanded gives us QS = 100 – 2PS + 0.5(40) + 0.2(100) or QS = 140 – 2PS. This is the equation for the demand curve for steel in France. When the price of steel is 10, the quantity of steel demanded is thus 120.

From equation (2.4) in the text, the price elasticity of demand for steel when the price is 10 is given by 𝜖𝑄,𝑃 = −2

10 = −0.167.

120

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Chapter 2 - 17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

2.15 A firm currently charges a price of $100 per unit of output, and its revenue (price multiplied by quantity) is $70,000. At that price it faces an elastic demand (€Q, P < −1). If the firm were to raise its price by $2 per unit, which of the following levels of output could the firm possibly expect to see? Explain. a) 400 b) 600 c) 800 d) 1000 Recall that for an elastic good, a higher price charged by the firm leads to a decrease in total revenue. Therefore, the firm should expect a level of output such that its revenue at a price of $102 is less than $70,000. Only if the output level is 400 or 600 is this possible (102*400 = $40,800) and (102*600 = $61,200). At the other quantities the revenue would rise.

2.16 Gina usually pays a price between $5 and $7 per gallon of ice cream. Over that range of prices, her monthly total expenditure on ice cream increases as the price decreases. What does this imply about her price elasticity of demand for ice cream?

Gina’s expenditure on ice-cream is P*Q, where P is the price and Q is the number of units of ice cream that she buys. We know that P*Q increases as P decreases which can only mean that Q increases at a faster rate than the rate at which P decreases. This is equivalent to saying that demand is very sensitive to price changes, or that her demand for ice cream is quite elastic (Q,P < –1) . More generally, recall that when price and total revenue (P*Q) move in opposite directions, it is because demand is elastic over that price range.

2.17 Consider the following demand and supply relationships in the market for golf balls: Qd = 90 − 2P − 2T and Qs = −9 + 5P − 2.5R, where T is the price of titanium, a metal used to make golf clubs, and R is the price of rubber. a) If R = 2 and T = 10, calculate the equilibrium price and quantity of golf balls. b) At the equilibrium values, calculate the price elasticity of demand and the price elasticity of supply. c) At the equilibrium values, calculate the cross-price elasticity of demand for golf balls with respect to the price of titanium. What does the sign of this elasticity tell you about whether golf balls and titanium are substitutes or complements? a) Substituting the values of R and T, we get Demand : Q d = 70 − 2 P Supply : Q s = −14 + 5 P

In equilibrium, 70 – 2P = –14 + 5P, which implies that P = 12. Substituting this value back, Q = 46.

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Chapter 2 - 18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) Elasticity of Demand = –2(12/46), or –0.52. Elasticity of Supply = 5(12/46) = 1.30. c) golf, titanium = ‒2(10/46) = -0.43. The negative sign indicates that titanium and golf balls are complements, i.e., when the price of titanium goes up the demand for golf balls decreases.

2.18 In Metropolis only taxicabs and privately owned automobiles are allowed to use the highway between the airport and downtown. The market for taxi cab service is competitive. There is a special lane for taxicabs, so taxis are always able to travel at 55 miles per hour. The demand for trips by taxi cabs depends on the taxi fare P, the average speed of a trip by private automobile on the highway E, and the price of gasoline G. The number of trips supplied by taxi cabs will depend on the taxi fare and the price of gasoline. a) How would you expect an increase in the price of gasoline to shift the demand for transportation by taxi cabs? How would you expect an increase in the average speed of a trip by private automobile to shift the demand for transportation by taxi cabs? b) Suppose the demand for trips by taxi is given by the equation Qd = 1000 + 50G - 4E 400P. The supply of trips by taxi is given by the equation Qs = 200 - 30G + 100P. On a graph draw the supply and demand curves for trips by taxi when G = 4 and E =30. Find equilibrium taxi fare. c) Solve for equilibrium taxi fare in a general case; that is, when you do not know G and E. Show how the equilibrium taxi fare changes as G and E change. a) When the price of gasoline goes up, it becomes more expensive to drive a private automobile; because private automobiles and taxis are substitutes, the demand for taxi service should increase (shift to the right). On the other hand, when the average speed of a trip by automobile increases, commuters are more likely to use their cars instead of public transportation; the demand for taxi service should shift to the left. b) Substituting G = 4 and E = 30 into equations for the supply and demand curves we have Q d = 1080 − 400  P, Q s = 80 + 100  P.

Solving equation Qd = Qs we have P = 2, Q = 280. Supply and demand curves are graphed below.

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Chapter 2 - 19

Besanko & Braeutigam – Microeconomics, 6th edition

P

Solutions Manual

Qs

$2.70

$2

Qd

80

280

1080

Q

c) In equilibrium Qd = Qs. When we P=

1 ( 200 − E + 20  G ) .

125

The equilibrium taxi fare goes up as gasoline price increases and goes down when it private automobiles can travel faster.

2.19 For the following pairs of goods, would you expect the cross-price elasticity of demand to be positive, negative, or zero? Briefly explain your answers. a) Tylenol and Advil b) DVD players and VCRs c) Hot dogs and buns a) Since the two goods are rather close substitutes for each other, you would expect that the demand for Tylenol would go up if the price of Advil increases and vice versa. Therefore, the cross price elasticity will be positive. b) Similar to part (a). Although VCRs and DVD players are not very close substitutes, if the price of VCRs were to go up substantially, potential buyers would probably decide to pay a little bit more and get the higher-end DVD player. Similarly if the latter becomes expensive, some consumers will not be able to afford it and will switch to the VCR instead. The elasticity will be positive. c) Since the two usually go together, a sharp increase in the price of one will lead to a decline in the demand for the other, and the cross-price elasticity will be negative.

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Solutions Manual

2.20 For the following pairs of goods, would you expect the cross-price elasticity of demand to be positive, negative, or zero? Briefly explain your answer. a) Red umbrellas and black umbrellas b) Coca-Cola and Pepsi c) Grape jelly and peanut butter d) Chocolate chip cookies and milk e) Computers and software a) Assuming red and black umbrellas are substitutes, we would expect the cross-price elasticity of demand to be positive. b) Coca-cola and Pepsi are substitutes. We would expect the cross-price elasticity of demand to be positive. c) Grape jelly and peanut butter are typically complements (people want both on their sandwiches!). We would expect the cross-price elasticity of demand to be negative. d) Chocolate chip cookies and milk are typically complements (people want to consume them together). We would expect the cross-price elasticity of demand to be negative. e) Computers and software are complements (consumers want to use them together). We would expect the cross-price elasticity of demand to be negative.

2.21. Suppose that the market for air travel between Chicago and Dallas is served by just two airlines, United and American. An economist has studied this market and has estimated that the demand curves for round-trip tickets for each airline are as follows: QdU = 10,000 − 100PU + 99PA (United’s demand) QdA = 10,000 − 100PA + 99PU (American’s demand) where PU is the price charged by United, and PA is the price charged by American. a) Suppose that both American and United charge a price of $300 each for a round-trip ticket between Chicago and Dallas. What is the price elasticity of demand for United flights between Chicago and Dallas? b) What is the market-level price elasticity of demand for air travel between Chicago and Dallas when both airlines charge a price of $300? (Hint: Because United and American are the only two airlines serving the Chicago–Dallas market, what is the equation for the total demand for air travel between Chicago and Dallas, assuming that the airlines charge the same price?)

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a)

Solutions Manual

QUd = 10000 − 100(300) + 99(300) QUd = 9700

Using PU = 300 and QUd = 9700 gives  300   = −3.09  9700 

 Q , P = −100 

b) Market demand is given by Qd = QUd + QAd . Assuming the airlines charge the same price we have

Q d = 10000 − 100 PU + 99 PA + 10000 − 100 PA + 99 PU Q d = 20000 − 100 P + 99 P − 100 P + 99 P Q d = 20000 − 2 P When P = 300, Q d = 19400 . This implies an elasticity equal to  300   = −.0309  19400 

 Q , P = −2 

2.22. You are given the following information: • • •

Price elasticity of demand for cigarettes at current prices is −0.5.

Current price of cigarettes is $0.05 per cigarette.

Cigarettes are being purchased at a rate of 10 million per year.

Find a linear demand that fits this information, and graph that demand curve.

We know that along a linear demand curve P  Q

 Q , P = −b 

Using the given information this implies

.05   −.5 = −b    10, 000, 000  b = 100, 000, 000 Plugging this result into a demand equation using the known price and quantity then implies

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Solutions Manual

Q d = A − bP 10, 000, 000 = A − 100, 000, 000(.05) A = 15, 000, 000

So a demand equation that fits this information is given by Q d = 15, 000, 000 − 100, 000, 000P

Graphically, the demand curve looks like P

0.15

Q 15,000,000

2.23 For each of the following, discuss whether you expect the elasticity (of demand or of supply, as specified) to be greater in the long run or the short run. a) The supply of seats in the local movie theater. b) The demand for eye examinations at the only optometrist in town. c) The demand for cigarettes. a) More elastic in the long run as the theatre owner can increase space or add another screen if the price remains high, but cannot easily adjust the number of seats at short notice. b) More elastic in the short run as people can be relatively flexible about when to undergo an eye exam, but in the long run the need for eye exams is fixed. c) More elastic in the long run. Cigarettes tend to be addictive and so smokers are less likely to be able to reduce their demand in response to short term fluctuations in price. However if the price remains high for a long time they will consider giving up the habit as it becomes too expensive.

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Solutions Manual

2.24 Suppose that in 2011, the global market for hard drives for notebook computers consists of a large number of producers. It is relatively easy for new producers to enter the industry, and when the market for notebook hard drives is booming, new producers do, in fact, enter.

In February 2011, there is an unexpected temporary surge in the demand for notebook hard drives, increasing the monthly demand for hard drives by 25 percent at any possible price. As a result of this, the price of notebook hard drives increased by $5 per megabyte by the end of February. This surge in demand ended in March 2011, and the price of notebook hard drives fell back to its level just before the temporary demand surge occurred.

Later that year, in August 2011, a permanent increase in the demand for notebook computers occurs, increasing the monthly demand for hard drives by 25 percent per month at any possible price. Nine months later, the price of notebook hard drives had increased, by $1 per unit.

In both circumstances, the market experienced a shift in demand of exactly the same magnitude. Yet, the change in the equilibrium price appears to have been different. Why? When demand surges temporarily, putting upward pressure on price, the quantity supplied expands along the short-run supply curve SS, as shown in the figure below. If demand increases by the identical rate, but the increase is permanent, the industry would expand along the long-run supply curve LS. The long-run supply curve is likely to be more price elastic than the short-run supply curve. If the demand increase and the resulting upward pressure on price is temporary, producers may be able to do very little to increase supply except to utilize their existing production facilities more intensively (perhaps by hiring some temporary labor). If the demand increase is permanent, industry supply can increase in response to upward pressure on price in a number of ways: existing firms can produce more output in their existing facilities; existing firms can expand their plants; and new firms can enter the industry and produce. Thus, over a longer horizon, the industry’s supply response when prices begin to rise is more flexible than it is over a shorter horizon.

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Solutions Manual

2.25. The demand for dinners in the only restaurant in town has a unitary price elasticity of demand when the current average price of a dinner is $8. At that price 120 people eat dinners at the restaurant every evening. a) Find a linear demand curve that fits this information and draw it on a clearly labeled graph. b) Do you need the information on the price elasticity of demand to find the curve? Why? a) In case of the linear demand Q = A - bP, we know that  Q ,P = −b

P = −1 Q

Using the values of P and Q given in the problem we have −1 = −b

8 120  b= = 15 .

120 8

Now we can solve for the second parameter of the linear demand curve 120 = a − 15(8)  a = 240 .

Hence the linear demand curve is given by equation Qd = 240 – 15P. b) There exist several linear demand curves for which the demand is equal to 120 at price of $8. Information about elasticity of demand lets us determine exactly one of those. More formally, we need second equation to solve for both parameters of the linear demand curve.

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Solutions Manual

2.26. In each of the following pairs of goods, identify the one which you would expect to have a greater price elasticity of demand. Briefly explain your answers. a) Butter versus eggs b) Trips by your congressman to Washington (say, to vote in the House) versus vacation trips by you to Hawaii c) Orange juice in general versus the Tropicana brand of orange juice a) Butter has some reasonably close substitutes such as margarine or cheese, while eggs have no immediate substitutes. Therefore we would expect the demand for butter to be more elastic. b) Vacation trips are sensitive to price because leisure travelers can be relatively flexible about when to fly. Your congressman, however, has fixed dates on which to be in Washington and would be prepared to pay more to ensure that he flies on the day of his choosing. Therefore, demand for vacation trips is likely to be more elastic (i.e. the price elasticity will be more negative) than the demand for trips by your congressman. c) As discussed in the chapter, market level elasticities tend to be lower (less negative) than the elasticity of a particular brand. Thus, expect the demand for Tropicana to be more elastic than the demand for generic orange juice.

2.27. In a city, the price for a trip on local mass transit (such as the subway or city buses) has been 10 pesos for a number of years. Suppose that the market for trips is characterized by the following demand curves: in the long run: Q = 30 − 2P; in the short run: Q = 15 − P/2. Verify that the long-run demand curve is “flatter” than the short-run curve. What does this tell you about the sensitivity of demand to price for this good? Discuss why this is the case.

First, consider each demand curve in its “inverse” form: long run demand is P = 15 – 0.5Q, and short run demand is P = 30 – 2Q. Thus, the slope of the long run demand is –0.5, which is closer to zero than that of the short run demand, –2. Thus, long run demand is flatter. Second, consider the graph below:

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Solutions Manual

P 30

Short run demand

15 10

Long run demand 15

Q

30

Again, long run demand is flatter and thus more sensitive to changes in price. Consider, for instance a price of $10. Quantity demanded is equal in both the long and short runs at P = 10.

However, consider increasing the price to, say, $15. Although this will reduce quantity demanded in the short run by a little, it would reduce quantity demanded all the way to zero in the long run.

2.28. Consider the following sequence of events in the U.S. market for strawberries during the years 1998–2000: • • •

1998: Uneventful. The market price was $5.00 per bushel, and 4 million bushels were sold.

1999: There was a scare over the possibility of contaminated strawberries from Michigan.

The market price was $4.50 per bushel, and 2.5 million bushels were sold.

2000: By the beginning of the year, the scare over contaminated strawberries ended when the media reported that the initial reports about the contamination were a hoax. A series of floods in the Midwest, however, destroyed significant portions of the strawberry fields in Iowa, Illinois, and Missouri. The market price was $8.00 per bushel, and 3.5 million bushels were sold.

Find linear demand and supply curves that are consistent with this information.

The scare in 1999 would shift demand to the left, identifying a second point on the supply curve.

The information implies that price fell $0.50 while quantity fell 1.5 million. This implies b=

− .5 1 = − .15 3

Using a linear supply curve we then have

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Solutions Manual

1 P = a + Qs 3 1 5 = a + (4) 3 11 a= 3

Finally, plugging these values for a and b into the supply equation results in 11 1 P = + Qs 3 3 3P = 11 + Q s

Q s = −11 + 3P The floods in 2000 will reduce supply. The shift in supply will identify a second point along the demand curve. Because the scare of 1999 is over, assume that demand has returned to its 1998 state. The change in price and quantity in 2000 imply that price increased $3.00 and that quantity fell 0.5 million.

Performing the same exercise as above we have −b =

3 = −6 −0.5

Using the 1998 price and quantity information along with this result yields

P = a − bQ d 5 = a − 6(4) a = 29 Finally, plugging these values for a and b into a linear demand curve results in P = 29 − 6Q d 6Q d = 29 − P Qd =

29 1 − P 6 6

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Solutions Manual

2.29 Consider the following sequence of changes in the demand and supply for cab service in some city. The price P is a price per mile, while quantity is the total length of cab rides over a month (in thousands of miles).

January: Initial demand and supply are given by the equations Qs = 30P - 30 (when P ≥ 1), and Qd = 120 - 20P February: Due to higher prices of gasoline, the supply of cab service changed to Qs = 30P 60 (when P ≥ 2).

March: Over the spring break, the demand for taxi service was higher and therefore demand curve was given by the equation Qd = 140 - 20P. a) For each month find equilibrium price and quantity. b) Illustrate your answer with a graph. Illustrate the equilibrium prices and quantities on the graph.

The equilibrium price in January is equal to P = 3 and equilibrium quantity is equal to Q = 60.

We find equilibrium price by solving Qs = Qd, which is 30∙P – 30 = 120 – 20∙P. When we have equilibrium price we can substitute it to either the demand function or supply function, since they have to give the same quantity at that price, and obtain equilibrium quantity equal to Q = 60.

After the supply decreases in February, new equilibrium price is per mile is equal to P = $3.60, while the demanded quantity is equal to Q = 48. When the demand goes up in March, the quantity in equilibrium is the same as in January but price is even higher and equal to P = $4. All those changes are illustrated on the graph below.

P

Qd

$6 Qs

$4 $3.60 $3

48 60

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120

140 Q

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Solutions Manual

2.30 Consider the demand curve for pomegranates in two countries. In one country, pomegranates are a critical part of the diet and are central to the preparation of many popular food recipes. For most of these dishes, there is no feasible substitute for pomegranates. In the second country, households will purchase pomegranates if the price is right, but they are not considered by consumers to be particularly special or unique, and few popular dishes rely on pomegranates in their recipes.

Suppose pomegranates are native to both countries. Suppose, further, that due to inherent limitations of shipping options, there is no inter-country trade in pomegranates. Each country’s market for pomegranates is independent of the other countries. Finally, suppose that in both countries, droughts and other weather-related shocks periodically cause unexpected changes in supply conditions.

The following graph shows the time paths of pomegranate prices over a 10-year period in each country (the blue (solid) line is the time path in one country; the red (dashed) line is the time path in the other country.) Based on the information provided, which is the time path for each country?

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Solutions Manual

The price path for Country A is the one in which there is substantial price variation over time, while the price path for Country B is the one on in which there is more modest price variation over time. Here is why.

Based on the information given, we can infer that the demand for pomegranates is probably less sensitive to price in Country A (where pomegranates have few good substitutes) than it is in country B. For a given shift in the supply curve in each country, the change in the equilibrium price in country A will be larger than the change in the equilibrium price in country B.

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Solutions Manual

Chapter 3 Consumer Preferences and the Concept of Utility Solutions to Review Questions 1.

What is a basket (or a bundle) of goods?

A basket is a collection of goods and services that an individual might consume.

2.

What does the assumption that preferences are complete mean about the consumer’s ability to rank any two baskets?

By requiring preferences to be complete, economists are ensuring that consumers will not respond indecisively when asked to compare two baskets. A consumer will always be able to state that either A is preferred B, B is preferred to A, or that she is indifferent between A and B. 3.

Consider Figure 3.1. If the more is better assumption is satisfied, is it possible to say which of the seven baskets is least preferred by the consumer?

Unfortunately, it is impossible to say definitively whether D, H, or J is the least preferred basket.

Since more is better, baskets to the northeast are more preferred and baskets to the southwest are less preferred. In this case, H has more clothing but less food than D, while J has more food but less clothing than D. Without more information regarding how the consumer feels about clothing relative to food, we cannot state which of these baskets is the least preferred.

4.

Give an example of preferences (i.e., a ranking of baskets) that do not satisfy the assumption that preferences are transitive.

If a consumer states that A is preferred to B and that B is preferred to C, but then states that C is preferred to A, she will be violating the assumption of transitivity. The third statement is inconsistent with the first two.

5.

What does the assumption that more is better imply about the marginal utility of a good?

If more is better, then the marginal utility of a good must be positive. That is, total utility must increase if the consumer consumes more of the good.

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6.

Solutions Manual

What is the difference between an ordinal ranking and a cardinal ranking?

An ordinal ranking simply orders the baskets, but does not give any indication as to how much better one basket is when compared with another; only that one is better. A cardinal ranking not only orders the baskets, but also provides information regarding the intensity of the preferences.

For example, a cardinal ranking might indicate that one basket is twice as good as another basket.

7.

Suppose Debbie purchases only hamburgers. Assume that her marginal utility is always positive and diminishing. Draw a graph with total utility on the vertical axis and the number of hamburgers on the horizontal axis. Explain how you would determine marginal utility at any given point on your graph.

Utility

Slope of this line measures marginal utility at this level of consumption, H’ Total Utility

H’

Hamburgers

Marginal utility would be measured as the slope of a line tangent to the total utility curve in the graph above.

8.

Why can’t you plot the total utility and marginal utility curves on the same graph?

The two cannot be plotted on the same graph because utility and marginal utility are not measured in the same dimensions. Total utility has the dimension 𝑈, while marginal utility has the dimension of utility per unit, or 𝛥𝑈/𝛥𝑦 where 𝑦 is the number of units purchased.

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9.

Solutions Manual

Adam consumes two goods: housing and food.

a) Suppose we are given Adam’s marginal utility of housing and his marginal utility of food at the basket he currently consumes. Can we determine his marginal rate of substitution of housing for food at that basket? b) Suppose we are given Adam’s marginal rate of substitution of housing for food at the basket he currently consumes. Can we determine his marginal utility of housing and his marginal utility of food at that basket? a)

𝑀𝑈

Yes, we can determine the MRS as 𝑀𝑅𝑆ℎ,𝑓 = 𝑀𝑈ℎ 𝑓

b) No, when we know the MRS, all we know is the ratio of the marginal utilities. We cannot “undo” that ratio to determine the individual marginal utilities. For example, if we know that MRSh,f = 5, it could be the case that MUh = 5 and MUf = 1, but it could equivalently be the case that MUh = 10 and MUf = 2. Clearly, there are countless combinations of MUh and MUf that could lead to some particular value of MRSh,f, and we have no way of inferring which is the right one.

10.

Suppose Michael purchases only two goods, hamburgers (H) and Cokes (C).

a) What is the relationship between MRSH,C and the marginal utilities MUH and MUC ? b) Draw a typical indifference curve for the case in which the marginal utilities of both goods are positive and the marginal rate of substitution of hamburgers for Cokes is diminishing. Using your graph, explain the relationship between the indifference curve and the marginal rate of substitution of hamburgers for Cokes. c) Suppose the marginal rate of substitution of hamburgers for Cokes is constant. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? d) Suppose that Michael always wants two hamburgers along with every Coke. Draw a typical indifference curve. In this case, are hamburgers and Cokes perfect substitutes or perfect complements? a)

𝑀𝑈

𝑀𝑅𝑆𝐻,𝐶 = 𝑀𝑈𝐻 𝐶

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b)

Solutions Manual

C

H

The indifference curve in this case will be convex toward the origin. The marginal rate of substitution is measured as the absolute value of the slope of a line tangent to the indifference curve. As can be seen in the graph above, this slope becomes less negative as we move down the indifference curve, implying a diminishing MRS. c) If the MRS was constant, this would imply that at any consumption level the consumer would be willing to trade a fixed amount of one good for a fixed amount of the other. This occurs with perfect substitutes. d) If the consumer wishes to always consume goods in a fixed ratio, then the goods are perfect complements. In this case, the indifference curves will be L-shaped.

C

1 2

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H

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Solutions Manual

11.

Suppose a consumer is currently purchasing 47 different goods, one of which is housing. The quantity of housing is measured by H. Explain why, if you wanted to measure the consumer’s marginal utility of housing (MUH) at the current basket, the levels of the other 46 goods consumed would be held fixed.

Marginal utility is defined as the change in total utility relative to a change in consumption for a particular good. In order to accurately measure the change in total utility, the levels of the other goods would need to be held constant. If they were not, the change in total utility would occur as a result of multiple goods changing and it would be impossible to determine what portion of the change in total utility should be assigned to each good.

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Solutions Manual

Solutions to Problems 3.1 Bill has a utility function over food and gasoline with the equation U = x2y, where x measures the quantity of food consumed and y measures the quantity of gasoline. Show that a consumer with this utility function believes that more is better for each good.

By plugging in ever higher numerical values of x and ever higher numerical values of y, it can be verified that U increases whenever x or y increases.

3.2 Consider the single-good utility function U(x) = 3x2, with a marginal utility given by MUx = 6x. Plot the utility and marginal utility functions on two separate graphs. Does this utility function satisfy the principle of diminishing marginal utility? Explain.

The two graphs are shown below. It can be seen from both graphs that this function does not satisfy the law of diminishing marginal utility. The first figure shows that utility increases with x, and moreover, that it increases at an increasing rate. For example, an increase in x from 2 to 3, increases utility from 12 to 27 (an increase of 15), while an increase in x from 3 to 4 induces an increase in utility from 27 to 48 (an increase of 21).

This fact is easier to see in the second figure. The marginal utility is an increasing function of x.

Higher values of x imply a greater marginal utility. Therefore, this function exhibits increasing marginal utility.

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Solutions Manual

3.3 Jimmy has the following utility function for hot dogs: U(H) = 10H − H2, with MUH = 10 − 2H a) Plot the utility and marginal utility functions on two separate graphs. b) Suppose that Jimmy is allowed to consume as many hot dogs as he likes and that hot dogs cost him nothing. Show, both algebraically and graphically, the value of H at which he would stop consuming hot dogs.

The first figure below shows Jimmy’s utility function for hotdogs. You can see that the point at which H = 5 corresponds to the flat portion of the utility function, i.e. the point at which the marginal utility of hotdogs is zero, and beyond which the marginal utility is negative.

Alternatively using the second graph it is clear that the point H = 5 is when the marginal utility intersects the x-axis, and beyond which it is negative. Both graphs tell you that to maximize his utility Jimmy should only consume 5 hotdogs and not more.

To answer this question algebraically, you should first recognize from the marginal utility function that Jimmy has a diminishing marginal utility of hotdogs. Therefore the point at which he should stop consuming hotdogs is the point at which 𝑀𝑈𝐻 = 0, or 10 − 2𝐻 = 0. This gives H = 5.

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Solutions Manual

3.4 Consider the utility function U(x, y) = y√x with the marginal utilities MUx = y/(2√x) and MUy = √x. a) Does the consumer believe that more is better for each good? b) Do the consumer’s preferences exhibit a diminishing marginal utility of x? Is the marginal utility of y diminishing? a) Since U increases whenever x or y increases, more of each good is better. This is also confirmed by noting that MUx and MUy are both positive for any positive values of 𝑥 and 𝑦. b)

𝑦

Since 𝑀𝑈𝑥 = 2 𝑥, as 𝑥 increases (holding 𝑦 constant), 𝑀𝑈𝑥 falls. Therefore the marginal √

utility of x is diminishing. However, 𝑀𝑈𝑦 = √𝑥. As y increases, MUy does not change.

Therefore the preferences exhibit a constant, not diminishing, marginal utility of y.

3.5 Carlos has a utility function that depends on the number of musicals and the number of operas seen each month. His utility function is given by U = xy2, where x is the number of movies seen per month and y is the number of operas seen per month. The corresponding marginal utilities are given by: MUx = y2 and MUy = 2xy. a) Does Carlos believe that more is better for each good? b) Does Carlos have a diminishing marginal utility for each good? a) By plugging in ever higher numerical values of x and ever higher numerical values of y, it can be verified that Carlos’ utility goes up whenever x or y increases.

First consider the marginal utility of x, MUx. Since x does not appear anywhere in the formula for MUx, MUx is independent of x. Hence, the marginal utility of movies is independent of the number of movies seen, and so the marginal utility of movies does not decrease as the number of movies increases. Next consider the marginal utility of y, MUy. Notice that MUy is an increasing function of y. Hence, the marginal utility of operas does not decrease in the number of operas seen. In this case, neither good, movies or operas, exhibits diminishing marginal utility.

3.6 For the following sets of goods draw two indifference curves, U1 and U2, with U2 > U1. Draw each graph placing the amount of the first good on the horizontal axis. a) Hot dogs and chili (the consumer likes both and has a diminishing marginal rate of substitution of hot dogs for chili) b) Sugar and Sweet’N Low (the consumer likes both and will accept an ounce of Sweet’N Low or an ounce of sugar with equal satisfaction) c) Peanut butter and jelly (the consumer likes exactly 2 ounces of peanut butter for every ounce of jelly) d) Nuts (which the consumer neither likes nor dislikes) and ice cream (which the consumer likes) e) Apples (which the consumer likes) and liver (which the consumer dislikes) Copyright © 2020 John Wiley & Sons, Inc.

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a)

Solutions Manual

Chili

U2 U1 Hot Dogs

b)

Sweet’N Low

Slopes = –1

U1

c)

U2 Sugar

Jelly

U2

2 U1

1 2

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4

Peanut Butter

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d)

Solutions Manual

Ice Cream

U2

U1 Nuts

e)

Liver U1 U2

Apples

3.7 Alexa likes ice cream, but dislikes yogurt. If you make her eat another gram of yogurt, she always requires two extra grams of ice cream to maintain a constant level of satisfaction. On a graph with grams of yogurt on the vertical axis and grams of ice cream on the horizontal axis, graph some typical indifference curves and show the directions of increasing utility.

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Solutions Manual

3.8 Joe has a utility function over hamburgers and hot dogs given by U = x + √y , where x is the quantity of hamburgers and y is the quantity of hot dogs. The marginal utilities for this utility function are MUx = 1 and MUy = 1/(2√y ). Does this utility function have the property that MRSx,y is diminishing?

This utility function does have the property of diminishing MRSx,y. One way to verify this is to graph several indifference curves. Another way to tell is to use algebra. Recall that 𝑀𝑈 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥 . Applying that general formula to this case gives us 𝑀𝑅𝑆𝑥,𝑦 = 2√𝑦. As we move 𝑦

“down” the indifference curve, x increases and y decreases. As y decreases, 2√𝑦 decreases.

Thus, MRSx,y decreases.

3.9

Julie and Toni consume two goods with the following utility functions:

𝑼𝑱𝒖𝒍𝒊𝒆 = (𝒙 + 𝒚)𝟐 , 𝑼𝑻𝒐𝒏𝒊 = 𝒙 + 𝒚,

𝑴𝑼𝑱𝒖𝒍𝒊𝒆 = 𝟐(𝒙 + 𝒚), 𝒙 𝑻𝒐𝒏𝒊 𝑴𝑼𝒙 = 𝟏,

𝑴𝑼𝑱𝒖𝒍𝒊𝒆 = 𝟐(𝒙 + 𝒚), 𝒚 𝑻𝒐𝒏𝒊 𝑴𝑼𝒚 = 𝟏

a) Graph an indifference curve for each of these utility functions. b) Julie and Toni will have the same ordinal ranking of different baskets if, when basket A is preferred to basket B by one of the functions, it is also preferred by the other. Do Julie and Toni have the same ordinal ranking of different baskets of x and y? Explain.

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Chapter 3-11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Indifference curves corresponding to U = 2 are shown for both Julie and Toni in the graph below. Notice that the indifference curves are parallel everywhere – indeed, MRSx,y = 1 for both Julie and Toni, for all values of x and y. Toni’s indifference curve for the utility level UToni = 2 is the same as Julie’s indifference curve for the utility level UJulie = 4. So whenever Julie ranks bundle A higher than bundle B, Toni would have the same ranking, and vice-versa. So Julie and Toni will have the same ordinal ranking of bundles of x and y. (Julie will associate each bundle with a higher utility level than Toni will, but that is a cardinal ranking.)

UToni = 2 UJulie = 2

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Chapter 3-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.10 The utility that Julie receives by consuming food F and clothing C is given by U(F, C) = FC. For this utility function, the marginal utilities are MUF = C and MUC = F. a) On a graph with F on the horizontal axis and C on the vertical axis, draw indifference curves for U = 12, U = 18, and U = 24. b) Do the shapes of these indifference curves suggest that Julie has a diminishing marginal rate of substitution of food for clothing? Explain. c) Using the marginal utilities, show that the MRSF,C = C/F. What is the slope of the indifference curve U = 12 at the basket with 2 units of food and 6 units of clothing? What is the slope at the basket with 4 units of food and 3 units of clothing? Do the slopes of the indifference curves indicate that Julie has a diminishing marginal rate of substitution of food for clothing? (Make sure your answers to parts (b) and (c) are consistent!) a)

b) Yes, since the indifference curves are bowed in toward the origin we know that MRSF,C declines as F increases and C decreases along an indifference curve. c)

𝑀𝑈

𝐶

𝑀𝑅𝑆𝐹,𝐶 = 𝑀𝑈𝐹 = 𝐹. When F = 2 and C = 6, MRSF,C = 3. The slope of the indifference 𝐶

curve is –3. When F = 4 and C = 3, MRSF,C = 0.75, so the slope of the indifference curve is -0.75. Since the marginal rate of substitution falls as F increases and C decreases, she has a diminishing marginal rate of substitution.

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Chapter 3-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.11 Sandy consumes only hamburgers (H) and milkshakes (M). At basket A, containing 2 hamburgers and 10 milkshakes, his MRSH,M is 8. At basket B, containing 6 hamburgers and 4 milkshakes, his MRSH,M is 1/2. Both baskets A and B are on the same indifference curve. Draw the indifference curve, using information about the MRSH,M to make sure that the curvature of the indifference curve is accurately depicted.

Milkshakes Slope = –8 A

10

Slope = –½ B

4

Hamburgers 2

6

3.12 Adam likes his café latte prepared to contain exactly 1/3 espresso and 2/3 steamed milk by volume. On a graph with the volume of steamed milk on the horizontal axis and the volume of espresso on the vertical axis, draw two of his indifference curves, U1 and U2, with U1 > U2.

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Chapter 3-14

Besanko & Braeutigam – Microeconomics, 6th edition

3.13

Solutions Manual

Draw indifference curves to represent the following types of consumer preferences.

a) I like both peanut butter and jelly, and always get the same additional satisfaction from an ounce of peanut butter as I do from 2 ounces of jelly. b) I like peanut butter, but neither like nor dislike jelly. c) I like peanut butter, but dislike jelly. d) I like peanut butter and jelly, but I only want 2 ounces of peanut butter for every ounce of jelly.

In the following graphs, U2 > U1. a)

Jelly 4 2

b)

U1

U2

1

2

Peanut Butter

Jelly U1

U2

Peanut Butter

c)

Jelly

U1 U2

Peanut Butter

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Chapter 3-15

Besanko & Braeutigam – Microeconomics, 6th edition

d)

Solutions Manual

Jelly U1

U2

2 1 2

Peanut Butter

4

3.14 Dr. Strangetaste buys only food (F) and clothing (C) out of his income. He has positive marginal utilities for both goods, and his MRSF,C is increasing. Draw two of Dr. Strangetaste’s indifference curves, U1 and U2, with U2 > U1.

Clothing

U1

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U2 Food

Chapter 3-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

The following exercises will give you practice in working with a variety of utility functions and marginal utilities and will help you understand how to graph indifference curves.

3.15

Consider the utility function U(x, y) = 3x + y, with MUx = 3 and MUy = 1.

a) Is the assumption that more is better satisfied for both goods? b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. c) What is MRSx, y? d) Is MRSx, y diminishing, constant, or increasing as the consumer substitutes x for y along an indifference curve? e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it needs to reflect accurately whether there is a diminishing MRSx, y). Also indicate on your graph whether the indifference curve will intersect either or both axes. Label the curve U1. f ) On the same graph draw a second indifference curve U2, with U2 > U1. a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 remains constant at 3 for all values of x.

c)

𝑀𝑅𝑆𝑥,𝑦 = 3

d)

The 𝑀𝑅𝑆𝑥,𝑦 remains constant moving along the indifference curve.

e & f) See figure below Y

U1 U2

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X

Chapter 3-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.16 Answer all parts of Problem 3.15 for the utility function U(x, y) = √xy. The marginal utilities are MUx = √y/(2√x) and MUy = √x/(2√y). a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 diminishes as the consumer buys more 𝑥.

c)

𝑀𝑅𝑆𝑥,𝑦 = (2√ 𝑥) ( √𝑥 ) = 𝑥

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish.

𝑦

2 𝑦

√

√

𝑦

e & f) See figure below. The indifference curves will not intersect either axis.

3.17 Answer all parts of Problem 3.15 for the utility function U(x, y) = xy + x. The marginal utilities are MUx = y + 1 and MUy = x. a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b) The marginal utility of 𝑥 remains constant as the consumer buys more x. 𝑦+1 c) 𝑀𝑅𝑆𝑥,𝑦 = 𝑥 d) As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish. e & f) See figure below. The indifference curves intersect the x-axis, since it is possible that U > 0 even if y = 0.

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Chapter 3-18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

20 18 16

U2

14

Y

12 10 U1 8 6 4 2 0 0

5

10

15

20

25

30

35

X

3.18 Answer all parts of Problem 3.15 for the utility function U(x, y) = x0.4y0.6. The marginal utilities are MUx = 0.4 (y0.6/x0.6) and MUy = 0.6 (x0.4/y0.4). a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 diminishes as the consumer buys more 𝑥.

c)

𝑀𝑅𝑆𝑥,𝑦 = .6(𝑥 0.4 /𝑦 0.4 ) = 0.6𝑥

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish.

.4(𝑦 0.6 /𝑥 0.6 )

0.4𝑦

e & f) See figure below. The indifference curves do not intersect either axis.

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Chapter 3-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.19 Answer all parts of Problem 3.15 for the utility function U = √x + 2√y. The marginal utilities for x and y are, respectively, MUx = 1/(2√x ) and MUy = 1/√y . a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 diminishes as the consumer buys more 𝑥.

c)

𝑀𝑅𝑆𝑥,𝑦 =

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish.

1/(2√𝑥) 𝑦 = 2√ 𝑥 1/√𝑦 √

e & f) See figure below. Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0 (and x > 0), the indifference curves intersect both axes.

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Chapter 3-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.20 Answer all parts of Problem 3.15 for the utility function U(x, y) = x2 + y2. The marginal utilities are MUx = 2x and MUy = 2y. a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 increases as the consumer buys more 𝑥.

c)

𝑀𝑅𝑆𝑥,𝑦 = 2𝑦 = 𝑦

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will increase.

2𝑥

𝑥

e & f) See figure below. Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0 (and x > 0), the indifference curves intersect both axes.

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Chapter 3-21

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.21 Suppose a consumer’s preferences for two goods can be represented by the Cobb– Douglas utility function U = Axαyβ , where A, α, and β are positive constants. The marginal utilities are MUx = αAxα−1yβ and MUy = βAxαyβ−1. Answer all parts of Problem 3.15 for this utility function. a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b) Since we do not know the value of 𝛼, only that it is positive, we need to specify three possible cases: 1. When 𝛼 < 1, the marginal utility of 𝑥 diminishes as 𝑥 increases.

2. When 𝛼 = 1, the marginal utility of 𝑥 remains constant as 𝑥 increases.

3. When 𝛼 > 1, the marginal utility of 𝑥 increases as 𝑥 increases. 𝛼𝐴𝑥 𝛼−1 𝑦 𝛽

𝛼𝑦

c)

𝑀𝑅𝑆𝑥,𝑦 = 𝛽𝐴𝑥 𝛼𝑦 𝛽−1 = 𝛽𝑥

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish.

e & f) The graph below depicts indifference curves for the case where 𝐴 = 1 and 𝛼 = 𝛽 = 0.5.

Thus 𝑈(𝑥, 𝑦) = 𝑥 0.5 𝑦 0.5. Regardless, the indifference curves will never intersect either axis.

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Chapter 3-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.22 Suppose a consumer has preferences over two goods that can be represented by the quasi-linear utility function U(x, y) = 2√x + y. The marginal utilities are MUx = 1/√x and MUy = 1. a) Is the assumption that more is better satisfied for both goods? b) Does the marginal utility of x diminish, remain constant, or increase as the consumer buys more x? Explain. c) What is the expression for MRSx,y? d) Is the MRSx,y diminishing, constant, or increasing as the consumer substitutes more x for y along an indifference curve? e) On a graph with x on the horizontal axis and y on the vertical axis, draw a typical indifference curve (it need not be exactly to scale, but it should accurately reflect whether there is a diminishing MRSx,y). Indicate on your graph whether the indifference curve will intersect either or both axes. f) Show that the slope of every indifference curve will be the same when x = 4. What is the value of that slope? a) Yes, the “more is better” assumption is satisfied for both goods since both marginal utilities are always positive. b)

The marginal utility of 𝑥 increases as the consumer buys more 𝑥.

1 √𝑥

1

c)

𝑀𝑅𝑆𝑥,𝑦 = 1 =

d)

As the consumer substitutes 𝑥 for 𝑦, the 𝑀𝑅𝑆𝑥,𝑦 will diminish.

√𝑥

e) Since it is possible to have U > 0 if either x = 0 (and y > 0) or y = 0 (and x > 0), the indifference curves intersect both axes.

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Chapter 3-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

The slope of a typical indifference curve at some basket (𝑥, 𝑦) is the 𝑀𝑅𝑆𝑥,𝑦 =

f)

𝑥 = 4, 𝑀𝑅𝑆𝑥,𝑦 =

1 √4

1 √𝑥

. At

= 0.5. Note that this holds regardless of the value of 𝑦. Therefore, the slope

of any indifference curve at 𝑥 = 4 will be −0.5.

3.23 Daniel and Will each consume two goods. When they consume the same basket, Daniel’s marginal utility of each good is higher than Will’s. But at any basket they both have the same marginal rate of substitution of one good for the other. Do they have the same ordinal ranking of different baskets?

Since the two consumers have the same marginal rate of substitution at any basket, the slopes of their indifference curves through that basket will always be the same. In other words, their willingness to trade off one good for the other will be identical at any basket. So their indifference curves will have the same shape. Their ordinal ranking for all baskets will be the same.

3.24 Claire consumes three goods out of her income, food (F) shelter (S), and clothing (C). At her current levels of consumption, her marginal utility of food is 3 and her marginal utility of shelter is 6. Her marginal rate of substitution of shelter for clothing is 4. Do you have enough information to determine her marginal rate of substitution of food for clothing? If so, what is it? If not, why not?

• • • •

We need to find MRSF,C = MUF/MUC We are given: MUF = 3, MUS = 6.

We are also given MRSS,C = 4, so we know that MUS/MUC = 6/MUC = 4.

Thus MUC = 6/4 = 1.5. Now we can determine MRSF,C = MUF/MUC = 3/1.5 = 2.

3.25 Suppose a person has a utility function given by U = [xρ + yρ]1/ρ where ρ is a number between -∞ and 1. This is called a constant elasticity of substitution (CES) utility function. You will encounter CES functions in Chapter 6, where the concept of elasticity of substitution will be explained. The marginal utilities for this utility function are given by MUx = [xρ + yρ]1/ρ−1xρ−1 MUy = [xρ + yρ]1/ρ−1 yρ−1 Does this utility function exhibit the property of diminishing MRSx,y? 𝑀𝑈

Recall that 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥 . Substituting in the marginal utilities given above yields 𝑀𝑅𝑆𝑥,𝑦 = 𝑦

𝑥 𝜌−1 𝑦 𝜌−1

. Now, because  < 1, x - 1 decreases as x increases. By the same logic, y - 1 increases as y

decreases. As we “slide down” an indifference curve, x increases and y decreases, so it follows that MRSx,y decreases. Thus, this utility function exhibits diminishing marginal rate of substitution of x for y.

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Chapter 3-24

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3.26 Annie consumes three goods out of her income, food (F) shelter (S), and clothing (C). At her current levels of consumption, her marginal rate of substitution of food for clothing is 2 and her marginal rate of substitution of clothing for shelter is 3. a) Do you have enough information to determine her marginal rate of substitution of food for shelter? If so, what is it? If not, why not? b) Do you have enough information to determine her marginal utility of shelter? If so, what is it? If not, why not? a)We do have enough information to determine MRSF,S = MUF/MUS. • • • •

We are given: MRSF,C = MUF/MUC = 2 and MRSC,S = MUC/MUS =3.

From MUF/MUC = 2, we know that MUC = MUF / 2.

From MUC/MUS = 3, we know that MUC = 3MUS. Thus MUF /2 = 3MUS, so MUF/MUS = 6. Thus MRSF,S = 6.

b) While the marginal rates of substitution provide quantitative information about the ratios of the marginal utilities, alone they do not allow us to determine the individual marginal utilities of any of the three goods.

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Chapter 3-25

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 4 Consumer Choice Solutions to Review Questions 1.

If the consumer has a positive marginal utility for each of two goods, why will the consumer always choose a basket on the budget line?

Relative to any point on the budget line, when the consumer has a positive marginal utility for all goods she could increase her utility by consuming some basket outside the budget line.

However, baskets outside the budget line are unaffordable to her, so she is constrained (as in “constrained optimization”) to choosing the most preferred basket that lies along the budget line.

2.

How will a change in income affect the location of the budget line?

An increase in income will shift the budget line away from the origin in a parallel fashion expanding the set of possible baskets from which a consumer may choose. A decrease in income will shift the budget line in toward the origin in a parallel fashion, reducing the set of possible baskets from which a consumer may choose.

3.

How will an increase in the price of one of the goods purchased by a consumer affect the location of the budget line?

If the price of one of the goods increases, the budget line will rotate inward on the axis for the good with the price increase. The budget line will continue to have the same intercept on the other axis. For example, suppose someone buys two goods, cups of coffee and doughnuts, and suppose the price of a cup of coffee increases. Then the budget line will rotate as in the following diagram: Doughnuts

BL2

BL1 Coffee

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Chapter 4-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.

What is the difference between an interior optimum and a corner point optimum in the theory of consumer choice?

With an interior optimum the consumer is choosing a basket that contains positive quantities of all goods, while with a corner point optimum the consumer is choosing a basket with a zero quantity for one of the goods. The tangency condition usually does not apply at corner optima.

5.

At an optimal interior basket, why must the slope of the budget line be equal to the slope of the indifference curve?

If the optimum is an interior solution, the slope of the budget line must equal the slope of the indifference curve. If these slopes are not equal at the chosen interior basket then the “bang for the buck” condition will not hold. This condition states that at the optimum the extra utility gained per dollar spent on good 𝑥 must be equal to the extra utility gained per dollar spent on good 𝑦. If this condition does not hold at the chosen basket, then the consumer could reallocate his income to purchase more of the good with the higher “bang for the buck” and increase his total utility while remaining within the given budget. Thus, if these slopes are not equal the basket cannot be optimal assuming an interior solution.

6.

At an optimal interior basket, why must the marginal utility per dollar spent on all goods be the same?

At an interior optimum, the slope of the budget line must equal the slope of the indifference 𝑀𝑈𝑦 𝑀𝑈 𝑃 𝑀𝑈 curve. This implies 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥 = 𝑃𝑥 . This can be rewritten as 𝑃 𝑥 = 𝑃 , which is known as 𝑦

𝑦

𝑥

𝑦

the “bang for the buck” condition. If this condition does not hold at the chosen interior basket, then the consumer can increase total utility by reallocating his spending to purchase more of the good with the higher “bang for the buck” and less of the other good.

7.

Why will the marginal utility per dollar spent not necessarily be equal for all goods at a corner point?

The “bang for the buck” condition will not necessarily hold at a corner solution optimum. The consumer could theoretically increase total utility by reallocating his spending to purchase more of the good with the higher “bang for the buck” and less of the other good. Since the basket is a corner point, however, he is already purchasing zero of one of the goods. This implies that he cannot purchase less of the good with a zero quantity (since negative quantities make no sense) and therefore cannot reallocate spending.

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Chapter 4-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

8.

Suppose that a consumer with an income of $1,000 finds that basket A maximizes utility subject to his budget constraint and realizes a level of utility U1. Why will this basket also minimize the consumer’s expenditures necessary to realize a level of utility U1?

In the utility maximization problem, the consumer maximizes utility subject to a fixed budget constraint. At the optimum the slope of the budget line will equal the slope of the indifference curve. If we now hold that indifference curve fixed, we can solve an expenditure minimization problem in which we ask “what is the minimum expenditure necessary to achieve that fixed level of utility?” Since the slope of the budget line and indifference curve have not changed, when the expenditure is minimized the budget line and indifference curve will be tangent at the same point as in the utility maximization problem. The same basket is optimal in both problems.

9.

What is a composite good?

First, consumers typically allocate income to more than two goods. Second, economists often want to focus on the consumer’s response to purchases of a single good or service. In this case it is useful to present the consumer choice problem using a two-dimensional graph. Since there are more than two goods the consumer is purchasing, however, an economist would need more than two dimensions to show the problem graphically. To reduce the problem to two dimensions, economists often group the expenditures on all other goods besides the one in question into a single good termed a “composite good.” When the problem is shown graphically, one axis represents the composite good while the other axis represents the single good in question. By creating this composite good, the problem can be illustrated using a two-dimensional graph.

10.

How can revealed preference analysis help us learn about a consumer’s preferences without knowing the consumer’s utility function?

By employing revealed preference analysis one can make inferences regarding a consumer’s preferences without knowing what the consumer’s indifference map looks like. For example, if a consumer chooses basket A over basket B when basket B costs at least as much as basket A, we know that basket A is at least as preferred as basket B. If the consumer chooses basket C, which is more expensive than basket D, then we know the consumer strictly prefers basket C to basket D. By observing enough of these choices, one can determine how the consumer ranks baskets even without knowing the exact shape of the consumer’s indifference map.

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Chapter 4-3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 4.1 Pedro is a college student who receives a monthly stipend from his parents of $1,000.

He uses this stipend to pay rent for housing and to go to the movies (you can assume that all of Pedro’s other expenses, such as food and clothing have already been paid for). In the town where Pedro goes to college, each square foot of rental housing costs $2 per month.

The price of a movie ticket is $10 per ticket. Let x denote the square feet of housing, and let y denote the number of movie tickets he purchases per month. a) What is the expression for Pedro’s budget constraint? b) Draw a graph of Pedro’s budget line. c) What is the maximum number of square feet of housing he can purchase given his monthly stipend? d) What is the maximum number of movie tickets he can purchase given his monthly stipend? e) Suppose Pedro’s parents increase his stipend by 10 percent. At the same time, suppose that in the college town he lives in, all prices, including housing rental rates and movie ticket prices, increase by 10 percent. What happens to the graph of Pedro’s budget line? a) 2x + 10y ≤ 1000 b)

c) The maximum amount of housing Pedro can purchase is his budget divided by the price of housing: $1,000/$2 per square feet = 500 square feet. d) The maximum number of movie tickets Pedro can purchase is his budget divided by the price of a movie ticket: $1,000/$10 per tickets = 100 tickets. e) His budget line does not change at all.

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Chapter 4-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Initially, the budget line (with x on the horizontal axis and y on the vertical axis) has a horizontal intercept equal to 1000/2 = 500 and a vertical intercept equal to 1000/10 = 100. The slope of the budget line is -2/10 = - 0.20 (the price of housing divided by the price of movie tickets).

With the increase in Pedro’s stipend and the increases in prices we have: • • •

Horizontal intercept of budget line: 1000(1.10)/(2(1.10)) = 500 Vertical intercept of budget line: 1000(1.10)/(10(1.10)) = 100 Slope of budget line: -2(1.10)/(10(1.10)) = - 0.20.

These are the same as before and thus the budget line does not change.

4.2 Sarah consumes apples and oranges (these are the only fruits she eats). She has decides that her monthly budget for fruit will be $50. Suppose that one apple costs $0.25, while one orange costs $0.50. Let x denote the quantity of apples and y denote the quantity of oranges that Sarah purchases. a) What is the expression for Sarah’s budget constraint? b) Draw a graph of Sarah’s budget line. c) Show graphically how Sarah’s budget line changes if the price of apples increases to $0.50. d) Show graphically how Sarah’s budget line changes if the price of oranges decreases to $0.25. e) Suppose Sarah decides to cut her monthly budget for fruit in half. Coincidentally, the next time she goes to the grocery store, she learns that oranges and apples are on sale for half price, will remain so for the next month, i.e., the price of apples falls from $0.25 per apple to $0.125 per apple and the price of oranges falls from $0.50 per orange to $0.25 per orange. What happens to the graph of Sarah’s budget line?

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a) 0.25x + 0.50y ≤ 50. b)

c)

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d)

e) Sarah’s budget line would not change.

• • •

Horizontal intercept of the budget line: (0.5)$50/((0.5)(0.25) = 200 Vertical intercept of the budget line: (0.5)$50/((0.5)(0.50) = 100 Slope of the budget line = -(0.5)(0.25)/((0.5)(0.50)) = 0.50

These are the same as before, and thus the budget line does not change.

4.3 Julie has preferences for food F and clothing C described by a utility function U(F,C) = FC. Her marginal utilities are MUF = C and MUC = F. Suppose that food costs $1 a unit and that clothing costs $2 a unit. Julie has $12 to spend on food and clothing. a) On a graph draw indifference curves corresponding to u=12, u=18, and u=24. Using the graph (and no algebra), find the optimal (utility-maximizing) choice of food and clothing.

Let the amount of food be on the horizontal axis and the amount of clothing be on the vertical axis. b) Using algebra (the tangency condition and the budget line), find the optimal choice of food and clothing. c) What is the marginal rate of substitution of food for clothing at her optimal basket?

Show this graphically and algebraically. d) Suppose Julie decides to buy 4 units of food and 4 units of clothing with her $12 budget (instead of the optimal basket). Would her marginal utility per dollar spent on food be greater than or less than her marginal utility per dollar spent on clothing? What does this tell you about how she should substitute food for clothing if she wanted to increase her utility without spending any more money?

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a)

b)

𝑀𝑈

𝑃

The tangency condition implies that 𝑀𝑈𝐹 = 𝑃𝐹 𝐶

𝐶

C 1 = Plugging in the known information results in F 2 2C = F

2C + 2C = 12 Substituting this result into the budget line, 𝐹 + 2𝐶 = 12, yields

4C = 12 C =3

Finally, plugging this result back into the tangency condition implies 𝐹 = 6. At the optimum the consumer choose 6 units of food and 3 units of clothing. c) At the optimum, 𝑀𝑅𝑆𝐹,𝐶 = 𝐶/𝐹 = 3/6 = 1/2. Note that this is equal to the ratio of the price of food to the price of clothing. The equality of the price ration and MRSF,C is seen in the graph above as the tangency between the budget line and the indifference curve for 𝑈 = 18. d) 𝑀𝑈𝐹 𝑃𝐹

If the consumer purchases 4 units of food and 4 units of clothing, then 4 𝑀𝑈 4 = 1 = 4 > 𝑃 𝐶 = 2 = 2. 𝐶

This implies that the consumer could reallocate spending by purchasing more food and less clothing to increase total utility. In fact, at the basket (4, 4) total utility is 16 and the consumer spent $12. By giving up one unit of clothing the consumer saves $2 which can then be used to purchase two units of food (they each cost $1). This will result in a new basket (6,3), total utility of 18, and spending of $12. By reallocating spending toward the good with the higher “bang for the buck” the consumer increased total utility while remaining within the budget constraint.

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4.4 The utility that Ann receives by consuming food F and clothing C is given by U(F, C) = FC + F. The marginal utilities of food and clothing are MUF = C + 1 and MUC = F. Food costs $1 a unit, and clothing costs $2 a unit. Ann’s income is $22. a) Ann is currently spending all of her income. She is buying 8 units of food. How many units of clothing is she consuming? b) Graph her budget line. Place the number of units of clothing on the vertical axis and the number of units of food on the horizontal axis. Plot her current consumption basket. c) Draw the indifference curve associated with a utility level of 36 and the indifference curve associated with a utility level of 72. Are the indifference curves bowed in toward the origin? d) Using a graph (and no algebra), find the utility maximizing choice of food and clothing. e) Using algebra, find the utility-maximizing choice of food and clothing. f ) What is the marginal rate of substitution of food for clothing when utility is maximized?

Show this graphically and algebraically. g) Does Ann have a diminishing marginal rate of substitution of food for clothing? Show this graphically and algebraically.

F + 2C = 22 a)

If Ann is spending all of her income then

8 + 2C = 22 2C = 14 C =7

b)

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c)

Solutions Manual

Yes, the indifference curves are convex, i.e., bowed in toward the origin.

Also, note that they intersect the F-axis.

d)

e)

The tangency condition requires that

Plugging in the known information yields

𝑀𝑈𝐹 𝑀𝑈𝐶

𝐶+1 𝐹

=

𝑃𝐹 𝑃𝐶

1

= 2 which means 2𝐶 + 2 = 𝐹

(2C + 2) + 2C = 22 Substituting this result into the budget line, 𝐹 + 2𝐶 = 22 results in

4C = 20 C =5

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Finally, plugging this result back into the tangency condition implies that 𝐹 = 2(5) + 2 = 12.

At the optimum the consumer chooses 5 units of clothing and 12 units of food. f)

𝑀𝑅𝑆𝐹,𝐶 =

𝐶+1 𝐹

5+1

1

= 12 = 2 The marginal rate of substitution is equal to the price ratio.

g) Yes, the indifference curves do exhibit diminishing 𝑀𝑅𝑆𝐹,𝐶 . We can see this in the graph in part c) because the indifference curves are bowed in toward the origin. Algebraically, 𝐶+1 𝑀𝑅𝑆𝐹,𝐶 = 𝐹 . As 𝐹 increases and 𝐶 decreases along an isoquant, 𝑀𝑅𝑆𝐹,𝐶 diminishes.

4.5 Consider a consumer with the utility function U(x, y) = min(3x, 5y), that is, the two goods are perfect complements in the ratio 3:5. The prices of the two goods are Px = $5 and Py = $10, and the consumer’s income is $220. Determine the optimum consumption basket.

This question cannot be solved using the usual tangency condition. However, you can see from the graph below that the optimum basket will necessarily lie on the “elbow” of some indifference curve, such as (5, 3), (10, 6) etc. If the consumer were at some other point, he could always move to such a point, keeping utility constant and decreasing his expenditure. The equation of all these “elbow” points is 3x = 5y, or y = 0.6x. Therefore the optimum point must be such that 3x = 5y.

The usual budget constraint must hold of course. That is, 5𝑥 + 10𝑦 = 220. Combining these two conditions, we get (x, y) = (20, 12).

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4.6 Jane likes hamburgers (H) and milkshakes (M). Her indifference curves are bowed in toward the origin and do not intersect the axes. The price of a milkshake is $1 and the price of a hamburger is $3. She is spending all her income at the basket she is currently consuming, and her marginal rate of substitution of hamburgers for milkshakes is 2. Is she at an optimum? If so, show why. If not, should she buy fewer hamburgers and more milkshakes, or the reverse?

From the given information we know that 𝑃𝐻 = 3, 𝑃𝑀 = 1, and 𝑀𝑅𝑆𝐻,𝑀 = 2. Comparing the 𝑃 3 MRSH,M to the price ratio, 𝑀𝑅𝑆𝐻,𝑀 = 2 < 𝑃 𝐻 = 1 𝑀

𝑃

Since these are not equal Jane is not currently at an optimum. In addition, we can say that 𝑃 𝐻 > 𝑀𝑈

𝑀𝑈

𝑀𝑈

𝑀

𝐻

𝑀

𝑀𝑅𝑆𝐻,𝑀 = 𝑀𝑈 𝐻 , which is equivalent to 𝑃 𝑀 > 𝑃 𝐻 . 𝑀

That is, the “bang for the buck” from milkshakes is greater than the “bang for the buck” from hamburgers. So Jane can increase her total utility by reallocating her spending to purchase fewer hamburgers and more milkshakes.

4.7 Ray buys only hamburgers and bottles of root beer out of a weekly income of $100.

He currently consumes 20 bottles of root beer per week, and his marginal utility of root beer is 6. The price of root beer is $2 per bottle. Currently, he also consumes 15 hamburgers per week, and his marginal utility of a hamburger is 8. Is Ray maximizing utility at his current consumption basket? If not, should he buy more hamburgers each week, or fewer?

Compare MUH/PH with MUR/PR, where the subscripts “H” and “R” refer respectively to hamburgers and root beer. We have all the information to make this comparison except for the price of a hamburger. But we can determine the price of a hamburger from Sam’s budget constraint: PHH + PRR = Income, or PH(15) + 2(20) = 100.

• • •

So PH = $4 per hamburger.

Now we can see that MUH/PH = 8/4 = 2 and MUR/PR = 6/2 = 3.

Since the “bang for the buck” is higher for root beer than for hamburgers, he should buy fewer hamburgers (and more root beer).

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4.8 Dave currently consumes 10 hot dogs and 6 sodas each week. At his current consumption basket, his marginal utility for hot dogs is 5 and his marginal utility for sodas is 3. If the price of one hot dog is $1 and the price of one soda is $0.50, is Dave currently maximizing his utility? If not, how should he reallocate his spending in order to increase his utility?

To determine if this situation is optimal, determine if the tangency condition holds. 𝑀𝑈 𝑀𝑈 5 3 𝑀𝑈 𝑀𝑈 Is 𝑃 𝐻 = 𝑃 𝑆 ? That is, is 1 = 0.50 ? No (5 ≠ 6). So 𝑃 𝐻 < 𝑃 𝑆 . 𝐻

𝑆

𝐻

𝑆

Since the tangency condition does not hold, Dave is not currently maximizing his utility. To increase his utility he should purchase more soda and fewer hot dogs (since the ‘bang for the buck’ for sodas is higher).

4.9 Helen’s preferences over CDs (C) and sandwiches (S) are given by U(S, C) = SC + 10(S + C), with MUC = S + 10 and MUS = C + 10. If the price of a CD is $9 and the price of a sandwich is $3, and Helen can spend a combined total of $30 each day on these goods, find Helen’s optimal consumption basket.

See the graph below. The fact that Helen’s indifference curves touch the axes should immediately make you want to check for a corner point solution.

To see the corner point optimum algebraically, notice if there was an interior solution, the tangency condition implies (S + 10)/(C +10) = 3, or S = 3C + 20. Combining this with the budget constraint, 9C + 3S = 30, we find that the optimal number of CDs would be given by 18𝐶 = −30 which implies a negative number of CDs. Since it’s impossible to purchase a negative amount of something, our assumption that there was an interior solution must be false.

Instead, the optimum will consist of C = 0 and Helen spending all her income on sandwiches: S = 10.

Graphically, the corner optimum is reflected in the fact that the slope of the budget line is steeper than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) = (0, 10) we have PC / PS = 3 > MRSC,S = 2. Thus, even at the corner point, the marginal utility per dollar spent on CDs is lower than on sandwiches. However, since she is already at a corner point with C = 0, she cannot give up any more CDs. Therefore the best Helen can do is to spend all her income on sandwiches: (C, S) = (0, 10). [Note: At the other corner with S = 0 and C = 3.3, PC / PS = 3 > MRSC,S = 0.75. Thus, Helen would prefer to buy more sandwiches and less CDs, which is of course entirely feasible at this corner point. Thus the S = 0 corner cannot be an optimum.]

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4.10 The utility that Corey obtains by consuming hamburgers (H) and steaks (S) is given 𝟎.𝟓 by 𝑼(𝑯, 𝑺) = √𝑯 + √𝑺 + 𝟒. The marginal utility of hamburgers is and the marginal utility of steaks is equal to

𝟎.𝟓

√𝑯

.

√𝑺+𝟒

a) Sketch the indifference curve corresponding to the utility level U = 12. b) Suppose that the price of hamburgers is $1 per hamburger, and the price of steak is $8 per steak. Moreover, suppose that Corey can spend $100 per month on these two foods.

Sketch Corey’s budget line for hamburgers and steak given this budget. c) Based on your answer to parts (a) and (b), what is Corey’s optimal consumption basket given his budget? a) Some points on the U = 12 indifference curve include S 0 5 12 21 32 45 60

H 100 81` 64 49 36 25 16

U 12 12 12 12 12 12 12

Connecting these points gives us the U = 12 indifference curve:

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b) The equation of the budget line is H + 8S = 100. Graphing this on the same axes as the U = 12 indifference curve gives us:

c) The optimal consumption basket is S = 0, H = 100, i.e., point R in the figure below. There are several ways to see this. One way is to sketch a few more indifference curves (each corresponding to a different level of utility). This picture strongly suggests that the point of maximum utility occurs at point R.

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Another way is to compare the marginal utility per dollar of spent on hamburger and the marginal utility per dollar spent on steak at point R. From the information given in the statement 0.5 0.5 of the problem, 𝑀𝑈𝐻 = and 𝑀𝑈𝑆 = , and so at point R √𝐻

√𝑆+4

0.5 𝑀𝑈𝑆 √0 + 4 1 = = 𝑃𝑠 8 32 0.5 𝑀𝑈𝐻 √100 1 = = 𝑃𝐻 1 20 Thus, at point R, the marginal utility per dollar spent on hamburger is greater than the marginal utility per dollar spent on steak, and so the consumer would like to purchase more hamburger and less steak. However, at point R, no further reduction in the quantity of steak is possible, and thus R is the optimal consumption basket.

4.11 This problem will help you understand what happens if the marginal rate of substitution is not diminishing. Dr. Strangetaste buys only French fries (F) and hot dogs (H) out of his income. He has positive marginal utilities for both goods, and his MRSH,F is increasing. The price of hot dogs is PH, and the price of French fries is PF. a) Draw several of Dr. Strangetaste’s indifference curves, including one that is tangent to his budget line. b) Show that the point of tangency does not represent a basket at which utility is maximized, given the budget constraint. Using the indifference curves you have drawn, indicate on your graph where the optimal basket is located.

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a)

F

Solutions Manual

Preference Directions

A B C H

b) At point A, Dr. Strangetaste’s indifference curve, which is bowed out from the origin, is tangent to his budget line. This point is not an optimum because, for example, Dr. Strangetaste could move to point B on his budget line and achieve a higher level of total utility. Point B, though, is not an optimum either because Dr. Strangetaste could move to point C, a corner point, to achieve an even higher level of total utility. When the MRS is increasing, a corner point optimum will occur (with F = 0 in this picture, though it could equivalently be with H = 0 for another set of indifference curves).

4.12 Julie consumes two goods, food and clothing, and always has a positive marginal utility for each good. Her income is 24. Initially, the price of food is 2 and the price of clothing is 2. After new government policies are implemented, the price of food falls to 1 and the price of clothing rises to 4. Suppose, under the initial budget constraint, her optimal choice is 10 units of food and 2 units of clothing. a) After the prices change, can you predict whether her utility will be higher, lower, or the same as under the initial prices? b) Does your answer require that there be a diminishing marginal rate of substitution of food for clothing? Explain.

As given, Julie consumes F = 10 and C = 2 with an income of 24.

Initially (with PF = PC = 2) she spends all her income: PFF + PCC = 2(10) + 2(2) = 24.

To buy her initial basket at the new prices, she would only need to spend PFF + PCC = 1(10) + 4(2) = 18.

Thus, her initial basket lies inside her new budget constraint (assuming her income stays at 24).

With her new budget line she would be able to choose a new basket to the “northeast” of (i.e., a basket involving more food and clothing than) her initial basket, making her better off.

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4.13 Toni likes to purchase round trips between the cities of Pulmonia and Castoria and other goods out of her income of $10,000. Fortunately, Pulmonian Airways provides air service and has a frequent-flyer program. Around trip between the two cities normally costs $500, but any customer who makes more than 10 trips a year gets to make additional trips during the year for only $200 per round trip. a) On a graph with round trips on the horizontal axis and “other goods” on the vertical axis, draw Toni’s budget line. (Hint: This problem demonstrates that a budget line need not always be a straight line.) b) On the graph you drew in part (a), draw a set of indifference curves that illustrates why Toni may be better off with the frequent-flyer program. c) On a new graph draw the same budget line you found in part (a). Now draw a set of indifference curves that illustrates why Toni might not be better off with the frequent-flyer program. a) The budget line will have a kink where round trips = 10 and other goods = 5,000.

Northwest of the kink, the budget line’s slope will be –500. Southeast of the kink, the slope will be –200.

Other 10,000

5,000 C

B Round Trips

10

20

35

b) With the indifference curves drawn on the above graph, Toni is better off with the frequent flyer program (at point B) than she would be without it (at point C). Without the frequent flyer program the best she could achieve is point C, which lies on the hypothetical budget line where the price of round trips is always $500. c) With the indifference curves drawn on graph below, Toni is no better off with the frequent flyer program than she would be without it (at point A). At this point, her indifference curve is tangent to a portion of the budget line where the frequent flyer program does not apply (less than 10 round trips).

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Other 10,000 A 5,000

Round Trips 10

20

35

4.14 A consumer has preferences between two goods, hamburgers (measured by H) and milkshakes (measured by M). His preferences over the two goods are represented by the utility function U = √H + √M. For this utility function MUH = 1/(2√H) and MUM = 1/(2√M). a) Determine if there is a diminishing MRSH,M for this utility function. b) Draw a graph to illustrate the shape of a typical indifference curve. Label the curve U1.

Does the indifference curve intersect either axis? On the same graph, draw a second indifference curve U2, with U2 > U1. c) The consumer has an income of $24 per week. The price of a hamburger is $2 and the price of a milkshake is $1. How many milkshakes and hamburgers will he buy each week if he maximizes utility? Illustrate your answer on a graph. a)

𝑀𝑈

1/(2√𝐻)

𝑀𝑅𝑆𝐻,𝑀 = 𝑀𝑈 𝐻 = 1/(2√𝑀) = 𝑀

√𝑀 √𝐻

This utility function has a diminishing marginal rate of substitution since 𝑀𝑅𝑆𝐻,𝑀 declines as H increases and M decreases. b)

Since it is possible to have U > 0 if either H = 0 (and M > 0) or M = 0 (and H > 0), the indifference curves will intersect both axes.

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c)

We know from the tangency condition that

Solutions Manual

M 2 = H 1 M = 4H

Substituting this into the budget line, 2𝐻 + 𝑀 = 24, yields

2 H + 4 H = 24 H =4

Finally, plugging this back into the tangency condition implies 𝑀 = 4(4) = 16. At the optimum the consumer will choose 4 hamburgers and 16 milkshakes. This can be seen in the graph above.

4.15 Justin has the utility function U = xy, with the marginal utilities MUx = y and MUy = x. The price of x is 2, the price of y is py, and his income is 40. When he maximizes utility subject to his budget constraint, he purchases 5 units of y. What must be the price of y and the amount of x consumed?

When Justin maximizes utility, his optimal consumption basket will be on the budget constraint and satisfy the tangency condition.

Any basket on the budget line will satisfy pxx + pyy = I, or 2x + 5py = 40.

The tangency condition requires that MUx / px = MUy / py, or that 5 / 2 = x / py. This implies that 5py = 2x.

Putting these two equations together reveals that 5py + 5py = 40; thus py = 4.

4.16 A student consumes root beer and a composite good whose price is $1. Currently, the government imposes an excise tax of $0.50 per six-pack of root beer. The student now purchases 20 six-packs of root beer per month. (Think of the excise tax as increasing the price of root beer by $0.50 per six-pack over what the price would be without the tax.) The government is considering eliminating the excise tax on root beer and, instead, requiring consumers to pay $10.00 per month as a lump sum tax (i.e., the student pays a tax of $10.00 per month, regardless of how much root beer is consumed). If the new proposal is adopted, how will the student’s consumption pattern (in particular, the amount of root beer consumed) and welfare be affected? (Assume that the student’s marginal rate of substitution of root beer for other goods is diminishing.)

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Solutions Manual

Composite Good BL1 B BL2 A Root Beer 20

Assume the student is initially at an interior optimum, point A. Denote the initial price of root beer by P and the student’s income as M. Point A then consists of RA = 20 units of root beer and YA = M – 20P units of the composite good. The effect of the proposal is to rotate the budget line outward (the price change) and then shift it inward (the lump sum tax), for a total movement from BL1 to BL2. Notice that BL2 intersects BL1 exactly at point A: under the proposal, (RA, YA) costs the student 20(P – 0.5) + M – 20P = M – 10, which is equal to her income under the proposal.

Because A was initially optimal, MRSR,Y = P at point A. Yet the price ratio along BL2 is (P – 0.5). Hence MRSR,Y > PR / PY, so the student can increase her utility by purchasing more root beer and less of the composite good, at a point such as B depicted in the graph above. Thus, the proposal will make the student better off.

4.17 When the price of gasoline is $2.00 per gallon, Joe consumes 1,000 gallons per year.

The price increases to $2.50, and to offset the harm to Joe, the government gives him a cash transfer of $500 per year. Will Joe be better off or worse off after the price increase and cash transfer than he was before? What will happen to his gasoline consumption? (Assume that Joe’s marginal rate of substitution of gasoline for other goods is diminishing.)

Other Goods B

A BL2 BL1 1000

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Gas

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Solutions Manual

Assume Joe is initially at an interior optimum, point A, and that the price of other goods is $1.

Let Joe’s income be M. Point A then consists of GA = 1000 units of gasoline and YA = M – 2000 units of other goods. The effect of the proposal is to rotate the budget line inward (the price change) and then shift it outward (the cash transfer), for a total movement from BL1 to BL2.

Notice that BL2 intersects BL1 exactly at point A: after the price increase, (GA, YA) costs Joe 1000*2.50 + M – 2000 = M + 500, which is equal to his income after the cash transfer.

Because A was initially optimal, MRSG,Y = 2 at point A. Yet the price ratio along BL2 is 2.5.

Hence MRSG,Y < PG / PY, so Joe can increase his utility by purchasing less gas and more of the composite good, at a point such as B depicted in the graph above. Thus, the proposal will make Joe better off.

4.18 Paul consumes only two goods, pizza (P) and hamburgers (H) and considers them to be perfect substitutes, as shown by his utility function: U(P, H) = P + 4H. The price of pizza is $3 and the price of hamburgers is $6, and Paul’s monthly income is $300. Knowing that he likes pizza, Paul’s grandmother gives him a birthday gift certificate of $60 redeemable only at Pizza Hut. Though Paul is happy to get this gift, his grandmother did not realize that she could have made him exactly as happy by spending far less than she did. How much would she have needed to give him in cash to make him just as well off as with the gift certificate?

Paul’s initial budget constraint is the line AC, allowing him to purchase at most 50 hamburgers or at most 100 pizzas. The $60 cash certificate shifts out his budget constraint without changing the maximum number of hamburgers that he can buy. The new budget constraint is ABD and he can now buy a maximum of 120 pizzas.

Hamburgers

60 55 50

B A

20

C 100

D 120

Pizza

Initially, Paul’s optimal basket contains all hamburgers and no pizza, at point A where (P, H) = (0, 50), because MUH /PH = 4/6 > MUP / PP = 1/3. His utility level at point A is U(0, 50) = 200.

When he gets the gift certificate, Paul’s optimal basket is at point B, spending all of his regular income on hamburgers and the $60 gift certificate on pizza. So point B is where (P, H) = (20, 50) with a utility of U(20, 50) = 220.

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Solutions Manual

However, Paul could also achieve a utility of 220 by consuming 220/4 = 55 hamburgers. To buy the extra 5 hamburgers he would require 5*6 = $30. So, if he had received a cash gift of $30 it would have made Paul exactly as well off as the $60 gift certificate for pizzas.

4.19 Jack makes his consumption and saving decisions two months at a time. His income this month is $1,000, and he knows that he will get a raise next month making his income $1,050. The current interest rate (at which he is free to borrow or lend) is 5 percent.

Denoting this month’s consumption by x and next month’s by y, for each of the following utility functions state whether Jack would choose to borrow, lend, or do neither in the first month. (Hint: In each case, start by assuming that Jack would simply spend his income in each month without borrowing or lending money. Would doing so be optimal?) a) U(x, y) = xy2, MUx = y2, MUy = 2xy b) U(x, y) = x2 y, MUx = 2xy, MUy = x2 c) U(x, y) = xy, MUx = y, MUy = x a)

𝑀𝑈

𝑦

In this case, 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥 = 2𝑥. If Jack neither borrows nor lends, then 𝑦

MRSx,y = 1050/(2*1000) = 0.525. Recall that if the interest rate is r, the slope of the budget line is –(1+r) = –1.05. Thus, if he neither borrows nor lends it will be the case that MRSx,y < 1 + r.

That is, the “bang for the buck” for spending this month (good x) is less than that for spending next month (good y). Thus, Jack should lend some of his income this month (so x < 1000) in order to earn interest and have higher spending next month (y > 1050). b) Now MRSx,y = 2y/x. If Jack neither borrows nor lends, MRSx,y = 2.1 > (1 + r). Thus, Jack could increase his utility by borrowing in the first month (so that x > 1000 and y < 1050). c) Now MRSx,y = y/x. If Jack neither borrows nor lends, MRSx,y = 1.05 = (1 + r). Thus, Jack’s utility is maximized when he neither borrows nor lends, simply spending all of his income in each month: (x, y) = (1000, 1050).

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Chapter 4-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.20 The figure below shows a budget set for a consumer over two time periods, with a borrowing rate rB and a lending rate rL, with rL < rB. The consumer purchases C1 units of a composite good in period 1 and C2 units in period 2. The following is a general fact about consumers making consumption decisions over two time periods: Let A denote the basket at which a consumer spends exactly his income each period (the point at the kink of the budget line). Then a consumer with a diminishing MRSC1,C2 will choose to borrow in the first period if at basket A MRSC1,C2 > 1 + rB and will choose to lend if at basket A MRSC1,C2 < 1 + rL. If the MRS lies between these two values, then he will neither borrow nor lend.

(You can try to prove this if you like. Keep in mind that diminishing MRS plays an important role in the proof.)

Using this rule, consider the decision of Meg, who earns $2,000 this month and $2,200 the next with a utility function given by U(C1, C2) = C1C2, where the C’s denote the value of consumption in each month. Suppose rL=0.05 (the lending rate is 5 percent) and rB = 0.12 (the borrowing rate is 12 percent). Would Meg borrow, lend, or do neither this month?

What if the borrowing rate fell to 8 percent?

The utility function implies that MRSC1,C2 = C2 / C1. At point A, MRSC1,C2 = 1.10, which lies between (1 + rL) = 1.05 and (1 + rB) = 1.12. Therefore Meg will neither borrow nor lend and will simply spend her entire income each month.

If the borrowing rate falls to 8%, then the lower part of the budget line pivots outward, as depicted in the graph below. Then at point A, MRSC1,C2 > (1+ rB) > (1 + rL) since 1.10 > 1.08 > 1.05. So Meg can increase her utility by moving away from point A to a point like B, where she borrows money, spending more than her income this month (C1 > 2000) and less than her income next month (C2 < 2200).

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Chapter 4-24

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.21 Sally consumes housing (denote the number of units of housing by h) and other goods (a composite good whose units are measured by y), both of which she likes. Initially she has an income of $100, and the price of a unit of housing (Ph) is $10. At her initial basket she consumes 2 units of housing. A few months later her income rises to $120; unfortunately, the price of housing in her city also rises, to $15. The price of the composite good does not change. At her later basket she consumes 1 unit of housing. Using revealed preference analysis (without drawing indifference curves), what can you say about how she ranks her initial and later baskets?

Other

B C A

BL2 BL1 H

With the initial budget line, BL1, Sally chooses point A, where (xA, yA) = (2, 80). When her income and the price of housing increase, the budget line becomes BL2 and she chooses point B, where (xB, yB) = (1, 105). Importantly, because the equation for BL1 is 10x+ y = 100 and that for BL2 is 15x + y = 120, we can solve to see that these lines intersect at x = 4, to the right of point A on BL1.

Consider a hypothetical basket C on BL2 but northeast of A. We can then deduce that 𝐵 ≻ 𝐴, because (i) B is at least as preferred as C (since B was chosen when C was affordable), and (ii) C is strictly preferred to A (since C lies to the northeast of A). By transitivity, B must be strictly preferred to A.

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Chapter 4-25

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.22 Samantha purchases food (F) and other goods (Y ) with the utility function U = FY. Her income is 12. The price of a food is 2 and the price of other goods 1. a) How many units of food does she consume when she maximizes utility? b) The government has recently completed a study suggesting that, for a healthy diet, every consumer should consume at least F = 8 units of food. The government is considering giving a consumer like Samantha a cash subsidy that would induce her to buy F = 8. How large would the cash subsidy need to be? Show her optimal basket with the cash subsidy on an optimal choice diagram with F on the horizontal axis and Y on the vertical axis. c) As an alternative to the cash subsidy in part (b), the government is also considering giving consumers like Samantha food stamps, that is, vouchers with a cash value that can only be redeemed to purchase food. Verify that if the government gives her vouchers worth $16, she will choose F = 8. Illustrate her optimal choice on an optimal choice diagram. (You may use the same graph you drew in part (b).)

a) MUY = F and MUF = Y, so MRSF,Y = Y/F, which diminishes as F increases along an indifference curve. Since the indifference curves do not intersect either axis, an optimal basket will be interior. At such an optimum two conditions must be satisfied: (1) tangency: MRSF,Y = PF / PY, or Y = 2F, and (2) budget line (“BL No subsidy” in the graph): 2F + Y = 12. This F = 3 and Y = 6. This optimum is depicted as point A in the graph. b) We need to find an interior optimum with F = 8. As income increases, the consumer chooses a basket along the Income Consumption Curve, which consists of the tangency points Y = 2F. So Y = 2(8) = 16. Total expenditure will then be 2F + Y = 2(8) + 16 = 32. So the consumer needs an income of 32 (“BL Cash” in the graph). Since the consumer has an income of 12, she needs an additional income of 20 (=32 – 12). So the subsidy needed is 20. This optimum is shown as point B in the graph. c) From part (b) we see that, with no restrictions on how the government subsidy can be spent, the consumer would like to buy 16 units of Y, more than her own budget (without subsidy) would permit. So we expect that with food stamps, she will use the voucher to purchase

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Chapter 4-26

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

the required 8 units of food and spend all of her own unrestricted income (12) on Y. In other words, this consumer will be at point C on the graph, at the kink on the budget constraint RCS (labeled “BL Food Stamps”).

We can verify that (F = 8, Y = 12) is her optimal choice by looking at the “bang for the buck” condition at C. MUF/price of food = Y/2 = 12/2 = 6. MUY/price of Y = F/1 = 8. So the consumer would like to substitute more Y for F, but cannot do so because at basket C she has purchased all the other goods she can, given her budget constraint.

4.23 As shown in the following figure, a consumer buys two goods, food and housing, and likes both goods. When she has budget line BL1, her optimal choice is basket A. Given budget line BL2, she chooses basket B, and with BL3, she chooses basket C.

a) What can you infer about how the consumer ranks baskets A, B, and C? If you can infer a ranking, explain how. If you cannot infer a ranking, explain why not. b) On the graph, shade in (and clearly label) the areas that are revealed to be less preferred to basket B, and explain why you indicated these areas. c) On the graph, shade in (and clearly label) the areas that are revealed to be (more) preferred to basket B, and explain why you indicated these areas.

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Chapter 4-27

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) From figure 4.21 we can infer that 𝐴 ≻ 𝐵, that 𝐵 ≻ 𝐶, and therefore by transitivity that 𝐴 ≻ 𝐶.

Housing A D B E C BL1

BL2

BL3 Food

First, A must be strictly preferred to B since A is at least as preferred as D and D must be strictly preferred to B. Second, B must be strictly preferred to C since B is at least as preferred as E and E is strictly preferred to C. So by transitivity A must be strictly preferred to C. b) A Baskets strictly preferred to B B C

Baskets less preferred to B

B will be strictly preferred to everything inside BL2. In addition, since B is strictly preferred to C, B will be strictly preferred to everything below BL3, including all the points along BL3 itself (since C is weakly preferred to everything on BL3). c) See the graph in part (b). Everything to the northeast of B is strictly preferred to B. In addition, since A is strictly preferred to B [see part (a)], everything to the northeast of A must also be strictly preferred to B. Notice, however, that there are points on BL1 (both northwest and southeast of A) about which we cannot infer anything.

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Chapter 4-28

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.24 The following graph shows the consumption decisions of a consumer over bundles of x and y, both of which he likes. When faced with budget line BL1, he chose basket A, and when faced with budget line BL2, he chose basket B. If he were to face budget line BL3, what possible set of baskets could he choose in order for his behavior to be consistent with utility maximization?

Let point E denote the intersection of BL1 and BL3, and point F denote the intersection of BL2 and BL3 (see the figure below).

E G A

BL3

B F BL1

BL2

First, any point to the northwest of E, including E, is in the consumer’s budget set when he faces budget constraint BL1. The fact that he chose A over these points implies that A is at least as preferred to E and strictly preferred to the points northwest of E. However, point G lies on BL3 and is northeast of A, so 𝐺 ≻ 𝐴. Therefore, by transitivity G is strictly preferred to E and all the points on BL3 northwest of E.

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Chapter 4-29

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Similarly, A is northeast of B so𝐴 ≻ 𝐵. Since B is at least as preferred as any point on BL2, including F, by transitivity we know that A is strictly preferred to F and all points on BL3 to the southeast of F. And since 𝐺 ≻ 𝐴, we know that G is strictly preferred to these points as well.

Therefore the consumer could choose any point between E and F on BL3, but neither E nor F themselves.

4.25 Darrell has a monthly income of $60. He spends this money making telephone calls home (measured in minutes of calls) and on other goods. His mobile phone company offers him two plans: • Plan A: pay no monthly fee and make calls for $0.50 per minute.

• Plan B: Pay a $20 monthly fee and make calls for $0.20 per minute.

Graph Darrell’s budget constraint under each of the two plans. If Plan A is better for him, what is the set of baskets he may purchase if his behavior is consistent with utility maximization? What baskets might he purchase if Plan B is better for him?

Let x denote the number of phone calls, and y denote spending on other goods. Under Plan A, Darrell’s budget line is JLM. Under Plan B, it is JKLN. These budget lines intersect at point L, or about x = 67.

y 60 40

J

K

L M 67

N 120

200

x

If we know that Darrell chooses Plan A, his optimal bundle must lie on the line segment JL. No point between L and M would be optimal under this plan because then Darrell could have chosen a point under Plan B, between L and N, that would have given him more minutes, and left him with more money to buy other goods. However, we cannot exclude point L itself (Darrell could, for instance, have perfect complements preferences with an “elbow” at point L). Thus, if Darrell chooses Plan A his optimal basket could be anywhere between J and L, including either of these points.

Similarly, if he chose Plan B then his optimal basket must lie between L and N. Any point between L and K (but not including point L) would be dominated by a point under Plan A between L and J. Thus, if Darrell chooses Plan B his optimal basket could be anywhere between L and N, including either of these points.

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Chapter 4-30

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Solutions Manual

4.26 Figure 4.17 illustrates the case in which a consumer is better off with a quantity discount. Can you draw an indifference map for a consumer who would not be better off with the quantity discount?

Other Goods

A

Electricity

With this set of indifference curves, the tangency with the budget line occurs on the portion of the budget line where the quantity discount has not taken effect. Therefore, the consumer does not receive any benefit from the quantity discount.

4.27 Angela has a monthly income of $120, which she spends on MP3s and a composite good (whose price you may assume is $1 throughout this problem). Currently, she does not belong to an MP3 club, so she pays the retail price of an MP3 of $2; her optimal basket includes 20 MP3s monthly.

For the past several months Asteroid, a media company, has offered her the chance to join their “Premium Club”; to join the club she would need to pay a membership fee of $60 per month, but then she could buy all the MP3s she wants at a price of $0.50. She has decided not to join the club.

Asteroid has now introduced an “Economy Club”; to join, Angela would need to pay a membership fee of $30 per month, but then she could buy all the MP3s she wants at a price of $1. Draw a graph illustrating (1) Angela’s budget line and optimal basket when she joins no club, (2) the budget line she would have faced had she joined the Premium Club, and (3) her budget line if she joins the Economy Club. Will Angela surely want to join the Economy Club? If she were to join the club, how many MP3s per month might she buy?

Show how you arrive at your answers using a revealed preference argument.

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Chapter 4-31

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The budget lines for the “no club,” “Economy Club,” and “Premium Club” opportunities available to Angela are drawn below.

She chooses A with “no club,” and chooses not to join the “Premium Club” when given the chance. Thus, A is weakly preferred to any basket on WZ, and strongly preferred basket on the segment CS (except for C). So, if she does join the Economy Club, she would not choose a basket on CS (except possibly for C).

With “no club” she chooses A when she could have afforded B; thus A > B. Since the rest of the segment RB lies inside the “no club” BL, these baskets are strongly inferior to A. So, if she does join the Economy Club, she would not choose a basket on RB (except possibly for B).

To summarize, if she joins the Economy Club, she might consume any basket on the segment BC, including B and C. So her consumption in that case would be 30 < MP3s < 60.

But she might not join the Economy Club at all! Although we have established that A > B and A > C, we cannot say how she ranks A with the baskets between B and C.

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Chapter 4-32

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.28 Alex buys two goods, food (F) and clothing (C). He likes both goods. His preferences for the goods do not change from month to month. The following table shows his income, the baskets he selected, and the prices of the goods over a two-month period.

Month 1 2

PF 3 2

PC 2 4

Income 48 48

Basket Chosen F=16, C=0 F=14, C=5

a) On the graph with F on the horizontal axis and C on the vertical axis, plot and clearly label the budget lines and consumption baskets during these two weeks. Label the consumption bundle in week 1 by point A on the graph and the consumption basket in week 2 by point B. Using revealed preference analysis, what can you say about Alex’s preferences for baskets A and B (i.e., how does he rank them)? b) In month 3 Alex’s income rises to 57. The prices of food and clothing are both 3.

Assuming his preferences do not change, describe the set of baskets he might consume in month 3 if he continues to maximize utility. Show this set of baskets in the graph. a)

C 24 22 20

BL1

18 16 14

BL3

12 10

BL2

E =(10,9)

8 6

B=(14,5) C

4 2 A 0

2

4

6

8

10

12

14

16

F 18

20

22

24

B is weakly preferred to C, which is strongly preferred to A. By transitivity, B is strongly preferred to A. b)

We know that he will choose a basket on BL3.

He will not choose any basket on or inside BL1. Why? He could choose B, and B>A (by part A).

Given BL1, he chose A instead of other baskets on BL1, all of which were affordable.

He will not choose a basket on BL3 to the southeast of B. Why? Given BL2 he chose B, and B is strictly preferred to any basket inside BL3, including those on BL2 to the southeast of B. Therefore, he might choose any basket on BL3 between B and E, not including E. This set of baskets is illustrated with the heavy line in the graph.

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Chapter 4-33

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Solutions Manual

4.29 Brian consumes units of electricity (E) and a composite good (Y), whose price is always $1. He likes both goods.

In period 1 the power company sets the price of electricity at $7 per unit, for all units of electricity consumed. Brian consumes his optimal basket, 20 units of electricity and 70 units of the composite good.

In period 2 the power company then revises its pricing plan, charging $10 per unit for the first 5 units and $4 per unit for each additional unit. Brian’s income is unchanged. Brian’s optimal basket with this plan includes 30 units of electricity and 60 units of the composite good. In period 3 the power company allows the consumer to choose either the pricing plan in period 1 or the plan in period 2. Brian’s income is unchanged. Which pricing plan will he choose? Illustrate your answer with a clearly labeled graph.

Brian’s income is I = (7)(20) + 1(70) = $210.

His initial budget line is the solid curve RJ. His initial optimal basket is A. His new budget line is the dotted, piecewise linear curve RST. He chooses basket B, which costs 10(5) + 4(30 – 5) + 1(60) = 210.

In period 2 Basket A costs (10)(5) + (4)(20 – 5) + 1(70) = $180, so Basket A lies inside the new budget line.

In period 3 Brian will choose the plan from the second period. B is weakly preferred to a basket like C, which, in turn is strictly preferred to A (because C lies to the northeast of A). Thus he strictly prefers B to A, and he can only reach B by choosing the plan from period 2.

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Chapter 4-34

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Solutions Manual

4.30 Carina consumes two goods, X and Y, both of which she likes. In month 1 she chooses basket A given budget line BL1. In month 2 she chooses B given budget line BL2, and in month 3 she chooses C given budget lineBL3. Assume her indifference map is unchanged over the three months. Use the theory of revealed preference to show whether her choices are consistent with utility maximizing behavior. If so, show how she ranks the three baskets. If it is not possible to infer how she ranks the baskets, explain why not.

From BL1 we would infer that A is weakly preferred to S, and that S is strongly preferred to C; by transitivity we conclude that A is strongly preferred to C. Y

BL2 B BL3 R A

BL1 S C

X

From BL3 we would infer that C is weakly preferred to R, which is strongly preferred to A; by transitivity we conclude that C is strongly preferred to A. It cannot be simultaneously true that A is strongly preferred to C and C is strongly preferred to A, for this would imply that A is strongly preferred to C, which is strongly preferred to A, or that A is strongly preferred to itself. The preferences are either intransitive, or else the consumer is failing to maximize utility in each of the three time periods.

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Chapter 4-35

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4.31 Julie’s preferences for food (measured by F) and clothing (measured by C) are described by the utility function U(F,C) = FC. Her marginal utilities are MUF = C and MUC = F . Suppose food costs $1 per unit and clothing costs $2 per unit. She has an income of $12 to spend on food and clothing. a) With Julie’s utility function, is it possible that the utility-maximizing basket of food and clothing will be at a corner point, with either F = 0 or C = 0? Do you expect the budget constraint to be binding? b) Using the method of Lagrange, find Julie’s optimal consumption basket (F,C) when income is $12. c) Find the numerical value of the Lagrange multiplier, λ, which measures Julie’s marginal ∆𝑼 utility of income (the rate of change ) when her income is exactly $12. ∆𝑰

d) Find Julie’s optimal consumption basket (F,C) and the value of λ if her income is $13. e) Show that the increase in Julie’s utility when her income rises from $12 to $13 is close to the values of 𝝀 you found in parts (c) and (d). a) Since Julie has a Cobb-Douglas utility function, we know that her optimal basket will be interior, with F > 0 and C > 0. Since both marginal utilities are positive, we know that the optimal basket will lie on the budget line. We therefore know that Julie would be able to increase utility if she had a higher income, so the Lagrange multiplier 𝜆 will be positive at the utilitymaximizing basket. b)

For this problem the Lagrangian function is:

Ʌ(𝐹, 𝐶, 𝜆) = 𝐹𝐶 − 𝜆(𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼) = 𝐹𝐶 − 𝜆(𝐹 + 2𝐶 − 𝐼) Since the optimal basket will be interior, the necessary conditions are: 𝑀𝑈𝐹 − 𝜆𝑃𝐹 = 0 => 𝐶 − 𝜆 = 0 𝑀𝑈𝐶 − 𝜆𝑃𝐶 = 0 => 𝐹 − 2𝜆 = 0 𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼 = 0 => 𝐹 + 2𝐶 − 12 = 0 We have three equations and three unknowns. When we combine the first two equations, we find that Julie should equate the marginal utility per dollar spent on each good. 𝜆=

𝑀𝑈𝐹 𝑀𝑈𝐶 = 𝑃𝐹 𝑃𝐶

=> 𝜆 =

𝐶 𝐹 = 1 2

𝑎𝑛𝑑 𝐹 = 2𝐶 .

Substituting 𝐹 = 2𝐶 into the budget constraint, we find that 𝐹 + 2𝐶 − 12 = 0 => 2𝐶 + 2𝐶 − 12 = 0

Copyright © 2020 John Wiley & Sons, Inc.

=> 𝐶 = 3 and 𝐹 = 6 .

Chapter 4-36

Besanko & Braeutigam – Microeconomics, 6th edition

𝐶

𝐹

Solutions Manual

3

6

∆𝑈

c) In part (b) we found that 𝜆 = 1 = 2 => 𝜆 = 1 = 2 = 3. This is the rate of change ∆𝐼 when Julie’s income is exactly 12. If her income were to rise by a dollar, we would expect to see her utility increase by about 3. d)

The two conditions that must be satisfied when income is 13 are:

(1) The tangency condition, 𝐹 = 2𝐶, and the budget constraint 𝐹 + 2𝐶 − 13 = 0.

Together these conditions tell us that Julie’s optimal basket will now contain F = 6.5 and C = 3.25. 𝐶

𝐹

Also, 𝜆 = 1 = 2 =

3.25 1

6.5

= 2 = 3.25.

e) When Julie’s income is 12, she chooses C = 3 and F = 6. Her utility is U = FC = (6)(3) = 18. When her income is 13, she chooses F = 6.5 and C = 3.25. Her utility is U = FC = (6.5)(4.25) = 21.25.

Julie’s utility increased by 3.25 (from 18 to 21.25) when her income increased by 1 (from 12 to 13). This is close to the values of 𝜆 we found in parts (c) and (d), as we expected.

4.32 The utility that Ann receives from consuming food (measured by F) and clothing (measured by C) is described by the utility function U(F,C) = FC + F. Her marginal utilities are MUF = C+1 and MUC = F. Suppose food costs $1 per unit and clothing costs $2 per unit.

She has an income of $22 to spend on food and clothing. a) Show that Ann has a diminishing marginal rate of substitution of food for clothing. Why is this important in applying the method of Lagrange to the consumer choice problem? b) With Ann’s utility function, is it possible that the utility-maximizing basket of food and clothing will be at a corner point, with either F = 0 or C = 0? Do you expect the budget constraint to be binding? c) Using the method of Lagrange, find Ann’s optimal consumption basket (F,C) when income is $22. d) Find the numerical value of the Lagrange multiplier, λ, which measures Ann’s marginal ∆𝑼 utility of income (the rate of change ∆𝑰 ) when her income is $22. e) Find Ann’s optimal consumption basket (F,C) and the value of λ if her income is $23. f) Show that the increase in Ann’s utility when her income rises from $22 to $23 is close to the values of 𝝀 you found in parts (d) and (e).

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Chapter 4-37

Besanko & Braeutigam – Microeconomics, 6th edition

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a) The marginal utilities for the two goods are MUF = C+1 and MUC = F. Since both marginal utilities are positive, we know that the indifference curves are negatively sloped.

When we increase F along an indifference curve, the level of C must therefore fall. 𝑀𝑈 𝐶+1 We know that 𝑀𝑅𝑆𝐹,𝐶 = 𝑀𝑈𝐹 = 𝐹 . As we increase F along an indifference curve (and C 𝐶

falls), the value of 𝑀𝑅𝑆𝐹,𝐶 falls. Therefore, we do have diminishing 𝑀𝑅𝑆𝐹,𝐶 .

This is important because it guarantees that the solution we find using the method of Lagrange will maximize utility while satisfying the budget constraint. b) Since U(F,C) = FC + F, it is possible to have a positive level of utility when C = 0. So it would be possible to have either an interior optimum (with F > 0 and C > 0) or a corner point solution (with F > 0 and C = 0). Since both marginal utilities are positive, we know that the optimal basket will lie on the budget line. We therefore know that Ann would be able to increase utility if she had a higher income, so the Lagrange multiplier 𝜆 will be positive. c)

For this problem the Lagrangian function is: Ʌ(𝐹, 𝐶, 𝜆) = (𝐹𝐶 + 𝐹) − 𝜆(𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼) = (𝐹𝐶 + 𝐹) − 𝜆(𝐹 + 2𝐶 − 22)

Since the budget constraint will be binding, the solution will be one with 𝜆 > 0. As noted in part (b), the utility-maximizing basket could be interior (with F > 0 and C > 0), but it might also be at a corner point. Let’s first assume that the optimum is interior. If we cannot find an interior optimum, we can then look for an optimum at a corner point. Assuming the optimum is interior, the necessary conditions are: 𝑀𝑈𝐹 − 𝜆𝑃𝐹 = 0 => 𝐶 + 1 − 𝜆 = 0 𝑀𝑈𝐶 − 𝜆𝑃𝐶 = 0 => 𝐹 − 2𝜆 = 0 𝑃𝐹 𝐹 + 𝑃𝐶 𝐶 − 𝐼 = 0 => 𝐹 + 2𝐶 − 22 = 0 We have three equations and three unknowns. When we combine the first two equations, we find that Ann should equate the marginal utility per dollar spent on each good. 𝜆=

𝑀𝑈𝐹 𝑀𝑈𝐶 = 𝑃𝐹 𝑃𝐶

=> 𝜆 =

𝐶+1 𝐹 = 1 2

𝑎𝑛𝑑 𝐹 = 2𝐶 + 2 .

Substituting 𝐹 = 2𝐶 + 2 into the budget constraint, we find that 𝐹 + 2𝐶 − 22 = 0 => 2𝐶 + 2 + 2𝐶 − 22 = 0

=> 𝐶 = 5 and 𝐹 = 12 .

So we have indeed found the optimum to be interior. d)

In part (c) we found that 𝜆 =

∆𝑈

𝐶+1 1

𝐹

=2

6

12

=> 𝜆 = 1 = 2 = 6. This is the rate of change

when Ann’s income is exactly 12. If her income were to rise by a dollar, we would expect to see her utility increase by about 6. ∆𝐼

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Solutions Manual

e) The two conditions we would need to satisfy when income is 23 are: (1) The tangency condition, 𝐹 = 2𝐶 + 2, and the budget constraint 23 − 𝐹 − 2𝐶 = 0.

Together these conditions tell us that Ann’s optimal basket will now contain F = 12.5 and C = 5.25.

Also, 𝜆 =

𝐶+1 1

𝐹

=2=

6.25 1

=

12.5 2

= 6.25.

f) When Ann’s income is 22, she chooses C = 5 and F = 12. Her utility is U = FC +F = (12)(5)+12 = 72.

When her income is 23, she chooses C = 5.25 and C = 12.5. Her utility is U = FC +F = (12.5)(5.25) + 12.5 = 78.125.

Her utility increased by 6.125 (from 72 to 78.125) when her income increased by 1 (from 22 to 23). This is close to the values of 𝜆 we found in parts (d) and (e), as we expected.

4.33 Omar consumes only two goods, whose quantities are measured by x and y. His preferences are described by the utility function U(x,y) = xy + 10(x + y). His marginal utilities are MUx = y+10 and MUy = x+10. The prices of the goods are PX = $9 and Py = $3.

He has a daily income of $30. a) Show that Omar has a diminishing marginal rate of substitution of x for y. Why is this important in applying the method of Lagrange to the consumer choice problem? b) With Omar’s utility function, is it possible that the utility-maximizing basket (x,y) will be at a corner point, with either x = 0 or y = 0? Do you expect the budget constraint to be binding? c) Using the method of Lagrange, find Omar’s optimal consumption basket (x,y) when his income is $30.

Also find the numerical value of the Lagrange multiplier, λ, which measures Omar’s ∆𝑼 marginal utility of income (the rate of change ∆𝑰 ) when his income is $30. d) Find Omar’s optimal consumption basket (x,y) and the value of λ if his income is $31. e) Show that the increase in Omar’s utility when his income rises from $30 to $31 is close to the values of 𝝀 you found in parts (d) and (e).

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Solutions Manual

a) The marginal utilities for the two goods are MUx = y+10 and MUy = x+10. Since both marginal utilities are positive, we know that the indifference curves are negatively sloped.

When we increase x along an indifference curve, the level of y must therefore fall. 𝑀𝑈

𝑦+10

We know that 𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈𝑥 = 𝑥+10 . As we increase x along an indifference curve (and y 𝑦

falls), the value of 𝑀𝑅𝑆𝑥,𝑦 falls. Therefore, we do have diminishing 𝑀𝑅𝑆𝑥,𝑦 .

This is important because it guarantees that the solution we find using the method of Lagrange will maximize utility while satisfying the budget constraint. b) Omar could achieve a positive level of utility when x > 0 and y = 0, in which case U(x,y) = 10x. So he might choose a corner point along the x axis.

He also could achieve a positive level of utility when x = 0 and y > 0, in which case U(x,y) = 10y. So he might choose a corner point along the y axis.

He might also choose an interior optimum, with U(x,y) = xy + 10(x + y).

Since both marginal utilities are positive, we know that the optimal basket will lie on the budget line. Since Omar would be able to increase utility if he had a higher income, the Lagrange multiplier 𝜆 will be positive. c)

For this problem the Lagrangian function is: Ʌ(𝑥, 𝑦, 𝜆) = 𝑥𝑦 + 10(𝑥 + 𝑦) − 𝜆(𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼) = 𝑥𝑦 + 10(𝑥 + 𝑦) − 𝜆(9𝑥 + 3𝑦 − 30)

One thing we know for sure, as noted in part (b), is that the budget constraint will be binding. So 𝜆 > 0 at the utility-maximizing solution. However, we do not yet know whether the utilitymaximizing basket is interior or at a corner point.

Let’s start by assuming that the solution will be interior, with x > 0 and y > 0. We can then see if the values of x, y, and λ calculated under that assumption are all positive. If so, we will have found that the solution is interior. If not, we can then find the corner point that maximizes utility.

If we do have an interior solution (with x > 0 and y > 0), the three necessary conditions for an optimum correspond to equations (4.17), (4.18), and (4.19) in the text: 𝑀𝑈𝑥 − 𝜆𝑃𝑥 = 0 => 𝑦 + 10 − 9 𝜆 = 0 𝑀𝑈𝑦 − 𝜆𝑃𝑦 = 0 => 𝑥 + 10 − 3 𝜆 = 0 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼 = 0 => 9𝑥 + 3𝑦 − 30 = 0

𝑥>0 𝑦>0 𝜆>0

We now have three equations and three unknowns. If Omar can do so, he should equate the marginal utility per dollar spent on each good. Combining the first two equations, we see that equating the marginal utility per dollar spent would require that:

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Besanko & Braeutigam – Microeconomics, 6th edition

𝜆=

𝑀𝑈𝑥 𝑀𝑈𝑦 = 𝑃𝑥 𝑃𝑦

=> 𝜆 =

𝑦 + 10 𝑥 + 10 = 9 3

Solutions Manual

𝑎𝑛𝑑 𝑦 = 3𝑥 + 20.

Substituting 𝑦 = 3𝑥 + 20 into the budget constraint, we find that 9𝑥 + 3𝑦 − 30 = 0 => 9𝑥 + 3(3𝑥 + 20) − 30 = 0

5

=> 𝑥 = − 3 and 𝑦 = 15 .

But x cannot be negative at an interior optimum, so our assumption that we have an interior optimum cannot be true! We tried to find an interior basket which equalizes the marginal utility per dollar spent on each good, and we could not find one. So the utility-maximizing basket must be at a corner point, with either x = 0 or y = 0.

Since we found a negative value for x when we tried to equate the marginal utility per dollar spent on each good, let’s next assume that the corner point solution is the one with x = 0 and y > 0. In that case the three conditions for an optimum are like the ones in equations (4.22), (4.23), and (4.24) in the text: 𝑀𝑈𝑥 − 𝜆𝑃𝑥 ≤ 0 𝑀𝑈𝑦 − 𝜆𝑃𝑦 = 0 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼 = 0

=> 𝑦 + 10 − 9𝜆 ≤ 0 => 𝑥 + 10 − 3𝜆 = 0 => 9𝑥 + 3𝑦 − 30 = 0

𝑥=0 𝑦>0 𝜆>0

With x = 0, these equations simplify to 𝑦 + 10 − 9𝜆 ≤ 0 10 − 3𝜆 = 0 3𝑦 − 30 = 0

𝑥=0 𝑦>0 𝜆>0

The third equation tells us that y = 10; Omar spends all of his $30 income to buy 10 units of y.

10 The second equation reveals that 𝜆 = 3 .

10

Finally, the first inequality is satisfied: 𝑦 + 10 − 9𝜆 = 10 + 10 − 9 ( 3 ) = −10 ≤ 0.

10

So we have found that Omar’s utility-maximizing basket contains (x,y) = (0,10), and that 𝜆 = 3 . d) Since the change in income is small (from $30 to $31), we will assume that Omar’s optimal basket will still be at a corner point with x = 0. If so, with an income of 31, the necessary conditions in (c) will be 𝑦 + 10 − 9𝜆 ≤ 0𝑥 = 0 10 − 3𝜆 = 0𝑦 > 0 3𝑦 − 31 = 0𝜆 > 0 31

31

The third equation tells us that y = 3 ; Omar spends all of his $31 income to buy 3 units of y.

10

The second equation reveals that 𝜆 = 3 , the same value we found when income was $30.

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Solutions Manual

31

10

29

Finally, the first inequality is now satisfied: 𝑦 + 10 − 9𝜆 = 3 + 10 − 9 ( 3 ) = − 3 ≤ 0.

31

10

So we have found that Omar’s utility-maximizing basket contains (x,y) = (0, 3 ), and that 𝜆 = 3 . e)

When Omar’s income is $30, he chooses x = 0 and y = 10.

300

His utility is U(x,y) = xy + 10(x + y) = 10y = 10(10) = 100 = 3 .

31

When his income is $31, he chooses x = 0 and y = 3 .

31

310

His utility is U(x,y) = xy + 10(x + y) = 10y = 10( 3 ) = 3 .

10

300

310

His utility increased by 3 (from 3 to 3 ) when his income increased by 1 (from 30 to 31). ∆𝑈

10

Thus, ∆𝐼 = 3 , which is exactly the value of 𝜆 we found in parts (c) and (d). In this problem the value of 𝜆 is constant as income changes. In other words, Omar’s marginal utility of income is constant.

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Solutions Manual

Chapter 5 The Theory of Demand Solutions to Review Questions 1.

What is a price consumption curve for a good?

The price consumption curve plots the set of optimal bundles for two goods, say X and Y, by changing the price of one good while holding the price of the other good and income constant.

2.

How does a price consumption curve differ from an income consumption curve?

The price consumption curve plots the set of optimal bundles for two goods as the price of one good changes while the price of the other good and income remain constant. The income consumption curve, on the other hand, plots the set of optimal bundles for two goods as the consumer’s income changes while holding the prices of both goods constant.

3.

What can you say about the income elasticity of demand of a normal good? of an inferior good?

With a normal good, when income increases, consumption of the good will increase. This implies the income elasticity for a normal good will be positive. With an inferior good, when income increases consumption of the good will decrease. This implies the income elasticity for an inferior good will be negative.

4.

If indifference curves are bowed in toward the origin and the price of a good drops, can the substitution effect ever lead to less consumption of the good?

If indifference curves are bowed in toward the origin and the price of, say, good X falls, consumption of X will always increase; so the substitution effect will always be positive. A decrease in the price of X implies that the slope of the budget line becomes flatter. When indifference curves are bowed in, a direct consequence of this change in relative prices is that any tangency will occur “southeast” of the original bundle along the initial indifference curve.

The only way for consumption to fall when price falls is for the income effect to be negative (an inferior good) and for its magnitude to more than offset the substitution effect. In this rare situation, the good is known as a Giffen good.

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Chapter 5-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.

Suppose a consumer purchases only three goods, food, clothing, and shelter. Could all three goods be normal? Could all three goods be inferior? Explain.

If the consumer purchases only three goods and income increases, it is possible that consumption of all three goods will increase. For example, the consumer might allocate one-third of the increase to each of the three goods. Thus, it is possible for all three goods to be normal. If the consumer purchases only three goods and income increases, it is not possible that consumption of all three goods will decrease. Recall that if consumption falls when income increases the good is inferior. If this were to occur, the consumer would be spending less income than he did prior to the income increase. Thus, it is not possible for all three goods to be inferior.

6.

Does economic theory require that a demand curve always be downward sloping? If not, under what circumstances might the demand curve have an upward slope over some region of prices?

Generally speaking demand curves are downward sloping. Economic theory, however, suggests a special case of an inferior good whose negative income effect is greater than its positive substitution effect. In this event, consumption of the good falls when the price of the good falls.

This type of good is known as a Giffen good. While economic theory suggests that such a good could exist, in practice no such good has been confirmed for humans (although the text suggests an experiment on rats where a good was determined to be a Giffen good).

7.

What is consumer surplus?

Consumer surplus is the difference between the maximum amount a consumer is willing to pay for a good and what he must actually pay when he purchases it in the marketplace. For example, if Joe is willing to pay $20 for a cap but purchases it at the store for only $5, Joe will receive $15 in consumer surplus. This measure indicates how much better off the consumer is after purchasing the good.

8. Two different ways of measuring the monetary value that a consumer would assign to the change in price of the good are (1) the compensating variation and (2) the equivalent variation. What is the difference between the two measures, and when would these measures be equal?

Compensating variation answers the question, “How much would the consumer be willing to give up after a price reduction to achieve the same level of satisfaction as she had before the price change?” Equivalent variation, on the other hand, answers the question, “How much money would we have to give the consumer before a price reduction to leave her level of satisfaction the same as it would be after the price reduction?” In essence, both of these are measures of the “distance” between the initial and final indifference curves after a price change.

Typically the compensating and equivalent variation measures will not be the same. In the case of quasi-linear utility functions, however, the compensating and equivalent variation measures will always be the same (they will be equal to the change in consumer surplus). In general, these two measures will be identical when there is no income effect associated with a price change.

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Solutions Manual

9. Consider the following four statements. Which might be an example of a positive network externality? Which might be an example of a negative network externality? (i) People eat hot dogs because they like the taste, and hot dogs are filling. (ii) As soon as Zack discovered that everybody else was eating hot dogs, he stopped buying them. (iii) Sally wouldn’t think of buying hot dogs until she realized that all her friends were eating them. (iv) When personal income grew by 10 percent, hot dog sales fell. (i) No network externality (ii) Negative network externality (iii) Positive network externality (iv) Since sales fall when income increases, this might be a negative network externality if some consumers stopped buying hot dogs not only because of a lower income, but also because other consumers bought fewer hot dogs.

10. Why might an individual supply less labor (demand more leisure) as the wage rate rises?

When the wage rate rises, the substitution effect will induce a worker to supply more hours of labor. The income effect, on the other hand, may induce the worker to increase the amount of leisure and decrease the amount of labor. If the income effect reduces the amount of labor supplied more than the substitution effect increases it, the worker will ultimately supply less hours of labor.

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Solutions Manual

Solutions to Problems 5.1 Figure 5.2(a) shows a consumer’s optimal choices of food and clothing for three values of weekly income: I1 = $40, I2 = $68, and I3 = $92. Figure 5.2(b) illustrates how the consumer’s demand curve for food shifts as income changes. Draw three demand curves for clothing (one for each level of income) to illustrate how changes in income affect the consumer’s purchases of clothing.

Py

D2 (I=68)

4 D3 (I=92) D1 (I=40) y 5

8

11

5.2 Use the income consumption curve in Figure 5.2(a) to draw the Engel curve for clothing, assuming the price of food is $2 and the price of clothing is $4.

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Chapter 5-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.3 Show that the following statements are true: a) An inferior good has a negative income elasticity of demand. b) A good whose income elasticity of demand is negative will be an inferior good. a)

%𝛥𝑄

𝛥𝑄/𝑄

𝛥𝑄

𝐼

𝜀𝑄,𝐼 = %𝛥𝐼 = 𝛥𝐼/𝐼 = ( 𝛥𝐼 ) (𝑄)

𝐼 and 𝑄 must be greater than zero. In addition, assume income increases, i.e., 𝛥𝐼 > 0. If the good is inferior, then 𝛥𝑄 < 0. Thus, the first term (𝛥𝑄/𝛥𝐼) < 0 and the second term (𝐼/𝑄) > 0.

Multiplying these two terms together implies 𝜀𝑄,𝐼 < 0. Inferior goods have a negative income elasticity of demand. b)

𝛥𝑄

𝐼

𝛥𝐼

𝑄

If income elasticity of demand is negative then 𝜀𝑄,𝐼 = ( ) ( ) < 0. 𝛥𝑄

Since 𝐼 and 𝑄 must be greater than zero, for 𝜀𝑄,𝐼 to be negative, we must have 𝛥𝐼 < 0.

This can only happen if either a) 𝛥𝑄 < 0 and 𝛥𝐼 > 0 or b) 𝛥𝑄 > 0 and 𝛥𝐼 < 0. In both instances, the change in quantity demanded moves in the opposite direction as the change in income implying the good is inferior.

5.4 If the demand for a product is perfectly price inelastic, what does the corresponding price consumption curve look like? Draw a graph to show the price consumption curve.

If demand for good X is perfectly price inelastic then the demand curve is a vertical line and quantity remains constant as price changes. Graphing the price consumption curve for good X on an optimal choice diagram would appear as Y

Price consumption curve

X

The price consumption curve is a straight line because the level of consumption of X is constant.

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Chapter 5-5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.5 Ann consumes five goods. The prices of all goods are fixed. The price of good x is px.

She spends 25 percent of her income on good x, regardless of the size of her income. a) Show that her income elasticity of demand of good x is the same for any level of income, and determine its value. b) Would the value of the income elasticity of demand for x be different if Ann always spends 60 percent of her income on good x? a) Since she spends 25% of her income on x, it must be true that pxx/I = 0.25. Thus x/I = 0.25/px.

This means that x/I is a constant. If I increases by 1%, x must also increase by 1%. Since the percentage increase in x is the same as the percentage increase in I, the income elasticity must be 1. b) The income elasticity of demand would still be 1. Now x/I = 0.6/px. This means that x/I is a constant. If I increases by 1%, x must also increase by 1%.

5.6 Suzie purchases two goods, food and clothing. She has the utility function U(x, y) = xy, where x denotes the amount of food consumed and y the amount of clothing. The marginal utilities for this utility function are MUx = y and MUy = x. a) Show that the equation for her demand curve for clothing is y = I/(2Py). b) Is clothing a normal good? Draw her demand curve for clothing when the level of income is I = 200. Label this demand curve D1. Draw the demand curve when I = 300 and label this demand curve D2. c) What can be said about the cross-price elasticity of demand of food with respect to the price of clothing?

a)

At the consumer’s optimum we must have

MU x MU y = Px Py y x = Px Py

  Py   Px  y    + Py y = I   Px   Substituting into the budget line, 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 = 𝐼, gives 2 Py y = I y=

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I 2 Py

Chapter 5-6

Besanko & Braeutigam – Microeconomics, 6th edition

b)

Solutions Manual

Yes, clothing is a normal good. Holding 𝑃𝑦 constant, if 𝐼 increases 𝑦 will also increase.

c) The cross-price elasticity of demand of food with respect to the price of clothing must be zero. Note in part a) that with this utility function the demand for y does not depend on the price of x. Similarly, you can show that the demand for food is x = I / (2Px), which does not depend on the price of y. In fact, the consumer divides her income equally between the two goods regardless of the price of either. Since the demands do not depend on the prices of the other goods, the cross-price elasticities must be zero.

5.7 Karl’s preferences over hamburgers (H) and beer (B) are described by the utility function: U(H, B) = min(2H, 3B). His monthly income is I dollars, and he only buys these two goods out of his income. Denote the price of hamburgers by PH and of beer by PB. a) Derive Karl’s demand curve for beer as a function of the exogenous variables. b) Which affects Karl’s consumption of beer more: a one dollar increase in PH or a one dollar increase in PB? a) Karl’s optimal bundle will always be such that 2H = 3B. If this were not true then he could decrease the consumption of one of the two goods, staying at the same level of utility and reducing expenditure. Also, at the optimal bundle, it must be true that 𝑃𝐻 𝐻 + 𝑃𝐵 𝐵 = 𝐼.

Substituting the first condition into the second we get 𝐵(1.5𝑃𝐻 + 𝑃𝐵 ) = 𝐼 which implies that the 𝐼 demand curve for beer is given by,𝐵 = (1.5𝑃 +𝑃 ) 𝐻

𝐵

b) You can answer this just by looking at the demand curve. Because it has a larger coefficient, the price of hamburgers affects the demand for beer more than the price of beer. A one dollar increase in 𝑃𝐻 decreases demand for beer more than a one dollar increase in 𝑃𝐵 .

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Chapter 5-7

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.8 David has a quasi-linear utility function of the form U(x, y) = √x + y, with associated marginal utility functions MUx = 1/(2√x) and MUy = 1. a) Derive David’s demand curve for x as a function of the prices, Px and Py. Verify that the demand for x is independent of the level of income at an interior optimum. b) Derive David’s demand curve for y. Is y a normal good? What happens to the demand for y as Px increases? a)

Denoting the level of income by I, the budget constraint implies that 𝑝𝑥 𝑥 + 𝑝𝑦 𝑦 = 𝐼 and 1

𝑝

𝑝𝑦 2

√

𝑦

𝑥

the tangency condition is 2 𝑥 = 𝑝𝑥 , which means that 𝑥 = 4𝑝 2. The demand for x does not depend on the level of income. b)

From the budget constraint, the demand curve for y is, 𝑦 =

𝐼−𝑝𝑥 𝑥 𝑝𝑦

𝐼

𝑝𝑦

𝑦

𝑥

= 𝑝 − 4𝑝 .

You can see that the demand for y increases with an increase in the level of income, indicating that y is a normal good. Moreover, when the price of x goes up, the demand for y increases as well.

5.9 Rick purchases two goods, food and clothing. He has a diminishing marginal rate of substitution of food for clothing. Let x denote the amount of food consumed and y the amount of clothing. Suppose the price of food increases from Px1 to Px2. On a clearly labeled graph, illustrate the income and substitution effects of the price change on the consumption of food. Do so for each of the following cases: a) Case 1: Food is a normal good. b) Case 2: The income elasticity of demand for food is zero. c) Case 3: Food is an inferior good, but not a Giffen good. d) Case 4: Food is a Giffen good.

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Chapter 5-8

Besanko & Braeutigam – Microeconomics, 6th edition

a)

Solutions Manual

Y

B C

A

X Substitution Effect Income Effect

b)

Y

B A C X Substitution Effect Income Effect = 0

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Chapter 5-9

Besanko & Braeutigam – Microeconomics, 6th edition

c)

Solutions Manual

Y

B A C X

Substitution Effect Income Effect

c)

Y

B A C X

Substitution Effect Income Effect

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Chapter 5-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.10 Reggie consumes only two goods: food and shelter. On a graph with shelter on the horizontal axis and food on the vertical axis, his price consumption curve for shelter is a vertical line. Draw a pair of budget lines and indifference curves that are consistent with this description of his preferences. What must always be true about Reggie’s income and substitution effects as the result of a change in the price of shelter?

A pair of possible indifference curves and budget lines are shown above. For the Price consumption curve to be a vertical line, it must be that Reggie’s demand for shelter does not change even when the price of shelter changes and the budget line rotates.

The fact that his optimal bundle stays the same, despite a price change, means that Reggie’s income and substitution effects as a result of a change in the price of shelter must cancel each other out so as to leave a net zero effect. For example, if the price of shelter were to decrease, the substitution effect would be positive and this would imply a negative income effect, just large enough to cancel out the substitution effect. In other words, the two effects have the same magnitude but opposite signs. This also implies that Reggie views shelter as an inferior good.

5.11 Ginger’s utility function is U(x, y) = x2y, with associated marginal utility functions MUx = 2xy and MUy = x2. She has income I = 240 and faces prices Px = $8 and Py = $2. a) Determine Ginger’s optimal basket given these prices and her income. b) If the price of y increases to $8 and Ginger’s income is unchanged, what must the price of x fall to in order for her to be exactly as well off as before the change in Py? 2𝑦

8

a) The budget constraint is 8𝑥 + 2𝑦 = 240 and the tangency condition is 𝑥 = 2 = 4.

Solving, the optimal bundle is (x, y)=(20, 40) with a utility of 202(40)=16,000.

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Chapter 5-11

Besanko & Braeutigam – Microeconomics, 6th edition

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b) Now py=8. We need to calculate px such that, with the new prices, Ginger reaches exactly the same indifference curve as before. The new optimal bundle (x,y) must be such that: 2𝑦 𝑝 𝑝𝑥 𝑥 + 8𝑦 = 240, and 𝑥 2 𝑦 = 16000. The tangency condition now implies that 𝑥 = 8𝑥 that is, 𝑝𝑥 𝑥 = 16𝑦. Substituting this into the budget constraint we find that y=10. Using the condition 𝑥 2 𝑦 = 16000, we find that x = 40. Finally, substituting the values of x and y back into the budget constraint, we can see that 𝑝𝑥 40 + 8(10) = 240, or px=4. Therefore, if the price of y were to increase to $8, Ginger would need the price of x to decrease to $4 in order to be exactly as well off as before.

5.12 Ann’s utility function is U(x, y) = x + y, with associated marginal utility functions MUx = 1 and MUy = 1. Ann has income I = 4. a) Determine all optimal baskets given that she faces prices Px = 1 and Py = 1. b) Determine all optimal baskets given that she faces prices Px = 1 and Py = 2. c) What is demand for y when Px = 1 and Py = 1? What is demand for y when Px = 1 and Py > 1? What is demand for y when Px = 1 and Py < 1? Plot Ann’s demand for y as a function of Py. d) Repeat the exercises in a), b) and c) for U(x, y) = 2x + y, with associated marginal utility functions MUx = 2 and MUy = 1, and with the same level of income. a) Notice that MUx / MUy = 1 for all x and y. In this case indifference curves are straight lines with slope 1. Therefore, when Px = 1 and Py = 1 all pairs of x and y such that x + y = 4 are optimal baskets. b) Optimal consumption in this case is at a corner point. Since the price of x is smaller than the price of y and marginal utility of each good is the same, consumer is better off purchasing only x. (Another way to see this is to note that MUX/PX = 1/1 > MUY/PY = ½.) Hence, the optimal basket consists of 4 units of x and zero units of y.

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Chapter 5-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

c) When price of y is lower than 1 there are zero units of x in the optimal basket. Hence, for Px = 1 and Py < 1 the demand for y equals to I / Py.

d) (i) Repeat of part (a): When U(x, y) = 2x + y and MUx = 2 and MUy = 1, then if Px = 1 and Py = 1, the marginal utility per dollar spent on x exceeds the marginal utility per dollar spent on y.

Thus, optimal consumption is at a corner point, with Ann purchasing only x when Px = 1 and Py = 1. The optimal basket is (x,y)=(4,0). (ii) Repeat of part (b): When Px = 1 and Py = 2, the marginal utility per dollar spent on x continues to exceed the marginal utility per dollar spent on y. Thus, optimal consumption continues to be at a corner point, with Ann purchasing only x when Px = 1 and Py = 2. The optimal basket is again (x,y)=(4,0) (iii) Repeat of part (c): As noted above, when Px = 1 and Py = 1, the demand for y is 0 units. For any Py >1, the marginal utility per dollar spent on x will continue to exceed the marginal utility per dollar spent on y, so the demand for y will continue to be zero.

This will also be the case as long as ½ < Py < 1, holding Px = 1. (For example, if Py = 0.75, then MUx/ Px = 2 and MUy/Py = 1/0.75= 1.33.)

When Px = 1 and Py = ½, MUx/ Px = MUy/Py = 2, and all baskets along the budget line are optimal for Ann. For example, it would be optimal to purchase no units of good y. It would also be optimal for Ann to spend all her income on good y, purchasing 4/(½) = 8 units.

When Px = 1 and Py < ½, MUx/ Px = 2 < MUy/Py. In this case, the optimal solution to the utility maximization problem occurs at a corner point, with Ann spending all her income on good y. In this case, the optimal basket is (x,y) = (0,4/Py).

Thus, to summarize, when Px = 1 and income = 4, the demand function for y is: 0 y* = anything between 0 and 8 4/Py

if Py > ½ if Py = ½ if Py < ½

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Chapter 5-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

This demand function has a shape similar to the one shown above, except the price at which the demand for y goes to 0 is ½ instead of 1, and the quantity at which the horizontal part of the demand function intersects the downward-sloping portion occurs at 8 instead of 4.

5.13 Some texts define a “luxury good” as a good for which the income elasticity of demand is greater than 1. Suppose that a consumer purchases only two goods. Can both goods be luxury goods? Explain.

Consider any change in income 𝛥𝐼. For the budget constraint to hold, it must be true that 𝛥𝐼 = 𝑃𝑥 𝛥𝑥 + 𝑃𝑦 𝛥𝑦.

(For example, if income increases then some of it may be spent on x and some on y, but the total new expenditures must be equal to the change in income.) Since we are interested in income elasticities, it helps to rewrite the previous equation as 1 = 𝑃𝑥

𝛥𝑥 𝛥𝑦 + 𝑃𝑦 𝛥𝐼 𝛥𝐼

Since 𝜀𝑥,𝐼 = (𝛥𝑥 ⁄𝛥𝐼 )(𝐼/𝑥) and 𝜀𝑦,𝐼 = (𝛥𝑦⁄𝛥𝐼 )(𝐼/𝑦), we can write this as 𝑥 𝑦 1 = 𝑃𝑥 𝜀𝑥,𝐼 + 𝑃𝑦 𝜀𝑦,𝐼 𝐼 𝐼 Or 𝐼 = (𝑃𝑥 𝑥)𝜀𝑥,𝐼 + (𝑃𝑦 𝑦)𝜀𝑦,𝐼 But if both goods are luxury goods, then 𝜀𝑥,𝐼 > 1 and 𝜀𝑦,𝐼 > 1 so that the previous equation implies 𝐼 > (𝑃𝑥 𝑥)(1) + (𝑃𝑦 𝑦)(1) Thus, if both x and y are luxury goods then I > I, which obviously is untrue! Therefore, both goods cannot simultaneously be luxury goods.

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Chapter 5-14

Besanko & Braeutigam – Microeconomics, 6th edition

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5.14 Scott consumes only two goods, steak and ale. When the price of steak falls, he buys more steak and more ale. On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption.

When the price of steak falls, the budget line rotates from BL1 to BL2. The consumer now maximizes utility on U2 at point B on BL2. The amounts of steak and ale consumed at point B are greater than the initial amounts consumed at point A. This is shown in the following figure.

Ale

B A

U2 U1 BL2 BL1

Steak

5.15 Dave consumes only two goods, coffee and doughnuts. When the price of coffee falls, he buys the same amount of coffee and more doughnuts. a) On an optimal choice diagram (with budget lines and indifference curves), illustrate this pattern of consumption. b) Is this purchasing behavior consistent with a quasi-linear utility function? Explain. a)

Doughnuts

B BL1 A

BL2 Coffee

In the diagram above, the consumer purchases the same amount of coffee and more doughnuts after the price of coffee falls.

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Chapter 5-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) No, this behavior is not consistent with a quasi-linear utility function. While it is true that there is no income effect with a quasi-linear utility function, the substitution effect would still induce the consumer to purchase more coffee when the price of coffee falls.

5.16 (This problem shows that an optimal consumption choice need not be interior and may be at a corner point.) Suppose that a consumer’s utility function is U(x, y) = xy + 10y.

The marginal utilities for this utility function are MUx = y and MUy = x + 10. The price of x is Px and the price of y is Py, with both prices positive. The consumer has income I. a) Assume first that we are at an interior optimum. Show that the demand schedule for x can be written as x = I/(2Px) − 5. b) Suppose now that I = 100. Since x must never be negative, what is the maximum value of Px for which this consumer would ever purchase any x? c) Suppose Py = 20 and Px = 20. On a graph illustrating the optimal consumption bundle of x and y, show that since Px exceeds the value you calculated in part (b), this corresponds to a corner point at which the consumer purchases only y. (In fact, the consumer would purchase y = I/Py = 5 units of y and no units of x.) d) Compare the marginal rate of substitution of x for y with the ratio (Px/Py) at the optimum in part (c). Does this verify that the consumer would reduce utility if she purchased a positive amount of x? e) Assuming income remains at 100, draw the demand schedule for x for all values of Px. Does its location depend on the value of Py? a)

If we are at an interior optimum the tangency condition must hold:

P y = x x + 10 Py Py y = Px ( x + 10) Substituting into the budget line, 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 = 𝐼, yields Px x + Px ( x + 10) = I 2 Px x + 10 Px = I 2 Px x = I − 10 Px x=

I −5 2 Px

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Chapter 5-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

If 𝐼 = 100, then 100 x= −5 2 Px

b)

x=

50 −5 Px

Since we must have 𝑥 ≥ 0, we must have 50 −5  0 Px 50 5 Px 50  5 Px Px  10

So the consumer would only purchase 𝑥 for prices less than 10. c)

Given 𝑃𝑥 = 𝑃𝑦 = 20, the slope of the budget line is –1. At the corner point optimum, the slope of the indifference curve is −

𝑀𝑈𝑥 𝑦 5 1 =− =− =− 𝑀𝑈𝑦 𝑥 + 10 10 2

Because the indifference curve has a flatter slope than the budget line, the consumer would like to substitute more 𝑦 for 𝑥, but has no more 𝑥 to give up at the corner point. d)

𝑀𝑈𝑥 𝑃𝑥

5

10

𝑀𝑈𝑦

= 20 < 20 = 𝑃 . If the consumer were to purchase any 𝑥, since the “bang for the 𝑦

buck” for 𝑥 is less than the “bang for the buck” for 𝑦, the consumer would reduce total utility by increasing 𝑥 above zero.

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Chapter 5-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

e)

As shown in part a), the demand for 𝑥 depends only on 𝐼 and 𝑃𝑥 . Therefore, the location of the demand curve does not depend on 𝑃𝑦 .

5.17 The accompanying figure illustrates the change in consumer surplus, given by Area ABEC, when the price decreases from P1 to P2. This area can be divided into the rectangle ABDC and the triangle BDE. Briefly describe what each area represents, separately, keeping in mind the fact that consumer surplus is a measure of how well off consumers are (therefore the change in consumer surplus represents how much better off consumers are).

(Hint: Note that a price decrease also induces an increase in the quantity consumed.)

P1

P2

As the figure shows, a decrease in the price from p1 to p2 induces an increase in quantity from q1 to q2. The resulting change in consumer surplus is due to two things: First, the consumer is paying a lower price, per unit, on all the units of the good that he was consuming before the price change. That is, for the q1 units he was earlier consuming, he now pays a lower price and therefore enjoys a higher consumer surplus, denoted by the area of the rectangle ABCD. Another way of putting this is that if he continued to consume q1 even after the price change his consumer surplus would increase by only area ABCD.

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Chapter 5-18

Besanko & Braeutigam – Microeconomics, 6th edition

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Second, the lower price induces him to consume more of the good in question. In fact he consumes (q2 – q1) more units. The additional benefit he gets from this is the area of triangle BDE. 5.18 The demand function for widgets is given by D(P) = 16 − 2P. Compute the change in consumer surplus when price of a widget increases for $1 to $3. Illustrate your result graphically.

For price of a widget equal to $1 consumer surplus is CS$1 = ½ ∙ (8 – 1) ∙ D(1) = ½ ∙ 7 ∙ 14 = 49.

When price is equal to $3 consumer surplus is CS$3 = ½ ∙ (8 – 3) ∙ D(3) = ½ ∙ 5 ∙ 10 = 25.

5.19 Jim’s preferences over cookies (x) and other goods (y) are given by U(x, y) = xy with associated marginal utility functions MUx = y. and MUy = x. His income is $20. a) Find Jim’s demand schedule for x when price of y is Py = $1. b) Illustrate graphically the change in consumer surplus when the price of x increases from $1 to $2. a) Jim’s optimal basket is a solution to equations MUx / MUy = Px / Py and Px x + Py y = I. Hence, we have y / x = Px and Px x + y = 20 with solution x = 10 / Px and y = 10. Demand schedule for x is D(Px) = 10 / Px.

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Chapter 5-19

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b)

The change in consumer surplus is area of region ABCD under the demand curve. The are of this region can be computed by simple integration: –∫[1,2] 10/p dp = – 10 ln(2).

5.20 Lou’s preferences over pizza (x) and other goods (y) are given by U(x, y) = xy, with associated marginal utilities MUx = y and MUy = x. His income is $120. a) Calculate his optimal basket when Px = 4 and Py = 1. b) Calculate his income and substitution effects of a decrease in the price of food to $3. c) Calculate the compensating variation of the price change. d) Calculate the equivalent variation of the price change. 𝑦

a) Using the tangency condition, 𝑥 = 4, and the budget constraint, 4𝑥 + 𝑦 = 120, Lou’s initial optimum is the basket (x, y) = (15, 60) with a utility of 900. b) First we need the decomposition basket. This would satisfy the new tangency condition, 𝑦 = 3 and would give him as much utility as before, i.e. 𝑥𝑦 = 900. This gives 𝑥 (𝑥, 𝑦) = (10√3, 30√3) or approximately (17.3,51.9). Now we need the final basket, which satisfies the same tangency condition as the decomposition basket and also the new budget constraint: 3𝑥 + 𝑦 = 120. Together, these conditions imply that (x, y) = (20, 60). The substitution effect is therefore 17.3 – 15 = 2.3, and the income effect is 20 – 17.3 = 2.7. c) The compensating variation is the amount of income Lou would be willing to give up after the price change to maintain the level of utility he had before the price change. This equals the difference between the consumer’s actual income, $120, and the income needed to buy the decomposition basket at the new prices. This latter income equals: 3*17.3 + 1*51.9 = 103.8. The compensating variation thus equals 120 – 103.8 = $16.2.

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Chapter 5-20

Besanko & Braeutigam – Microeconomics, 6th edition

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d) The equivalent variation is the amount of income that Lou would need to be given before the price change in order to leave him as well off as he would be after the price change. After the price change his utility level is 20(60)=1200. Therefore the additional income should be such 𝑦 that it allows Lou to purchase a bundle (x, y) satisfying the initial tangency condition, 𝑥 = 4, and also such that 𝑥𝑦 = 1200. This implies that (𝑥, 𝑦) = (10√3, 40√3) or approximately (17.3, 69.2). How much income would Lou need to purchase this bundle under the original prices? He would need 4(17.3) + 69.2 = 138.4. That is he would need to increase his income by (138.4 – 120) dollars in order to be as well off as if the price of pizza were to decrease instead. Therefore his equivalent variation is $18.4.

5.21 Catrina buys two goods, food F and clothing C, with the utility function U = FC + F. Her marginal utility of food is MUF = C + 1 and her marginal utility of clothing is MUC = F. She has an income of 20. The price of clothing is 4. a) Derive the equation representing Catrina’s demand for food, and draw this demand curve for prices of food ranging between 1 and 6. b) Calculate the income and substitution effects on Catrina’s consumption of food when the price of food rises from 1 to 4, and draw a graph illustrating these effects. Your graph need not be exactly to scale, but it should be consistent with the data. c) Determine the numerical size of the compensating variation (in monetary terms) associated with the increase in the price of food from 1 to 4. a) MUF = C + 1 MUC = F Tangency: MUF/MUC = PF / PC. (C + 1)/ F = PF/4 => 4C + 4 = PFF. (Eq 1) Budget Line: PFF + PCC = I . PFF + 4C = 20. (Eq 2) Substituting (Eq 1) into (Eq 2): 4C + 4 + 4C = 20. Thus C = 2, independent of PF. From the budget line, we see that PFF + 4(2) = 20, so the demand for F is F = 12/PF .

b) Initial Basket: From the demand for food in (a), F = 12/1 = 12, and C = 2.

Also, the initial level of utility is U = FC + F = 12(2) + 12 = 36.

Final Basket: From the demand for food in (a), we know that F = 12/4 = 3, and C = 2. (Also, U = 3(2) + 3 = 9.)

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Chapter 5-21

Besanko & Braeutigam – Microeconomics, 6th edition

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Decomposition Basket: Must be on initial indifference curve, with U = FC + F = 36 (Eq 5) Tangency condition satisfied with final price: MUF/MUC = PF / PC. (C + 1)/ F = 4/4 => C + 1 = F. (Eq 3) Eq 5 can be written as F(C + 1) = 36. Using Eq 3, (C + 1)2 = 36, and thus, C = 5. Also, by Eq 3, F = 6.

So the decomposition basket is F = 6, C = 5.

Income effect on F: Ffinal basket – Fdecomposition basket = 3 – 6 = -3.

Substitution effect on F: F decomposition basket – Finitial basket = 6 – 12 = -6. c) PFF + PCC = 4(6) + 4(5) = 44. So she would need an additional income of 24 (plus her actual income of 20).

The compensating variation associated with the increase in the price of food is -24.

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Chapter 5-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.22 Suppose the market for rental cars has two segments, business travelers and vacation travelers. The demand curve for rental cars by business travelers is Qb = 35 − 0.25P, where Qb is the quantity demanded by business travelers (in thousands of cars) when the rental price is P dollars per day. No business customers will rent cars if the price exceeds $140 per day.

The demand curve for rental cars by vacation travelers is Qv = 120 − 1.5P, where Qv is the quantity demanded by vacation travelers (in thousands of cars) when the rental price is P dollars per day. No vacation customers will rent cars if the price exceeds $80 per day. a) Fill in the table to find the quantities demanded in the market at each price.

b) Graph the demand curves for each segment, and draw the market demand curve for rental cars. c) Describe the market demand curve algebraically. In other words, show how the quantity demanded in the market Qm depends on P. Make sure that your algebraic equation for the market demand is consistent with your answers to parts (a) and (b). d) If the price of a rental car is $60, what is the consumer surplus in each market segment? a) Price ($/day) 100 90 80 70 60 50

Business (000 cars/Week) 10.0 12.5 15.0 17.5 20.0 22.5

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Vacation (000 cars/Week) 15.0 30.0 45.0

Market Demand (000 cars/Week) 10.0 12.5 15.0 32.5 50.0 67.5

Chapter 5-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b)

c) For price greater than $80, vacation traveler’s demand will be zero. So above 𝑃 = 80, market demand is 𝑄𝑏 = 35 − 0.25𝑃.

For price between $0 and $80, market demand is the sum of the vacation and business demand, 𝑄𝑚 = 𝑄𝑏 + 𝑄𝑣 , or 𝑄𝑚 = 35 − 0.25𝑃 + 120 − 1.5𝑃 𝑄𝑚 = 155 − 1.75𝑃 Above a price of $140, no purchases will be made so market demand is zero. In summary, 0, when 𝑃 ≥ 140 𝑄𝑚 = {35 − 0.25𝑃, when 80 ≤ 𝑃 < 140 155 − 1.75𝑃, when 𝑃 < 80 d)

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Chapter 5-24

Besanko & Braeutigam – Microeconomics, 6th edition

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5.23 There are two types of customers in a market for sheet metal. Let P represent the market price.

The total quantity demanded by Type I consumers is Q1 = 100 - 2P, for 0< P < 50.

The total quantity demanded by Type II consumers is Q2 = 40 - P, for 0< P < 40.

Draw the total market demand on a clearly labeled graph.

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Chapter 5-25

Besanko & Braeutigam – Microeconomics, 6th edition

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5.24 There are two consumers on the market: Jim and Donna. Jim’s utility function is U(x, y) = xy, with associated marginal utility functions MUx = y and MUy = x. Donna’s utility function is U(x,y) = x2y, with associated marginal utility functions MUx = 2xy and MUy = x2. Income of Jim is IJ = 100 and income of Donna is ID = 150. a) Find optimal baskets of Jim and Donna when price of y is Py = 1 and price of x is P. b) On separate graphs plot Jim’s and Donna’s demand schedule for x for all values of P. c) Compute and plot aggregate demand when Jim and Donna are the only consumers. d) Plot aggregate demand when there is one more consumer that has identical utility function and income as Donna. a)

Jim’s optimal basket is a solution to equations

MUx / MUy = P / Py and P x + Py y = IJ. Hence, we have 2xy / x2 = P and P x + y = 100 with solution x = 200 / (3P) and y = 100 / 3.

Analogous system of equations for Donna is y / x = P and P x + y = 150 with solution x = 75 / P and y = 75. b)

Approximate shape of the demand curve for Jim and Donna is depicted below.

c)

Aggregate demand is

Dx(P) = 200 / (3P) + 75 / P = 445 / (3P). d) When there is one more consumer that has preferences identical to Donna’s then her demand is also 75 / P and hence aggregate demand is Dx(P) = 200 / (3P) + 75 / P + 75 / P = 650 / (3P).

Shape of the demand curve in this case is the same as in part b).

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Chapter 5-26

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.25 One million consumers like to rent movie videos in Pulmonia. Each has an identical demand curve for movies. The price of a rental is $P. At a given price, will the market demand be more elastic or less elastic than the demand curve for any individual. (Assume there are no network externalities.)

The market demand and individual demand will have the same price elasticity given any particular price. Denote an individual’s demand curve by Qi(P). With 1,000,000 identical individuals the market demand curve will be Qm(P) = 1,000,000Qi(P). At a given price P, an individual’s demand curve will have elasticity 𝜀𝑄𝑖 ,𝑃 = (𝛥𝑄𝑖 ⁄𝛥𝑃 )(𝑃/𝑄𝑖 ). Since Qm(P) = 1,000,000Qi(P), it must also be true that 𝛥𝑄𝑚 𝛥𝑄𝑖 = 1,000,000 𝛥𝑃 𝛥𝑃 The elasticity for the market demand curve will be 𝜀𝑄𝑚,𝑃 =

𝛥𝑄𝑚 𝑃 𝛥𝑄𝑖 𝑃 𝛥𝑄𝑖 𝑃 = 1,000,000 = = 𝜀𝑄𝑖 ,𝑃 𝛥𝑃 𝑄𝑚 𝛥𝑃 1,000,000𝑄𝑖 𝛥𝑃 𝑄𝑖

In other words, with identical consumers the elasticity of the market demand curve will equal the elasticity of the individual demand curve at any price P. 5.26 Suppose that Bart and Homer are the only people in Springfield who drink 7-UP. Moreover their inverse demand curves for 7-UP are, respectively, P = 10 − 4QB and P = 25 − 2QH, and, of course, neither one can consume a negative amount. Write down the market demand curve for 7-UP in Springfield, as a function of all possible prices.

Bart will only consume when the price is less than 10. Therefore his demand curve for 7-UP is 10−𝑃 𝑄𝐵 = 4 , when P<10 and zero otherwise. Homer will only consume if the price is less than 25 so his demand curve is 𝑄𝐻 =

25−𝑃 2

, when P < 25 and zero otherwise.

Therefore the market demand curve for 7-UP as a function of all possible values of price is: 𝑄 𝑀 = 0, if 𝑃 > 25 25 − 𝑃 𝑄𝑀 = , if 10 < 𝑃 < 25 2 60 − 3𝑃 𝑄𝑀 = , if 𝑃 < 10 4

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Chapter 5-27

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.27 Joe’s income consumption curve for tea is a vertical line on an optimal choice diagram, with tea on the horizontal axis and other goods on the vertical axis. a) Show that Joe’s demand curve for tea must be downward sloping. b) When the price of tea drops from $9 to $8 per pound, the change in Joe’s consumer surplus (i.e., the change in the area under the demand curve) is $30 per month. Would you expect the compensating variation and the equivalent variation resulting from the price decrease to be near $30? Explain. a) If the income consumption curve is vertical the utility function has no income effect. This will occur, for example, with a quasi-linear utility function. This utility function will have the same marginal rate of substitution for any particular value of tea regardless of the level of total utility. If the price of tea falls, flattening the budget line, the consumer will reach a new optimum where the marginal rate of substitution is equal to the slope of the new budget line. Since the budget line has flattened, this cannot occur at the previous optimum amount of tea. The substitution effect implies that this new optimum level of tea will be greater than the previous level. Thus, when the price of tea falls, the quantity of tea demanded increases, implying a downward sloping demand curve. This can be seen in the following figure.

Other Income consumption curve Level of tea consumption increases

Tea Price of tea falls

b) Yes, the values will be exactly $30. When the income consumption curve is vertical, the consumer’s utility function has no income effect. As stated in the text, when there is no income effect, compensating and equivalent variation will be identical and these will also equal the change in consumer surplus measured as the change in the area under the demand curve.

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Chapter 5-28

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.28 Consider the optimal choice of labor and leisure discussed in the text. Suppose a consumer works the first 8 hours of the day at a wage rate of $10 per hour, but receives an overtime wage rate of $20 for additional time worked. a) On an optimal choice diagram, draw the budget constraint. (Hint: It is not a straight line.) b) Draw a set of indifference curves that would make it optimal for him to work 4 hours of overtime each day. a)

Because the wage rate changes for any hours worked over eight (leisure less than sixteen) the budget line has a kink at sixteen hours of leisure. b)

With this set of indifference curves, the consumer reaches an optimum at 12 hours of leisure and 12 hours of labor, or $160 of income.

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Chapter 5-29

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.29 Terry’s utility function over leisure (L) and other goods (Y ) is U(L, Y ) = Y + LY. The associated marginal utilities are MUY = 1 + L and MUL = Y. He purchases other goods at a price of $1, out of the income he earns from working. Show that, no matter what Terry’s wage rate, the optimal number of hours of leisure that he consumes is always the same. What is the number of hours he would like to have for leisure?

If Terry’s wage rate is w, then the income he earns from working is (24 – L)w. Since PY = 1, the number of units of other goods he purchases is Y = (24 – L)w.

Now at an optimal bundle, Terry’s 𝑀𝑅𝑆𝐿,𝑌 must equal the price ratio w/PY = w. Therefore, the 𝑌 tangency condition tells us that 1+𝐿 = 𝑤. The two conditions imply 𝑤(1 + 𝐿) = (24 − 𝐿)𝑤.

This means that the optimal amount of leisure is L = 11.5. You can see that this does not depend on the wage rate.

5.30 Consider Noah’s preferences for leisure (L) and other goods (Y ), U(L, Y) = √L + √Y. The associated marginal utilities are MUL = 1/(2√L) and MUY = 1/(2√Y). Suppose that PY = $1. Is Noah’s supply of labor backward bending?

If Noah’s wage rate is w, then the income he earns from working is (24 – L)w. Since PY = 1, the number of units of other goods he purchases is Y = (24 – L)w. Also, the tangency condition gives 𝑌

24

us √𝐿 = 𝑤. Combining the two conditions, 𝑤 2 𝐿 = (24 − 𝐿)𝑤, or 𝐿 = 𝑤+1. Clearly, the amount of leisure that Noah consumes decreases with an increase in the wage rate, and this is true no matter what the wage rate is. Since the amount of labor that Noah supplies equals (24 – L), we see that his supply of labor always increases with an increase in the wage rate. So, his labor supply curve is always positively sloped – that is, it is not backward bending.

5.31 Raymond consumes leisure (L hours per day) and other goods (Y units per day), with preferences described by 𝑼(𝑳, 𝒀) = 𝑳 + 𝟐√𝒀. The associated marginal utilities are 𝑴𝑼𝒀 = 𝟏 and 𝑴𝑼𝑳 = 𝟏/√𝑳. The price of other goods is 1 euro per unit. The wage rate is w Euros per hour. a) Show how the number of units of leisure Raymond chooses depends on the wage rate? b) How does Raymond’s daily income depend on the wage rate? c) Does Raymond work more when the wage rate rises? a) For this utility function, it turns out that the amount of leisure can be determined from the tangency condition alone. The tangency condition for an optimum is Thus w2 = 1/L, or L = 1/w2.

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𝑀𝑈𝐿 𝑤

=

𝑀𝑈𝑌

, or 1

1/√𝐿 𝑤

1

= 1.

Chapter 5-30

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) When Raymond consumes L units of leisure, he works (24 – L) hours, and receives an income of w(24 – L) Euros per day. His expenditure on other goods is Y Euros per day. His budget constraint will have income equal to expenditures, or w(24 – L) = Y. In (a) we learned from the tangency condition that L = 1/w2; substituting this into the budget equation reveals that w(24 – [1/w2]) = Y, which can be rewritten as Y = 24w – (1/w). c) We can answer this in two ways. First, from part (a) we see that Raymond consumes less leisure as the wage rate rises. Thus he works more as the wage rate rises. Alternatively, Raymond works (24 – L) hours per day, i.e., (24 – [1/w2]) hours per day; this increases as w rises.

5.32 Julie buys food and other goods. She has an income of 400 per month. The price of food is initially $1.00 per unit. It then rises to $1.20 per unit. The prices of other goods do not change. To help Julie out, her mother offers to send her a check each month to supplement her income. Julie tells her mother, “Thanks, Mom. If you would send me a check for $50 per month, I would be exactly as happy paying $1.20 per unit as I would have been paying $1.00 per unit and not receiving the $50 from you.” Which of the following statements is true? Explain.

The increased price of food has: a) an income effect of + $50 per month. b) an income effect of – $50 per month. c) a compensating variation of +$50 per month d) a compensating variation of –$50 per month e) an equivalent variation of +$50 per month f) an equivalent variation of –$50 per month The answer is (d). Let’s call Julie’s initial basket A; this is the basket she chooses when she faces a price of food of $1.00 per unit and has a monthly income of $400. Julie’s statement to her mother indicates that she is indifferent between her initial basket A and a different basket (call it basket B) which she would purchase if she had to pay $1.20 per unit of food, but received an extra monthly income of $50. (Basket B would be the decomposition basket if we were analyzing income and substitution effects associated with the increase in the price of food.) So her compensating variation is - $50 per month.

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Chapter 5-31

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.33 Gina lives in Chicago and very much enjoys traveling by air to see her mother in Italy. On the graph below, x denotes her number of round trips to Italy each year. The composite good y measures her annual consumption of other goods; the price of the composite good is py, which is constant in this problem. Several indifference curves from her preference map are drawn below, with levels of utility U1 < U2 < U3 < U4 < U5. If she spends all her income on the composite good, she can purchase y* units, as shown in the graph below. When the initial price of air travel is 1000, she could purchase as many as 18 round trips if she spends all her income on air travel to Italy. a) Make a copy of the graph below, and use it to determine the income and substitution effects on the number of round trips Gina makes as the price of a round trip increases from $1000 to $3000. Clearly label these effects on the graph. b) Using the graph, estimate the numerical size of the compensating variation associated with the price increase? You may refer to the graph to explain your answer. c) Will the consumer surplus measured using Gina’s demand for air travel to Italy provide an exact measure of the monetary value she associates with the price increase? In a sentence explain why or why not.

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Chapter 5-32

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a)

With the initial budget line (labeled “Initial” on the graph), Gina’s income is $18,000 (she could make a maximum of 18 trips, each costing $1000). She chooses initial basket A, making 6 round trips. With the final budget line (labeled “Final” on the graph), she could use her $18,000 to purchase only 6 trips (the horizontal intercept of the final budget line). She chooses basket C, making 2 round trips.

The decomposition budget line (labeled “Decomposition”) is parallel to the final budget line, but tangent to the initial indifference curve (U4) at basket B (with 4 trips).

The income effect is XC – XB = (2 – 4) trips = - 2 trips.

The substitution effect is XB – XA = (4 – 6) trips = -2 trips. b) The compensating variation is the additional amount of money we would have to give Gina so that she can be as well off as she is initially (U4), but making purchases at the final price, in which case she would choose basket B. The decomposition budget line has a horizontal intercept of 9 trips, which represents an income of $27,000 at the final price of $3,000 per trip. Thus, we would have to compensate Gina by giving her and extra $9,000 (above her initial income of $18,000) to keep her as well off as she was initially. Because the price is rising, the compensating variation is negative. So the compensating variation is -$9,000. c) No. From the graph we can see that the income effect is nonzero (for example, baskets B and C are associated with different numbers of round trips). So consumer surplus will not provide an exact measure of the monetary value of the price increase for Gina.

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Chapter 5-33

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

5.34 Emily purchases food (measured by x) and clothing (measured by y). Her preferences 𝟏𝟔 are described by the quasilinear utility function 𝑼(𝒙, 𝒚) = 𝟑𝟐√𝒙 + 𝟐𝒚, with 𝑴𝑼𝒙 = 𝒙 and √

𝑴𝑼𝒚 = 𝟐. Throughout this problem the prices of the two goods are Px = 2, Py = 1. Emily has a monthly income of 𝑰. a) Show that Emily has a diminishing marginal rate of substitution of food for clothing. b) Show that, as long as 𝑰 ≥ 𝟑𝟐, show that Emily always purchases the same amount of food each month. When 𝑰 ≥ 𝟑𝟐, how many units does she buy? c) Fill in the following table with the amounts of food and clothing and the level of utility she realizes at the three levels of income shown.

Income

x, food

y, other goods

Utility U(x,y)

40 42 44 ∆𝑼

d What do the numbers in the table tell you about the marginal utility of income ( ∆𝑰 ) as income rises from 40 to 42? What is the marginal utility of income when income increases from 42 to 44? e) In chapter 4 we showed how to use the method of Lagrange to solve the consumer choice problem: 𝐦𝐚𝐱 𝑼(𝒙, 𝒚) (𝒙,𝒚)

subject to: 𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 ≤ 𝑰 We also demonstrated that the value of the Lagrange multiplier, 𝝀, measures the consumer’s marginal utility of income. Assuming Px = 2, Py = 1, and 𝑰 ≥ 𝟑𝟐, find the value of the Lagrange multiplier. Show that it is does not vary with income, and verify that the value of the Lagrange multiplier is the same as the values of the marginal utility of income you calculated in part (c). a) Since both marginal utilities are positive, the indifference curves are negatively sloped. 𝑀𝑈𝑥

16 √𝑥

𝑀𝑅𝑆𝑥,𝑦 = 𝑀𝑈 = 2 = 𝑦

8 √𝑥

. As we move along an indifference curve in the direction of

increasing x and decreasing y, the value of 𝑀𝑅𝑆𝑥,𝑦 falls. Emily has a diminishing marginal rate of substitution of x for y.

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Chapter 5-34

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) With this utility function, it is possible to have a corner point optimum or an interior optimum.

Since we have a diminishing𝑀𝑅𝑆𝑥,𝑦 , let’s look for the conditions under which we have an interior optimum. At an interior optimum, the tangency condition requires that 𝑀𝑈𝑥 𝑃 8 2 = 𝑃𝑥 => 𝑥 = 1 => 𝑥 = 16. 𝑀𝑈 𝑦

𝑦

√

If we have an interior optimum, Emily will buy 16 units of food at a price of 2. Therefore, she will need an income of at least 32 to reach an interior optimum, and she would then always buy 16 units of food. She would then spend the rest of her income on clothing. 𝐼

𝐼

If she had an income below 32, she would spend all her income on food (buying 𝑥 = 𝑃 = 2 units 𝑥

of food and y=0). c) At the levels of income in the table, we know from (b) that Emily will buy 16 units of food.

Income

x, food

40 42 44

16 16 16

y, other goods 8 10 12

Utility U(x,y) 144 148 152

We can find the number of units of y that she buys from the budget constraint, 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 = 𝐼 .

To determine the amount of y in the table, we calculate 𝑦 = 𝐼 − 2𝑥 = 𝐼 − 2(16). Thus, 𝑦 = 𝐼 − 32.

The utility is then 𝑈(𝑥, 𝑦) = 32√16 + 2𝑦 = 128 + 2𝑦. ∆𝑈

148−144

d) When 𝐼 = 40, 𝑈 = 144. When 𝐼 = 42, 𝑈 = 148. Over that range of income ∆𝐼 = 42−40 = 2. ∆𝑈

152−148

When 𝐼 = 42, 𝑈 = 148. When 𝐼 = 44, 𝑈 = 152. Over that range of income ∆𝐼 = 44−42 = 2 .

Observation: The marginal utility of income is 2 in both cases, not changing with income. e) For this problem the Lagrangian function is: Ʌ(𝑥, 𝑦, 𝜆) = 𝑈(𝑥, 𝑦) − 𝜆(𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼) Ʌ(𝑥, 𝑦, 𝜆) = 32√𝑥 + 2𝑦 − 𝜆(2𝑥 + 𝑦 − 𝐼)

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Chapter 5-35

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Since we know that x, y, and 𝜆 will all be positive at an interior optimum, the three necessary conditions (corresponding to equations (4.17), (4.18), and (4.19)) simplify to the following: 16

𝑀𝑈𝑥 − 𝜆𝑃𝑥 = 0

=>

𝑀𝑈𝑦 − 𝜆𝑃𝑦 = 0 𝑃𝑥 𝑥 + 𝑃𝑦 𝑦 − 𝐼 = 0

=> 2 − 𝜆 = 0 => 2𝑥 + 𝑦 − 𝐼 = 0

√𝑥

− 2𝜆 = 0

From the second condition, we see that 𝜆 = 2. (We could also observe that 𝜆 = 2 from the first condition, using the information that x = 16 at any interior optimum.)

As we showed in chapter 4, 𝜆 is the marginal utility of income. Since the utility function in this problem is quasilinear, the marginal utility of income will be constant at an interior optimum.

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Chapter 5-36

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 6 Inputs and Production Functions Solutions to Review Questions 1. We said that the production function tells us the maximum output that a firm can produce with its quantities of inputs. Why do we include the word maximum in this definition?

The production function tells us the maximum volume of output that may be produced given a combination of inputs. It is possible that the firm might produce less than this amount of output due to inefficient management of resources. While it is possible to produce many levels of output with the same level of inputs, some of which are less technically efficient than others, the production function gives us the upper bound on (the maximum of) the level of output.

2. Suppose a total product function has the “traditional shape” shown in Figure 6.2. Sketch the shape of the corresponding labor requirements function (with quantity of output on the horizontal axis and quantity of labor on the vertical axis).

The labor requirements function, which is the inverse of the production function, tells us the minimum amount of labor that is required to produce a given amount of output.

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Chapter 6-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

3. What is the difference between average product and marginal product? Can you sketch a total product function such that the average and marginal product functions coincide with each other?

The average product of labor is the average amount of output per unit of labor. 𝐴𝑃𝐿 =

Total Product 𝑄 = Quantity of Labor 𝐿

The marginal product of labor is the rate at which total output changes as the firm changes its quantity of labor. 𝑀𝑃𝐿 =

Change in Total Product 𝛥𝑄 = Change in Quantity of Labor 𝛥𝐿

The total product function in the graph below 𝑄 = 3𝐿, which is linear, would have the average and marginal products coincide. In particular, for all values of 𝑄 we would have 𝐴𝑃𝐿 = 𝑀𝑃𝐿 = 3.

4. What is the difference between diminishing total returns to an input and diminishing marginal returns to an input? Can a total product function exhibit diminishing marginal returns but not diminishing total returns?

With diminishing total returns to an input, increasing the level of the input will decrease the level of total output holding the other inputs fixed. Diminishing marginal returns to an input means that as the use of that input increases holding the quantities of the other inputs fixed, the marginal product of that input will become less and less. Essentially, diminishing total returns implies that output is decreasing while with diminishing marginal returns we could have output increasing, but at a decreasing rate as the amount of the input increases.

It is entirely plausible to have a total product function exhibit diminishing marginal returns but not diminishing total returns. This would occur when each additional unit of an input increased the total level of output, but increased the level of output less than the previous unit of the input

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Chapter 6-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

did. Essentially, this occurs when output is increasing at a decreasing rate as the level of the input increases.

5. Why must an isoquant be downward sloping when both labor and capital have positive marginal products?

If the marginal product of labor is positive, then when we increase the level of labor holding everything else constant this will increase total output. To keep the level of output at the original level, we need to stay on the same isoquant. To do so, since the marginal product of capital is positive we would then need to reduce the amount of capital being used. So, to keep output constant, when the level of one input increases the level of the other input must decrease. This negative relationship between the inputs implies the isoquant will have a negative slope, i.e., be downward sloping.

6. Could the isoquants corresponding to two different levels of output ever cross?

No, as with indifference curves, isoquants can never cross. For example, suppose we draw isoquants for two levels of output 𝑄1 and 𝑄2 with 𝑄2 > 𝑄1. In addition, suppose that these isoquants crossed at some point A as in the following diagram.

Capital

C B

A Q2 Q1 Labor

Because A and B are on Q2, both achieve the same level of output. Since A and C are on Q1, both achieve the same level of output. This would imply that B and C achieve the same level of output. However, this is not possible since point C contains more of both inputs which would achieve a higher level of output. Therefore, isoquants cannot cross.

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Chapter 6-3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

7. Why would a firm that seeks to minimize its expenditures on inputs not want to operate on the uneconomic portion of an isoquant?

By operating on the uneconomic portion of an isoquant, the firm would be using a combination of inputs in which one of the inputs has a negative marginal product, i.e., increasing the input decreases the level of total output. At a point such as this, the firm could increase output by decreasing the level of the input. By decreasing the level of the input, the firm could decrease total cost. Thus, if a firm were operating on the uneconomic region of an isoquant it could simultaneously increase output and decrease total cost. Thus, a cost-minimizing firm would never operate on this portion of an isoquant because it would always take advantage of this opportunity.

8. What is the elasticity of substitution? What does it tell us?

The elasticity of substitution measures how the marginal rate of technical substitution of labor for capital changes as we move along an isoquant. Essentially this value tells us the level of substitutability between capital and labor, i.e., how easily the firm can substitute capital for labor to maintain the same level of total output.

9. Suppose the production of electricity requires just two inputs, capital and labor, and that the production function is Cobb–Douglas. Now consider the isoquants corresponding to three different levels of output: Q = 100,000 kilowatt-hours, Q = 200,000 kilowatt-hours, and Q = 400,000 kilowatt-hours. Sketch these isoquants under three different assumptions about returns to scale: constant returns to scale, increasing returns to scale, and decreasing returns to scale.

With decreasing returns to scale the firm needs to more than double inputs to double output.

Equivalently, doubling inputs less than doubles output.

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Chapter 6-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

With constant returns to scale the firms needs to double inputs to double output.

With increasing returns to scale the firm needs to less than double the inputs to double the output. Equivalently, doubling inputs more than doubles output.

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Chapter 6-5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 6.1 A firm uses the inputs of fertilizer, labor, and hothouses to produce roses. Suppose that when the quantity of labor and hothouses is fixed, the relationship between the quantity of fertilizer and the number of roses produced is given by the following table:

a) What is the average product of fertilizer when 4 tons are used? b) What is the marginal product of the sixth ton of fertilizer? c) Does this total product function exhibit diminishing marginal returns? If so, over what quantities of fertilizer do they occur? d) Does this total product function exhibit diminishing total returns? If so, over what quantities of fertilizer do they occur? 𝑄

a) 𝐴𝑃𝐹 = 𝐹 = 𝛥𝑄

2200

b) 𝑀𝑃𝐹 = 𝛥𝐹 =

4

= 550.

2600−2500 6−5

= 100.

c) Diminishing marginal returns set in when 𝑀𝑃𝐹 for some unit is lower than 𝑀𝑃𝐹 for the previous unit. This occurs for 𝐹 > 3. d) Diminishing total returns set in at the point where total output begins to fall. This occurs for 𝐹 > 6.

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Chapter 6-6

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.2. A firm is required to produce 100 units of output using quantities of labor and capital (L, K) = (7, 6). For each of the following production functions, state whether it is possible to produce the required output with the given input combination. If it is possible, state whether the input combination is technically efficient or inefficient. a) Q = 7L + 8K b) Q = 20√KL c) Q = min(16L, 20K) d) Q = 2(KL + L + 1) a) The input combination gives Q = 97 so it is infeasible. b) Q = 129.6 which is greater than 100, so feasible, but inefficient. c) Q = 112 so again feasible but inefficient. d) Q = 100 therefore the required output is feasible and the input combination is efficient.

6.3. For the production function Q = 6L2 − L3, fill in the following table and state how much the firm should produce so that: a) average product is maximized b) marginal product is maximized c) total product is maximized d) average product is zero

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Chapter 6-7

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

The completed table is shown below: L 0 1 2 3 4 5 6

Q 0 5 16 27 32 25 0

a) You can calculate the average product at each point by just dividing total output by L. The values obtained are 0,5,8,9,8,5,0. Therefore Average Product is maximized when L = 3. b) The marginal product at values 1 through 6 are respectively: 5, 11, 11, 5,–7,–25. Therefore both the second and the third unit of L give the greatest marginal increase in output [if you use calculus techniques it can be seen that marginal product is maximized when L = 2]. c) From the Table it is clear that total product is maximized when L = 4. d) Average Product will be zero only when Total Product is zero. This happens when L = 6.

6.4. Suppose that the production function for DVDs is given by Q = KL2 − L3, where Q is the number of disks produced per year, K is machine-hours of capital, and L is man-hours of labor. a) Suppose K = 600. Find the total product function and graph it over the range L = 0 to L = 500. Then sketch the graphs of the average and marginal product functions. At what level of labor L does the average product curve appear to reach its maximum? At what level does the marginal product curve appear to reach its maximum? b) Replicate the analysis in (a) for the case in which K = 1200. c) When either K = 600 or K = 1200, does the total product function have a region of increasing marginal returns?

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Chapter 6-8

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a)

Based on the figure above it appears that the average product reaches its maximum at L = 300.

The marginal product curve appears to reach its maximum at L = 200. b)

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Chapter 6-9

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Based on the figure above it appears that the average product curve reaches its maximum at L = 600. The marginal product curve appears to reach its maximum at L = 400. c) In both instances, for low values of 𝐿 the total product curve increases at an increasing rate.

For K = 600, this happens for L < 200. For K = 1200, it happens for L < 400.

6.5. Are the following statements correct or incorrect? a) If average product is increasing, marginal product must be less than average product. b) If marginal product is negative, average product must be negative. c) If average product is positive, total product must be rising. d) If total product is increasing, marginal product must also be increasing. a) Incorrect. When 𝑀𝑃 > 𝐴𝑃 we know that 𝐴𝑃 is increasing. When 𝑀𝑃 < 𝐴𝑃 we know that 𝐴𝑃 is decreasing. b) Incorrect. If 𝑀𝑃 is negative, MP < 0. But𝐴𝑃 = Q / L can never be negative since total product Q and the level of input L can never be negative. Thus, MP < 0 < AP, which only implies that 𝐴𝑃 is falling. c) Incorrect. Average product is always positive, so this tells us nothing about the change in total product. Whether or not total product is rising depends on whether or not marginal product is positive. d) Incorrect. If total product is increasing we know that 𝑀𝑃 > 0. If diminishing marginal returns have set in, however, marginal product will be positive but decreasing, but that does not preclude MP > 0.

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Chapter 6-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.6. Economists sometimes “prove” the law of diminishing marginal returns with the following exercise: Suppose that production of steel requires two inputs, labor and capital, and suppose that the production function is characterized by constant returns to scale.

Then, if there were increasing marginal returns to labor, you or I could produce all the steel in the world in a backyard blast furnace. Using numerical arguments based on the production function shown in the following table, show that this (logically absurd) conclusion is correct. The fact that it is correct shows that marginal returns to labor cannot be everywhere increasing when the production function exhibits constant returns to scale.

To develop the answer, suppose that we were initially producing 64 units of steel. According to the table, we could do this with 8 units of labor and 100 units of capital.

Now, since we have constant returns to scale, if we double the amount of labor and capital, i.e., L = 16 and K = 200, we can double output, i.e., produce Q = 128 units of steel.

But notice from the table that the input combination L = 16 and K = 100 results in an even greater output of Q = 256 units of steel. Thus, by reducing the amount of capital it uses (from K = 200 to K = 100), holding the quantity of labor fixed, the firm can produce more output! That is, the marginal product of capital is negative over this range.

We can see the same thing if we start with any other input combination. For example, suppose the firm is initially producing 4 units of steel using 2 units of labor and 100 units of capital.

Because of constant returns to scale, if we double the amount of labor and capital, i.e., L = 4 and K = 200, we can double output, i.e., produce Q = 8 units of steel.

But notice from the table that the input combination L = 4 and K = 100 results in an even greater output of Q = 16 units of steel. Again, by reducing the amount of capital it uses (holding the quantity of labor fixed), the firm can produce more output. Again, we see that the marginal product of capital is negative.

The above calculations illustrate that a two-input production function with (a) constant returns to scale and (b) increasing marginal returns to labor must necessarily imply that the marginal product of capital is negative. And, of course, if the marginal product of capital is negative, the firm can expand output by reducing the amount of capital it uses. It could, theoretically, produce an enormous amount of steel in a backyard blast furnace.

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Chapter 6-11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Because this conclusion is absurd, the point of the illustration is that with constant returns to scale, marginal returns to labor cannot be everywhere increasing. Eventually the law of diminishing marginal returns must set in.

6.7. The following table shows selected input quantities, total products, average products, and marginal products. Fill in as much of the table as you can:

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Chapter 6-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

The correct answers are shown in bold face red type.

Labor, L 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Total product, Q 0 19 72 153 256 375 504 637 768 891 1000 1089 1152 1183 1176 1125

APL 0 19 36 51 64 75 84 91 96 99 100 98 96 91 84 75

MPL ---19 53 81 103 119 129 133 131 123 109 89 63 31 -7 -51

6.8. Widgets are produced using two inputs, labor, L, and capital, K. The following table provides information on how many widgets can be produced from those inputs:

a) Use data from the table to plot sets of input pairs that produce the same number of widgets. Then, carefully, sketch several of the isoquants associated with this production function. b) Find marginal products of K and L for each pair of inputs in the table. c) Does the production function in the table exhibit decreasing, constant, or increasing returns to scale?

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Chapter 6-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) Three sample isoquants: red for production of 4 widgets (Q = 4), green for production of 8 widgets (Q = 8), and blue for production of 12 widgets (Q = 12). The dots represent particular combinations of inputs.

b) Recall that marginal product of an input, say labor, is given by Q/L. If we compute the marginal product of labor and capital at any point in the table, we find that it always equals 2.

For example, in moving from input combination (2,2) to (3,2), we increase output from 8 to 10.

Hence, MPL = (10 – 8)/(3 – 2) = 2. c) From the table, we see that as we increase the quantity of each input by a given proportion, the quantity produced increases by the same proportion. Hence, in moving from input combination (1,1) to (3,3), we are tripling the quantity of labor and capital used. As a result, the quantity of output produced triples as well.

6.9 Suppose the production function for automobiles is 𝑸 = 𝑳𝑲 where Q is the quantity of automobiles produced per year, L is the quantity of labor (man-hours) and K is the quantity of capital (machine-hours). a) Sketch the isoquant corresponding to a quantity of Q = 100? b) What is the general equation for the isoquant corresponding to any level of output Q? c) Does the isoquant exhibit diminishing marginal rate of technical substitution?

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Chapter 6-14

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) The Q = 100 isoquant looks like this:

b) We find the general equation of an isoquant for this production function by starting with the production function and solving for K in terms of L. Thus, since Q = LK, the general equation of 𝑄 an isoquant for this production function is 𝐾 = 𝐿 . c) The isoquants for this production function exhibit diminishing marginal rate of technical substitution. We can see this from the graph above which shows that the isoquant is convex to the origin.

6.10. Suppose the production function is given by the equation Q = L√K. Graph the isoquants corresponding to Q = 10, Q = 20, and Q = 50. Do these isoquants exhibit diminishing marginal rate of technical substitution?

Because these isoquants are convex to the origin they do exhibit diminishing marginal rate of technical substitution.

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Chapter 6-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.11. Consider again the production function for DVDs: Q = KL2 − L3. a) Sketch a graph of the isoquants for this production function. b) Does this production function have an uneconomic region? Why or why not? a)

b) Because each of these isoquants has an upward-sloping portion beyond some level of labor, each one does indeed have an uneconomic region.

6.12. Suppose the production function is given by the following equation (where a and b are positive constants): Q = aL + bK. What is the marginal rate of technical substitution of labor for capital (MRTSL,K) at any point along an isoquant?

For this production function 𝑀𝑃𝐿 = 𝑎 and 𝑀𝑃𝐾 = 𝑏. The 𝑀𝑅𝑇𝑆𝐿,𝐾 is therefore 𝑀𝑃𝐿 𝑎 𝑀𝑅𝑇𝑆𝐿,𝐾 = = 𝑀𝑃𝐾 𝑏 6.13. You might think that when a production function has a diminishing marginal rate of technical substitution of labor for capital, it cannot have increasing marginal products of capital and labor. Show that this is not true, using the production function Q = K2L2, with the corresponding marginal products MPK = 2KL2 and MPL = 2K2L. First, note that MRTSL,K = L/K, which diminishes as L increases and K falls as we move along an isoquant. So MRTSL,K is diminishing. However, the marginal product of capital MPK is increasing (not diminishing) as K increases (remember, the amount of labor is held fixed when we measure MPK.) Similarly, the marginal product of labor is increasing as L grows (again, because the amount of capital is held fixed when we measure MPL). This exercise demonstrates that it is possible to have a diminishing marginal rate of technical substitution even though both of the marginal products are increasing.

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Chapter 6-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.14 Consider the following production functions and their associated marginal products.

For each production function, determine the marginal rate of technical substitution of labor for capital, and indicate whether the isoquants for the production function exhibit diminishing marginal rate of substitution.

Completed table is shown below:

Production function 𝑄 =𝐿+𝐾 𝑄 = √𝐿𝐾

MPL 𝑀𝑃𝐿 = 1 1 √𝐾 𝑀𝑃𝐿 = 2 √𝐿 1 1 𝑀𝑃𝐿 = 2 √𝐿

MPK 𝑀𝑃𝐾 = 1 1 √𝐿 𝑀𝑃𝐾 = 2 √𝐾 1 1 𝑀𝑃𝐾 = 2 √𝐾

𝑄 = 𝐿3 𝐾 3

𝑀𝑃𝐿 = 3𝐿2 𝐾 3

𝑀𝑃𝐾 = 3𝐿3 𝐾 2

𝑄 = 𝐿2 + 𝐾 2

𝑀𝑃𝐿 = 2𝐿

𝑀𝑃𝐾 = 2𝐾

𝑄 = √𝐿 + √𝐾

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MRTSL,K 𝑀𝑅𝑇𝑆𝐿,𝐾 = 1 𝐾 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝐿 𝐾 𝐿 𝐾 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝐿 𝐿 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝐾

Diminishing marginal product of labor?

No Yes

Diminishing marginal product of capital?

No Yes

Diminishing marginal rate of technical substitution No Yes

Yes

Yes

Yes

No

No

Yes

No

No

No

𝑀𝑅𝑇𝑆𝐿,𝐾 = √

Chapter 6-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.15. Suppose that a firm’s production function is given by Q = KL + K, with MPK = L + 1 and MPL = K. At point A, the firm uses K = 3 units of capital and L = 5 units of labor. At point B, along the same isoquant, the firm would only use 1 unit of capital. a) Calculate how much labor is required at point B. b) Calculate the elasticity of substitution between A and B. Does this production function exhibit a higher or lower elasticity of substitution than a Cobb–Douglas function over this range of inputs? a) At point A, the firm produces 18 units of output. Therefore, since B is on the same isoquant, it must be that L = 17 at B. b) The capital-to-labor ratio at A is 3/5 and MRTSL,K = ½. At B, the capital-to-labor ratio is 1/17, and MRTSL,K = 1/18. Therefore the elasticity of substitution is: (1 / 17 − 3 / 5) /(3 / 5) 69 = .

(1 / 18 − 1 / 2) /(1 / 2) 68

A Cobb-Douglas production function has an elasticity of substitution of 1. Therefore this production function has a slightly higher elasticity of substitution, indicating a slightly greater ease of substitutability of inputs.

6.16. Two points, A and B, are on an isoquant drawn with labor on the horizontal axis and capital on the vertical axis. The capital–labor ratio at B is twice that at A, and the elasticity of substitution as we move from A to B is 2. What is the ratio of the MRTSL,K at A versus that at B? Since the capital-labor ratio at B is twice that at A, it implies that the percent change in this ratio as we move from A to B is 100%. If we denote the percent change in the MRTS over these two points as x then using the definition of elasticity of substitution, 100 = 2, which means that %Δ𝑀𝑅𝑇𝑆𝐿,𝐾 = 50. %𝛥𝑀𝑅𝑇𝑆𝐿,𝐾 Equivalently,

𝑀𝑅𝑇𝑆𝐵 −𝑀𝑅𝑇𝑆𝐴 𝑀𝑅𝑇𝑆𝐴

𝑀𝑅𝑇𝑆

2

× 100 = 50. Solving, 𝑀𝑅𝑇𝑆𝐴 = 3.

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𝐵

Chapter 6-18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.17. Let B be the number of bicycles produced from F bicycle frames and T tires. Every bicycle needs exactly two tires and one frame. a) Draw the isoquants for bicycle production. b) Write a mathematical expression for the production function for bicycles. a) This isoquants for this situation will be L-shaped as in the following diagram Tires

6

Q=3

4

Q=2

2

Q=1 Frames 1

2

3

These L-shaped isoquants imply that once you have the correct combination of inputs, say 2 frames and 4 tires, additional units of one resource without more units of the other resource will not result in any additional output.

1

b) Mathematically this production function can be written 𝑄 = 𝑚𝑖𝑛( 𝐹, 2 𝑇) where 𝐹 and 𝑇 represent the number of frames and tires.

6.18. To produce cake, you need eggs E and premixed ingredients I. Every cake needs exactly one egg and one package of ingredients. When you add two eggs to one package of ingredients, you produce only one cake. Similarly, when you have only one egg, you can’t produce two cakes even though you have two packages of ingredients. a) Draw several isoquants of the cake production function. b) Write a mathematical expression for this production function. What can you say about returns to scale of this function?

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Chapter 6-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a)

b) The formula is # of cakes = Min{# of eggs, # of ingredients’ packages}. This production function has constant returns to scale. To see why, let x and y denoted the quantities of eggs and mix, respectively, and let Q denote the number of cakes produced. The equation of our production function is: Q = Min(x,y). If we increase each input by a factor of a, we have the following quantity of cake: min{ax, ay} = a min{x, y} = aQ. Hence, increasing the quantities of inputs by a given proportion results in the same proportionate increase in output, and the production function thus exhibits constant returns to scale.

6.19. What can you say about the returns to scale of the linear production function Q = aK + bL, where a and b are positive constants?

If we were to scale up all inputs by a factor  (that is, replace K by K, and L by L), the resulting output would equal Q. Therefore a linear production function has constant returns to scale.

6.20. What can you say about the returns to scale of the Leontief production function Q = min(aK, bL), where a and b are positive constants?

A general fixed proportions production function is of the form 𝑄 = 𝑚𝑖𝑛( 𝑎𝐾, 𝑏𝐿). If we were to scale up all inputs by a factor  (that is, replace K by K, and L by L), the resulting output would be 𝑚𝑖𝑛( 𝑎𝜆𝐾, 𝑏𝜆𝐿) = 𝜆 𝑚𝑖𝑛( 𝑎𝐾, 𝑏𝐿) = Q. Therefore the production function has constant returns to scale.

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Chapter 6-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.21. A firm produces a quantity Q of breakfast cereal using labor L and material M with the production function Q = 50√ML + M+ L. The marginal product functions for this production function are

a) Are the returns to scale increasing, constant, or decreasing for this production function? b) Is the marginal product of labor ever diminishing for this production function? If so, when? Is it ever negative, and if so, when? a) To determine the nature of returns to scale, increase all inputs by some factor 𝜆 and determine if output goes up by a factor more than, less than, or the same as 𝜆.

Q = 50  M  L +  M +  L Q = 50 ML +  M +  L Q =  50 ML + M + L  Q =  Q

By increasing the inputs by a factor of 𝜆 output goes up by a factor of 𝜆. Since output goes up by the same factor as the inputs, this production function exhibits constant returns to scale. 𝑀

b) The marginal product of labor is 𝑀𝑃𝐿 = 25√ 𝐿 + 1 Suppose 𝑀 > 0. Holding 𝑀 fixed, increasing 𝐿 will have the effect of decreasing 𝑀𝑃𝐿 . The marginal product of labor is decreasing for all levels of 𝐿. The 𝑀𝑃𝐿 , however, will never be negative since both components of the equation above will always be greater than or equal to zero. In fact, for this production function, 𝑀𝑃𝐿 ≥ 1.

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Chapter 6-21

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.22 Consider a production function whose equation is given by the formula 𝑸 = 𝑳𝑲𝟐 , which has corresponding marginal products, 𝑴𝑷𝑳 = 𝑲𝟐 and 𝑴𝑷𝑲 = 𝟐𝑳𝑲. Show that the elasticity of substitution for this production function is exactly equal to 1, no matter what the values of K and L are. 𝐾2

𝑀𝑃

First note that 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝑀𝑃 𝐿 . In this case that implies, 𝑀𝑅𝑇𝑆𝐿,𝐾 = 2𝐿𝐾 , 𝐾

1𝐾

This simplifies to 𝑀𝑅𝑇𝑆𝐿,𝐾 = 2 𝐿 . 𝐾

%∆( )

𝐿 Now recall that the definition of the elasticity of substitution is 𝜎 = %∆ 𝑀𝑅𝑇𝑆 . 𝐿,𝐾

1𝐾

𝐾

Since 𝑀𝑅𝑇𝑆𝐿,𝐾 = 2 𝐿 , it follows that %∆ 𝑀𝑅𝑇𝑆𝐿,𝐾 will be exactly equal to %∆ ( 𝐿 ). (You can 𝐾

𝐾

verify this by plugging in particular values for 𝐿 .) Suppose, for example, 𝐿 = 2, and then 𝐾

1𝐾

changes to 𝐿 = 3, an increase of 66.66 percent. Since 𝑀𝑅𝑇𝑆𝐿,𝐾 = 2 𝐿 , it follows that MRTSL,K changes from 1 to 1.5, (also an increase of 66.66 percent.) In other words, since the marginal rate of substitution of labor for capital is proportional to the capital-labor ratio, the percentage change in the marginal rate of substitution of labor for capital must equal the percentage change in the 𝐾 capital-labor ratio. Since %∆ 𝑀𝑅𝑇𝑆𝐿,𝐾 = %∆ ( 𝐿 ), then using the definition of the elasticity of substitution, it follows that,

𝜎=

𝐾 %∆ ( 𝐿 ) 𝐾 %∆ ( 𝐿 )

= 1.

6.23. A firm’s production function is Q = 5L2/3K1/3 with MPK = (5/3)L2/3K−2/3 and MPL = (10/3)L−1/3K1/3. a) Does this production function exhibit constant, increasing, or decreasing returns to scale? b) What is the marginal rate of technical substitution of L for K for this production function? c) What is the elasticity of substitution for this production function? a) Notice that (aK)1/3(aL)2/3 = a1/3a2/3 K1/3 L2/3 = a K1/3 L2/3 = aQ. This production function exhibits constant returns to scale. b) MRTSL,K = MPL / MPK = 2 K/L .

Because this is a Cobb-Douglas production function, its elasticity of substitution equals 1.

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Chapter 6-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.24. Consider a CES production function given by Q = (K0.5 + L0.5)2. a) What is the elasticity of substitution for this production function? b) Does this production function exhibit increasing, decreasing, or constant returns to scale? c) Suppose that the production function took the form Q = (100 + K0.5 + L0.5)2. Does this production function exhibit increasing, decreasing, or constant returns to scale? 𝜎−1 𝜎

𝜎 𝜎−1 𝜎−1 𝜎

a) For a CES production function of the form 𝑄 = [𝑎𝐿 + 𝑏𝐾 ] , the elasticity of substitution is 𝜎. In this example we have a CES production function of the form 𝑄 = [𝐾 0.5 + 𝐿0.5 ]2 .

To determine the elasticity of substitution, either set (𝜎 − 1)/𝜎 = 0.5 or 𝜎/(𝜎 − 1) = 2 and solve for 𝜎.

 −1 = 0.5   − 1 = 0.5 0.5 = 1  = 2.

In either case, the elasticity of substitution is 2. b) 𝑄𝜆 = [(𝜆𝐾)0.5 + (𝜆𝐿)0.5 ]2 𝑄𝜆 = [(𝜆0.5 )(𝐾 0.5 + 𝐿0.5 )]2 𝑄𝜆 = 𝜆[𝐾 0.5 + 𝐿0.5 ]2 𝑄𝜆 = 𝜆𝑄.

Since output goes up by the same factor as the inputs, this production function exhibits constant returns to scale.

Q = 100 + ( K )0.5 + ( L)0.5 

c)

Q = 100 +  0.5 ( K 0.5 + L0.5 ) 

2

2

2

 100  Q =   0.5 + K 0.5 + L0.5   Q.  

When the inputs are increased by a factor of 𝜆, where 𝜆 > 1 output goes up by a factor less than 𝜆 implying decreasing returns to scale.

Intuitively, in this production function, while you can increase the 𝐾 and 𝐿 inputs, you cannot increase the constant portion. So output cannot go up by as much as the inputs.

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Chapter 6-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.25 Consider the following production functions and their associated marginal products.

For each production function, indicate whether (a) the marginal product of each input is diminishing, constant, or increasing in the quantity of that input; (b) the production function exhibits decreasing, constant, or increasing returns to scale.

Production function

MPL

MPK

𝑄 =𝐿+𝐾 𝑄 = √𝐿𝐾

𝑀𝑃𝐿 = 1 𝑀𝑃𝐿 1 √𝐾 = 2 √𝐿 𝑀𝑃𝐿 1 1 = 2 √𝐿 𝑀𝑃𝐿 = 3𝐿2 𝐾 3 𝑀𝑃𝐿 = 𝐾

𝑀𝑃𝐾 = 1 𝑀𝑃𝐾 1 √𝐿 = 2 √𝐾 𝑀𝑃𝐾 1 1 = 2 √𝐾 𝑀𝑃𝐾 = 3𝐿3 𝐾 2 𝑀𝑃𝐾 = 𝐿

𝑄 = √𝐿 + √𝐾 𝑄 = 𝐿3 𝐾 3 𝑄 = 𝐿𝐾

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Marginal product of labor?

Constant in L Diminishing in L

Marginal product of capital?

Constant in K Diminishing in K

Returns to scale?

Diminishing in L

Diminishing in K

Decreasing

Increasing in L

Increasing in K

Increasing

Constant in L

Constant in K

Increasing

Constant Constant

Chapter 6-24

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.26. The following table presents information on how many cookies can be produced from eggs and a mixture of other ingredients (measured in ounces):

Recently, you found a new way to mix ingredients with eggs. The same amount of ingredients and eggs produces different numbers of cookies, as shown in the following table:

a) Verify that the change to the new production function represents technological progress. b) For each production fuction find the marginal products of eggs when mixed ingredients is held fixed at 8. Verify that when mixed ingredients is held fixed at 8, the technological progress increases the marginal product of eggs. a) For each pair of inputs, except those where there are no eggs or no other ingredients, new recipe produces more cookies. Hence, the new recipe represents technological progress. b) MPE =Q/E, where E denotes the quantity of eggs. With mixed ingredients held fixed at 8, we have: MPE = (8 – 0)/(1 – 0) = 8, when E goes from 0 to 1.

MPE = (16 – 8)/(2 – 1) = 8, when E goes from 1 to 2.

MPE = (16 – 16)/(3 – 2) = 0, when E goes from 2 to 3.

MPE is zero for all subsequent changes in E. After the technological progress we have: MPE = (10 – 0)/(1 – 0) = 10, when E goes from 0 to 1.

MPE = (19 – 10)/(2 – 1) = 9, when E goes from 1 to 2.

MPE = (22 – 19)/(3 – 2) = 3, when E goes from 2 to 3.

MPE = (23 – 22)/(4 – 3) = 1, when E goes from 3 to 4.

Comparing the marginal products, we see that MPE (when mix equals 8 is higher after the technological progress.

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Chapter 6-25

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6.27. Suppose a firm’s production function initially took the form Q = 500(L + 3K).

However, as a result of a manufacturing innovation, its production function is now Q = 1000(0.5L + 10K). a) Show that the innovation has resulted in technological progress in the sense defined in the text. b) Is the technological progress neutral, labor saving, or capital saving? a) It is possible to write the two production functions as Q1 = 500 L + 1,500 K Q2 = 500 L + 10, 000 K

Since 𝑄2 > 𝑄1 for given quantities of 𝐾 and 𝐿, the firm can achieve more output for a given combination of inputs. This innovation has therefore resulted in technological progress as defined in the text. b) Initially 𝑀𝑃𝐾 = 1,500 and 𝑀𝑃𝐿 = 500 implying the 𝑀𝑅𝑇𝑆𝐿,𝐾 = 0.33. After the innovation the 𝑀𝑃𝐾 = 10,000 and 𝑀𝑃𝐿 = 500 implying the 𝑀𝑅𝑇𝑆𝐿,𝐾 = 0.05. Since the marginal rate of technical substitution of labor for capital has decreased after the innovation this is labor-saving technological progress.

6.28. A firm’s production function is initially Q = KL, with MPK = 0.5(√L/√K) and MPL = 0.5(√K/√L). Over time, the production function changes to Q = KL, with MPK = L and MPL = K. a) Verify that this change represents technological progress. b) Is this change labor saving, capital saving, or neutral? a) With any positive amounts of K and L, √𝐾𝐿 < 𝐾𝐿 so more Q can be produced with the final production function. So there is indeed technological progress. b)

𝑀𝑃

𝐾

With the initial production function 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝑀𝑃 𝐿 = 𝐿 𝐾

𝑀𝑃

𝐾

With the final production function 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝑀𝑃 𝐿 = 𝐿 𝐾

For any ratio of capital to labor, MRTSL,K is the same for the two production functions. Thus, the technological progress is neutral.

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Chapter 6-26

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Solutions Manual

6.29. A firm’s production function is initially Q = KL, with MPK = 0.5(√L/√K) and MPL = 0.5(√K/√L). Over time, the production function changes to Q = K√L, with MPK = √L and MPL = 0.5(K/√L). a) Verify that this change represents technological progress. b) Is this change labor saving, capital saving, or neutral? a) With any positive amounts of K and L, √𝐾𝐿 < 𝐾√𝐿 so more Q can be produced with the final production function. So there is indeed technological progress. 𝑀𝑃

𝐾

b) With the initial production function 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝑀𝑃 𝐿 = 𝐿 𝐾

𝑀𝑃

With the final production function 𝑀𝑅𝑇𝑆𝐿,𝐾 = 𝑀𝑃 𝐿 = 𝐾

0.5𝐾 𝐿

For any ratio of capital to labor, MRTSL,K is lower with the second production function. Thus, the technological progress is labor-saving.

6.30 Suppose that in the 21st century the production of semiconductors requires two inputs: capital (denoted by K) and labor (denoted by L). The production function takes the form 𝑸 = √𝑲𝑳. However, in the 23rd century, suppose the production function for semiconductors will take the form 𝑸 = 𝑲. In other words, in the 23rd century it will be possible to produce semiconductors entirely with capital (perhaps because of robots). a) Does this change in the production function change the returns to scale? b) Is this change in the production function an illustration of technological progress?

No. In both the 21st and 23rd centuries, the production function for this good exhibited constant returns to scale. In both cases, increasing inputs by a given proportion increases output by the same proportion.

The change in the production function is not an example of technological progress. This is because we do not get more output from a given combination of inputs. For example, if L = 100, K=100, in the 21st century 𝑄 = √100 × 100 = 100. In the 21st century, with the same input combination, we would get the same output, Q = 100.

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Chapter 6-27

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 7 Costs and Cost Minimization Solutions to Review Questions 1. A biotechnology firm purchased an inventory of test tubes at a price of $0.50 per tube at some point in the past. It plans to use these tubes to clone snake cells. Explain why the opportunity cost of using these test tubes might not equal the price at which they were acquired.

Acquisition cost and opportunity cost are not necessarily the same. As the text points out, opportunity costs are forward looking. The opportunity cost is the payoff associated with the best of the alternatives that are not chosen. Once the test tubes are purchased, the decision is to use the tubes to clone snake cells or something else. It is possible that someone values the tubes for some purpose at higher (or lower) than $0.50 so that selling the tubes would earn the firm something more (or less) than $0.50 per tube. The opportunity cost then is different than the acquisition cost.

2. You decide to start a business that provides computer consulting advice for students in your residence hall. What would be an example of an explicit cost you would incur in operating this business? What would be an example of an implicit cost you would incur in operating this business?

Since the business is computer consulting, an explicit cost, a cost involving a direct monetary outlay, might be the cost of paper and ink used to advertise your service. An implicit cost, a cost not involving a direct monetary outlay, might be the opportunity cost of your time, e.g., to earn money working at the student fitness center or to study for your own classes.

3. Why does the “sunkness” or “nonsunkness” of a cost depend on the decision being made?

Whether or not a particular cost is sunk or not depends on the decision being made. If the cost does not change as a result of the decision the cost is sunk, while if the cost does change the cost is not sunk.

4. How does an increase in the price of an input affect the slope of an isocost line? 𝑇𝐶

𝑤

A firm’s total costs are TC = rK + wL, so the equation for a typical isocost line is 𝐾 = 𝑟 − 𝑟 𝐿.

Since the slope of the isocost line is given by −𝑤/𝑟, if the price of labor increases the isocost line will become steeper and if the price of capital increases the isocost line will become flatter.

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Chapter 7-1

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Solutions Manual

5. Could the solution to the firm’s cost-minimization problem ever occur off the isoquant representing the required level of output?

The solution to the firm’s cost minimization problem must lie on an isoquant. While the firm could produce a given output with a combination of inputs not on the isoquant, say by using more labor and more capital than necessary, a combination such as this would not be efficient and therefore not cost minimizing.

6. Explain why, at an interior optimal solution to the firm’s cost-minimization problem, the additional output that the firm gets from a dollar spent on labor equals the additional output from a dollar spent on capital. Why would this condition not necessarily hold at a corner point optimal solution?

To understand why at an interior optimum the additional output the firm gets from a dollar spent on labor must equal the additional output the firm gets from a dollar spent on capital, assume these were not equal. For example, suppose the firm could get more output from a dollar spent on labor than on a dollar spent on capital. Then the firm could take one dollar away from capital and reallocate it to labor. Since the firm gets more output from a dollar of labor than from a dollar of capital, it will require the firm to spend less than one dollar on labor to offset the decline in output from taking one dollar away from capital. This implies the firm can keep output at the same level but do so at a lower cost. Therefore, if these amounts are not equal the firm is not minimizing cost.

This requirement does not necessarily hold at a corner solution. While the firm could potentially reduce cost by reallocating spending to the more productive input, at a corner solution, by definition, the firm is not using one of the inputs. There is no further opportunity to reallocate spending if the firm is spending nothing on one of the inputs, i.e., the firm cannot move to a point where one of the inputs is negative.

7. What is the difference between the expansion path and the input demand curve?

The expansion path traces out the cost minimizing combinations of all inputs as the level of output is increased (expanded) holding the prices of the inputs fixed. An input demand curve traces out a firm’s cost minimizing quantity of one input as the price of that input varies holding the level of output and the prices of the other inputs fixed.

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Chapter 7-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

8. In Chapter 5 you learned that, under certain conditions, a good could be a Giffen good: An increase in the price of the good could lead to an increase, rather than a decrease, in the quantity demanded. In the theory of cost minimization, however, we learned that, an increase in the price of an input will never lead to an increase in the quantity of the input used. Explain why there cannot be “Giffen inputs.”

Giffen goods arise when the income effect is so severely negative that it offsets the substitution effect. This can happen because in consumer choice, income was an exogenous variable – therefore, changes in price affect both the relative substitutability of goods (via the tangency condition) as well as the consumer’s purchasing power (via the budget constraint). By contrast, in the cost minimization problem output is exogenous while the expenditure is the objective function. Thus, a change in an input price affects only the relative substitutability of inputs (via the tangency condition) – there is no corresponding effect on the production constraint, since prices do not appear there. So while there is a “substitution effect” in cost minimization, there is no corresponding “income effect” as in consumer choice. Therefore, increases in input prices will always lead to decreases in the use of that input (except at corner solutions, where there might be no change). So there cannot be a Giffen input.

9. For a given quantity of output, under what conditions would the short-run quantity demanded for a variable input (such as labor) equal the quantity demanded in the long run?

Assuming quantity is fixed, the short-run demand for a variable input would equal its long-run demand if the level of the fixed input in the short run was cost minimizing for the quantity of output being produced in the long run.

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Chapter 7-3

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Solutions Manual

Solutions to Problems 7.1. A computer-products retailer purchases laser printers from a manufacturer at a price of $500 per printer. During the year the retailer will try to sell the printers at a price higher than $500 but may not be able to sell all of the printers. At the end of the year, the manufacturer will pay the retailer 30 percent of the original price for any unsold laser printers. No one other than the manufacturer would be willing to buy these unsold printers at the end of the year. a) At the beginning of the year, before the retailer has purchased any printers, what is the opportunity cost of laser printers? b) After the retailer has purchased the laser printers, what is the opportunity cost associated with selling a laser printer to a prospective customer? (Assume that if this customer does not buy the printer, it will be unsold at the end of the year.) c) Suppose that at the end of the year, the retailer still has a large inventory of unsold printers. The retailer has set a retail price of $1,200 per printer. A new line of printers is due out soon, and it is unlikely that many more old printers will be sold at this price. The marketing manager of the retail chain argues that the chain should cut the retail price by $1,000 and sell the laser printers at $200 each. The general manager of the chain strongly disagrees, pointing out that at $200 each, the retailer would “lose” $300 on each printer it sells. Is the general manager’s argument correct? a) $500 b) 30% of $500, or $150 c) By not lowering the price and assuming the firm cannot sell any more printers, the best the firm can hope for is the $150 the firm can receive from the manufacturer. If the firm drops the price to $200 and sells the printers on their own they can actually “profit” an additional $50 over their best available alternative.

7.2. A grocery shop is owned by Mr. Moore and has the following statement of revenues and costs: Revenues Supplies Electricity Employee salaries Mr. Moore’s salary

$250,000 $25,000 $6,000 $75,000 $80,000

Mr. Moore always has the option of closing down his shop and renting out the land for $100,000. Also, Mr. Moore himself has job offers at a local supermarket at a salary of $95,000 and at a nearby restaurant at $65,000. He can only work one job, though. What are the shop’s accounting costs? What are Mr. Moore’s economic costs? Should Mr. Moore shut down his shop?

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Chapter 7-4

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Solutions Manual

The accounting costs are simply the sum: 25,000 + 75,000 + 80,000 + 6,000 = $186,000 and the shop’s accounting profit is $64,000 which means that Mr. Moore’s total gain from this venture is 80,000 + 64,000 = $144,000.

The economic costs also include the opportunity cost of the land rental ($100,000) and of Mr. Moore’s next best alternative, which in this case is $95,000. That is, Mr. Moore loses $15,000 by not choosing his next best alternative. Therefore Mr. Moore’s total economic costs are 186,000 + 100,000 + 15,000 = $301,000, which exceeds his revenues by $51,000.

If he were to shut down the shop, Mr. Moore would earn 100,000 + 95,000 = $195,000 which is more than the $144,000 he currently earns (by precisely the $51,000 figure from above).

Therefore he should shut down the shop.

7.3. Last year the accounting ledger for an owner of a small drugstore showed the following information about her annual receipts and expenditures. She lives in a tax-free country (so don't worry about taxes).

Revenues Wages paid to hired labor (other than herself) Utilities (fuel, telephone, water) Purchases of drugs and other supplies for the store Wages paid to herself

$1,000,000 $300,000 $ 20,000 $500,000 $100,000

She pays a competitive wage rate to her workers, and the utilities and drugs and other supplies are all obtained at market prices. She already owns the building, so she has no cash outlay for its use. If she were to close the business, she could avoid all of her expenses, and, of course, would have no revenue. However, she could rent out her building for $200,000. She could also work elsewhere herself. Her two employment alternatives include working at another drugstore, earning wages of $100,000, or working as a freelance consultant, earning $80,000. Determine her accounting profit and her economic profit if she stays in the drug store business. If the two are different, explain the difference between the two values you have calculated.

Her accounting profit equals revenues less all of the expenses reflected in the ledger: $1,000,000 - $300,000 - $20,000 - $500,000 - $100,000 = $80,000.

All of the accounting costs are also economic costs. The first three expense items (wages paid to hired labor, utilities, and purchases of drugs and supplies) are expenses in competitive markets, so the opportunity cost is reflected in the market prices. Further, the wages she pays herself are the same as the opportunity cost of her time, because the most she could earn if she closes her business is $100,000 working at another drugstore.

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Chapter 7-5

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Solutions Manual

The economic costs of the business include all of the accounting costs, plus the $200,000 opportunity cost of the building because she could earn that if she exits the drug store business.

Her economic profit is her accounting profit ($80,000) less the additional opportunity cost ($200,000) not included in the accounting cost. So her economic profit is actually -$120,000.

We can look at this another way. If she continues to work at the grocery store, she earns an accounting profit of $80,000, plus the salary she pays herself ($100,000). But if she exits the business, her salary working for another drugstore would be $100,000, and she would receive $200,000 rent for the building. She would therefore be better off by $120,000 if she takes the job working for another drugstore.

7.4. A consulting firm has just finished a study for a manufacturer of wine. It has determined that an additional man-hour of labor would increase wine output by 1,000 gallons per day. Adding another machine-hour of fermentation capacity would increase output by 200 gallons per day. The price of a man-hour of labor is $10 per hour. The price of a machine-hour of fermentation capacity is $0.25 per hour. Is there a way for the wine manufacturer to lower its total costs of production and yet keep its output constant? If so, what is it?

At the optimum we must have

𝑀𝑃𝐾 𝑟

=

𝑀𝑃𝐿 𝑤

.

200 1000  In this problem we have 0.25 10 .

800  100

This implies that the firm receives more output per dollar spent on an additional machine hour of fermentation capacity than for an additional hour spent on labor. Therefore, the firm could lower cost while achieving the same level of output by using fewer hours of labor and more hours of fermentation capacity.

7.5. A firm uses two inputs, capital and labor, to produce output. Its production function exhibits a diminishing marginal rate of technical substitution. a) If the price of capital and labor services both increase by the same percentage amount (e.g., 20 percent), what will happen to the cost-minimizing input quantities for a given output level? b) If the price of capital increases by 20 percent while the price of labor increases by 10 percent, what will happen to the cost-minimizing input quantities for a given output level? a) If the price of both inputs change by the same percentage amount, the slope of the isocost line will not change. Since we are holding the level of output fixed, the isocost line will be tangent to the isoquant at the same point as prior to the price increase. Therefore, the cost-minimizing quantities of the inputs will not change.

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Chapter 7-6

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Solutions Manual

b) If the price of capital increases by a larger percentage than the price of labor, then, relatively speaking, the price of labor has become cheaper. The firm will substitute away from capital and add labor until either the tangency condition holds or a corner solution is reached.

7.6. A farmer uses three inputs to produce vegetables: land, capital, and labor. The production function for the farm exhibits diminishing marginal rate of technical substitution. a) In the short run the amount of land is fixed. Suppose the prices of capital and labor both increase by 5 percent. What happens to the cost-minimizing quantities of labor and capital for a given output level? Remember that there are three inputs, one of which is fixed. b) Suppose only the cost of labor goes up by 5 percent. What happens to the costminimizing quantity of labor and capital in the short run? a) The amount of land used in production is fixed in the short-run. Hence, in the short-run the farmer chooses amount of capital and labor. It follows that cost-minimizing quantities of labor and capital have to satisfy equation MPL / MPK = w/r where w and r denote prices of labor and capital. Notice that w/r = (1.05 w)/ (1.05 r). The cost-minimizing quantities of inputs, for each level of output, do not change when prices of both inputs go up by 5% and quantity of land is fixed. b) For a given output level, the cost-minimizing farmer uses more capital and less labor.

7.7. The text discussed the expansion path as a graph that shows the cost-minimizing input quantities as output changes, holding fixed the prices of inputs. What the text didn’t say is that there is a different expansion path for each pair of input prices the firm might face. In other words, how the inputs vary with output depends, in part, on the input prices.

Consider, now, the expansion paths associated with two distinct pairs of input prices, (w1, r1) and (w2, r2). Assume that at both pairs of input prices, we have an interior solution to the cost-minimization problem for any positive level of output. Also assume that the firm’s isoquants have no kinks in them and that they exhibit diminishing marginal rate of technical substitution. Could these expansion paths ever cross each other at a point other than the origin (L = 0, K = 0)?

Imagine that two expansion paths did cross at some point. Recall that the expansion path traces out the cost- minimizing combinations of inputs as output increases. Essentially the expansion path traces out all of the tangencies between the isocost lines and isoquants. These tangencies 𝑀𝑃 𝑤 occur at the point where 𝑀𝑃 𝐿 = 𝑟 . 𝐾

If the expansion paths cross at some point then the cost minimizing combination of inputs must 𝑀𝑃 𝑀𝑃 𝑀𝑃 𝑀𝑃 be identical with both sets of prices. This would require that 𝑟 𝐾 = 𝑤 𝐿 and 𝑟 𝐾 = 𝑤 𝐿.

1

1

2

2

Unless the input prices are proportional, i.e. unless w1 / r1 = w2 / r2, it is not possible for both of these equations to hold. Therefore, it is not possible for the expansion paths to cross unless the prices are proportional, in which case the two expansion paths will be identical.

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Chapter 7-7

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Solutions Manual

7.8. Suppose the production of airframes is characterized by a CES production function: Q = (L½ + K½)2. The marginal products for this production function are MPL = (L½ + K½)L−½ and MPK = (L½+ K½)K−½. Suppose that the price of labor is $10 per unit and the price of capital is $1 per unit. Find the cost-minimizing combination of labor and capital for an airframe manufacturer that wants to produce 121,000 airframes.

The tangency condition implies:  L1/ 2 + K 1/ 2  K −1/ 2  L1/ 2 + K 1/ 2  L−1/ 2 = r w 1 1 = r K w L w L =r K K w2 = L r2

Given that 𝑤 = 10 and 𝑟 = 1, this implies K L 100 L = K 100 =

Returning to the production function and assuming 𝑄 = 121,000 yields 121, 000 =  L1/ 2 + K 1/ 2 

2

121, 000 =  L1/ 2 + (100 L)1/ 2  121, 000 =  L1/ 2 + 10 L1/ 2  121, 000 = 11L1/ 2 

2

2

2

121, 000 = 121L 1, 000 = L

Since 𝐾 = 100𝐿, 𝐾 = 100(1000) = 100,000. The cost minimizing quantities of capital and labor to produce 121,000 airframes is 𝐾 = 100,000 and 𝐿 = 1,000.

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Chapter 7-8

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Solutions Manual

7.9. Suppose the production of airframes is characterized by a Cobb–Douglas production function: Q = LK. The marginal products for this production function are MPL = K and MPK = L. Suppose the price of labor is $10 per unit and the price of capital is $1 per unit.

Find the cost-minimizing combination of labor and capital if the manufacturer wants to produce 121,000 airframes.

The tangency condition implies K L 10 L = K 10 =

Substituting into the production function yields 121, 000 = LK 121, 000 = L(10 L) 121, 000 = 10 L2 12,100 = L2 110 = L

Since 𝐾 = 10𝐿, 𝐾 = 1,100. The cost-minimizing quantities of labor and capital to produce 121,000 airframes are 𝐾 = 1,100 and 𝐿 = 110.

7.10. The processing of payroll for the 10,000 workers in a large firm can either be done using 1 hour of computer time (denoted by K) and no clerks or with 10 hours of clerical time (denoted by L) and no computer time. Computers and clerks are perfect substitutes; for example, the firm could also process its payroll using 1/2 hour of computer time and 5 hours of clerical time. a) Sketch the isoquant that shows all combinations of clerical time and computer time that allows the firm to process the payroll for 10,000 workers. b) Suppose computer time costs $5 per hour and clerical time costs $7.50 per hour. What are the cost-minimizing choices of L and K? What is the minimized total cost of processing the payroll? c) Suppose the price of clerical time remains at $7.50 per hour. How high would the price of an hour of computer time have to be before the firm would find it worthwhile to use only clerks to process the payroll?

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Chapter 7-9

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a)

K and L are perfect substitutes, meaning that the production function is linear and the isoquants are straight lines. We can write the production function as Q = 10,000K + 1000L, where Q is the number of workers for whom payroll is processed. b) If 𝑟 = 5 and 𝑤 = 7.50, the slope of a typical isocost line will be −7.5/5.0 = −1.5. This is steeper than the isoquant implying that the firm will employ only computer time (𝐾) to minimize cost. The cost minimizing combination is 𝐾 = 1 and 𝐿 = 0. This outcome can be seen in the graph below. The isocost lines are the dashed lines.

The total cost to process the payroll for 10,000 workers will be 𝑇𝐶 = 5(1) + 7.5(0) = 5. c) The firm will employ clerical time only if MPL / w > MPK / r. Thus we need 0.1 / 7.5 > 1/r or r > 75.

7.11. A firm produces an output with the production function Q = KL, where Q is the number of units of output per hour when the firm uses K machines and hires L workers each hour. The marginal products for this production function are MPK = L and MPL = K. The factor price of K is 4 and the factor price of L is 2. The firm is currently using K = 16 and just enough L to produce Q = 32. How much could the firm save if it were to adjust K and L to produce 32 units in the least costly way possible?

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Chapter 7-10

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Solutions Manual

Currently the firm must be using L = Q/K = 32/16 = 2 units of labor. Let the factor prices of capital and labor be, respectively, r and w.

Its total expenditure is C = wL + rK = 2(2) + 4(16) = 68.

If it were to minimize cost, it would hire L and K so that (1) MPK/r = MPL/w, or L/4 = K/2, or L = 2K and (2) Q = LK. (1) And (2) imply that Q = 2K2, or 32 = 2K2, and thus K = 4 and L = 8.

So Q = 32 can be produced efficiently with a cost of C = wL + rK = 2(8) + 4(4) = 32.

The firm could save 68 – 32 = 36 by producing efficiently.

7.12. A firm operates with the production function Q = K2L. Q is the number of units of output per day when the firm rents K units of capital and employs L workers each day.

The marginal product of capital is 2KL, and the marginal product of labor is K2. The manager has been given a production target: Produce 8,000 units per day. She knows that the daily rental price of capital is $400 per unit. The wage rate paid to each worker is $200 day. a) Currently the firm employs at 80 workers per day. What is the firm’s daily total cost if it rents just enough capital to produce at its target? b) Compare the marginal product per dollar sent on K and on L when the firm operates at the input choice in part (a). What does this suggest about the way the firm might change its choice of K and L if it wants to reduce the total cost in meeting its target? c) In the long run, how much K and L should the firm choose if it wants to minimize the cost of producing 8,000 units of output day? What will the total daily cost of production be? a) Suppose that the firm is operating in the short run, with L = 80. To produce Q = 8,000, how much K will it require? From the production function we observe that 8,000 = K2 (80) => K = 10.

The total cost would be C = wL + rK = $200(80) + $400(10) = $2,000 per day. b) Let’s examine the “bang for the buck” for K and L when K = 10 and L = 80.

For capital: MPK / r = 2KL / 400 = 2(10)(80) / 400 = 4 For labor: MPL / w = K2 / 200 = 102 / 200 = 0.5 So the marginal product per dollar spent on capital exceeds that of labor. The firm would like to rent more capital and hire fewer workers.

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Solutions Manual

c) Because the production function is Cobb-Douglas, we know that it has diminishing MRTSL,K and that the isoquants do not intersect either the K or L axis. Thus the cost reducing basket (K,L) will be interior (with K > 0 and L > 0). To find the optimum, we use the two conditions: (1) Tangency condition: MPK / MPL = r / w => 2KL/K2 = 400 / 200 => K = L (2) Production Requirement: K2L = 8,000 Together equations (1) and (2) tell us that K = 20 and L = 20.

The total cost would be C = wL + rK = $200(20) + $400(20) = $12,000 per day.

7.13. Consider the production function Q = LK, with marginal products MPL = K and MPK = L. Suppose that the price of labor equals w and the price of capital equals r. Derive expressions for the input demand curves. 𝐾

𝑤

𝑤

From the tangency condition, we get 𝐿 = 𝑟  𝐾 = ( 𝑟 ) 𝐿 Substituting into the production function yields 𝑄 = 𝐿𝐾 𝑤 𝑄 = 𝐿( )𝐿 𝑟 𝑤 2 𝑄 = ( )𝐿 𝑟 𝑟𝑄 1/2 𝐿=( ) 𝑤 𝑤

This represents the input demand curve for 𝐿. Since 𝐾 = ( 𝑟 ) 𝐿 we have 𝑤 𝑟𝑄 1/2 𝐾 = ( )( ) 𝑟 𝑤 𝑤𝑄 1/2 𝐾=( ) 𝑟 This represents the input demand curve for 𝐾.

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Chapter 7-12

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Solutions Manual

7.14. A cost-minimizing firm’s production function is given by Q = LK, where MPL = K and MPK = L. The price of labor services is w and the price of capital services is r. Suppose you know that when w = $4 and r = $2, the firm’s total cost is $160. You are also told that when input prices change such that the wage rate is 8 times the rental rate, the firm adjusts its input combination but leaves total output unchanged. What would the cost-minimizing input combination be after the price changes? 𝐾

𝑤

Using the tangency condition, with the original input prices: 𝐿 = 𝑟 = 2. So, K = 2L. Also, using the information on total costs, 4𝐿 + 2𝐾 = 160. Combining these two equations, we get (L, K) = (20, 40). Therefore the firm produces 20*40 = 800 units of output.

After the prices change, even though we don’t know the numerical values of the input prices, we can still answer the question using the fact that we’re told w = 8r. The tangency condition 𝐾 implies that 𝐿 = 8, so K = 8L. Also, we have 𝐾𝐿 = 800. This implies that the optimal input combination is (L, K) = (10, 80).

7.15. Ajax, Inc. assembles gadgets. It can make each gadget either by hand or with a special gadget-making machine. Each gadget can be assembled in 15 minutes by a worker or in 5 minutes by the machine. The firm can also assemble some of the gadgets by hand and some with machines. Both types of work are perfect substitutes, and they are the only inputs necessary to produce the gadgets. a) It costs the firm $30 per hour to use the machine and $10 per hour to hire a worker. The firm wants to produce 120 gadgets. What are the cost-minimizing input quantities?

Illustrate your answer with a clearly labeled graph. b) What are the cost-minimizing input quantities if it costs the firm $30 per hour to use the machine, and $10 per hour to hire a worker? Illustrate your answer with a graph. c) Write down the equation of the firm’s production function for the firm. Let G be the number of gadgets assembled, M the number of hour the machines are used, and L the number of hours of labor.

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Solutions Manual

a) Isoquants for the production function are straight lines. At the given input prices slope of an isoquant is equal to the ratio of the input prices. Hence, all positive input quantities (measured in work hours) such that 4L + 12M = 120 are cost-minimizing.

b) When one hour of the machine’s work costs $20 cost-minimizing firm does not use manual work at all. The cost-minimizing quantity of the machine’s work necessary to produce 120 widgets is equal to M = 120/12 = 10 hours. The firm spends $200. (Note that if the firm were to use only manual labor, the cost would be $300 = 30 hours x $10 per hour).

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

7.16. A construction company has two types of employees: skilled and unskilled. A skilled employee can build 1 yard of a brick wall in one hour. An unskilled employee needs twice as much time to build the same wall. The hourly wage of a skilled employee is $15. The hourly wage of an unskilled employee is $8. a) Write down a production function with labor. The inputs are the number of hours of skilled workers, LS, the number of hours worked by unskilled employees, LU, and the output is the number of yards of brick wall, Q. b) The firm needs to build 100 yards of a wall. Sketch the isoquant that shows all combinations of skilled and unskilled labor that result in building 100 yards of the wall. c) What is the cost-minimizing way to build 100 yards of a wall? Illustrate your answer on the graph in part (b). a) The production function is Q = LS + ½ LU where LS denotes hours worked by skilled workers and LU denotes hours worked by unskilled workers. Both types of labor are perfect substitutes. b) The isoquant is a straight line.

c) MPLs/ws = 1/15; MPLu/wu = 0.5/8 = 1/16. Thus, the “bang for the buck” is higher for skilled labor, and the firm will use only skilled labor.

Note that the total cost of building 100 yards with skilled labor is (100 hours)($15/ hour) = $1500.

The total cost of building 100 yards with unskilled labor is (200 hours)($8/ hour) = $1600.

The isocost line representing a $1500 expenditure is drawn as a dotted line in the graph in (b).

The isocost line is more steeply sloped than the isoquant in the graph because the marginal rate of technical substitution of unskilled labor for unskilled labor is equal to ½, while the ratio of input prices is equal to 8/15.

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Solutions Manual

7.17. A paint manufacturing company has a production function Q = K + √L. For this production function MPK = 1 and MPL = 1/(2√L). The firm faces a price of labor w that equals $1 per unit and a price of capital services r that equals $50 per unit. a) Verify that the firm’s cost-minimizing input combination to produce Q = 10 involves no use of capital. b) What must the price of capital fall to in order for the firm to use a positive amount of capital, keeping Q at 10 and w at 1? c) What must Q increase to for the firm to use a positive amount of capital, keeping w at 1 and r at 50? a) First, note that this production function has diminishing MRSL,K. The tangency condition would imply that 1/2√𝐿 = 1/50 or L = 625. Substituting this back into the production function we see that K = 10 – 25 = –15. Since the firm cannot use a negative amount of capital, the tangency condition is not valid in this case.

Looking at the corner with K = 0, since Q = 10 the firm requires L = Q2 = 100 units of labor. At this point, MPL / w = (1/20)/1 = 0.05 > MPK / r = 1/50 = 0.02. Since the marginal product per dollar is higher for labor, the firm will use only labor and no capital. b) The firm will use a positive amount of capital when

𝑀𝑃𝐿 𝑤

=

𝑀𝑃𝐾 𝑟

, or 2√𝐿 = 𝑟. Thus L = 0.25r2.

From the production constraint K = 𝑄 − √𝐿 = 10 – 0.5r. So if K > 0 then we must have 10 – 0.5r > 0, or r < 20. c) Again, using the tangency condition we must have 2√𝐿 = 𝑟. Therefore, since r = 50, L = 625.

From the production constraint, the input demand for capital is K = 𝑄 − √𝐿 = Q – 25. So if K > 0 then we must have Q > 25.

7.18. A researcher claims to have estimated input demand curves in an industry in which the production technology involves two inputs, capital and labor. The input demand curves he claims to have estimated are L = wr2Q and K = w2rQ. Are these valid input demand curves? In other words, could they have come from a firm that minimizes its costs?

No, these are not valid input demand curves. In both cases the quantity of the input is positively related to the input’s price. Such upward-sloping input demand curves cannot exist.

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Solutions Manual

7.19. A manufacturing firm’s production function is Q = KL + K + L. For this production function, MPL = K + 1 and MPK = L + 1. Suppose that the price r of capital services is equal to 1, and let w denote the price of labor services. If the firm is required to produce 5 units of output, for what values of w would a cost-minimizing firm use a) only labor? b) only capital? c) both labor and capital?

If K = 0, then the firm must hire L = 5 units of labor. For this to be optimal, it must be that MPL / w > MPK / r, or 1/w > 6. In other words, w < 1/6.

If L = 0, then the firm must hire K = 5 units of capital. For this to be optimal, it must be that MPL / w < MPK / r, or 6/w > 1. In other words, w > 6.

For the firm to use both capital and labor, it must be that 1/6 < w < 6. To see why, notice that the indifference curves will have diminishing MRTSL,K. In particular, MRTSL,K = 6 where the Q = 5 indifference curve intersects the K-axis (where L = 0). Diminishing MRTSL,K implies that the Q = 5 indifference curve will gradually flatten out until it intersects the L-axis (where K = 0), at which point MRTSL,K = 1/6.

7.20. Suppose a production function is given by Q = min(L, K)—that is, the inputs are perfect complements. Draw a graph of the demand curve for labor when the firm wants to produce 10 units of output (Q = 10).

The input demand curves will be vertical lines, representing the fact that the demand by firms for such inputs is inelastic. If the firm’s production function is 𝑄 = 𝑚𝑖𝑛( 𝐿, 𝐾) then, holding fixed the quantity of production and the price of capital, if the wage rate were to increase it would not change the firm’s requirement for labor. Therefore, the demand for each input is independent of price and the demand curves are vertical lines.

7.21. A firm’s production function is Q = min(K , 2L), where Q is the number of units of output produced using K units of capital and L units of labor. The factor prices are w = 4 (for labor) and r = 1 (for capital). On an optimal choice diagram with L on the horizontal axis and K on the vertical axis, draw the isoquant for Q = 12, indicate the optimal choices of K and L on that isoquant, and calculate the total cost.

The isoquant Q = 12 is shown for this Leontief technology. To produce Q = 12, the firm will need at least K = 12 and L = 6. This will cost the firm C = wL + rK = 4(6) + 1(12) = 36. The isocost line representing an expenditure of 36 is drawn below. The optimal basket of inputs is A.

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K

Solutions Manual

B

A Q = 12

12

Isocost C = 36 Isocost line has slope -w/r = -4 L 6

7.22. Suppose a production function is given by Q = K + L—that is, the inputs are perfect substitutes. For this production function, MPL = 1 and MPK = 1. Draw a graph of the demand curve for labor when the firm wants to produce 10 units of output and the price of capital services is $1 per unit (Q = 10 and r = 1).

Recall that with a linear production function we are usually going to get corner point solutions.

In this case, the firm will employ only labor and no capital if labor is cheap enough or, 𝑀𝑃𝐿 𝑀𝑃 𝑏𝑟 > 𝑟 𝐾 i.e. if 𝑤 < 𝑎 . Similarly it will use just capital if the rental rate is low enough 𝑤 𝑎𝑤

𝑄

i.e. 𝑟 < 𝑏 . If the firm uses only labor, it will use 𝐿 = 𝑎 units regardless of the price, and 𝑄

similarly it will use 𝐾 = 𝑏 units of capital if it uses any capital at all. The input demand curve for labor for a given price, r, of capital, is shown below.

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

7.23. Suppose a production function is given by Q = 10K + 2L. The factor price of labor is 1.

Draw the demand curve for capital when the firm is required to produce Q = 80.

With this production function the firm views K and L as perfect substitutes.

The firm will be at a corner point with K = 0 when MPK/r < MPL/w, or when 10/r <2/1, or when r > 5.

The firm will be at a corner point with L = 0 when MPK/r > MPL/w, or when 10/r >2/1, or when r < 5. When the firm needs to produce Q = 80, how much capital will it need? The production function shows that 80 = 10K, or K = 8 units.

When r = 5, the firm might use any combination of K and L along the isoquant 80 = 10K + 2L. The firm might therefore use any K such that 0 < K < 8.

The graph of the demand for labor is as shown.

7.24. Consider the production function Q = K + √L. For this production function, MPL = 1/(2√L) and MPK = 1. Derive the input demand curves for L and K, as a function of the input prices w (price of labor services) and r (price of capital services). Show that at an interior optimum (with K > 0 and L > 0) the amount of L demanded does not depend on Q. What does this imply about the expansion path?

1

𝑤

𝑟

The tangency condition implies that 2√𝐿 = 𝑟 , or √𝐿 = 2𝑤. Clearly the demand curve for L is not a function of the level of output, Q. Therefore, as the level of output changes, the amount of labor is constant. Therefore, if we were to graph isoquants with labor on the horizontal axis, the expansion path for labor would just be a straight, vertical line.

The demand curve for capital can be derived by substituting the demand curve for labor into the 𝑟 𝑟 production function. That is, 𝐾 + = 𝑄, so 𝐾 = 𝑄 − .

2𝑤

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2𝑤

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Solutions Manual

7.25. A firm has the production function Q = LK. For this production function, MPL = K and MPK = L. The firm initially faces input prices w = $1 and r = $1 and is required to produce Q = 100 units. Later the price of labor w goes up to $4. Find the optimal input combinations for each set of prices and use these to calculate the firm’s price elasticity of demand for labor over this range of prices. 𝐾

Using the tangency condition, initially 𝐿 = 1, implying that K = L. Since KL = 100, we get K = L = 10.

Under the new prices, the tangency condition implies that K=4L. This means that the optimal input combination is (L, K) = (5, 20).

The percent change in price is (4 – 1)*100 = 300%. While the percent change in the demand for labor is [(5 – 10)/10]*100 = –50%. Therefore the price elasticity of demand over this range of prices is –50/300 = –1/6.

7.26 A bicycle is assembled out of a bicycle frame and two wheels. a) Write down a production function of a firm that produces bicycles out of frames and wheels. No assembly is required by the firm, so labor is not an input in this case. Sketch the isoquant that shows all combinations of frames and wheels that result in producing 100 bicycles. b) Suppose that initially the price of a frame is $100 and the price of a wheel is $50. On the graph you drew for part (a), show the choices of frames and wheels that minimize the cost of producing 100 bicycles, and draw the isocost line through the optimal basket. Then repeat the exercise if the price of a frame rises to $200, while the price of a wheel remains $50. a) The production function is Q = min(F, ½ W), where F denotes the number of frames and W denotes the number of wheels.

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Solutions Manual

b) To produce 100 bicycles in the least costly manner, the firm always needs to choose basket A, with 200 wheels and 100 frames.

Initially, when the price of a frame is $100 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $20,000.

Later, when the price of a frame is $200 and the price of a wheel is $50, the isocost line is the lighter one shown in the graph; all points on the isocost line indicate an expenditure of $30,000.

7.27. Suppose that the firm’s production function is given by Q = 10KL1/3. The firm’s capital is fixed at K. What amount of labor will the firm hire to solve its short-run costminimization problem?

With just two inputs, there is no tangency condition to worry about in the short run. To find the short-run cost-minimizing quantity of labor, we need only solve the production function for 𝐿 in 1

terms of 𝑄 and 𝐾: 𝑄 = 10𝐾𝐿3 .

This gives us 𝐿 =

𝑄3 1000𝐾

3

.

This is the cost-minimizing quantity of labor in the short run.

7.28. A plant’s production function is Q = 2KL + K. For this production function, MPK = 2L + 1 and MPL = 2K. The price of labor services w is $4 and of capital services r is $5 per unit. a) In the short run, the plant’s capital is fixed at K = 9. Find the amount of labor it must employ to produce Q = 45 units of output. b) How much money is the firm sacrificing by not having the ability to choose its level of capital optimally? a) Since 𝐾 = 9, we get 18𝐿 + 9 = 45which implies that L = 36/18 = 2. Therefore the firm’s total cost with this input combination is 4(2) + 5(9) = $53. b) If the firm could operate optimally, it would choose labor and capital to satisfy the tangency 2𝐾 4 condition: 2𝐿+1 = 5, implying that 10𝐾 = 8𝐿 + 4. Also, 2𝐾𝐿 + 𝐾 = 45. Combining these two conditions, 𝐾 = √18= 4.24 and L = 4.8. Now the firm’s expenditure would be 4(4.24) + 5(4.8) = $41 approximately. Therefore the firm loses about $12 because of its constraint on capital.

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Solutions Manual

7.29. Suppose that the firm uses three inputs to produce its output: capital K, labor L, and materials M. The firm’s production function is given by Q = K1/3L1/3M1/3. For this production function, the marginal products of capital, labor, and materials are MPK = 1/3K−2/3L1/3M1/3, MPL = 1/3K1/3L−2/3M1/3, and MPM = 1/3K1/3L1/3M−2/3. The prices of capital, labor, and materials are r = 1, w = 1, and m = 1, respectively. a) What is the solution to the firm’s long-run cost minimization problem given that the firm wants to produce Q units of output? b) What is the solution to the firm’s short-run cost minimization problem when the firm wants to produce Q units of output and capital is fixed at K? c) When Q = 4, the long-run cost-minimizing quantity of capital is 4. If capital is fixed at K = 4 in the short run, show that the short-run and long-run cost-minimizing quantities of labor and materials are the same. a) Here we have two tangency conditions and the requirement that 𝐿, 𝐾, and 𝑀 produce 𝑄 units of output.

1

−2

1

1

−2

1

1 K 3L 3M 3 1 MPL w =  3 1 1 −2 =  M = L 1 3 3 3 MPM m 1 3 K L M 1 K 3L 3M 3 1 MPL w =  3 −2 1 1 =  K = L 1 3 MPK r L3 M 3 1 3 K 1

1

1

Q = K 3 L3 M 3

This is a system of three equations in three unknowns. The solution to this system gives us the long-run cost-minimizing input combination: L=Q M =Q K =Q 𝑀𝑃

𝑤

b) The tangency condition 𝑀𝑃 𝐿 = 𝑚 is 𝑀

1

1 3 −23 13 3 𝐾 𝐿 𝑀 = 1, 1 1 3 13 −23 1 𝐾 3 𝐿 𝑀 which implies 𝑀 = 𝐿 To find the short-run cost-minimizing quantity of labor, we plug this back into the production function and solve for 𝐿 in terms of 𝑄 and 𝐾.

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Solutions Manual

1 1 1 𝑄 = 𝐾 3 𝐿3 𝐿3 3

which when we solve for 𝐿 gives us the short-run cost-minimizing quantity of labor 𝐿 =

𝑄2 1

.

𝐾2 3

Since 𝑀 = 𝐿, the short-run cost-minimizing quantity of materials is 𝑀 =

𝑄2 1

.

𝐾2

c) Plugging 𝑄 = 4 into the expressions for the long-run cost-minimizing quantities of labor and materials gives us

L=4 M =4 Plugging 𝑄 = 4 and 𝐾 = 4 into the expressions for the short-run cost-minimizing quantities of labor and materials gives us 3

L=

42 4

1 2

(3−1)

=4 2 2 =4

3

M=

42 4

1 2

(3−1)

=4 2 2 =4

7.30. Consider the production function in Learning-By-Doing Exercise 7.6: Q = √L + √K + √M. For this production function, the marginal products of labor, capital, and materials are MPL = 1/(2√L), MPK = 1/(2√K), and MPM = 1/(2√M). Suppose that the input prices of labor, capital, and materials are w = 1, r = 1, and m = 1, respectively. a) Given that the firm wants to produce Q units of output, what is the solution to the firm’s long-run cost minimization problem? b) Given that the firm wants to produce Q units of output, what is the solution to the firm’s short-run cost minimization problem when K = 4? Will the firm want to use positive quantities of labor and materials for all levels of Q? (c) Given that the firm wants to produce 12 units of output, what is the solution to the firm’s short-run cost minimization problem when K = 4 and L = 9? Will the firm want to use a positive quantity of materials for all levels of Q? a) With three inputs, we need two tangency conditions to ensure that the marginal product per dollar spent is equal across all inputs. (We could write down a third tangency condition, but it would be redundant.) Equating the “bang for the buck” between labor and capital implies 1⁄2√𝐿 = 1⁄2√𝐾 or L = K. Similarly, equating the “bang for the buck” between labor and materials implies 1⁄2√𝐿 = 1⁄2√𝑀 or L = M. Then using the production constraint to find the

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Solutions Manual

input demand for labor yields 𝑄 = √𝐿 + √𝐿 + √𝐿 or L = (1/3)Q2. Since L = M = K from the tangency conditions, we also have K = (1/3)Q2 and M = (1/3)Q2. b) First, note that with K = 4, the firm can produce up to Q = √0 + √4 + √0 = 2 units of output without hiring any labor or materials. To produce more than Q = 2, the firm still balances the marginal product per dollar spent on labor and materials; in part (a), we saw this implied L = M. Substituting this and K = 4 into the production constraint, we have Q = √𝐿 + √4 + √𝐿 which yields L = (1/4)(Q – 2)2 as the input demand for labor. Then L = M implies that the input demand for materials is M = (1/4)(Q – 2)2. Therefore, the input demand functions are 0 𝐿(𝑄) = 𝑀(𝑄) = {1 (𝑄 − 2)2 4

𝑄≤2 𝑄>2

c) Again, with K = 4 and L = 9, the firm can produce up to Q = 5 units of output without hiring any materials. Should it desire to produce greater levels of output, it can hire materials according to Q = √9 + √4 + √𝑀, or M = (Q – 5)2. Therefore, the input demand for materials is 𝑀(𝑄) = {

0 (𝑄 − 5)2

𝑄≤5 𝑄>5

7.31. Acme, Inc. has just completed a study of its production process for gadgets. It uses labor and capital to produce gadgets. It has determined that 1 more unit of labor would increase output by 200 gadgets. However, an additional unit of capital would increase output by 150 gadgets. If the current price of capital is $10 and the current price of labor is $25, is the firm employing the optimal input bundle for its current output? Why or why not? If not, which input’s usage should be increased?

The information in the problem tells us that MPL = 200 and MPK = 150 while w = 25 and r = 10.

So MPL/w = 8 < MPK/r = 15. Thus Acme could maintain its current level of output while reducing costs by employing more capital and less labor. So it is not employing the optimal input bundle.

7.32. A firm operates with a technology that is characterized by a diminishing marginal rate of technical substitution of labor for capital. It is currently producing 32 units of output using 4 units of capital and 5 units of labor. At that operating point the marginal product of labor is 4 and the marginal product of capital is 2. The rental price of a unit of capital is 2 when the wage rate is 1. Is the firm minimizing its total long-run cost of producing the 32 units of output? If so, how do you know? If not, show why not and indicate whether the firm should be using (i) more capital and less labor, or (ii) less capital and more labor to produce an output of 32.

We have MPL/w = 4/1 = 4 > MPK/r = 2/2 = 1. Thus the firm cannot be minimizing its long-run total cost. By employing more labor and less capital, it could maintain 32 units of output while lowering total costs.

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Solutions Manual

7.33. Suppose that in a given production process a blueprint (B) can be produced using either an hour of computer time (C) or 4 hours of a manual draftsman’s time (D). (You may assume C and D are perfect substitutes. Thus, for example, the firm could also produce a blueprint using 0.5 hour of C and 2 hours of D.) a) Write down the production function corresponding to this process (i.e., express B as a function of C and D). b) Suppose the price of computer time (pc) is 10 and the wage rate for a manual draftsman (pD) is 5. The firm has to produce 15 blueprints. What are the cost minimizing choices of C and D? On a graph with C on the horizontal axis and D on the vertical axis, illustrate your answer showing the 15-blueprint isoquant and isocost lines. a) Computers are four times as productive as draftsmen; an alternative way of saying this is that MPC = 4MPD. Since C and D are perfect substitutes, we know the production function has the form B = aC + bD, where a and b are positive constants. Thus we can write the production function as B = C + (1/4)D. Note that this is consistent with generating one blueprint (B = 1) from the following combinations of inputs: (C, D) = (1, 0), (C, D) = (0, 4), and (C, D) = (0.5, 2). b) Notice that 𝑀𝑃𝐶 ⁄𝑝𝐶 = 1/10 > 𝑀𝑃𝐷 ⁄𝑝𝐷 = 0.25/5 = 1/20. That is, the marginal product per dollar spent on computer time is always higher than the marginal product per dollar spent on draftsman time. So the optimal input combination involves D = 0 and C = 15. The graph below illustrates the (dotted) isocost lines with slope = –pC / pD = –2, along with the (solid) B = 15 isoquant with slope = –MPC / MPD = –4.

D 60

B = 15 isoquant slope = –4 Isocost lines slope = –pC / pD = –2 15

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C

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Solutions Manual

7.34. This problem will enable you to apply a revealed preference argument to see if a firm is minimizing the total cost of production. The firm produces output with a technology characterized by a diminishing marginal rate of technical substitution of labor for capital.

It is required to produce a specified amount of output, which does not change in this problem. When faced with input prices w1 and r1, the firm chooses the basket of inputs at point A on the graph below, and it incurs the total cost on the isocost line IC1. When the factor prices change to w2 and r2 the firm’s choice of inputs is at basket B, on isocost line IC2. Basket A lies on the intersection of the two isocost lines. Are these choices consistent with cost minimizing behavior?

Since the firm’s production remains unchanged, it must be producing the same level of output at both points A and B. That is, the isoquant through A also passes through B. Now, suppose that the firm is minimizing costs at point B. Then the isoquant through B is tangent to isocost line IC2 .

Since we are told that the MRTS is diminishing, there is no way the isoquant passing through B can also pass through A. You can see this easily from the graph. And you can also reach this conclusion using the property that a line tangent to a curve does not intersect the curve at any point other than the point of tangency. Similarly, if the firm were minimizing costs at A, then isocost line IC1 would be tangent to the isoquant; but then it would be impossible for isocost line IC2 also to be tangent. Thus, it is not possible for both A and B to be cost minimizing input combinations.

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Chapter 7-26

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7.35 A firm uses two inputs, labor services whose quantity is denoted by 𝑳 and capital services whose quantity is denoted by 𝑲. The production function is given by 𝑸 = 𝟏𝟎𝟎𝑳𝑲, and the price of labor 𝒘 is $2 per unit and the price of capital 𝒓 is $1 per unit. For this production function, the marginal products of labor and capital, respectively are 𝑴𝑷𝑳 = 𝟏𝟎𝟎𝑲 and 𝑴𝑷𝑲 = 𝟏𝟎𝟎𝑳. a) Using the method of Lagrange, find the firm’s cost-minimizing input combination (𝑳, 𝑲) when it seeks to produce 5,000 units per year. What is the minimized level of total cost 𝑻𝑪 when output equals 5,000? b) Find the numerical value of the Lagrange multiplier, λ, which measures the firm’s ∆𝑻𝑪 marginal cost (the rate of change ∆𝑸 ) when output is 5,000. c) Find the firm’s cost-minimizing input (𝑳, 𝑲), minimized total cost, and the value of λ if output is 5,001. d) Verify that the increase in the firm’s total cost when output increases from 5,000 to 5,001 is close to the values of 𝝀 you found in parts (c) and (d). a) Since the firm has a Cobb-Douglas production function, we know that it cost-minimizing input combination will be interior, with 𝐿 > 0 and 𝐾 > 0. Since both marginal utilities are positive, we know that the cost-minimizing input combination will lie on the 5,000 unit isoquant.

Now, for this problem the Lagrangian function is: Ʌ(𝐹, 𝐶, 𝜆) = 2𝐿 + 𝐾 − 𝜆(100𝐿𝐾 − 5000).

Since the optimal basket will be interior, the necessary conditions are those in equations (7.10), (7.11), (7.12) of the text. Adapting those conditions to the specifics of this problem we get: 𝑤 − 𝜆𝑀𝑃𝐿 = 0 ⟹ 2 − 100𝐾𝜆 = 0. 𝑟 − 𝜆𝑀𝑃𝐾 = 0 ⟹ 1 − 100𝐿𝜆 = 0. 𝑓(𝐿, 𝐾) − 𝑄 = 0 ⟹ 100𝐿𝐾 − 5,000 = 0.

We have three equations and three unknowns. When we combine the first two equations, we find that 𝜆=

𝑤 𝑟 2 1 = ⟹ 𝜆= = ⟹ 𝐿 = 0.5𝐾. 𝑀𝑃𝐿 𝑀𝑃𝐾 100𝐾 100𝐿

Substituting 𝐿 = 0.5𝐾 into the production constraint, we find that 100(0.5𝐾)𝐾 = 5,000 ⟹ 50𝐾 2 = 5,000 ⟹ 𝐾 = 10. This, in turn, implies 𝐿 = 5. The minimized level of total cost is ($2 per unit  5 units per year) + ($1 per unit  10 units per year) = $20 per year.

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2

b) We can use the expression for 𝜆 in the preceding equation to conclude that 𝜆 = 100(10) = 1

1

100(5)

= 500 = 0.002.

c) To solve this part of the problem, we replicating the analysis in parts (a) and (b) except we make 𝑄 = 5,001. 𝑤 − 𝜆𝑀𝑃𝐿 = 0 ⟹ 𝑤 − 100𝐾𝜆 = 0. 𝑟 − 𝜆𝑀𝑃𝐾 = 0 ⟹ 𝑟 − 100𝐿𝜆 = 0. 𝑓(𝐿, 𝐾) − 𝑄 = 0 ⟹ 100𝐿𝐾 − 5,001 = 0.

We have three equations and three unknowns. When we combine the first two equations, we find that 𝜆=

𝑤 𝑟 2 1 = ⟹ 𝜆= = ⟹ 𝐿 = 0.5𝐾. 𝑀𝑃𝐿 𝑀𝑃𝐾 100𝐾 100𝐿

Substituting 𝐿 = 0.5𝐾 into the production constraint, we find that 100(0.5𝐾)𝐾 = 5,001 ⟹ 50𝐾 2 = 5,001 ⟹ 𝐾 = 10.001. This, in turn, implies 𝐿 = 5.0005 The minimized level of total cost is ($2 per unit  5.0005 units per year) + ($1 per unit  10.001 units per year) = $20.002 per year. We can use the expression for 𝜆 in the preceding equation to conclude that 2 1 𝜆 = 100(10.001) = 100(5.0005) = 0.0019998. This is very close to, but slightly different than, the value of 𝜆 than we obtained in part (b). d) As we move from 𝑄 = 5,000 to 𝑄 = 5,001, total cost changes from $20 per year to ∆𝑇𝐶 0.002 $20.002, so ∆𝑄 = 1 = 0.002, which is very close to the two values of 𝜆 we obtained.

7.36 A firm has a production function 𝑸 = 𝟏𝟎𝑳 + 𝟑𝟎𝑲. Suppose the price of labor services 𝒘 is $5 per unit, and the price of capital services 𝒓 is also $5 per unit. Using the method of Lagrange, find the optimal input combination given that the firm wishes to produce 900 units of output. Verify that the solution satisfies the conditions for an optimum at a corner point, equations (7.10), (7.11), (7.12) in the text. Then verify that the solution satisfies the conditions for an optimum at a corner point, equations (7.16), (7.17), (7.18) in the text.

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For this problem the Lagrangian function is: Ʌ(𝑥, 𝑦, 𝜆) = 𝑤𝐿 + 𝑟𝐾 − 𝜆(𝑓(𝐿, 𝐾) − 𝑄) = 5𝐿 + 5𝐾 − 𝜆(10𝐿 + 30𝐾 − 900).

Because the marginal products of both inputs are positive, we know that the production constraint 𝑓(𝐿, 𝐾) ≥ 𝑄, will be binding. Thus 10𝐿 + 30𝐾 = 900, or in other words, the firm will operate along the 900-unit isoquant. However, we do not yet know whether the utilitymaximizing basket is interior or at a corner point.

Let’s start by assuming that the solution will be interior, with 𝐿 > 0 and 𝐾 > 0. We can then see if the values of 𝐿, 𝐾, and λ calculated under that assumption are all positive. If so, we will have found that the solution is interior. If not, we can then find the corner point along the 900 unit isoquant that minimizes total cost.

If we do have an interior solution (with 𝐿 > 0 and 𝐾 > 0), the three necessary conditions for an optimum (corresponding to equations (7.10), (7.11), and (7.12) in the text) simplify to the following: 𝑤 − 𝜆𝑀𝑃𝐿 = 0 𝑟 − 𝜆𝑀𝑃𝐾 = 0 𝑓(𝐿, 𝐾) − 𝑄 = 0

⟹ 5 − 10𝜆 = 0. ⟹ 5 − 30𝜆 = 0. ⟹ 10𝐿 + 2𝐾 = 200.

These are three equations and three unknowns. But we can see immediately that the first two of 1 these equations could not hold simultaneously as they imply 𝜆 = 0.5 and 𝜆 = 6 at the same time.

Both cannot simultaneously be true. Therefore, we must have a corner solution.

But which corner will it be? One approach is to identify the two corners of the 200-unit isoquant and directly check which input combination gives us the lowest total cost. One corner of the isoquant is where 𝐿 = 0 and thus 30𝐾 = 900, or 𝐾 = 30. Another corner is where 𝐾 = 0 and 10𝐿 = 900, or 𝐿 = 90. Thus, we have: 𝑇𝐶 at 𝐿 = 0, 𝐾 = 30 is ($5 per unit  0) + ($5 per unit  30) = $150 per year. 𝑇𝐶 at 𝐿 = 90, 𝐾 = 0 is ($5 per unit  90) + ($5 per unit  0) = $450 per year.

In this case, we see that total cost is minimized by utilizing the input combination 𝐿 = 0, 𝐾 = 30.

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To verify that this satisfies the optimality conditions for a corner solution we can use conditions (7.16), (7.17), and (7.18) in the text. Because 𝐿 = 0 and 𝐾 > 0, these conditions reduce to: 5 − 10𝜆 ≥ 0; 5 − 30𝜆 ≥ 0; 10𝐿 + 30𝐾 = 900;

𝐿 = 0. 𝐾 > 0. 𝜆 > 0.

The third equation above tells us that the firm operates on its 200-unit isoquant. The second 1 equation reveals that 𝜆 = 6. Finally, the inequality in the first equation is satisfied: 1

1

5 − 10𝜆 = 2 − 10 (6) = 3 ≥ 0. This verifies that the firm’s cost-minimizing input combination 1

(𝐿, 𝐾) = (0,30), and that 𝜆 = 6.

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Chapter 8 Cost Curves Solutions to Review Questions 1. What is the relationship between the solution to the firm’s long-run cost-minimization problem and the long-run total cost curve?

The long-run total cost curve plots the minimized total cost for each level of output holding input prices fixed. In other words, for a given set of input prices, the long-run total cost curve represents the total cost associated with the solution to the long-run cost minimization problem for each level of output.

2. Explain why an increase in the price of an input typically causes an increase in the longrun total cost of producing any particular level of output.

When the price of one input increases, the isocost line for a particular level of total cost will rotate in toward the origin. Assuming the isocost line was tangent to the isoquant for the firm’s selected level of output, when the isocost line rotates it will no longer touch the original isoquant.

In order for an isocost line to reach a tangency with the original isoquant, the firm would need to move to an isocost line associated with a higher level of cost, i.e. an isocost line further to the northeast.

3. If the price of labor increases by 20 percent, but all other input prices remain the same, would the long-run total cost at a particular output level go up by more than 20 percent, less than 20 percent, or exactly 20 percent? If the prices of all inputs went up by 20 percent, would long-run total cost go up by more than 20 percent, less than 20 percent, or exactly 20 percent?

If the price of a single input goes up leaving all other input prices the same and the level of output constant, total cost will rise but by a smaller percentage than the increase in the input price. This occurs because the firm will substitute away from the now relatively more expensive labor to the now relatively less expensive other inputs. So, if the price of labor rises by 20% holding all other input prices constant, total cost will rise by less than 20%.

If the prices of all inputs go up by the same percentage, total cost will rise by exactly that same percentage. So, if input prices rise by 20%, total cost will also rise by 20%.

4. How would an increase in the price of labor shift the long-run average cost curve?

An increase in the price of labor would result in a long-run total cost curve that lies above the initial long-run total cost curve at every quantity except 𝑄 = 0. Since 𝐴𝐶 = 𝑇𝐶/𝑄, increasing total cost will raise average cost at every quantity except 𝑄 = 0. Therefore, the long-run average cost curve will shift up.

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5. a) If the average cost curve is increasing, must the marginal cost curve lie above the average cost curve? Why or why not? b) If the marginal cost curve is increasing, must the marginal cost curve lie above the average cost curve? Why or why not? a) When 𝑀𝐶 > 𝐴𝐶, average cost is increasing, and when 𝑀𝐶 < 𝐴𝐶, average cost is decreasing.

So, if the average cost curve is increasing it must lie below the marginal cost curve. b) If the marginal cost curve is increasing, it may lie above or below the average cost curve. The only determining factor here is whether or not marginal cost lies above or below average cost. If it lies above, average cost will be increasing and if it lies below, average cost will be decreasing.

Knowing that marginal cost is increasing or decreasing tells us nothing about average cost.

6. Sketch the long-run marginal cost curve for the “flat-bottomed” long-run average cost curve shown in Figure 8.11.

TC MC AC

Q

When average cost is falling, marginal cost will lie below average cost, and when average cost is increasing, marginal cost will lie above average cost. Over the flat-bottomed portion where average cost is neither increasing nor decreasing, marginal cost and average cost will be equal.

7. Could the output elasticity of total cost ever be negative? 𝑀𝐶

The output elasticity of total cost, when simplified can be written as 𝜀𝑇𝐶,𝑄 = 𝐴𝐶 .

Since 𝐴𝐶 = 𝑇𝐶/𝑄, and since 𝑇𝐶 and 𝑄 must always be positive, 𝐴𝐶 will always be positive.

Marginal cost, MC, represents the change in total cost associated with an increase in output.

When output increases, total cost must always rise for a given set of input prices, implying that 𝑀𝐶 is also always positive. Therefore, the output elasticity of total cost must always be positive.

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8. Explain why the short-run marginal cost curve must intersect the average variable cost curve at the minimum point of the average variable cost curve.

Because fixed cost does not change, marginal costs reflect the change in variable costs. Thus, as with the relationship between any average and marginal, if average variable cost is decreasing, marginal cost must be below average variable cost, and if average variable cost is increasing, marginal cost must lie above average variable cost. This implies marginal cost will intersect average variable cost at the minimum of average variable cost.

9. Suppose the graph of the average variable cost curve is flat. What shape would the shortrun marginal cost curve be? What shape would the short-run average cost curve be?

If the average variable cost curve is flat, average variable cost is neither increasing nor decreasing. Marginal cost will therefore be equal to average variable cost and the marginal cost curve will therefore also be flat. Since average fixed cost is always declining, and since average total cost is the vertical sum of average variable and average fixed costs, average total cost must also be declining at all levels of 𝑄 if average variable cost is constant. Graphically, average total cost will be declining and asymptotic to the average variable cost curve.

10. Suppose that the minimum level of short-run average cost was the same for every possible plant size. What would that tell you about the shapes of the long-run average and long-run marginal cost curves?

The long-run average cost curve is the envelope to the short-run average cost curves associated with each level of output. If each of these short-run average cost curves has the same minimum point, the long-run average cost curve will be a horizontal line tangent to all of these minimum points. Because the long-run average cost curve will be flat, long-run average cost is neither increasing nor decreasing, and the long-run marginal cost curve will also be flat and equal to long-run average cost.

11. What is the difference between economies of scope and economies of scale? Is it possible for a two-product firm to enjoy economies of scope but not economies of scale? Is it possible for a firm to have economies of scale but not economies of scope?

Economies of scale refer to a situation when average total cost for a single product declines as the level of output for that product increases. These economies of scale might occur, for example, because workers can specialize in tasks as the level of output increases and the workers’ productivity may increase. Economies of scope refer to efficiencies that arise when a firm produces more than one product. In particular, economies of scope exist if one firm producing N products does so at a lower total cost than N separate firms producing the same quantities of each product individually.

The notion of economies of scale can actually be applied to a multi-product firm as well. We can use this extension to further refine the distinction between economies of scale and scope.

Suppose a firm is producing 𝑁 products, with output levels measured by 𝑄1 , 𝑄2 , … , 𝑄𝑁 . If it operates with economies of scale, the total cost of production will rise by less than 1% when

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production of all outputs increases by 1%. If it operates with diseconomies of scale, the total cost of production will rise by more than 1% when production of all outputs increases by 1%. By contrast, economies of scope exist if it is less costly to have the outputs produced by one firm instead of by 𝑁 firms, each specializing in the production of one of the outputs.

Note that information about economies of scope does not tell us whether the firm has economies of scale. If a production process has economies of scope, there may not be economies of scale.

Further, information about economies of scale does not tell us whether the firm has economies of scope. If a production process has economies of scale, there may not be economies of scope.

12. What is an experience curve? What is the difference between economies of experience and economies of scale?

The experience curve represents the relationship between average variable cost and cumulative production volume over time. One would expect that as cumulative production volume increased, average variable cost would fall. Economies of scale refer to a situation when average cost declines as the level of output for that product increases within a given time frame.

In general, economies of scale would occur if the average cost curve declined as the level of output increased. Economies of experience would occur if, as cumulative production volume increased, the average cost curve shifted downward for all levels of output. So, economies of scale refer to lower average costs that occur as output increases and economies of experience refer to lower average costs for all levels of output as cumulative production volume increases.

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Chapter 8-4

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Solutions to Problems 8.1. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

The table is reproduced below. First, since fixed costs are independent of quantity, the entire TFC column can be easily filled in. Proceeding through the table row by row, for Q = 1 it is easy to see that TVC = TC – TFC = 80, and the rest of the row is similarly straightforward. For Q = 2, TC = TVC + TFC = 180, and the rest of the follows easily. For Q = 3, all we have is TFC = 20; thus, we cannot infer anything else. For Q = 4, TC = Q*AC = 380. It’s then possible to get TVC and AVC; however, we cannot find MC since we don’t know TC or TVC for Q = 3. For Q = 5, the important step is to use MC(5) = TC(5) – TC(4) to find TC(5) = 550. For Q = 6, the important step is TVC = AVC*Q = 720.

Q 1 2 3 4 5 6

TC 100 180 380 550 740

TVC 80 160 360 530 720

TFC 20 20 20 20 20 20

AC 100 90 95 110 123.3

MC 80 80 170 190

AVC 80 80 90 106 120

8.2. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

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It helps to rewrite this table adding an extra column for Total Fixed Costs at each level of output.

The TFC for Q = 2 is just 2*30 = 60, and this is also the TFC value for every other output level.

Then for Q = 1, we know TC = AC*Q = 100, TVC = TC – TFC = 40 and the rest is straightforward. Similarly we can fill in the rows for Q = 2, 3, 4, and 6. For Q = 5, we need to use the fact that MC(6) = TC(6) – TC(5) to infer TC(5) = 250. The rest is straightforward.

Q 1 2 3 4 5 6

TC 100 110 120 180 250 330

TVC 40 50 60 120 190 270

AFC 60 30 20 15 12 10

AC 100 55 40 45 50 55

MC 40 10 10 60 70 80

AVC 40 25 20 30 38 45

TFC 60 60 60 60 60 60

8.3. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

Q 1 2 3 4 5 6

TC 18 30 46 66 90 118

TVC 8 20 36 56 80 108

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TFC 10 10 10 10 10 10

AC 18 15 46/3 66/4 18 118/6

MC 8 12 16 20 24 28

AVC 8 10 12 14 16 18

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8.4. The following incomplete table shows a firm’s various costs of producing up to 6 units of output. Fill in as much of the table as possible. If you cannot determine the number in a box, explain why it is not possible to do so.

Q 1 2 3 4 5 6

TC 20 36 55 82 112 144

TVC 10 26 45 72 102 134

TFC 10 10 10 10 10 10

AC 20 18 18.33 20.5 22.4 24

MC 10 16 19 27 30 32

AVC 10 13 15 18 20.4 22.33

8.5. A firm produces a product with labor and capital, and its production function is described by Q = LK. The marginal products associated with this production function are MPL = K and MPK = L. Suppose that the price of labor equals 2 and the price of capital equals 1. Derive the equations for the long-run total cost curve and the long-run average cost curve.

Starting with the tangency condition, we have MPL w = MPK r K 2 = L 1 K = 2L

Substituting into the production function yields 𝑄 = 𝐿𝐾 𝑄 = 𝐿(2𝐿) 𝐿=√

𝑄 2

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𝑄

Plugging this into the expression for 𝐾 above gives 𝐾 = 2√ 2 .

Finally, substituting these into the total cost equation results in 𝑄 𝑄 𝑇𝐶 = 2 (√ ) + 2 (√ ) 2 2 𝑄 𝑇𝐶 = 4 (√ ) 2 𝑇𝐶 = √8𝑄 𝑇𝐶

and average cost is given by 𝐴𝐶 = 𝑄 =

𝐴𝐶 = √

√8𝑄 𝑄

8 𝑄

8.6 A firm’s long-run total cost curve is 𝑻𝑪(𝑸) = 𝟏𝟎𝟎𝟎𝑸𝟐 . Derive the equation for the corresponding long-run average cost curve, 𝑨𝑪(𝑸). Given the equation of the long-run average cost curve, which of the following statements is true: a) The long-run marginal cost curve 𝑴𝑪(𝑸) lies below 𝑨𝑪(𝑸) for all positive quantities 𝑸. b) The long-run marginal cost curve 𝑴𝑪(𝑸) is the same as the 𝑨𝑪(𝑸) for all positive quantities 𝑸. c) The long-run marginal cost curve 𝑴𝑪(𝑸) lies above the 𝑨𝑪(𝑸) for all positive quantities 𝑸. d) The long-run marginal cost curve 𝑴𝑪(𝑸) lies below 𝑨𝑪(𝑸) for some positive quantities 𝑸 and above the 𝑨𝑪(𝑸) for some positive quantities 𝑸.

C is correct. The equation of the AC curve is AC(Q) = 1000Q. It is increasing in Q. Given the relationship between AC and MC curves, the fact that the AC curve is increasing means that the MC curve must lie above the AC curve.

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𝟏

8.7 A firm’s long-run total cost curve is 𝑻𝑪(𝑸) = 𝟏𝟎𝟎𝟎𝑸𝟐 . Derive the equation for the corresponding long-run average cost curve, 𝑨𝑪(𝑸). Given the equation of the long-run average cost curve, which of the following statements is true: a) The long-run marginal cost curve 𝑴𝑪(𝑸) lies below 𝑨𝑪(𝑸) for all positive quantities 𝑸. b) The long-run marginal cost curve 𝑴𝑪(𝑸) is the same as the 𝑨𝑪(𝑸) for all positive quantities 𝑸. c) The long-run marginal cost curve 𝑴𝑪(𝑸) lies above the 𝑨𝑪(𝑸) for all positive quantities 𝑸. d) The long-run marginal cost curve 𝑴𝑪(𝑸) lies below 𝑨𝑪(𝑸) for some positive quantities 𝑸 and above the 𝑨𝑪(𝑸) for some positive quantities 𝑸.

A is correct. The equation of the AC curve is AC(Q) = TC(Q)/Q = 1000Q1/2/Q = 1000Q-(1/2). This is a decreasing function of Q. Given the relationship between AC and MC curves, the fact that the AC curve is decreasing means that the MC curve must lie below the AC curve.

8.8. A firm’s long-run total cost curve is TC(Q) = 1000Q − 30Q2 + Q3. Derive the expression for the corresponding long-run average cost curve and then sketch it. At what quantity is minimum efficient scale? 𝑇𝐶 1000𝑄 − 30𝑄 2 + 𝑄 3 𝐴𝐶 = = 𝑄 𝑄 𝐴𝐶 = 1000 − 30𝑄 + 𝑄 2 Graphically, average cost is

Minimum efficient scale occurs where the average cost curve reaches a minimum, 𝑄 = 15 for this cost function.

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8.9. A firm’s long-run total cost curve is TC(Q) = 40Q − 10Q2 + Q3, and its long-run marginal cost curve is MC(Q) = 40 − 20Q + 3Q2. Over what range of output does the production function exhibit economies of scale, and over what range does it exhibit diseconomies of scale?

From the total cost curve, we can derive the average cost curve, 𝐴𝐶(𝑄) = 40 − 10𝑄 + 𝑄 2. The minimum point of the AC curve will be the point at which it intersects the marginal cost curve, i.e. 40 − 10𝑄 + 𝑄 2 = 40 − 20𝑄 + 3𝑄 2 . This implies that AC is minimized when Q = 5. By definition, there are economies of scale when the AC curve is decreasing (i.e. Q < 5) and diseconomies when it is rising (Q > 5).

8.10. For each of the total cost functions, write the expressions for the total fixed cost, average variable cost, and marginal cost (if not given), and draw the average total cost and marginal cost curves. a) TC(Q) = 10Q b) TC(Q) = 160 + 10Q c) TC(Q) = 10Q2, where MC(Q) = 20Q d) TC(Q) = 10√Q, where MC(Q) = 5/√Q e) TC(Q) = 160 + 10Q2, where MC(Q) = 20Q Green = AC, Red = MC a) TFC = 0, AVC = 10, MC = 10.

Here, AC=MC=10

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b) TFC = 160, AVC = 10, MC = 10.

c) TFC = 0, AVC = 10Q.

d) TFC = 0, AVC = 10⁄√𝑄 .

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Solutions Manual

e) TFC = 160, AVC = 10Q.

8.11. A firm produces a product with labor and capital as inputs. The production function is described by Q = LK. The marginal products associated with this production function are MPL = K and MPK = L. Let w = 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q. b) Solve the firm’s short-run cost-minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s short-run total cost curve as a function of quantity Q and graph it together with the long-run total cost curve. c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 4? d) How do the graphs of the long-run and short-run total cost curves change when w = 4 and r = 1?

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a) Cost-minimizing quantities of inputs are equal to L = √Q √(r/w) and K = √Q / √(r/w). Hence, in the long-run the total cost of producing Q units of output is equal to TC(Q) = 10 + 2√(Qrw).

For w = 1 and r = 1 we have TC(Q) = 2√Q. b) When capital is fixed at a quantity of 5 units (i.e., K = 5) we have Q = K*L = 5 L. Hence, in the short-run the total cost of producing Q units of output is equal to STC(Q) = 5 + Q/5.

c) We have L = √Q √(r/w) and K = √Q / √(r/w). Hence, TC(Q) = 2√(Qrw) and STC(Q) = 5r + wQ/5. When w = 1 and r = 4 we have TC(Q) = 4√Q and STC(Q) = 20 + Q/5.

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d) When w = 4 and r = 1 we have TC(Q) = 4√Q and STC(Q) = 4Q/5.

8.12. A firm produces a product with labor and capital. Its production function is described

by Q = min(L, K). Let w and r be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q and input prices, w and r. b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5). Derive the equation for the firm’s short-run total cost curve as a function of quantity Q. Graph this curve together with the long-run total cost curve for w = 1 and r = 1. c) How do the graphs of the long-run and short-run total cost curves change when w = 1 and r = 2? d) How do the graphs of the long-run and short-run total cost curves change when w = 2 and r = 1? a) The inputs are complementary and the cost-minimizing firm uses them in proportions 1:1.

Hence, we have TC(Q) = Q(w + r). b) If w = r = 1, then TC(Q) = 2Q. In the short run, it is impossible to produce more than 5 units.

This is because min(L,5) cannot be any greater than 5. To produce Q  5 units, we set L = Q. With w = r = 1, this implies STC(Q) = 15 + Q. (10 is the fixed cost of the indivisible input, 5 is the fixed cost of labor, and Q is the variable cost of labor.) The diagram below shows TC(Q) and STC(Q) for Q  5 when w = r = 1. c) For w = 1 and r = 2 we have TC(Q) = 3Q and STC(Q) = Q + 10 for Q not larger than 5.

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Chapter 8-14

Besanko & Braeutigam – Microeconomics, 6th edition

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Solutions Manual

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Solutions Manual

d) For w = 2 and r = 1 we have TC(Q) = 3Q and STC(Q) = 2Q + 5 for Q not larger than 5.

8.13. A firm produces a product with labor and capital. Its production function is described

by Q = L + K. The marginal products associated with this production function are MPL = 1 and MPK = 1. Let w = 1 and r = 1 be the prices of labor and capital, respectively. a) Find the equation for the firm’s long-run total cost curve as a function of quantity Q when the prices labor and capital are w = 1 and r = 1. b) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 5 units (i.e., K = 5), and w = 1 and r = 1. Derive the equation for the firm’s short-run total cost curve as a function of quantity Q and graph it together with the longrun total cost curve. c) How do the graphs of the short-run and long-run total cost curves change when w = 1 and r = 2? d) How do the graphs of the short-run and long-run total cost curves change when w = 2 and r = 1? a) With a linear production function, the firm operates at a corner point depending on whether w < r or w > r. If w < r, the firm uses only labor and thus sets L = Q. In this case, the total cost (including the fixed cost) is wQ. If w > r, the firm uses only capital and thus sets K = Q. in this case, the total cost is rQ. When w = r = 1, the firm is indifferent among combination of L and K that make L + K = 10. Thus, we have TC(Q) = Q.

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b) When capital is fixed at 5 units, the firm’s output would be given by Q = 5 + L. If the firm wants to produce Q < 5 units of output, it must produce 5 units and throw away 5 – Q of them.

The total cost of producing fewer than 5 units is constant and equal to $5, the cost of the fixed capital. For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q – 5 units of labor, for a cost of Q – 5, and 5 units of capital, for a cost of 5. Thus, STC(Q) = Q – 5 + 5 = Q

c) In the long run, since w < r, the firm produces its output entirely with labor. Thus, TC(Q) = Q, just as in part (b). In the short-run, with capital fixed at 5 units, the firm’s output would be given

by Q = 5 + L. If the firm wants to produce Q < 5 units of output, it must produce 5 units of output and throw away 5 – Q of them. It can produce this output using its fixed stock of 5 units of capital and no labor. The total cost of producing Q < 5 units of output when the price of capital is $2 per unit is $10.

For Q > 5 units, the firm increases its output by increasing its use of labor. In particular, to produce Q units of output, the firm uses Q – 5 units of labor, for a cost of Q – 5, and 5 units of capital, for a cost of 10. Thus, STC(Q) = (Q – 5) + 10 = Q + 5.

Notice that when 𝐾̄ = 5, w = 1, and r = 2, the STC curve strictly lies above the TC curve. This is because K = 5 is never an optimal capital choice for the firm when w = 1 and r = 2. As a result the firm’s total costs are always higher in the short run than they are in the long run.

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d) The total cost curve is the same as in part (b), i.e. TC(Q) = Q. This is because the cheaper input (in this case capital) continues to have a price of $1 per unit. In the short run, with capital being fixed at 5 units, the cost of producing Q < 5 is $5. To produce more than Q units, the firm uses Q – 5 units of labor at a total cost of 2(Q – 5) = 2Q – 10. It also uses 5 units of capital at a total cost of 5. Thus, for Q > 5, STC(Q) = 2Q – 10 + 5 = 2Q – 5.

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Solutions Manual

8.14. Consider a production function of two inputs, labor and capital, given by Q = (√L + √K)2. The marginal products associated with this production function are as follows:

Let w = 2 and r = 1. a) Suppose the firm is required to produce Q units of output. Show how the costminimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of capital depends on the quantity Q. b) Find the equation of the firm’s long-run total cost curve. c) Find the equation of the firm’s long-run average cost curve. d) Find the solution to the firm’s short-run cost minimization problem when capital is fixed at a quantity of 9 units (i.e., K = 9). e) Find the short-run total cost curve, and graph it along with the long-run total cost curve. f ) Find the associated short-run average cost curve. a) Starting with the tangency condition we have MPL w = MPK r  L1/ 2 + K 1/ 2  L−1/ 2 2 =  L1/ 2 + K 1/ 2  K −1/ 2 1 K =4 L K = 4L

Plugging this into the total cost function yields 𝑄 = [𝐿1/2 + (4𝐿)1/2 ] 𝑄 = [3𝐿1/2 ] 𝑄 = 9𝐿 𝑄 𝐿= 9

2

2

Inserting this back into the solution for 𝐾 above gives 𝐾=

4𝑄 9

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Besanko & Braeutigam – Microeconomics, 6th edition

𝑄

Solutions Manual

4𝑄

b) 𝑇𝐶 = 2 ( 9 ) + 9 𝑇𝐶 =

2𝑄 3 𝑇𝐶

c) 𝐴𝐶 = 𝑄 = 𝐴𝐶 =

(

2𝑄 ) 3

𝑄

2 3

d) When 𝑄 ≤ 9 the firm needs no labor. If 𝑄 > 9 the firm must hire labor, setting 𝐾̄ = 9 and plugging in for capital in the production function yields Q =  L1/ 2 + 91/ 2 

2

Q1/ 2 = L1/ 2 + 3 L1/ 2 = Q1/ 2 − 3 L = Q1/ 2 − 3

2

Thus, 1

2

2 𝐿 = {[𝑄 − 3] if 𝑄 > 9

0if 𝑄 ≤ 9 2

1/2 e) 𝑇𝐶 = {2(𝑄 − 3) + 9when 𝑄 > 9 9when 𝑄 ≤ 9

Graphically, short-run and long-run total costs are shown in the following figure.

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Solutions Manual

2

2(𝑄 1/2 −3) +9 𝑇𝐶

f) 𝐴𝐶 = 𝑄 = {

𝑄 9 𝑄

if 𝑄 > 9

if 𝑄 ≤ 9

8.15. Tricycles must be produced with 3 wheels and 1 frame for each tricycle. Let Q be the number of tricycles, W be the number of wheels, and F be the number of frames. The price of a wheel is PW and the price of a frame is PF. a) What is the long-run total cost function for producing tricycles, TC(Q,PW, PF)? b) What is the production function for tricycles, Q(F,W )? a) Each tricycle requires the purchase of three wheels at price PW and one frame at price PF. Thus, TC(Q, PW, PF) = Q(3PW + PF). b) Three wheels and one frame are perfect complements in production. Thus the production function is Q(F, W) = min{F, (1/3)W}. Notice that (F, W) = (1, 3) yields Q = 1, (F, W) = (2, 6) yields Q = 2, etc.

8.16. A hat manufacturing firm has the following production function with capital and labor being the inputs: Q = min(4L, 7K )—that is it has a fixed-proportions production function. If w is the cost of a unit of labor and r is the cost of a unit of capital, derive the firm’s long-run total cost curve and average cost curve in terms of the input prices and Q. The fixed proportions production function implies that for the firm to be at a cost minimizing optimum, 4𝐿 = 7𝐾and both of these equal Q. Therefore, L = Q/4 and K = Q/7. So the firm’s 𝑤 𝑟 total cost is 𝑤𝐿 + 𝑟𝐾 = 𝑤𝑄/4 + 𝑟𝑄/7 = [ 4 + 7]𝑄. 𝑤

𝑟

The average cost curve is 𝐿𝑅𝐴𝐶 = 𝑇𝐶/𝑄 = 4 + 7. Note that this average cost curve is independent of Q and is simply a straight line.

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Chapter 8-21

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Solutions Manual

8.17. A packaging firm relies on the production function Q = KL + K, with MPL = K and MPK = L + 1. Assume that the firm’s optimal input combination is interior (it uses positive amounts of both inputs). Derive its long-run total cost curve in terms of the input prices, w and r. Verify that if the input prices double, then total cost doubles as well.

Since we can assume an interior solution, the tangency condition must hold. Therefore the 𝐾 𝑤 𝑟𝐾 optimal bundle must be such that 𝐿+1 = 𝑟 . This means 𝐿 + 1 = 𝑤 . Substituting this back into the 𝑟𝐾2

𝑄𝑤

production function, we see that 𝑄 = 𝑤 , so 𝐾 = √ 𝑟 . 𝑄𝑟

This implies that 𝐿 = √ 𝑤 − 1. The total cost curve is then 𝑇𝐶 = 𝑤𝐿 + 𝑟𝐾 = 2√𝑤𝑟𝑄 − 𝑤. If we substitute 2w and 2r in the place of w and r respectively, we get TC2 = 2√(2𝑤)(2𝑟)𝑄 − (2𝑤) = 4√𝑤𝑟𝑄 − 2𝑤 = 2 ∗ 𝑇𝐶, so total cost does indeed double when input prices double.

8.18. A firm has the linear production function Q = 3L + 5K, with MPL = 3 and MPK = 5.

Derive the expression for the 1ong-run total cost that the firm incurs, as a function of Q and the factor prices, w and r.

As we saw in Chapter 7, linear production functions usually have corner solutions. In this case, the firm will use only labor if 𝑀𝑅𝑇𝑆𝐿,𝐾 >

𝑤 𝑤 𝑟 , or < 𝑟 3 5 𝑤

𝑟

Similarly, it will use only capital if 3 > 5 . 𝑄

If the firm does use labor, then it will use 𝐿 = 3 with a total cost of wQ/3. Similarly if it uses 𝑄

capital it will use 𝐾 = 5 with a total cost of rQ/5. Therefore, the firm’s total cost curve can be 𝑤 𝑟

expressed as 𝑇𝐶 = 𝑚𝑖𝑛{ 3 , 5}𝑄.

8.19 A firm uses two inputs: labor and capital. The price of labor is 𝒘 and the price of 𝟏

𝟒

capital is 𝒓. The firm’s long-run total cost is given by the equation 𝑻𝑪(𝑸) = 𝒘𝟓 𝒓𝟓 𝑸. Based on this equation, which change would cause the greater upward rotation in the long-run total cost curve: a 10 percent increase in 𝒘 or a 10 percent increase in 𝒓? Based on your answer, is the firm’s production operation more capital intensive or labor intensive?

Explain your answer.

A 10 percent increase in r would cause the TC curve to rotate upward more than a 10 increase in w. (In fact, a 10 percent increase in r would cause a (4/5)*10 = 8 percent increase in TC for any positive level of output Q, while a 10 percent increase in w would cause only a (1/5)*10 = 2 percent increase in TC for any positive level of output Q. The fact that total costs are more

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responsive to a change in the price of capital than to a change in the price of labor, suggests that the firm’s production operation is more capital intensive than labor intensive.

8.20. When a firm uses K units of capital and L units of labor, it can produce Q units of output with the production function Q = K√L. Each unit of capital costs 20, and each unit of labor costs 25. The level of K is fixed at 5 units. a) Find the equation of the firm’s short-run total cost curve. b) On a graph, draw the firm’s short-run average cost. a) From the production function we see that 𝑄 = 5√𝐿, so the amount of labor required to 𝑄2

𝑄2

produce Q is given by 𝐿 = 25 . The short run total cost function is 𝐶 = 25𝐿 + 20𝐾 = 25 [ 25 ] + 20(5) = 100 + 𝑄 2 . b) AC

20

10

Q

8.21. When a firm uses K units of capital and L units of labor, it can produce Q units of output with the production function Q = √L + √K . Each unit of capital costs 2, and each unit of labor costs 1. a) The level of K is fixed at 16 units. Suppose Q ≤ 4. What will the firm’s short-run total cost be? (Hint: How much labor will the firm need?) b) The level of K is fixed at 16 units. Suppose Q > 4. Find the equation of the firm’s shortrun total cost curve. a) Even if the firm hires zero units of labor, with K fixed at 16 it can still produce up to Q = √0 + √16= 4 units of output. So for 𝑄 ≤ 4, L = 0 is the cost-minimizing choice of labor and the short-run total cost function is just the cost of capital: C = rK + wL = 2(16) + 1(0) = 32. b) For Q > 4, the firm needs to hire positive amounts of labor, according to 𝑄 = √𝐿 + √16 or L = (Q – 4)2. So for Q > 4, the short-run total cost function is C(Q) = rK + wL = 2(16) + 1(Q – 4)2 = 32 + (Q – 4)2.

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8.22. Consider a production function of three inputs, labor, capital, and materials, given by Q = LKM. The marginal products associated with this production function are as follows: MPL = KM, MPK = LM, and MPM = LK. Let w = 5, r = 1, and m = 2, where m is the price per unit of materials. a) Suppose that the firm is required to produce Q units of output. Show how the costminimizing quantity of labor depends on the quantity Q. Show how the cost minimizing quantity of capital depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q. b) Find the equation of the firm’s long-run total cost curve. c) Find the equation of the firm’s long-run average cost curve. d) Suppose that the firm is required to produce Q units of output, but that its capital is fixed at a quantity of 50 units (i.e., K = 50). Show how the cost-minimizing quantity of labor depends on the quantity Q. Show how the cost-minimizing quantity of materials depends on the quantity Q. e) Find the equation of the short-run total cost curve when capital is fixed at a quantity of 50 units (i.e., K = 50) and graph it along with the long-run total cost curve. f) Find the equation of the associated short-run average cost curve. a) Equating the bang for the buck between labor and capital implies

MPL w = MPK r KM 5 = LM 1 K = 5L 𝑀𝑃

𝑤

Equating the bang for the buck between labor and materials implies 𝑀𝑃 𝐿 = 𝑚 𝑀

𝐾𝑀 5 = 𝐾𝐿 2 5𝐿 𝑀= 2 5𝐿

Plugging these into the production function yields 𝑄 = 𝐿(5𝐿) ( 2 ) 25𝐿3 𝑄= 2 2𝑄 3 𝐿 = 25 2𝑄 1/3 𝐿=( ) 25 Copyright © 2020 John Wiley & Sons, Inc.

Chapter 8-24

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Solutions Manual

2𝑄 1/3

Substituting into the tangency condition results above implies 𝐾 = 5 ( 25 ) 2𝑄 1/3

b) 𝑇𝐶 = 5 ( 25 )

2𝑄 1/3

+ 5 ( 25 )

5

5 2𝑄 1/3

and 𝑀 = 2 ( 25 )

.

2𝑄 1/3

+ 2 (2) ( 25 )

2𝑄 1/3 𝑇𝐶 = 15 ( ) 25 𝑇𝐶

15 2𝑄 1/3

c) 𝐴𝐶 = 𝑄 = 𝑄 ( 25 )

𝑀𝑃

𝑤

d) Beginning with the tangency condition 𝑀𝑃 𝐿 = 𝑚 𝑀

𝐾𝑀 5 = 𝐾𝐿 2 5𝐿 𝑀= 2 5𝐿

Setting 𝐾̄ = 50 and substituting into the production function yields

𝑄 = 𝐿(50) ( 2 )

𝑄 = 125𝐿2 𝐿=√

𝑄 125

Substituting this result into the tangency condition result above implies 𝑀 =

5√

𝑄 125

2

. This reduces

𝑄

to 𝑀 = √20. 𝑄

𝑄

𝑄

e) In the short run, 𝑇𝐶 = 5√125 + 50 + 2√20 which reduces to 𝐶 = 2√ 5 + 50.

Graphically, short-run and long-run total cost curves are shown in the following figure.

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Chapter 8-25

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𝑇𝐶

f) Short run average cost is given by 𝐴𝐶 = 𝑄 =

Solutions Manual

𝑄 5

2√ +50 𝑄

.

8.23. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the long run. What is the long-run total cost of producing 400 units of output?

First, notice that if the firm uses L it must necessarily use K and vice versa; there is no point using a positive amount of one of these inputs and zero of the other. Thus, there are three possible solutions to the long-run cost minimization problem: (i) interior, using positive amounts of K, L, and M; (ii) a corner with K = L = 0 and M > 0; or (iii) a corner with M = 0 but positive amounts of both K and L. Using approach (i), we equate MPK/r = MPM/s and MPL/w = MPM/s to get K = 16 and L = 4.

Using the production constraint then yields M = Q – 64. Total cost using this approach will be CKLM(Q) = 16(4) + 4(16) + 1(Q – 64) = Q + 64.

Using approach (ii), we have K = L = 0 and the input demand for M comes from the production constraint: M = Q. Total cost will be CM(Q) = Q. Using approach (iii), M = 0 and the tangency condition between K and L yields MPL/w = MPK/r, or K = 4L. Combined with the production function, we get the input demand functions 𝐾 = 2√𝑄 1 and 𝐿 = 2 √𝑄. Total cost will be 1

CKL(Q) = 16 (2 √𝑄) + 4(2√𝑄) + 1(0) = 16√𝑄.

Comparing the three approaches, it is easy to see that CM(Q) < CKLM(Q) for all values of Q. Hence, a cost-minimizing firm will never use K, L, and M simultaneously; it could produce the same output at less cost by just using M. Furthermore, CM(Q) < CKL(Q) only for Q < 256. So for Q = 400, the firm should set M = 0 and, following approach (ii), set K = 40 and L = 10. Total cost will be CKL = 320.

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8.24. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the short run, with K fixed at 20 units. What is the short-run total cost of producing 400 units of output?

With K fixed at 20 units, the production function becomes Q = 20L + M. Thus, L and M are perfect substitutes. Since MPL/w = 1.25 > MPM/s = 1, the marginal product per dollar spent on labor is always higher than that on materials. So the cost-minimizing input combination is M = 0 with L solving 400 = 20L + 0, or L = 20. The short run total cost is C = 16(20) + 4(20) + 1(0) = 400.

8.25. The production function Q = KL + M has marginal products MPK = L, MPL = K, and MPM = 1. The input prices of K, L, and M are 4, 16, and 1, respectively. The firm is operating in the short run, with K fixed at 20 units and M fixed at 40. What is the short-run total cost of producing 400 units of output?

With K fixed at 20 and M fixed at 40, the production function becomes Q = 20L + 40. To produce 400 units, the firm needs to hire labor until 400 = 20L + 40, or L = 18. The short-run total cost is C = 16(18) + 4(20) + 1(40) = 408.

8.26. A short-run total cost curve is given by the equation STC(Q) = 1000 + 50Q2. Derive expressions for, and then sketch, the corresponding short-run average cost, average variable cost, and average fixed cost curves.

STC (Q ) = 1000 + 50Q 2 STC (Q ) 1000 = + 50Q Q Q AVC (Q ) = 50Q SAC (Q ) =

AFC (Q ) =

1000 Q

Graphing 𝑆𝐴𝐶(𝑄), 𝐴𝑉𝐶(𝑄), and 𝐴𝐹𝐶(𝑄) yields

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Solutions Manual

8.27. A producer of hard disk drives has a short-run total cost curve given by STC(Q) = K + Q2/K. Within the same set of axes, sketch a graph of the short-run average cost curves for three different plant sizes: K = 10, K = 20, and K = 30. Based on this graph, what is the shape of the long-run average cost curve?

Since each of these short-run average cost curves reaches a minimum at an average cost of 2.0, the long-run average cost curve associated with these short-run curves will be a horizontal line, tangent to the bottom of each of these curves, at a long-run average cost of 2.0.

8.28. Figure 8.18 shows that the short-run marginal cost curve may lie above the long-run marginal cost curve. Yet, in the long run, the quantities of all inputs are variable, whereas in the short run, the quantities of just some of the inputs are variable. Given that, why isn’t short-run marginal cost less than long-run marginal cost for all output levels?

With some inputs fixed, it is likely that the fixed level is not optimal given the firm’s size.

Therefore, it may be more expensive to produce additional units in the short run than in the long run when the firm can employ the optimal, i.e., cost minimizing, quantity of the fixed input.

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8.29. The following diagram shows the long-run average and marginal cost curves for a firm. It also shows the short-run marginal cost curve for two levels of fixed capital: K = 150 and K = 300. For each plant size, draw the corresponding short-run average cost curve and explain briefly why that curve should be where you drew it and how it is consistent with the other curves.

The SRAC curves are shown below. Each curve must satisfy two requirements- (i) the SRAC curve must be tangent to the LRAC curve at the output level at which the SRMC curve for that particular plant size intersects the LRMC curve; and (ii) the SRAC curve must reach a minimum at the output level at which it intersects its own SRMC curve.

For the plant size of 300 this is easily achieved by just drawing a curve tangent to the LRAC curve at its minimum point, since this is also the point at which the LRMC and the corresponding SRMC curves intersect (at the output level Q = 5).

For a plant size of 150, these two points must be kept in mind and the curve must be drawn carefully to comply with both. First, SRAC is tangent to LRAC at Q = 2, where LRMC intersects SRMC for K = 150. Second, SRAC reaches its minimum where it intersects SRMC, near Q = 2.4.

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8.30. Suppose that the total cost of providing satellite television services is as follows:

where Q1 and Q2 are the number of households that subscribe to a sports and movie channel, respectively. Does the provision of satellite television services exhibit economies of scope?

Economies of scope exist if 𝑇𝐶(𝑄1 , 𝑄2 ) − 𝑇𝐶(𝑄1 , 0) < 𝑇𝐶(0, 𝑄2 ) − 𝑇𝐶(0,0).

In this case

TC (Q1 , Q2 ) = 1000 + 2Q1 + 3Q2 TC (Q1 , 0) = 1000 + 2Q1 TC (0, Q2 ) = 1000 + 3Q2 TC (0, 0) = 0 So, economies of scope exist if (1000 + 2Q1 + 3Q2 ) − (1000 + 2Q1 )  1000 + 3Q2 3Q2  1000 + 3Q2 0  1000

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which is certainly true. Thus, in this case the cost of adding a movie channel when the firm is already providing a sports channel is less costly (by $1000) than a new firm supplying a movie channel from scratch. Economies of scope exist for this satellite TV company.

8.31 A railroad has two types of services: freight service and passenger service. The standalone cost for freight service is 𝑻𝑪𝟏 = 𝟓𝟎𝟎 + 𝑸𝟏 , where 𝑸𝟏 equals the number of ton-miles of freight hauled each day and 𝑻𝑪𝟏 is the total cost in thousands of dollars per day. The stand-alone cost for passenger service is 𝑻𝑪𝟐 = 𝟏𝟎𝟎𝟎 + 𝟐𝑸𝟐 , where 𝑸𝟐 equals the number of passenger-miles per day and 𝑻𝑪𝟐 is the total cost in thousands of dollars per day. When a railroad offers both services jointly its total is 𝑻𝑪(𝑸𝟏 , 𝑸𝟐 ) = 𝟐𝟎𝟎𝟎 + 𝑸𝟏 + 𝟐𝑸𝟐 . Do the provision of passenger and freight service exhibit economies of scope?

The provision of passenger and freight service do not exhibit economies of scope.

Economies of scope would be present if, for all positive values of Q1 and Q2, the total cost of offering both services together is less than the sum of the stand-alone costs of offering each service.

However, in this case, the reverse is true: the sum of the stand-alone costs of freight and passenger services is 1,500 + Q1 + 2Q2 which is less than the total cost if both services are offered together, which is 2,000+ Q1 + 2Q2 8.32 Suppose that the experience curve for the production of a certain type of semiconductor has a slope of 80 percent. Suppose over a five-year period, cumulative production experience increases by a factor of 8. Input prices over this period did not change. At the beginning of the period, average variable cost was $10 per unit. This cost is independent of the level of output at any particular point in time. What is your best estimate of average variable cost at the end of this five-year period?

The problem tells us that AVC is independent of output at any point in time and that factor prices did not change. It is plausible to conclude that the level of AVC over the five-year period is affected by changes in cumulative experience.

If cumulative experience has increased by a factor of 8, this means that cumulative experience has doubled three times (from N to 2N, then from 2N to 4N, and then again from 4N to 8N). With a slope of 80 percent: The first doubling reduced AVC to 80 percent of what they had been, i.e., from $10 to $8.

The second doubling reduced AVC to 80 percent of the new level, i.e., from $8 to (0.8)*$8 = $6.4 The third doubling reduced AVC to 80 percent of this new level, i.e., from $6.4 to (0.8)*$6.4 = $5.12

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8.33. A railroad provides passenger and freight service. The table shows the long-run total annual costs TC(F, P), where P measures the volume of passenger traffic and F the volume of freight traffic. For example, TC(10,300) = 1,000. Determine whether there are economies of scope for a railroad producing F = 10 and P = 300. Briefly explain.

TC(10, 300) = 1000 while TC(10, 0) + TC(0, 300) = 500 + 400 = 900. Thus TC(10, 300) > TC(10, 0) + TC(0, 300) so economies of scope do not exist at this output level.

8.34. A researcher has claimed to have estimated a long-run total cost function for the production of automobiles. His estimate is that TC(Q, w, r) = 100w−½r½Q3, where w and r are the prices of labor and capital. Is this a valid cost function—that is, is it consistent with long-run cost minimization by the firm? Why or why not? 𝑇𝐶 =

100𝑄 3 √𝑟 √𝑤

This 𝑇𝐶 function implies that for a fixed 𝑄 and 𝑟, increasing 𝑤 would lower long-run total cost.

If the firm were minimizing cost in the long run, by using the optimal combination of 𝐾 and 𝐿, it would not be possible to reduce total cost when 𝑤 is increased.

As Figures 8.4 and 8.5 in the text illustrate, when one input price increases, the total long-run cost will increase. Therefore, this long-run total cost function is not consistent with long-run cost minimization by the firm.

8.35. A firm owns two production plants that make widgets. The plants produce identical products and each plant (i) has a production function given by Qi = √KiLi, for i = 1, 2. The plants differ, however, in the amount of capital equipment in place in the short run. In particular, plant 1 has K1 = 25, whereas plant 2 has K2 = 100. Input prices for K and L are w = r = 1. a) Suppose the production manager is told to minimize the short-run total cost of producing Q units of output. While total output Q is exogenous, the manager can choose how much to produce at plant 1(Q1) and at plant 2(Q2), as long as Q1 + Q2 = Q. What percentage of its output should be produced at each plant? b) When output is optimally allocated between the two plants, calculate the firm’s shortrun total, average, and marginal cost curves. What is the marginal cost of the 100th widget? Of the 125th widget? The 200th widget?

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c) How should the entrepreneur allocate widget production between the two plants in the long run? Find the firm’s long-run total, average, and marginal cost curves. a)

Essentially, the firm’s production function is

Q = Q1 + Q2 = 25L1 + 100L2 = 5 L1 + 10 L2 That is, the firm has two variable inputs, L1 and L2. The marginal products are MPL1 = 2.5(L1)–0.5 and MPL2 = 5(L2)–0.5. Using the tangency condition, we see that L2 = 4L1. Using the production functions at each plant, we see 𝑄2 = 10√𝐿2 = 20√𝐿1 = 4𝑄1 Since total output Q = Q1 + Q2, we have Q2 = 4(Q – Q2) or Q2 = .8Q. Similarly, Q1 = .2Q. So the firm should produce 80 percent of output at plant 2 and 20 percent at plant 1. b) Combining the above tangency condition and the production constraint, we find the input demands are L1 = Q2/625 and L2 = 4Q2/625. Including the cost of capital, total cost is then C = 125 + (Q2/125). Average cost is AC = (125/Q) + (Q/125). Marginal cost is MC = 2Q/125.

MC(100) = 1.6, MC(125) = 2, and MC(200) = 3.2. c) In the long run, the plants are identical so the entrepreneur should split production equally between the two plants (i.e. Q1 = Q2). Thus the total production function can be written as Q = Q1 + Q2 = 2Q1 = 2 K1 L1

Again, we can view total output as depending on the choice of only two inputs. (Since the plants are identical the entrepreneur will hire equal amounts of capital at each plant; and similarly for labor.) Minimizing cost implies 𝑀𝑅𝑇𝑆𝐿1,𝐾1 = 𝑤/𝑟 or K1 = L1. Input demands are then L1 = K1 = 0.5Q so that total cost is C = w(L1 + L2) + r(K1 + K2) = 2Q. Long-run average cost is AC = 2 and long-run marginal cost is MC = 2.

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Chapter 9 Perfectly Competitive Markets Solutions to Review Questions 1. What is the difference between accounting profit and economic profit? How could a firm earn positive accounting profit but negative economic profit?

The difference between accounting profit and economic profit is in how total cost is measured.

With accounting profit, total cost is measured as total accounting cost while with economic profit, total cost is measured as total economic cost. Accounting cost measures the historical expenses the firm incurred to produce and sell its product while economic cost measured the opportunity cost of the resources that the firm uses to produce and sell its product.

If a firm chose to produce and sell a product it could earn a positive accounting profit but negative economic profit. This would occur if the economic cost of the resources used was greater than the accounting cost of the resources used. For example, the firm might purchase resources for $1 million and use these to produce a product when instead the firm could have resold the resources for $2 million. In this case the economic cost exceeds the accounting cost and economic profit would be less than accounting profit.

2. Why is the marginal revenue of a perfectly competitive firm equal to the market price?

The law of one price ensures that all transactions will take place at a single market price. A perfectly competitive firm cannot affect the market price by increasing or decreasing production.

Therefore, for each unit produced and sold, the firm will receive the market price as revenue.

Revenue will increase with each unit sold by the market price, implying the market price is equal to marginal revenue.

3. Would a perfectly competitive firm produce if price were less than the minimum level of average variable cost? Would it produce if price were less than the minimum level of shortrun average cost?

A perfectly competitive firm would not produce if the market price is below the minimum of its average variable cost. If the firm shuts down, the “bad news” is that it loses revenue. The “good news” is that the firm avoids non-sunk costs (including variable costs). If the market price is below the minimum of the firm’s average variable cost, the good news from shutting down outweighs the bad news.

If the market price is below the minimum of the firm’s short-run average cost, the decision as to whether the firm should shut down depends on how much of the fixed costs are non-sunk (avoidable). Suppose first that all of the fixed costs are non-sunk. If the firm shuts down, the “bad news” is that it loses revenue. The “good news” is that the firm avoids variable costs, as

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well as all of the fixed costs. If the market price is below the minimum of the firm’s short-run average cost, the good news from shutting down outweighs the bad news.

Now suppose that some of the fixed costs are sunk. Then for at least some levels of market price below the minimum of short-run average cost, the revenue lost may be greater than the costs that can be avoided if the firm shuts down. For such a market price, the firm would be better off continuing to operate in the short-run, because its losses from operating would be less than the losses it would sustain if it were to shut down.

4. What is the shutdown price when all fixed costs are sunk? What is the shutdown price when all fixed costs are nonsunk?

When all fixed costs are sunk, the shut-down price is the minimum level of average variable cost.

When all fixed costs are non-sunk, the shut-down price is the minimum level of short-run average cost.

5. How does the price elasticity of supply affect changes in the short-run equilibrium price that results from an exogenous shift in the market demand curve?

The supply elasticity can be used to determine the extent to which the equilibrium price will change when demand shifts exogenously. If supply is elastic, then a shift in demand will have a smaller impact on the equilibrium price than when supply is inelastic.

6. Consider two perfectly competitive industries—Industry 1 and Industry 2. Each faces identical demand and cost conditions except that the minimum efficient scale output in Industry 1 is twice that of Industry 2. In a long-run perfectly competitive equilibrium, which industry will have more firms?

Because the minimum efficient scale is higher in industry 1 than in industry 2, and since in a perfectly competitive market each firm must produce at minimum efficient scale in the long-run, there will be fewer firms in industry 1 since each firm is producing more units.

7. What is economic rent? How does it differ from economic profit?

Economic rent measures the economic surplus that is attributable to an extraordinarily productive input whose supply is limited. It is equal to the difference between the maximum amount a firm is willing to pay for the services of the input and the input’s reservation value.

Economic rent and economic profit are closely related. A firm’s economic profit depends on the price it pays for the extraordinarily productive input. If the firm pays the input its reservation value, the firm can earn a positive economic profit, but if the firm pays the input the reservation value plus the economic rent, as defined above, the firm will earn zero profit. Economic rent is thus the potential increase in the firm’s economic profit from employing the productive input, and the firm’s economic profit will depend on how much of the economic rent gets allocated to the firm and how much gets allocated to the input.

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8. What is the producer surplus for an individual firm? What is the producer surplus for a market when the number of firms in the industry is fixed and input prices do not vary as industry output changes? When is producer surplus equal to economic profit (for either a firm or an industry)? When producer surplus and economic profit are not equal, which is bigger?

The producer surplus for an individual firm is the difference between the total revenue the firm receives and the non-sunk cost. In general, it is the area below price and above the supply curve.

Producer surplus for a market of firms when the number of firms is fixed is the sum of the producer surplus for each of the individual firms.

Producer surplus for a firm will equal economic profit if the firm has no sunk fixed costs.

If producer surplus and economic profit are not equal, producer surplus equals the difference between total revenue and total non-sunk costs while economic profit equals the difference between total revenues and all total costs. Therefore, producer surplus will exceed economic profit if the two are not equal.

9. In the long-run equilibrium in an increasing-cost industry, each firm earns zero economic profits. Yet there is a positive area between the long-run industry supply curve and the long-run equilibrium price. What does this area represent?

In a market in which the long-run industry supply curve is upward sloping, the area between the price and the long-run supply curve measures the economic rents of inputs that are in scarce supply and whose price is bid up as more firms enter the industry.

10. Explain the difference between the following concepts: producer surplus, economic profit, and economic rent.

Producer surplus and economic profit may be equal in the short run. In particular, in the short run, Producer Surplus = Economic Profit + Sunk Fixed Costs.

Thus, in the short run producer surplus and economic profits differ by the level of sunk fixed costs. However, in the long run, since no fixed costs are sunk, producer surplus and economic profit will be equal. In general, producer surplus measures the difference between total revenue and total non-sunk costs while economic profit measures the difference between total revenue and all total costs.

These two measures differ from economic rent. Economic rent is the economic surplus that is attributable to an extraordinarily productive input whose supply is limited. Essentially, economic rent measures the potential increase in economic profit attributable to the scarce input above and beyond the economic profit the firm would enjoy if the firm paid suppliers of the input an amount equal to their reservation value. However, it is possible that the input, because of its scarcity, can extract the economic rent from the firm so that the firm still earns zero economic profit.

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Solutions to Problems 9.1. The annual accounting statement of revenues and costs for a local flower shop shows the following: Revenues Supplies Employee Salaries

$250,000 $25,000 $170,000

If the owners of the firm closed its operations, they could rent out the land for $100,000.

They would then avoid incurring any of the expenses for employees and supplies. Calculate the shop’s accounting profit and its economic profit. Would the owners be better off operating the shop or shutting it down? Explain.

The accounting costs are Supplies $25,000 Employee Salaries $170,000 Total Accounting Cost $195,000 Accounting profit = Revenue – Accounting Cost = $250,000 - $195,000 = $55,000 The economic costs are Supplies $25,000 Employee Salaries $170,000 Opportunity cost of land $100,000 Total Economic Cost $295,000 Economic profit = Revenue – Economic Cost = $250,000 - $295,000 = - $45,000 The negative economic profit indicates that the owners would be better off by $45,000 if they shut down the shop and rent out the land.

9.2. Last year, the accounting ledger for an owner of a small drugstore showed the following information about her annual receipts and expenditures (she lives in a tax-free country, so don’t worry about taxes): Revenues $1,000,000 Wages paid to hired labor $300,000 (other than herself ) Utilities (fuel, telephone, water) $20,000 Purchases of drugs and other $500,000 supplies for the store Wages paid to herself $100,000 She pays a competitive wage rate to her workers, and the utilities and drugs and other supplies are all obtained at market prices. She already owns the building, so she pays no money for its use. If she were to close the business, she could avoid all of her expenses and,

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of course, would have no revenue. However, she could rent out her building for $200,000.

She could also work elsewhere herself. Her two employment alternatives include working as a lawyer, earning wages of $100,000, or working at a local restaurant, earning $20,000.

Determine her accounting profit and her economic profit if she stays in the drugstore business. If the two are different, explain the difference.

Her accounting profit equals revenues less all of the expenses reflected in the ledger: $1,000,000 - $300,000 - $20,000 - $500,000 - $100,000 = $80,000.

All of the accounting costs are also economic costs. The first three expense items (wages paid to hired labor, utilities, and purchases of drugs and supplies) are expenses in competitive markets, so the opportunity cost is reflected in the market prices. Further, the wages she pays herself are the same as the opportunity cost of her time, because the most she could earn if she exits the drug store business is $100,000 working as a lawyer.

The economic costs of the business include all of the accounting costs, plus the $200,000 opportunity cost of the building because she could earn that if she exits the drug store business.

Her economic profit is her accounting profit ($80,000) less the additional opportunity cost ($200,000) not included in the accounting cost. So her economic profit is actually -$120,000.

We can look at this another way. If she continues to work at the grocery store, she earns an accounting profit of $80,000, plus the salary she pays herself ($100,000). But if she exits the business, her salary as a lawyer would be $100,000, and she would receive $200,000 rent for the building. She would therefore be better off by $120,000 if she worked as a lawyer.

9.3. A firm sells a product in a perfectly competitive market, at a price of $50. The firm has a fixed cost of $30. Fill in the following table and indicate the level of output that maximizes profit. How would the profit-maximizing choice of output change if the fixed cost increased from $40 to $60? More generally, explain how the level of fixed cost affects the choice of output.

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The table is as follows: Output (Units)

Total Revenue ($/unit)

Total Cost ($/unit)

Profit ($)

Marginal Revenue ($/unit)

Marginal Cost ($/unit)

0 1 2 3 4 5 6

0 50 100 150 200 250 300

30 80 100 130 172 226 296

-30 -30 0 20 28 24 4

50 50 50 50 50 50 50

50 20 30 42 54 70

When the firm is producing a positive amount of output, profit is maximized when Q = 4, regardless of the fixed cost. The firm will produce another unit when MR > MC, and cut back production when MR < MC. The relationship between MR and MC is unaffected by fixed cost.

9.4. A firm can sell its product at a price of $150 in a perfectly competitive market. Below is an incomplete table of a firm’s various costs of producing up to 6 units of output. Fill in the remaining cells of the table, and then calculate the profit the firm earns when it maximizes profit.

The table is as follows: Q 1 2 3 4 5 6

TC 200 220 240 360 500 660

TVC 80 100 120 240 380 540

AFC 120 60 40 30 24 20

AC 200 110 80 90 100 110

MC 80 20 20 120 140 160

AVC 80 50 40 60 76 90

The firm should produce 5 units. (Up to that level of output P > MC, but P < MC for the sixth unit.) Profit = PQ – C = 150(5) – 500 = 250.

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9.5. A competitive, profit-maximizing firm operates at a point where its short-run average cost curve is upward sloping. What does this imply about the firm’s economic profits?

Briefly explain.

If the firm operates at a point where its SRAC curve is rising, it must mean that the SRMC curve lies above the SRAC curve. And since the firm will choose an output such that price=SRMC, it means that price is greater than SRAC. Therefore the firm is earning positive economic profit.

9.6. A bicycle-repair shop charges the competitive market price of $10 per bike repaired.

The firm’s short-run total cost is given by STC(Q) = Q2/2, and the associated marginal cost curve is SMC(Q) = Q. a) What quantity should the firm produce if it wants to maximize its profit? b) Draw the shop’s total revenue and total cost curves, and graph the total profit function on the same diagram. Using your graph, state (approximately) the profit-maximizing quantity in each case. a) Since the firm is producing in a perfectly competitive market, the firm views the output price as exogenous. It should produce up to the point at which P = SMC(Q), that is, so that 10 = Q. So it should produce 10 units of output. b) TC

TR

Total Profit

The total cost function increases in Q, and at an increasing rate. Total Profit at first increases in Q and then decreases. From the graph, it appears that Profit is maximized when Q is about 10, which we found in (a).

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9.7. A producer operating in a perfectly competitive market has chosen his output level to maximize profit. At that output, his revenue and costs are as follows: Revenue $200 Variable costs $120 Sunk fixed costs $60 Nonsunk fixed costs $40 Calculate his producer surplus and his profits. Which (if either) of these should he use to determine whether he should exit the market in the short run? Briefly explain.

Producer surplus equals revenue less all non-sunk costs. Thus: Producer surplus = 200 – 160 = 40 The non-sunk costs of 160 include the variable cost of 120 and the non-sunk fixed cost of 40.

Profit = Revenue minus all costs = 200 – 120 – 60 – 40 = –20.

To decide whether to operate or shut down, the firm should look at producer surplus (rather than profit). Producer surplus (40) shows how much better off he would be operating (with a profit = –20) than shutting down (with a profit = –60). So he should stay in business in the short run; he will lose money, but not as much as if her were to shut down.

9.8 Dave’s Fresh Catfish is a northern Mississippi farm that operates in the perfectly competitive catfish farming industry. Dave’s short-run total cost curve is 𝑺𝑻𝑪(𝑸) = 𝟒𝟎𝟎 + 𝟐𝑸 + 𝟎. 𝟓𝑸2, where 𝑸 is the number of catfish harvest per month. The corresponding short-run marginal cost curve is 𝑺𝑴𝑪(𝑸) = 𝟐 + 𝑸. All of the fixed costs are sunk. a) What is the equation for the average variable cost (𝑨𝑽𝑪)? b) What is the minimum level of average variable costs? c) What is Dave’s short-run supply curve? 𝑇𝑉𝐶 = 2𝑄 + 0.5𝑄 2 so 𝐴𝑉𝐶 = 𝑇𝑉𝐶/𝑄 = 2 + 0.5𝑄.

The minimum level of 𝐴𝑉𝐶 occurs at the 𝑄 where 𝑆𝑀𝐶 = 𝐴𝑉𝐶, or 2 + 𝑄 = 2 + 0.5𝑄, or 𝑄 = 0. The minimum level of AVC is thus 2.

Since all fixed costs are sunk, the firm will not produce if the price is below the minimum level of 𝐴𝑉𝐶., or 2. For prices above 2, the quantity supplied is found by equation price to marginal cost, or 2 + 𝑄 = 𝑃, which implies 𝑄 = 𝑃 − 2. Thus, the firm’s short-run supply curve is: 𝑠(𝑃) = 0, 𝑖𝑓 𝑃 < 2 𝑠(𝑃) = 𝑃 – 2 𝑖𝑓 𝑃  2

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9.9. Ron’s Window Washing Service is a small business that operates in the perfectly competitive residential window washing industry in Evanston, Illinois. The short-run total cost of production is STC(Q) = 40+ 10Q + 0.1Q2, where Q is the number of windows washed per day. The corresponding short-run marginal cost function is SMC(Q) = 10 + 0.2Q. The prevailing market price is $20 per window. a) How many windows should Ron wash to maximize profit? b) What is Ron’s maximum daily profit? c) Graph SMC, SAC, and the profit-maximizing quantity. On this graph, indicate the maximum daily profit. d) What is Ron’s short-run supply curve, assuming that all of the $40 per day fixed costs are sunk? e) What is Ron’s short-run supply curve, assuming that if he produces zero output, he can rent or sell his fixed assets and therefore avoid all his fixed costs? a) In order to maximize profit Ron should operate at the point where 𝑃 = 𝑀𝐶.

20 = 10 + 0.20Q Q = 50

b) Ron’s profit is given by 𝜋 = 𝑇𝑅 − 𝑇𝐶. 𝜋 = 20(50) − (40 + 10(50) + 0.10(50)2 ) 𝜋 = 210 c) The firm’s profit is equal to the shaded area in the graph below. It is a rectangle whose height is the market price and the average cost of the 50th unit, and whose width is the 50 units being produced.

d) If all fixed costs are sunk, then ANSC = AVC = (10Q + 0.1Q2)/Q = 10 + 0.1Q. So the first step is to find the minimum of ANSC by setting ANSC = SMC, or 10 + 0.1Q = 10 + 0.2Q

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which occurs when Q = 0. The minimum level of ANSC is thus 10. For prices below 10 the firm will not produce and for prices above 10, its supply curve is found by setting P = SMC: P = 10 + .2Q Q = 5 P − 50

The firm’s short-run supply curve is thus 0 if 𝑃 < 10 𝑠(𝑃) = { 5𝑃 − 50 if 𝑃 ≥ 10 e) If all fixed costs are non-sunk, as in this case, then ANSC = ATC = (40/Q) + 10 + 0.1Q. The minimum point of ANSC occurs where ANSC = SMC: 40 + 10 + .1Q = 10 + .2Q Q Q = 20

The minimum level of ANSC is thus 14. For prices below 14 the firm will not produce and for prices above 14, its supply curve is found by setting P = SMC as before.

0 if 𝑃 < 14 𝑠(𝑃) = { 5𝑃 − 50 if 𝑃 ≥ 14 9.10. The bolt-making industry currently consists of 20 producers, all of whom operate with the identical short-run total cost curve STC(Q) = 16 + Q2, where Q is the annual output of a firm. The corresponding short-run marginal cost curve is SMC(Q) = 2Q. The market demand curve for bolts is D(P) = 110 − P, where P is the market price. a) Assuming that all of each firm’s $16 fixed cost is sunk, what is a firm’s short-run supply curve? b) What is the short-run market supply curve? c) Determine the short-run equilibrium price and quantity in this industry. a) First, find the minimum of 𝐴𝑉𝐶 by setting 𝐴𝑉𝐶 = 𝑆𝑀𝐶.

TVC Q 2 = Q Q AVC = Q AVC =

Q = 2Q Q=0

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The minimum level of 𝐴𝑉𝐶 is thus 0. When the price is 0 the firm will produce 0, and for prices above 0 find supply by setting 𝑃 = 𝑆𝑀𝐶. 𝑃 = 2𝑄 1 𝑄= 𝑃 2 Thus, 𝑠(𝑃) =

1 𝑃 2

b) Market supply is found by horizontally summing the supply curves of the individual firms.

Since there are 20 identical producers in this market, market supply is given by 𝑆(𝑃) = 20𝑠(𝑃) 𝑆(𝑃) = 10𝑃 c) Equilibrium price and quantity occur at the point where 𝑆(𝑃) = 𝐷(𝑃).

10 P = 110 − P P = 10 Substituting 𝑃 = 10 back into 𝐷(𝑃) implies equilibrium quantity is 𝑄 = 100. So at the equilibrium, 𝑃 = 10 and 𝑄 = 100.

9.11. Newsprint (the paper used for newspapers) is produced in a perfectly competitive market. Each identical firm has a total variable cost TVC(Q) = 40Q + 0.5Q2, with an associated marginal cost curve SMC(Q) = 40 + Q. A firm’s fixed cost is entirely nonsunk and equal to 50. a) Calculate the price below which the firm will not produce any output in the short run. b) Assume that there are 12 identical firms in this industry. Currently, the market demand for newsprint is D(P) = 360 − 2P, where D(P) is the quantity consumed in the market when the price is P. What is the short-run equilibrium price? a) The firm will not produce any output when the price falls below the point where SMC = ANSC, i.e. the minimum of the ANSC curve. Therefore 50/𝑄 + 40 + 0.5𝑄 = 40 + 𝑄 This implies Q = 10. The corresponding price, below which the firms will not produce, is equal to MC(10) = ANSC(10) = 50. b) Each firm will produce according to the relation, P = MC, or 𝑃 = 40 + 𝑄. This means that each firm’s supply curve is 𝑄 = 𝑃 − 40 if P > 50 and zero if P < 50. Therefore market supply equals 12(𝑃 − 40) and in equilibrium this must equal market demand, 360 − 2𝑃. Therefore the equilibrium price is P = 60. At this price, each firm produces 20 units of output. The firm’s profit is 𝑃𝑄 − 𝑉(𝑄) − 𝐹and this equals 30. Substituting Q = 20 and P = 60, we get total fixed costs, F = 170. Since non-sunk fixed costs are 50, sunk fixed costs must total up to 120.

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9.12. The oil drilling industry consists of 60 producers, all of whom have an identical shortrun total cost curve, STC(Q) = 64 + 2Q2, where Q is the monthly output of a firm and $64 is the monthly fixed cost. The corresponding short-run marginal cost curve is SMC(Q) = 4Q. Assume that $32 of the firm’s monthly $64 fixed cost can be avoided if the firm produces zero output in a month. The market demand curve for oil drilling services is D(P) = 400 − 5P, where D(P) is monthly demand at price P. Find the market supply curve in this market, and determine the short-run equilibrium price.

The firm’s ANSC curve is given by 32/Q + 2Q. To find the shut-down price, we find the minimum level of ANSC. This occurs at the quantity at which ANSC equals MC, or 32/Q + 2Q = 4Q. Solving for Q yields Q = 4, and substituting this into the expression for ANSC tells us that the minimum level of ANSC is equal to 32/4 + 2(4) = $16. At prices below $16, a firm’s supply is 0. At prices above $16, a firm produces a quantity at which P = SMC: P = 4Q, or Q = P/4. Thus, the short-run supply curve for a firm is: 0 if P < 16 s(P) ={ P/4 if P  16 Since there are 60 identical producers, each with this supply curve, the short-run market supply curve S(P) is 60 times s(P), or: 0 if P < 16 S(P) ={ 15P if P  16 To find the equilibrium price, we equate market supply to market demand and solve for P: 15P = 400 – 5P, or P = 20.

9.13. There are currently 10 identical firms in the perfectly competitive gadget manufacturing industry. Each firm operates in the short run with a total fixed cost of F and total variable cost of 2Q2, where Q is the number of gadgets produced by each firm.

The marginal cost for each firm is MC = 4Q. Each firm also has non sunk fixed costs of 128.

Each firm would just break even (earn zero economic profit) if the market price were 40.

(Note: The equilibrium price is not necessarily 40 when there are 10 firms in the market.)

The market demand for gadgets is QM = 180 − 2.5P, where QM is the amount purchased in the entire market. a) How large are the total fixed costs for each firm? Explain. b) What would be the shutdown price for each firm? Explain. c) Draw a graph of the short-run supply schedule for this firm. Label it clearly. d) What is the equilibrium price when there are 10 firms currently in the market? e) With the cost structure assumed for each firm in this problem, how many firms would be in the market at an equilibrium in which every firm’s economic profits are zero?

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a) C = F + 2Q2. MC = 4Q. Breakeven price = 40. When P = 40, the firm would produce Q so that MC = P; 40 = 4Q; Q = 10.

Profit = PQ – F – 2Q2 = 40(10) – F – 2(10)2 = 200 – F = 0. So F = 200. b) The total nonsunk fixed cost is NSC = 128 + 2Q2 .

The firm will shut down if the market price is less than the minimum of ANSC. ANSC = (128 + 2Q)2/Q. At minimum of ANSC, we know that ANSC = MC, or that [128 + 2Q2]/Q = 4Q, so the quantity at the shutdown price is Q = 8.

The shutdown price will be where Q = 8; MC = 4Q = 4(8) = 32. So the shutdown price is P = 32.

(Alternatively, you can verify that when Q = 8, then ANSC = 32.) c) The firm’s supply schedule will be Q = 0 when P < 32.

When P > 32, the firm will supply according to the optimal quantity choice rule P = MC; thus P = 4Q, so that Q = P/4.

When P = 32, the firm will be indifferent between shutting down (Q = 0) or operating with Q = 8. 𝑃/4 when 𝑃 > 32 To summarize, 𝑄 = {0 when 𝑃 < 32 0 or 8 when 𝑃 = 32

d) With 10 firms in the market, total market supply will be 10(P/4) = 2.5P. Market demand is 180 – 2.5P. In equilibrium 2.5P = 180 – 2.5P, so P = 36 (note: P > 32, so the firms do produce). e) For profits to be zero, the price would be P = 40, and each firm would produce 40/4 = 10 units.

The quantity demanded in the market would be 180 – 2.5(40) = 80 units. Thus, there is room for only 80/10 = 8 firms.

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9.14. A perfectly competitive industry consists of two types of firms: 100 firms of type A and 30 firms of type B. Each type A firm has a short-run supply curve sA(P) = 2P. Each type B firm has a short-run supply curve sB(P) = 10P. The market demand curve is D(P) = 5000 − 500P. What is the short-run equilibrium price in this market? At this price, how much does each type A firm produce, and how much does each type B firm produce?

Total industry supply is the sum of the supply curves of the individual firms. Since we have 100 type A firms, total supply from type A firms is 100sA(P) = 200P, and since we have 30 type B firms, total supply from type B firms is 30sB(P) = 300P. The short-run industry supply curve is thus S(P) = 200P + 300P = 500P. The short-run market equilibrium occurs at the price at which quantity supplied equals quantity demanded, or 5000 – 500P = 500P, or P = 5. At this price, a type A firm supplies 10 units, while a type B firm supplies 50 units.

9.15. A market contains a group of identical price taking firms. Each firm has a marginal cost curve SMC(Q) = 2Q, where Q is the annual output of each firm. A study reveals that each firm will produce if the price exceeds $20 per unit and will shut down if the price is less than $20 per unit. The market demand curve for the industry is D(P) = 240 − P/2, where P is the market price. At the equilibrium market price, each firm produces 20 units.

What is the equilibrium market price, and how many firms are in this industry?

To determine the quantity supplied for a given price, set 𝑃 = 𝑆𝑀𝐶. 𝑃 = 2𝑄 1 𝑄= 𝑃 2 1

Thus the supply curve for each firm is 𝑠(𝑃) = 2 𝑃. If each firm is producing 20 units, then 20 = 12 P P = 40

So the market price is 40. Substituting into demand reveals D( P) = 240 − 12 (40) D( P) = 220 220

If each firm is producing 20 units, the market will have 20 = 11 firms.

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9.16. The wood-pallet market contains many identical firms, each with the short-run total cost function STC(Q) = 400 + 5Q + Q2, where Q is the firm’s annual output (and all of the firm’s $400 fixed cost is sunk). The corresponding marginal cost function is SMC(Q) = 5 + 2Q. The market demand curve for this industry is D(P) = 262.5 − P/2, where P is the market price. Each firm in the industry is currently earning zero economic profit. How many firms are in this industry, and what is the market equilibrium price?

Since each firm is earning zero economic profit, we know 𝑃 = 𝑆𝐴𝐶. Since each firm supplies where 𝑃 = 𝑆𝑀𝐶, set 𝑆𝐴𝐶 = 𝑆𝑀𝐶.

400 + 5 + Q = 5 + 2Q Q Q = 20

Since 𝑃 = 5 + 2𝑄, 𝑃 = 45. If market price is 𝑃 = 45, 𝐷(𝑃) = 240. Finally, if total market 240 demand is 𝑄 = 240 and each firm is producing 20 units, there will be 20 = 12 firms in the market.

9.17. Suppose a competitive, profit-maximizing firm operates at a point where its short-run average cost curve is upward sloping. What does this imply about the firm’s economic profits? If the profit-maximizing firm operates at a point where its short-run average cost curve is downward sloping, what does this imply about the firm’s economic profits?

If the firm operates at a point where its SAC curve is rising, it must mean that the marginal cost curve is above the SAC curve. And since the firm must set price=MC, it means that price is greater than average cost. Therefore the firm earns positive economic profit.

If it operates at a point where the SAC curve is falling, it means SMC < SAC and therefore price is less than average cost. Therefore the firm is making negative economic profit in the short run.

However, the fact that the firm is still operating means that marginal cost must be above the average non-sunk cost curve, so that it is better for the firm to continue operating, albeit at a loss, than to shut down.

9.18. A firm in a competitive industry produces its output in two plants. Its total cost of producing Q1 units from the first plant is TC1 = (Q1)2, and the marginal cost at this plant is MC1 = 2Q1. The firm’s total cost of producing Q2 units from the second plant is TC2 = 2(Q2)2; the marginal cost at this plant is MC2 = 4Q2. The price in the market is P. What fraction of the firm’s total supply will be produced at plant 2?

Given a market price P, the firm will produce from each plant so that MC = P. The profit maximizing quantity supplied at plant 1 will be 2Q1 = P, or Q1 = P/2.

The profit maximizing quantity supplied at plant 2 will be 4Q2 = P, or Q2 = P/4.

The quantity supplied by the whole firm will QFirm = Q1 + Q2. Thus QFirm = 3P/4.

So 1/3 of the firm’s total production will come from plant 2.

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9.19. A competitive industry consists of 6 type A firms and 4 type B firms. Each firm of type A operates with the supply curve:

Each firm of type B operates with the supply curve:

a) Suppose the market demand is At the market equilibrium, which firms are producing, and what is the equilibrium price? b) Suppose the market demand is At the market equilibrium, which firms are producing, and what is the equilibrium price? a) When P <10, only Type B firms will operate, and the market supply will be 4(2P) = 8P. When P >10, both types of firms will operate, and the market supply will be 4(2P) + 6(-10 + P) = -60 + 14P. −60 + 14𝑃, when 𝑃 > 10 𝑆𝑢𝑝𝑝𝑙𝑦 To summarize, the market supply will be 𝑄𝑀𝑎𝑟𝑘𝑒𝑡 = { 8𝑃, when 𝑃 ≤ 10 Let’s first assume the equilibrium price exceeds 10, so that all firms are producing. If this is true, setting market supply equal to market demand: -60 + 14P = 108 – 10P, so that P = 7; however, the market supply we have used is valid for P>10, but not valid for P = 7.

So the equilibrium price must be less than 10, with only Type B firms producing (and Type A firms not producing).

Setting market supply equal to market demand: 8P = 108 – 10P, so that P = 6.

We have found that in equilibrium, only Type B firms produce, and the equilibrium price is 6. b) Let’s first assume the equilibrium price exceeds 10, so that all firms are producing. If this is true, setting market supply equal to market demand: -60 + 14P = 228 – 10P, so that P = 12; the market supply we have used is valid for P=12. At this equilibrium both types of firms will be producing.

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9.20. A firm’s short-run supply curve is given by

What is the equation of the firm’s marginal cost curve SMC(Q)?

We know that if a firm produces positive output, it produces where P=SMC. In this case, when 𝑄 the firm produces positive output, 𝑄 = 3𝑃 − 30, or 𝑃 = 3 + 10. This means that the equation of 𝑄

the firm’s short run marginal cost is 𝑆𝑀𝐶(𝑄) = 3 + 10.

9.21. Consider a point on a supply curve where price and quantity are positive. Determine the numerical value of the price elasticity of supply at that point when the supply curve is a) vertical at a positive quantity b) horizontal at a positive price c) a straight line through the origin, with a positive slope a) When the supply curve is vertical at a positive quantity, the quantity supplied is unresponsive to price, and the price elasticity of supply is equal to 0 (supply is perfectly inelastic). b) When the supply curve is horizontal at a positive quantity, price elasticity of supply is infinite (supply is perfectly elastic). c) When the supply curve is a straight line going through the origin, the price elasticity of supply must equal 1. We determine this as follows. The equation of a straight line supply curve through the origin takes the form Q = aP, where a is the slope of the supply curve. Thus Q/P = a. The price elasticity of supply is equal to (Q/P )(P/Q) = a(P/Q) = a(P/aP) = 1.

9.22. During the week of February 9–15, 2001, the U.S. rose market cleared at a price of $1.00 per stem, and 4,000,000 stems were sold that week. During the week of June 5–11, 2001, the U.S. rose market cleared at a price of $0.20 per stem, and 3,800,000 stems were sold that week. From this information, what would you conclude about the price elasticity of supply in the U.S. rose market?

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As with example 9.4 in the text, we can estimate the slope of the supply curve as

Q 4, 000, 000 − 3,800, 000 = P 1.00 − 0.20 Slope = 250, 000 Elasticity can then be estimated as 𝜀𝑄,𝑃 =

𝛥𝑄 𝑃 ( ) 𝛥𝑃 𝑄

1.00 ) 𝜀𝑄,𝑃 = 250,000 ( 4,000,000 1 𝜀𝑄,𝑃 = 16 This implies the market supply of roses is quite inelastic.

9.23. The global cobalt mining industry is perfectly competitive. Each existing firm and every potential entrant faces an identical U-shaped average cost curve. The minimum level of average cost is $5 per ton and occurs when a firm produces 2 million tons of cobalt per year. The market demand curve for cobalt is D(P) = 205 − P, where D(P) is the demand for cobalt in millions of tons per year when the market price is P dollars per ton. What is the long-run equilibrium price for cobalt? How much cobalt does each producer make at this equilibrium price? How many active cobalt producers will be in the market?

The long-run equilibrium price in a perfectly competitive equilibrium equals the minimum level of long-run average cost. This is given as $5 per ton. Each producer supplies a quantity of output equal to the point at which long-run average is minimized. This is given as 2 million tons per year. Market demand at the long-run equilibrium price of $5 per ton is equal to 205 – 5 = 200 million tons per year. This implies that there must be 100 active firms in the long-run equilibrium because (200 million tons per year)/(2 million tons per year per firm) = 100.

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9.24. The global propylene industry is perfectly competitive, and each producer has the long-run marginal cost function MC(Q) = 40 − 12Q + Q2. The corresponding long-run average cost function is AC(Q) = 40 − 6Q + Q2/3. The market demand curve for propylene is D(P) = 2200 − 100P. What is the long-run equilibrium price in this industry, and at this price, how much would an individual firm produce? How many active producers are in the propylene market in a long-run competitive equilibrium?

In a long-run equilibrium all firms earn zero economic profit implying 𝑃 = 𝐴𝐶 and each firm produces where 𝑃 = 𝑀𝐶. Thus, 40 − 12Q + Q 2 = 40 − 6Q + 13 Q 2 Q=9

So each individual firm produces 𝑄 = 9, and the long-run equilibrium price must be𝑃 = 40 − 12(9) + 92 = 13. Since 𝐷(𝑃) = 2200 − 100𝑃, 𝐷(𝑃) = 2200 − 100(13) 𝐷(𝑃) = 900 If each firm produces 9 units, the market will have 100 firms in equilibrium.

9.25 The raspberry growing industry in the U.S. is perfectly competitive, and each producer has a long-run marginal cost curve given by 𝑴𝑪(𝑸) = 𝟐𝟎 + 𝟐𝑸. The 𝟏𝟒𝟒 corresponding long-run average cost function is given by 𝑨𝑪(𝑸) = 𝟐𝟎 + 𝑸 + . The 𝑸

market demand curve is 𝑫(𝑷) = 𝟐𝟒𝟖𝟖 – 𝟐𝑷. What is the long-run equilibrium price in this industry, and at this price, how much would an individual firm produce? How many active producers are in the raspberry growing industry in a long-run competitive?

The long-run competitive equilibrium satisfies the following three equations: Profit-maximization: 𝑃 = 𝑀𝐶, or 𝑃 = 20 + 2𝑄.

Zero profit: 𝑃 = 𝐴𝐶, or 𝑃 = 20 + 𝑄 + 144/𝑄.

Supply equals demand: 𝐷(𝑃) = 𝑛𝑄, or 2488 – 2𝑃 = 𝑛𝑄.

Solving these equations gives us: 𝑄 = 12. 𝑃 = 44 𝑛 = 200.

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9.26. Suppose that the world market for calcium is perfectly competitive and that, as a first approximation, all existing producers and potential entrants are identical. Consider the following information about the price of calcium: • Between 1990 and 1995, the market price was stable at about $2 per pound.

• In the first three months of 1996, the market price doubled, reaching a high of $4 per pound, where it remained for the rest of 1996.

• Throughout 1997 and 1998, the market price of calcium declined, eventually reaching $2 per pound by the end of 1998.

• Between 1998 and 2002, the market price was stable at about $2 per pound.

Assuming that the technology for producing calcium did not change between 1990 and 2002 and that input prices faced by calcium producers have remained constant, what explains the pattern of prices that prevailed between 1990 and 2002? Is it likely that there are more producers of calcium in 2002 than there were in 1990? Fewer? The same number? Explain your answer.

The scenario described in the problem can be explained as a constant-cost perfectly competitive industry that experienced an increase in demand (i.e., rightward shift in the demand curve) in early 1996 as shown in the figure below. The price between 1990-1995 reflects a market that is in long-run equilibrium. The increase in price in early 1996 reflects the movement to a short-run equilibrium following the increase in demand. Once price stabilizes at the new short-run equilibrium, firms earn positive economic profits, which attracts new entry. As new entry occurs during 1997 and 1998, the short-run supply curve shifts rightward, causing price to fall. Entry is no longer profitable once price is reestablished at the minimum level of long-run average cost for a typical firm. As a result of the increase in demand, the market now contains more active producers in 2002 than it did in 1990.

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Typical Firm

Industry

$/lb

$/lb SSpre-1996 SSpost-1998

SMC $4 AC

$2

$4

$2

Dpre-1996 units per year

Dpost-1996

units per year

9.27 It is 2017, and you work for a prestigious management consultant firm whose client is a large agribusiness company that is considering acquiring an ownership stake in several U.S. yellow perch farming operations. (The yellow perch is a fresh fish found in the U.S. and raised commercially for sale as food.) As a member of the consulting team working on this project, you have been assigned the task of understanding why the U.S. farm-raised perch industry has evolved as it has over the last 6 years.

Between 2010-2013, the farm-raised yellow perch market was stable. However, in 2013 an unexpected exogenous shock occurred that affected prices and quantities in the market.

You don’t know much about the details of the industry, and since the industry is not covered extensively in the press, it is hard to find articles on the Web about what happened to the industry. From talking to the client, you learn that the shock might have had something to do with either a change in the market demand for yellow perch or a change in the price of corn (which affects the price of perch feed). But you do not know for sure, nor do you know whether the shock was a permanent change or merely a temporary one.

However, you do have data (obtained from the client), shown in the table below, on yellow perch prices, market demand, quantity supplied, and the number of producers. The data pertains to 2010-2013, 2014 (within one year of the shock), and 2016 (three years after the shock). You also know (from the client) that yellow perch farms are virtually identical, with U-shaped long-run average cost curves. You also learn from the client that the minimum efficient scale of a typical yellow perch farm occurs at a rate of production of about 1,000 pounds per month (and this is unaffected by changes in the prices of key inputs such as feed or labor).

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a) Based on the data in the table, what type of shock most likely explains the evolution of the yellow perch farming industry from 2010-2013 to 2016?

b) How would your answer change if the number of active yellow perch farms in 2016 was 100? c) How would your answer change if the data in the table looked like this?

The data are consistent with the adjustment to a long-run equilibrium in a constant-cost industry as a result of a permanent rightward shift in the market demand curve, i.e., a permanent increase in the demand for yellow perch.

If number of active yellow perch farms in 2016 was 100 rather than 150, the data are consistent with an adjustment to a new short-run equilibrium and back again to a temporary shift in demand.

The data in the table are consistent with the adjustment to a short-run equilibrium, then to a longrun equilibrium, as a result of a permanent increase in the price of feed (caused by a permanent increase in the price of corn).

This one is harder than the other two, so here is the reasoning: In the short-run, an increase in the price of feed, would shift each firm’s SMC upward, thus shifting the industry short-run supply curve to the left. The price would rise, and total quantity demanded would fall. For a fixed number of firms, each firm would necessarily produce less.

The increase in the price of feed would also shift the minimum level of long-run average cost upward. Thus, the new long-run equilibrium price would have to be higher than the price that prevailed in 2010-2013, as well, which is what the data in the table show. Because the long-run

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equilibrium price in 2016 is higher than the short-run equilibrium price in 2014, the number of yellow-perch farms would have to have fallen between 2014 and 2016, which is what we see in the data. Summing up that data is consistent with a permanent increase in the price of feed.

The data are not consistent with a permanent or transitory increase in demand. The only way the price could have gone up in the short-run is for the market demand curve to have shifted to the right (i.e., market demand increased.) If that had happened and the shift had been temporary, the number of active farms would not have changed. If the rightward shift had been permanent, then in the long run, entry would have occurred in the industry, increasing the number of activeproducing farms in the long run. But we actually see a decrease in the number of farms, which is inconsistent with either a temporary or permanent increase in demand.

9.28. The long-run total cost function for producers of mineral water is TC(Q) = cQ, where Q is the output of an individual firm expressed as thousands of liters per year. The market demand curve is D(P) = a − bP. Find the long-run equilibrium price and quantity in terms of a, b, and c. Can you determine the equilibrium number of firms? If so, what is it? If not, why not?

For this total cost function, 𝑀𝐶 = 𝑐. Since each firm will supply where 𝑃 = 𝑀𝐶, in equilibrium 𝑃 = 𝑐. If in equilibrium 𝑃 = 𝑐, 𝐷(𝑃) = 𝑎 − 𝑏𝑐 Equilibrium market quantity is 𝑎 − 𝑏𝑐.

In order to determine the number of firms we need to know the quantity that each individual firm will produce. In this case marginal cost is constant implying perfectly elastic supply. Thus, at 𝑃 = 𝑐 a firm may produce any quantity. Therefore, the number of firms cannot be determined.

9.29. Support or refute the following: “In the long run the firm’s producer surplus and profits will be equal.”

This statement is true.

Profit = Revenue – Total Cost = Revenue – Nonsunk Cost – Sunk Cost Producer Surplus = Revenue – Nonsunk Cost In the long run, there are no sunk costs, so profit is the same as producer surplus.

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9.30. Each firm in the perfectly competitive widget industry produces with the levels of marginal cost (MC) and total variable cost (TVC) at various levels of output Q shown in the following table. Each firm has a total fixed cost of 64 and a sunk fixed cost of 48. a) Draw a clearly labeled graph of the short-run supply schedule for this firm. Be sure to indicate the shutdown price for each firm and to explain your reasoning for the shape of the supply curve. b) What is each firm’s producer surplus when the market price is 16? c) What is the breakeven price for each firm?

a)

We know that the supply curve for the firm is just the marginal cost curve for all prices greater than the shut down price. At the shut down price: Producer surplus = Revenue – V – FNonsunk = 0.

FNonsunk = FTotal – FSunk = 64 – 48 = 16.

Simple calculations from the table show that the shut down price is P = 10. If the firm elects to produce when P = 10, it chooses Q so that MC = P, that is, Q = 4.

Producer surplus = Revenue – V – FNonsunk = 10(4) – 24 – 16 = 0.

The graph of the firm’s supply function is as follows. Note that it is the same as the firm’s supply function when P > 10.

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Solutions Manual

b) When P = 16, the firm chooses Q so that MC = P, that is, Q = 7.

Producer surplus = Revenue – V – FNonsunk.

Producer surplus = Revenue – V – FNonsunk = 16(7) – 63 – 16 = 33. c) Again, calculations from the table show that the breakeven price is P = 18.

When P = 18, the firm chooses Q so that MC = P, that is, Q = 8.

Profit = Revenue – V – FTotal = 18(8) – 80 – 64 = 0.

9.31. In a constant-cost industry in which firms have U-shaped average cost curves, the long-run market supply curve is a horizontal line. This market supply curve is not the horizontal sum of individual firms’ long-run supply curves. In this respect, the long-run market supply curve differs from the short-run market supply curve, which, in a constantcost industry, will equal the horizontal sum of individual firms’ short-run supply curves.

Why does the derivation of the long-run market supply curve differ from the derivation of the short-run market supply curve?

In the short run, as demand increases, price is driven up and firms can earn positive economic profits. In the short run, however, the number of firms is fixed, so total market supply is simply the sum of the supply of each individual firm. In the long run, though, the firms cannot continue to earn positive economic profit. New firms will enter, driving the price back down until economic profit is zero. In a constant cost industry this occurs at the same equilibrium price as prior to the increase in market demand. Thus, in the long run, any quantity will be supplied and the number of firms will adjust so that each firm earns zero economic profit. The primary difference in the derivation then is that in the short run the number of firms is fixed, but in the long run the number of firms will adjust to maintain zero economic profit.

9.32. The long-run average cost for production of hard-disk drives is given by AC(Q) = √wr(120−20Q + Q2), where Q is the annual output of a firm, w is the wage rate for skilled assembly labor, and r is the price of capital services. The corresponding long-run marginal cost curve is MC(Q) = √wr(120 − 40Q + 3Q2). The demand for labor for an individual firm is

The price of capital services is fixed at r = 1. a) In a long-run competitive equilibrium, how much output will each firm produce? b) In a long-run competitive equilibrium, what will be the market price? Note that your answer will be expressed as a function of w. c) In a long-run competitive equilibrium, how much skilled labor will each firm demand?

Again, your answer will be in terms of w. d) Suppose that the market demand curve is given by D(P) = 10,000/P. What is the market equilibrium quantity as a function of w? e) What is the long-run equilibrium number of firms as a function of w?

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f ) Using your answers to parts (c) and (e), determine the overall demand for skilled labor in this industry as a function of w. g) Suppose that the supply curve for the skilled labor used in this industry is Γ(w) = 50w. At what value of w does the supply of skilled labor equal the demand for skilled labor? h) Using your answer from part (g), go back through parts (b), (d), and (e) to determine the long-run equilibrium price, market demand, and number of firms in this industry. i) Repeat the analysis in this problem, now assuming that the market demand curve is given by D(P) = 20,000/P. a) In a long-run competitive equilibrium 𝑃 = 𝑀𝐶 and 𝑃 = 𝐴𝐶, implying 𝑀𝐶 = 𝐴𝐶.

wr (120 − 40Q + 3Q 2 ) = wr (120 − 20Q + Q 2 ) Q = 10 b) In a long-run competitive equilibrium 𝑃 = 𝑀𝐶 so that (with 𝑟 = 1 and 𝑄 = 10) 𝑃 = √𝑤(1)(120 − 40(10) + 3(10)2 ) 𝑃 = 20√𝑤 c) Given demand for labor and setting 𝑟 = 1 and 𝑄 = 10 √𝑟(120𝑄 − 20𝑄 2 + 𝑄 3 ) 2√𝑤 100

𝐿(𝑄, 𝑤, 𝑟) = 𝐿(𝑄, 𝑤) =

√𝑤

d) Given market demand and setting 𝑟 = 1 10000 P 10000 Q= 20 w 500 Q= w

D( P) =

e) Since each firm will produce 10 units, 500 𝑤 𝑁= √ 10 50 𝑁= √𝑤

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Besanko & Braeutigam – Microeconomics, 6th edition

f) From part c), the labor demand for an individual firm is 𝐿(𝑄, 𝑤) =

Solutions Manual

100 √𝑤

. Overall demand for

labor is then 50 100 ( ) √𝑤 √𝑤 5000 Demand for Labor = 𝑤 Demand for Labor =

g) Setting the supply of skilled labor equal to the demand for skilled labor, 5000 w w = 10

50 w =

h) Plugging 𝑤 = 10 into the solution for price implies 𝑃 = 63.25; plugging 𝑤 = 10 in market demand implies 𝑄 = 158.10; and plugging 𝑤 = 10 into the solution for the number of firms and rounding down to the nearest integer implies 𝑁 = 15. i) If 20000 P 20000 Q= 20 w 1000 Q= w

D( P) =

The number of firms will be 1000 𝑤 𝑁= √ 10 100 𝑁= √𝑤 Overall labor demand will be 100 100 ( ) √𝑤 √𝑤 10000 Labor = 𝑤 Labor =

Setting the supply of labor equal to the demand for labor implies

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Chapter 9-27

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10000 w 2 w = 200

50 w =

w = 14.14

Plugging 𝑤 = 14.14 into the solution for price implies 𝑃 = 75.21; plugging 𝑤 = 14.14 into market demand implies 𝑄 = 265.92; and plugging 𝑤 = 14.14 into the solution for the number of firms and rounding down to the nearest integer implies 𝑁 = 26.

9.33. A price-taking firm’s supply curve is s (P) = 10P. What is the producer surplus for this firm if the market price is $20? By how much does producer surplus change when the market price increases from $20 to $21?

The solution is shown in the figure below. The producer surplus at a price of $20 is equal to the area of triangle A, or (1/2)(20)(200) = $2,000. When the price increases to $21, producer surplus increases by area B ($200) plus area C ($5), or $205.

Price ($/unit) s(P) = 10P $21 B

C

$20 A

200

210

Quantity (units)

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Solutions Manual

9.34. The semiconductor market consists of 100 identical firms, each with a short-run marginal cost curve SMC(Q) = 4Q. The equilibrium price in the market is $200. Assuming that all of the firm’s fixed costs are sunk, what is the producer surplus of an individual firm and what is the overall producer surplus for the market?

Since an individual firm will supply where 𝑃 = 𝑆𝑀𝐶, 𝑃 = 4𝑄 1 𝑄= 𝑃 4 1

Assuming a firm will supply for any positive price this implies 𝑠(𝑃) = 4 𝑃. Graphically we have

Producer surplus for an individual firm is given by area A in the figure above which is 1 (200)50 = 5000. Since all firms are identical, overall producer surplus will be 100(5000) = 2 500,000.

9.35. Consider an industry in which chief executive officers (CEOs) run firms. There are two types of CEOs: exceptional and average. There is a fixed supply of 100 exceptional CEOs and an unlimited supply of average CEOs. Any individual capable of being a CEO in this industry is willing to work for a salary of $144,000 per year. The long-run total cost of a firm that hires an exceptional CEO at this salary is

where Q is annual output in thousands of units and total cost is expressed in thousands of dollars per year. The corresponding long-run marginal cost curve is MCE(Q) = Q, where marginal cost is expressed as dollars per unit. The long-run total cost for a firm that hires an average CEO for $144,000 per year is TCA(Q) = 144 + Q2. The corresponding marginal cost curve is MCA(Q) = 2Q. The market demand curve in this market is D(P) = 7200 − 100P, where P is the market price and D(P) is the market quantity, expressed in thousands of units per year.

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Chapter 9-29

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Solutions Manual

a) What is the minimum efficient scale for a firm run by an average CEO? What is the minimum level of long-run average cost for such a firm? b) What is the long-run equilibrium price in this industry, assuming that it consists of firms with both exceptional and average CEOs? c) At this price, how much output will a firm with an average CEO produce? How much output will a firm with an exceptional CEO produce? d) At this price, how much output will be demanded? e) Using your answers to parts (c) and (d), determine how many firms with average CEOs will be in this industry at a long-run equilibrium. f ) What is the economic rent attributable to an exceptional CEO? g) If firms with exceptional CEOs hire them at the reservation wage of $144,000 per year, how much economic profit do these firms make? h) Assuming that firms bid against each other for the services of exceptional CEOs, what would you expect their salaries to be in a long-run competitive equilibrium? a) Minimum efficient scale occurs at the point where average cost reaches a minimum. This point occurs where 𝑀𝐶 = 𝐴𝐶.

144 +Q Q Q = 12

2Q =

At 𝑄 = 12, 144 +𝑄 𝑄 𝐴𝐶 = 24 𝐴𝐶 =

b) In the long-run, the equilibrium price will be determined by the minimum level of average cost for firms with average CEOs, thus 𝑃 = 24. At this price, firms having average CEOs will earn zero economic profit and firms with exceptional CEOs will earn positive economic profit. c) At the price, the firms with an average CEO will produce where 𝑃 = 𝑀𝐶 24 = 2Q Q = 12

The firms with an exceptional CEO will also produce where 𝑃 = 𝑀𝐶 𝑄 = 24

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

d) At this price 𝐷(𝑃) = 7200 − 100𝑃 𝐷(𝑃) = 4800 e) Since there are 100 exceptional CEOs and assuming they are all employed, the total supply from exceptional CEO firms will be 𝑆𝐸 = 100(24) 𝑆𝐸 = 2400 This leaves 𝑄 = 4800 − 2400 = 2400 units to be supplied by firms with average CEOs. Thus, 2400 12 𝑁𝐴 = 200 𝑁𝐴 =

f) To calculate the exceptional CEO’s economic rent we must compute the highest salary the firm would pay this CEO. This salary is the amount that would drive economic profit to zero.

Call this amount 𝑆 ∗ . Since the exceptional CEO firm is producing 𝑄 = 24, the firm’s average cost is 144 1 + (24) 24 2 𝐴𝐶 = 18 𝐴𝐶 =

Since 𝑃 = 24, the exceptional CEO has produced a $6 per unit cost advantage. This implies

S * 144 − =6 24 24 S * = 288 Economic rent is the difference between this salary, $288,000, and the reservation wage of $144,000. Thus, the exceptional CEO’s economic rent is $144,000. g) Firms that hire exceptional CEOs for $144,000 will gain all of the CEO’s economic rent and will therefore earn economic profit of $144,000. h) In a long-run competitive equilibrium, exceptional CEO salaries should be bid up as other firms compete for the exceptional CEOs. This should bid up the salary of the CEOs until economic profits for firms with exceptional CEOs are driven to zero. Thus, exceptional CEO salaries should approach $288,000 in a long-run equilibrium.

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Chapter 9-31

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 10 Competitive Markets: Applications Solutions to Review Questions 1. What is the significance of the “invisible hand’’ in a competitive market?

In a long-run equilibrium, a competitive market allocates resources efficiently. As Adam Smith wrote over 200 years ago, it is as though there is an “Invisible Hand” guiding a competitive market to the efficient level of production and consumption. This occurs through producers and consumers acting in their own self-interest to maximize profits and utility.

2. What is the size of the deadweight loss in a competitive market with no government intervention?

In a competitive market with no government intervention there is no deadweight loss.

3. What is meant by the incidence of a tax? How is the incidence of an excise tax related to the elasticities of supply and demand in a market?

The incidence of the tax refers to who ultimately pays the tax. In general, when the government imposes a tax, consumers and producers share the tax burden to some degree. The incidence of the tax refers to how the tax is shared.

The incidence of the tax depends on the shapes of the supply and demand curves. If demand is relatively inelastic compared with supply, consumers will bear most of the burden of the tax, whereas if supply is relatively inelastic compared with demand, producers will bear most of the burden of the tax.

4. In the competitive market for hard liquor, the demand is relatively inelastic and the supply is relatively elastic. Will the incidence of an excise tax of $T be greater for consumers or producers?

Since demand is relatively inelastic compared with supply, most of the burden of the tax will rest with the consumers.

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Chapter 10-1

Besanko & Braeutigam – Microeconomics, 6th edition

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5. Gizmos are produced and sold in a competitive market. When there is no tax, the equilibrium price is $100 per gizmo. The own-price elasticity of demand for gizmos is known to be about –0.9 and the own-price elasticity of supply is about 1.2. In commenting on a proposed excise tax of $10 per gizmo, a newspaper article states that “the tax will probably drive the price of gizmos up by about $10.” Is this a reasonable conclusion?

No, this is not reasonable. The incidence of the tax can be summarized quantitatively as 𝛥𝑃𝑑 𝜀𝑄𝑠 ,𝑃 = 𝛥𝑃 𝑠 𝜀𝑄𝑑 ,𝑃 𝜀 𝑠

1.2

𝜀 𝑠

Using the figures given in this problem, 𝜀 𝑄 ,𝑃 = −0.9 so 𝜀 𝑄 ,𝑃 = −1.33. 𝑄𝑑 ,𝑃

𝑄𝑑 ,𝑃

This implies the price consumers pay will increase by about 1.33 times as much as the decrease in the price producers receive. Thus, if the tax is $10, consumers will bear $5.70 of the tax and producers will bear $4.30 of the tax. We would therefore expect the market price to rise by $5.70, not the $10 the newspaper suggested.

6. The cheese-making industry in Castoria is competitive, with an upward-sloping supply curve and a downward-sloping demand curve. The government gives cheese producers a subsidy of $T for each kilogram of cheese they make. Will consumer surplus increase? Will producer surplus increase? Will there be a deadweight loss?

If the government provides a subsidy, consumer surplus will increase and producer surplus will increase. The market will have a deadweight loss, however, because these increases will be outweighed by the impact on the government treasury.

7. Will a price ceiling always increase consumer surplus? Will a price floor always increase producer surplus?

Price ceilings and price floors will not always make consumers and producers better off. In particular, if the price ceiling is set above the equilibrium price or if the price floor is set below the equilibrium price, the price ceilings and floors will have no effect. In addition, depending on which consumers or producers are able to purchase in or supply to the market, consumer surplus or producer surplus may be lower after the imposition of the price ceiling or price floor.

8. Will a production quota in a competitive market always increase producer surplus?

Producer surplus may increase. If the most efficient producers serve the market, producer surplus will increase for some levels of the quota. However, if the quota is too low (for example, close to zero), producer surplus could actually decrease.

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Chapter 10-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

9. Why are agricultural price support programs, such as acreage limitation and government purchase programs, often very costly to implement?

One outcome of these programs is to increase farmers’ producer surplus. As the text states, a government must often spend much more than one dollar to increase farmers’ surplus by one dollar. Since these programs tend to be more politically palatable than direct cash payments to farmers, however, governments tend to implement these types of programs.

10. If an import tariff and an import quota lead to the same price in a competitive market, which one will lead to a larger domestic deadweight loss?

Because the government collects no revenues from a quota, deadweight loss will be greater with the quota by the amount of the revenues collected from a tariff.

11. Why does a market clear when the government imposes an excise tax of $T per unit?

When the government imposes an excise tax, the price consumers pay increases, reducing the quantity demanded, and the price suppliers receive falls, decreasing the quantity supplied. These amounts fall to the point where the number of units supplied equals the number of units demanded by consumers, clearing the market.

12. Why does a market clear when the government gives producers a subsidy of $S per unit?

When the government provides a subsidy to suppliers, the price suppliers receive increases, increasing the quantity supplied, and the price consumers pay falls, increasing the quantity demanded. These amounts increase to the point where the number of units supplied equals the number of units demanded, clearing the market.

13. Why does the market not clear with a production quota?

With a production quota the market quantity is held arbitrarily below the equilibrium quantity raising the price above the equilibrium price. At this higher price, suppliers would like to supply more than the quota will allow, creating an excess supply. Because of the excess supply the market does not clear.

14. With a price floor, will the most efficient producers necessarily be the ones supplying the market?

No, it is not clear which producers will supply the market. Because the price is held above the equilibrium price there are many producers who will compete to supply the market, even more than would exist in equilibrium. There is no reason to believe that the most efficient producers will necessarily be the ones supplying the market.

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Chapter 10-3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 10.1. In a competitive market with no government intervention, the equilibrium price is $10 and the equilibrium quantity is 10,000 units. Explain whether the market will clear under each of the following forms of government intervention: a) The government imposes an excise tax of $1 per unit. b) The government pays a subsidy of $5 per unit produced. c) The government sets a price floor of $12. d) The government sets a price ceiling of $8. e) The government sets a production quota, allowing only 5000 units to be produced. a) The market will clear. The excise tax will alter equilibrium price and quantity, but there will be no excess demand or excess supply. b) The market will clear. The subsidy will alter equilibrium price and quantity, but there will be no excess demand or excess supply. c) The market will not clear. A price floor set above the equilibrium price will create excess supply. d) The market will not clear. A price ceiling set below the equilibrium price will create excess demand. e) The market will not clear. A quota limiting output below the equilibrium level will create excess supply since the price will be driven above the equilibrium price.

10.2. In Learning-By-Doing Exercise 10.1 we examined the effects of an excise tax of $6 per unit. Repeat that exercise for an excise tax of $3.

With a $3 tax, setting 𝑄 𝑑 = 𝑄 𝑠 implies

10 − .5( P s + 3) = −2 + P s Ps = 7

.

Substituting into the equation for 𝑃𝑑 implies 𝑃𝑑 = 10. Substituting this price into the equation for quantity demanded implies 𝑄 = 5million. At these prices and quantities, consumer surplus is $25 million, producer surplus is $12.5 million, and government tax receipts are $15 million. The deadweight loss is $1.5 million. The deadweight loss measures the difference between potential net benefits ($54 million) and the net benefits that are actually achieved ($25 + $12.5 + $15 = $52.5 million).

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Chapter 10-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.3. Gadgets are produced and sold in a competitive market. When there is no tax, the equilibrium price is $20 per gadget. The own-price elasticity of demand for gadgets is −0.5.

If an excise tax of $4 leads to an increase in the price of gadgets to $24, what must be true about the own-price elasticity of supply for gadgets? 𝛥𝑃 𝑑

𝜀 𝑠

The incidence of a tax can be summarized quantitatively by 𝛥𝑃𝑠 = 𝜀 𝑄 ,𝑃 . 𝑄𝑑 ,𝑃

From the given information, 𝛥𝑃𝑑 = 4, 𝛥𝑃 𝑠 = 0, and 𝜀𝑄𝑑 ,𝑃 = −0.5. These price changes imply that 100% of the burden of the tax is borne by the consumer, implying the elasticity of supply must be infinite. Supply is perfectly elastic.

10.4. When gasoline prices recently reached a price of $2.00 per gallon, public policy makers considered cutting excise taxes by $0.10 per gallon to lower prices for the consumer. In discussing the effects of the proposed tax reduction, a news commentator stated that the effect of tax reduction should lead to a price of about $1.90 per gallon, and, that if the price did not drop by as much, it would be evidence that oil companies are somehow conspiring to keep gasoline prices high. Evaluate this claim.

The incidence of a tax depends on the relative price elasticities of supply and demand. For example, if the elasticity of demand is –0.5 and the elasticity of supply is +0.5, one would expect the incidence of a tax reduction of $0.10 to be equally split between producers and consumers, leading to a reduction of about $0.05 per gallon at the pump. The fact that the price does not fall by $0.10 is expected in a perfectly competitive market, and does not, by itself, provide evidence that producers are “conspiring to keep prices high.”

10.5. Consider the market for crude oil. Suppose the demand curve is described by Qd = 100 – P, where Qd is the quantity buyers will purchase when the price they pay is P (measured in dollars per barrel). The equation representing the supply curve is QS= P/3, where QS is the quantity that producers will supply when the price they receive is P. The market for crude oil is initially in equilibrium, with no tax and no subsidy. Because it regards the price of oil as too high, the government wishes to help buyers by announcing that it will give producers a subsidy of 4 dollars per barrel. A local television station reporter announces that the subsidy should lower the price consumers pay by 4 dollars per barrel. Analyze the reporter’s claim by determining the price buyers pay before and after the subsidy, and provide intuition to explain why the reporter is correct or incorrect.

Before the subsidy, the price buyers pay is the same as the price producers receive; call this price P. The equilibrium can be found by setting supply equal to demand: P/3 =100 – P. Thus, in equilibrium buyers pay P = $75 / barrel.

In an equilibrium with the subsidy, the price producers receive (Ps) will be $4 per barrel more than price buyers pay (Pd); thus PS=Pd+4. The market will clear when the quantity bought at a price Pd (i.e., 100 – Pd) equals the quantity purchased at a price PS = (i.e., PS/3). Thus, 100 – Pd =

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Chapter 10-5

Besanko & Braeutigam – Microeconomics, 6th edition

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PS/3, or 100 – Pd = (Pd +4)/3. In equilibrium the price paid by buyers is $74, and the price received by producers is $78. So the reporter’s claim is false; the subsidy reduces the price buyers pay by only $1 per barrel (from $75 to $74 per barrel). The reason that the price does not fall by $4 per barrel is that neither the demand nor the supply is totally inelastic. Thus the incidence of the subsidy (akin to the incidence of a tax) is shared by buyers and sellers. The price buyers pay falls by $1, while the price producers receive rises by $3.

10.6. The table in Application 10.1 indicates that revenues from gasoline taxes will increase by about $9.5 billion (from $61.5 billion to about $71 billion per year) if the gasoline tax is raised from $0.43 to $0.50 per gallon. Using the supply and demand curves in Application 10.1, show that the equilibrium quantity, price consumers pay, price producers receive, and tax receipts are as indicated in the table when the tax is $0.50 per gallon. Draw a graph illustrating the equilibrium when the tax is $0.50 per gallon.

Using the supply and demand curves from Example 10.1 and an excise tax of $0.50 implies 214.5 – 28.26(PS + 0.5) = 85.8 + 27.24PS PS = $2.06 Substituting into the equation Pd = PS + 0.5 implies. Pd = $2.56. Substituting 𝑃 𝑠 into the supply equation implies Q =85.8 + 27.24(2.06) = 142 billion gallons. Finally, the government tax receipts will be tQ = 0.5(142) = $71 billion. These values correspond with those in Table 10.1.

Graphically, the solution is as follows:

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Chapter 10-6

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.7. In a competitive market, there is currently no tax, and the equilibrium price is $40.

The market has an upward-sloping supply curve. The government is about to impose an excise tax of $5 per unit. In the new equilibrium with the tax, what price will producers receive and consumers pay if the demand curve is a) Perfectly elastic b) Perfectly inelastic Illustrate your answers graphically. a) With perfectly elastic demand, a $5 excise tax shifts the supply curve up as pictured below.

Producers will bear the entire burden of the tax. The end result is that the equilibrium quantity falls from Q1 to Q2, the price consumers pay remains the same ($40), and the price producers receive falls from $40 to $35.

P

S+ $5

S D

$40 $35

Q2

Q1

Q

b) When demand is perfectly inelastic, consumers will bear the entire burden of the tax. The $5 tax has no effect on the equilibrium quantity. However, the price consumers pay rises from $40 to $45. The price producers receive remains constant at $40.

P $45

D

S+ $5 S

$40

Q1

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Q

Chapter 10-7

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.8. In a competitive market, there is currently no tax, and the equilibrium price is $60.

The market has a downward-sloping demand curve. The government is about to impose an excise tax of $4 per unit. In the new equilibrium with the tax, what price will producers receive and consumers pay if the supply curve is a) Perfectly elastic b) Perfectly inelastic Illustrate your answers graphically. a) With perfectly elastic supply, a $4 excise tax shifts the supply curve up as pictured below.

Consumers will bear the entire burden of the tax. The end result is that the equilibrium quantity falls from Q1 to Q2, the price producers receive remains the same ($60), and the price consumers pay rises from $60 to $64.

P $64

S + $4 S

$60

D Q2

Q1

Q

b) When supply is perfectly inelastic, producers will bear the entire burden of the tax. Since supply is inelastic, it is easiest to think about this problem graphically as shifting down the demand curve. [Convince yourself you’d get the same result had we done this in part (a).] The $4 tax has no effect on the equilibrium quantity. However, the price sellers receive falls from $60 to $56. The price consumers pay remains constant at $60.

P

S

$60 $56

D D – $4 Q1

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Q

Chapter 10-8

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.9. The current equilibrium price in a competitive market is $100. The price elasticity of demand is −4 and the price elasticity of supply is +2. If an excise tax of $3 per unit is imposed, how much would you expect the equilibrium price paid by consumers to change?

How much would you expect the equilibrium price received by producers to change? 𝛥𝑃 𝑑

𝜀 𝑠

𝛥𝑃 𝑑

1

Using the formula, 𝛥𝑃𝑠 = 𝜀 𝑄 ,𝑃 , we can see that 𝛥𝑃𝑠 = − 2 (the negative sign is because the two 𝑄𝑑 ,𝑃

elasticities have opposite signs). Therefore, 𝛥𝑃 𝑠 = −2𝛥𝑃𝑑 . Also, after the tax, the price paid by consumers must rise, so the change in the price paid by them is positive. Similarly, the change in the price received by producers is negative. Finally, the sum of the magnitude of these two changes must equal the tax collected by the government, so that 𝛥𝑃𝑑 − 𝛥𝑃 𝑠 = 3. Combining the two conditions we see that 𝛥𝑃𝑑 = 1 and 𝛥𝑃 𝑠 = 2. That is, the price paid by consumers increases by $1, and that received by suppliers decreases by $2. Producers bear most of the tax, which is to be expected given that supply is less elastic than demand.

10.10. Suppose that the market for cigarettes in a particular town has the following supply and demand curves: QS = P; QD = 50 − P, where the quantities are measured in thousands of units. Suppose that the town council needs to raise $300,000 in revenue and decides to do this by taxing the cigarette market. What should the excise tax be in order to raise the required amount of money?

Suppose that the required tax is $T. Then in equilibrium, 𝑃𝐷 = 𝑃 𝑆 + 𝑇. This implies that 50 − 𝑄 = 𝑄 + 𝑇, or Q = 25 – 0.5T. Since the required amount is $300,000, we must have T*Q = 600.

(Remember that Q is measured in thousands of units). So, T(25 – 0.5T) = 600. Solving this equation we get two possible values for the tax: T = $20 or T = $30. Either one would generate $300,000 in tax revenues, though of course T = $20 would do so with a smaller deadweight loss.

10.11. Assume that a competitive market has an upward-sloping supply curve and a downward-sloping demand curve, both of which are linear. A tax of size $T is currently imposed in the market. Suppose the tax is doubled. By what multiple will the deadweight loss increase? (You may assume that at the new tax, the equilibrium quantity is positive.)

Since the demand and supply curves are assumed to be linear, assume they take the form

Q d = a − bP d Q s = e + fP s and assume that initially the tax imposed on the market is 𝑇.

In equilibrium without the tax the solution is

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a − bP = e + fP P=

a−e b+ f

Substituting into either the supply equation implies 𝑄 = 𝑒+𝑓(

𝑎−𝑒 ) 𝑏+𝑓

When the tax is imposed, the new equilibrium solution becomes a − b( P s + T ) = e + fP s Ps =

a − bT − e b+ f

Substituting this result into the equation for 𝑃𝑑 implies 𝑃𝑑 =

𝑎 − 𝑒 + 𝑓𝑇 𝑏+𝑓

Finally, substituting the price into the supply equation implies 𝑎 − 𝑏𝑇 − 𝑒 ) 𝑄𝑇 = 𝑒 + 𝑓 ( 𝑏+𝑓 where 𝑄𝑇 represents the equilibrium quantity when the tax is 𝑇. Given these equilibrium prices and quantities, the deadweight loss with this tax is 1 𝑎−𝑒 𝑎 − 𝑏𝑇 − 𝑒 )) − (𝑒 + 𝑓 ( ))] 𝐷𝑊𝑇 = 𝑇 [(𝑒 + 𝑓 ( 2 𝑏+𝑓 𝑏+𝑓 1 𝑓𝑏𝑇 ) 𝐷𝑊𝑇 = 𝑇 ( 2 𝑏+𝑓 Now, to determine the deadweight loss when the tax doubles, repeat this exercise for a tax of 2𝑇.

Completing the same steps implies a − 2bT − e b+ f a − e + 2 fT Pd = b+ f Ps =

 a − 2bT − e  Q2T = e + f    b+ f 

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Solving for the deadweight loss as before implies 𝐷𝑊2𝑇 =

1 2𝑓𝑏𝑇 ) (2𝑇) ( 2 𝑏+𝑓

Finally, taking the ratio of 𝐷𝑊2𝑇 to 𝐷𝑊𝑇 gives  2 fbT  1 (2T )   DW2T 2  b+ f  = DWT 1  fbT  T 2  b + f  DW2T =4 DWT

Therefore, doubling the tax will quadruple the size of the deadweight loss if the supply and demand curves are linear.

10.12. Refer to the accompanying diagram depicting a competitive market. If the government imposes a price ceiling of P1, using the areas in the graph below, identify a) The most that consumers can gain from such a move. b) The most that consumers can lose from such a move.

In other words, provide a maximum and a minimum limit to the possible change in consumer surplus from the imposition of this price ceiling.

If the government does not impose a price ceiling then the market will clear and the consumer surplus will be area A + C. The highest that consumer surplus can be after the price ceiling is imposed will be when the consumers with the greatest willingness to pay are able to buy the

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good. With the price ceiling, the most that producers will supply is 100 units. If these 100 units go the consumers with the greatest willingness to pay then consumer surplus will be area A + B. However, if these 100 units go to the consumers with the least willingness to pay (but who are still willing to buy the good at this price) then the new consumer surplus will be area F. Therefore, the most that CS can increase by is (A + B) – (A + C) = B – C. The most that CS can decrease by is F – (A + C).

10.13. In a perfectly competitive market, the market demand curve is given by Qd = 200 − 5Pd, and the market supply curve is given by Qs = 35Ps. a) Find the equilibrium market price and quantity demanded and supplied in the absence of price controls. b) Suppose a price ceiling of $2 per unit is imposed. What is the quantity supplied with a price ceiling of this magnitude? What is the size of the shortage created by the price ceiling? c) Find the consumer surplus and producer surplus in the absence of a price ceiling. What is the net economic benefit in the absence of the price ceiling? d) Find the consumer surplus and producer surplus under the price ceiling. Assume that rationing of the scarce good is as efficient as possible. What is the net economic benefit in this case? Does the price ceiling result in a deadweight loss? If so, how much is it? e) Find the consumer surplus and producer surplus under the price ceiling, assuming that the rationing of the scarce good is as inefficient as possible. What is the net economic benefit in this case? Does the price ceiling result in a deadweight loss? If so, how much is it? a) Pd = Ps = $5; Qd = Qs = 175 units. b) Qs= 70 units. Qd = 190 units. The shortage is therefore 120 units.

The surplus implications of a price ceiling are shown below.

With No Price Ceiling Consumer surplus Producer surplus Net benefits (consumer surplus + producer surplus) Deadweight loss

A+B+C+I ($3,062.50) G+F+E+H ($437.50) A+B+C+I+G+F+E+H ($3,500)

With Price Ceiling: Efficient Rationing A+B+F ($2,170) G ($70) A+B+F+G ($2,240)

Impact of Price Ceiling F–C–I (-$892.5) -F-E-H (-367.50) -C-I-E-H (-$1,260)

Zero

C+E+I+H ($1,260)

C+E+I+H ($1,260)

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Besanko & Braeutigam – Microeconomics, 6th edition

With No Price Ceiling Consumer surplus Producer surplus Net benefits (consumer surplus + producer surplus) Deadweight loss

A+B+C+I ($3,062.50) G+F+E+H ($437.50) A+B+C+I+G+F+E+H ($3,500) Zero

Solutions Manual

With Price Ceiling: Inefficient Rationing I+H+K+L ($490) G ($70) I+H+K+L+G ($560)

Impact of Price Ceiling H+K+L-A-B-C(-$2572.5) -F-E-H (-367.50) K+L-A-B-C-F-E (-$2,940)

A+B+C+F+E -K-L ($2,940)

A+B+C+F+E -K-L ($2,940)

For the next three questions, use the following information. The market for gizmos is competitive, with an upward sloping supply curve and a downward sloping demand curve. With no government intervention, the equilibrium price would be $25 and the equilibrium quantity would be 10,000 gizmos. Consider the following programs of government intervention: Program I: The government imposes an excise tax of $2 per gizmo Program II: The government provides a subsidy of $2 per gizmo for gizmo producers.

Program III: The government imposes a price floor of $30.

Program IV: The government imposes a price ceiling of $20.

Program V: The government allows no more than 8,000 gizmos to be produced.

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10.14. Which of these programs would lead to a less than 10,000 units exchanged in the market? Briefly explain.

Program I: The excise tax will increase the price consumers pay to a level above $25, and lower the price producers receive to a level below $25; thus, the quantity exchanged in the market will fall below 10,000 units.

Program II: With the subsidy, the price producers receive will increase to a level above $25; the price consumers receive will fall below $25. Thus, the equilibrium quantity exchanged will rise to a level above 10,000.

Program III. With the price floor of $30, consumers will buy less than 10,000 gizmos, so fewer than 10,000 will be exchanged in the market.

Program IV. With the price ceiling of $20, producers will supply less than 10,000 gizmos, so fewer than 10,000 will be exchanged in the market.

Program V. By government decree fewer than 10,000 gizmos will be exchanged.

10.15. Under which of these programs will the market clear? Briefly explain.

With the excise tax or the subsidy, the market will clear (Programs I and II).

With the price floor (Program III) there will be excess supply, so the market will not clear.

With the price ceiling (Program IV) there will be excess demand, so the market will not clear.

With the production quota (Program V) the price consumers pay will exceed $25, so there will be excess supply. The market will not clear.

10.16. Which of these programs would surely lead to an increase in consumer surplus?

Briefly explain.

With the excise tax (Program I) the price consumers pay will rise, so consumer surplus will surely fall.

With the subsidy (Program II) the price consumers pay will fall, so consumer surplus will surely rise.

With the price floor (Program III) the price consumers pay will rise, so consumer surplus will surely fall.

With the price ceiling (Program III) the price consumers pay will fall, but so will the quantity produced. Consumer surplus may fall. This can occur if the price floor is so low that very few units are produced (and thus available for purchase by consumers).

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With the production quota (Program V) the price consumers pay will rise, so consumer surplus will surely fall.

10.17. Suppose the market for corn in Pulmonia is competitive. No imports and exports are possible. The demand curve is Qd = 10 − Pd, where, Qd is the quantity demanded (in millions of bushels) when the price consumers pay is Pd. The supply curve is

where Qs is the quantity supplied (in millions of bushels) when the price producers receive is Ps. a) What are the equilibrium price and quantity? b) At the equilibrium in part (a), what is consumer surplus? producer surplus? deadweight loss? Show all of these graphically. c) Suppose the government imposes an excise tax of $2 per unit to raise government revenues. What will the new equilibrium quantity be? What price will buyers pay? What price will sellers receive? d) At the equilibrium in part (c), what is consumer surplus? producer surplus? the impact on the government budget (here a positive number, the government tax receipts)? deadweight loss? Show all of these graphically. e) Suppose the government has a change of heart about the importance of corn revenues to the happiness of the Pulmonian farmers. The tax is removed, and a subsidy of $1 per unit is granted to corn producers. What will the equilibrium quantity be? What price will the buyer pay? What amount (including the subsidy) will corn farmers receive? f) At the equilibrium in part (e), what is consumer surplus? producer surplus? What will be the total cost to the government? deadweight loss? Show all of these graphically. g) Verify that for your answers to parts (b), (d), and (f) the following sum is always the same: consumer surplus + producer surplus + budgetary impact + deadweight loss. Why is the sum equal in all three cases? a) Setting 𝑄 𝑑 = 𝑄 𝑠 results in 10 − P = −4 + P P = $7 per bushel

Substituting this result into the demand equation gives 𝑄 = 3million bushels.

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1

1

b) At the equilibrium, consumer surplus is 2 (10 − 7)3 = 4.5 and producer surplus is 2 (7 − 4)3 = 4.5. There is no deadweight loss in this case and total net benefits equal $9 million.

In the graph above, area A represents consumer surplus and area B represents producer surplus. c) If the government imposes an excise tax of $2, the new equilibrium will be

10 − ( P s + 2) = −4 + P s P = $6 per bushel Substituting back into the equation for 𝑃𝑑 yields 𝑃𝑑 = 8, and substituting 𝑃 𝑠 into the supply equation implies 𝑄 = 2 million.

1

1

d) Now the consumer surplus is 2 (10 − 8)2 = 2, the producer surplus is 2 (6 − 4)2 = 2, the tax 1

receipts are 2(2) = 4, and the deadweight loss is 2 (8 − 6)(3 − 2) = 1 (all measured in millions of dollars).

In the graph above, area A represents consumer surplus, area B represents producer surplus, areas C+D represent government tax receipts, and area E represents the deadweight loss.

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e) If the government provides a subsidy of $1, the new equilibrium will be

10 − ( P s − 1) = −4 + P s P s = $7.5 per bushel Substituting back into the equation for 𝑃𝑑 yields 𝑃𝑑 = 6.5, and substituting 𝑃 𝑠 into the supply equation implies 𝑄 = 3.5 million.

1

1

f) Now the consumer surplus is 2 (10 − 6.5)3.5 = 6.125, the producer surplus is 2 (7.5 − 4)3.5 = 6.125, the subsidy paid is −1(3.5) = −3.5 (negative since the government is paying 1 this amount), and the deadweight loss is 2 (7.5 − 6.5)(3.5 − 3) = 0.25 (all measured in millions of dollars).

In the graph above, areas A+B+E represent consumer surplus, areas B+C+F represent producer surplus, areas B+C+D+E represent the government subsidy payment, and area D represents the deadweight loss. g) For part (b), the sum of consumer surplus, producer surplus, budgetary impact, and deadweight loss is 4.5 + 4.5 + 0 + 0 = 9; for part (d), the sum is 2 + 2 + 4 + 1 = 9; and for part (f) it is 6.125 + 6.125 – 3.5 + 0.25 = 9. (As above, all are measured in millions of dollars.)

These sums are all the same because the deadweight loss measures the difference between net benefits (in terms of CS, PS, and budgetary impact) under the competitive outcome and net benefits under a form of government intervention.

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10.18. In a perfectly competitive market, the market demand and market supply curves are given by Qd = 1000 −10Pd and Qd = 30Ps. Suppose the government provides a subsidy of $20 per unit to all sellers in the market. a) Find the equilibrium quantity demanded and supplied; find the equilibrium market price paid by buyers; find the equilibrium after-subsidy price received by firms. b) Find the consumer surplus and producer surplus in the absence of the subsidy. What is the net economic benefit in the absence of a subsidy? c) Find the consumer surplus and producer surplus in the presence of the subsidy. What is the impact of the subsidy on the government budget? What is the net economic benefit under the subsidy program? d) Does the subsidy result in a deadweight loss? If so, how much is it?

In this case, the after-subsidy price received by sellers is Ps = Pd + 20. The market-clearing condition is: 1000 – 10P = 30(P + 20), where P denotes the market price. This implies P = 10 and Q = 900. Since sellers receive the subsidy, P = Pd = 10 and Ps = Pd + 20 = 30. The surplus implications of the subsidy are shown below:

Consumer surplus Producer surplus Government spending on subsidy Net benefits (consumer surplus + producer surplus – government spending) Deadweight loss

With No Subsidy

With Subsidy

A+B ($28,125) C+E ($9,375) Zero A+B+C+E ($37,500)

A+B+C+F+G ($40,500) C+E+B+I ($13,500) B+C+F+G+H+I ($18,000) A+B+C+E-H ($36,000)

Impact of the Subsidy C+F+G ($12,375) B+I ($4,125) -B-C-F-G-H-I (-$18,000) -H (-$1,500)

Zero

H ($1,500)

H ($1,500)

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10.19. In a perfectly competitive market, the market demand curve is Qd = 10 − Pd, and the market supply curve is Qs = 1.5Ps. a) Verify that the market equilibrium price and quantity in the absence of government intervention are Pd = Ps = 4 and Qd = Qs = 6. b) Consider two possible government interventions: (1) A price ceiling of $1 per unit; (2) a subsidy of $5 per unit paid to producers. Verify that the equilibrium market price paid by consumers under the subsidy equals $1, the same as the price ceiling. Are the quantities supplied and demanded the same under each government intervention? c) How will consumer surplus differ in these different government interventions? d) For which form of intervention will we expect the product to be purchased by consumers with the highest willingness to pay? e) Which government intervention results in the lower deadweight loss and why? a) 10 – P = 1.5P  P = 4 and Q = 10 – 4 = 6. b) Under a $5 subsidy paid to producer, market price P = Pd and the after-subsidy price received by producers is Ps = Pd+5. Thus: 10 – P = 1.5(P + 5)  P = 1. The quantities demanded each intervention are the same, and consumers pay the same price. However, the quantity supplied under the subsidy will be higher than it is under the price ceiling. c) Consumer surplus under the subsidy will be greater than the consumer surplus under a price ceiling. Under both interventions, consumers pay the same price, but under subsidies consumers are supplied as much as they demand at the $1 market price, while under price ceilings, consumers get less than they demand at the $1 ceiling price. d) Subsidies. Under subsidies, because consumers get what they demand at the market price, there is no possibility of consumers with a lower willingness to pay getting the good while consumers with a higher willingness to pay do not get the good. This is a possibility with a price ceiling. e) The subsidy has the smaller deadweight loss. The deadweight loss under the price ceiling (assuming efficient rationing) is area C+H+I, which equals 16.875. The deadweight loss under the subsidy is area L, which equals 7.5.

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10.20. Consider a perfectly competitive market in which the market demand curve is given

by Qd = 20 − 2Pd and the market supply curve is given by Qs = 2Ps. a) Find the equilibrium price and quantity in the absence of government intervention. b) Suppose the government imposes a price ceiling of $3 per unit. How much is supplied? c) Suppose, as an alternative, the government imposes a production quota limiting the quantity supplied to 6 units. What is the market price under this type of intervention? Is the quantity supplied under the price ceiling greater than, less than, or the same as the quantity under the production quota? d) Assuming that under price controls rationing is as efficient as possible and under the quota, the allocation is as efficient as possible, under which program is the deadweight loss larger: the price ceiling or the production quota? e) Assuming that under price controls rationing is as inefficient as possible, while under the quota the allocation is as efficient as possible, under which program is the deadweight loss larger: the price ceiling or the production quota? f) Assuming that under price controls rationing is as inefficient as possible, while under the quota the allocation is as inefficient as possible, under which program is the deadweight loss larger: the price ceiling or the production quota? a) Letting P = Pd = Ps denote the market price in the absence of government intervention, we have: 20 – 2P = 2P  P = 5. The equilibrium quantity is these 10 units. b) The quantity supplied under a price ceiling of $3 per unit is 6 units, as shown in the first picture below. c) The market-clearing price when a production quota of 6 is imposed is given by 6 = 20 – 2P or P = 7.

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d) Referring to the graph below, the deadweight loss under a 6 unit production quota (assuming efficient allocation of quotas) and the deadweight loss under a $3 per unit price ceiling (assuming efficient rationing) are the same and equal area C + F. e) The deadweight loss under a $3 price ceiling with inefficient rationing is equal to areas B+C+E+F, which works to be $32. This is necessarily bigger than the deadweight loss under the production quota because inefficient rationing entails an additional deadweight loss (area B + E) that is not present with a price ceiling with efficient rationing. f) In this case, the deadweight loss under a production quota in which allocation is as inefficient as possible is the same as the deadweight under a price ceiling in which the rationing is as inefficient as possible. Here’s why. In the second picture below, producer surplus when the quota is rationed to the highest cost producers willing to produce at the market-clearing price with a 6unit quota is given by the shaded triangle. The area of this triangle is equal to the area of triangle G. Consumer surplus under the quota is area A, for a total surplus of A+G (this is more easily seen in the first picture below). In the absence of a quota, total surplus is A+B+C+E+F+G. The deadweight loss is thus B+C+E+F, the same as it would be with a price ceiling of $3 and the most inefficient rationing possible.

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10.21. Figure 10.18 below shows the supply and demand curves for cigarettes. The equilibrium price in the market is $2 per pack if the government does not intervene, and the quantity exchanged in the market is 1,000 million packs. Suppose the government has decided to discourage smoking and is considering two possible policies that would reduce the quantity sold to 600 million packs. The two policies are (i) a tax on cigarettes and (ii) a law setting a minimum price for cigarettes. Analyze each of the policies, using the graph and filling in the table on the next page.

a) Based on the graph, the government would need to set a tax of $2.00 per unit to achieve the government’s target of 600 million units sold. By setting a tax at $2.00, the supply curve will shift upward by $2.00 and intersect the demand curve at 𝑃 = $3.00 and 𝑄 = 600, the new market equilibrium. Alternatively, the government could set a minimum price (price floor) at P = $3.00, at which point consumers would only demand Q = 600 million units.

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b) What price per unit would consumers pay?

What price per unit would producers receive?

What area represents consumer surplus?

What area represents the largest producer surplus under the policy?

What area represents the smallest producer surplus under the policy?

What area represents government receipts?

What area represents smallest deadweight loss possible under the policy?

Tax $3.00 $1.00 F B

Minimum Price $3.00 $3.00 F B+C+E

B

G+H+L+T

C+E G+L

Zero G+L

10.22. Consider a market with an upward sloping supply curve and a downward sloping demand curve. Under a government purchase program, which of the following statements are true, and which are false? (a) The increase in producer surplus will exceed the size of the government expenditure. (b) Consumer surplus will increase. (c) The size of the government expenditure will exceed the size of the deadweight loss.

The areas referred to in the answer below are those in Figure 10.14 in the text (a) False. The figure illustrates how the increase in producer surplus ($14 billion in the example) can be less than the size of the government expenditure ($30 billion). (b) False. Consumer surplus will decrease because the price is higher with the support. (In the figure, consumer surplus decreases by $11 billion). (c) True. The size of the government expenditure will exceed the size of the deadweight loss by area G in the figure.

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10.23. The market demand for sorghum is given by Qd = 500 − 10Pd, while the market supply curve is given by Qs = 40Ps. The demand and supply curve are shown below. The government would like to increase the income of farmers and is considering two alternative government interventions: an acreage limitation program and a government purchase program.

a) What is the equilibrium market price in the absence of government intervention? b) The government’s goal is to increase the price of sorghum to $15 per unit. This is the support price. How much would be demanded at a price of $15 unit? How much would farmers want to supply at a price of $15 per unit? How much would the government need to pay farmers in order for them to voluntarily restrict their output of sorghum to the level demanded at $15 per unit? c) Fill in the following table for the acreage limitation program:

d) As an alternative way to support a price of $15, suppose the government purchases the difference between the quantity demanded at a price of $15 and the quantity supplied. How much does the government spend on this price support program? e) Fill in the following table for the government purchases program:

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Chapter 10-24

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Solutions Manual

a) Let P = Pd = Ps denote the market equilibrium price in the absence of government intervention. The market equilibrium price is found by solving 500 – 10P = 40P, which gives us P = $10. The equilibrium quantity is 500 – 10(10) = 400 units.

If the government designates a support price of $15, the quantity demanded would be 500 – 10(15) = 350 units, while the quantity supplied would be 40(15) = 600 units. c) (See figure below)

Consumer surplus Producer surplus Impact on the government budget Net benefits (consumer surplus + producer surplus-government expenditure Deadweight loss

With no program A+B+C ($8,000) G+F ($2,000) Zero A+B+C+G+F ($10,000)

$0

With acreage limitation program A ($6,125) G+F+B+C+E ($4,500) -F-C-E (-$781.25) A+B+G ($9843.75)

Impact of program -B–C (-$1,875) B+C+E ($2,500) -F-G-E (-$781.25) -C-F ($156.25)

C+F ($156.25)

C+F ($156.25)

d) As shown in the figure below, the quantity supplied at a price of $15 per unit is 4(15) = 600 units, while the quantity demanded is 500 – 10(15) = 350 is the quantity supplied. Thus the government will support the price of $15 by purchasing 250 units at a price of $15 or $3,750.

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e) The entries in the table refer to the figure below:

Consumer surplus Producer surplus Impact on the government budget Net benefits (consumer surplus + producer surplusgovernment expenditure Deadweight loss

Copyright © 2020 John Wiley & Sons, Inc.

With no program A+B+C ($8,000) G+F ($2,000) zero A+B+C+G+F ($10,000) zero

With government purchase program A ($6,125) G+F+B+C+E ($4,500) -F- C- E- J- I- H (-$3,750) A+B+G –J – I – H ($6,875)

Impact of program -B-C (-$1,875) B+C+E ($2,500) - F-C-E- J- I- H (-$3,750) - F- C- J- I- H (-$3,125)

F+C+J+I+H ($3,125)

F+C+J+I+H ($3,125)

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.24. Suppose that in the domestic market for computer chips the demand is Pd = 110 − Qd, where Qd is the number of units of chips demanded domestically when the price is Pd. The domestic supply is Ps = 10 + Qs, where Qs is the number of units of chips supplied domestically when domestic suppliers receive a price Ps. Foreign suppliers would be willing to supply any number of chips at a price of $30. The government is contemplating three possible policies: Policy I: The government decides to ban imports of chips.

Policy II: Foreign suppliers are allowed to import chips (with no tariff ).

Policy III: The government allows imports, but imposes a tariff of $10 per unit.

Fill in the table below, giving numerical answers.

Chips consumed domestically?

Chips produced domestically?

What is the size of the producer surplus?

What is the size of the consumer surplus?

What is the size of government receipts?

Policy I 50 50 1250 1250 0

Policy II 80 20 200 3200 0

Policy III 70 30 450 2450 400

10.25. The domestic demand curve for portable radios is given by Qd = 5000 − 100P, where Qd is the number of radios that would be purchased when the price is P. The domestic supply curve for radios is given by Qs = 150P, where Qs is the quantity of radios that would be produced domestically if the price were P. Suppose radios can be obtained in the world market at a price of $10 per radio. Domestic radio producers have successfully lobbied Congress to impose a tariff of $5 per radio. a) Draw a graph illustrating the free trade equilibrium (with no tariff). Clearly illustrate the equilibrium price. b) By how much would the tariff increase producer surplus for domestic radio suppliers? c) How much would the government collect in tariff revenues? d) What is the deadweight loss from the tariff?

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Solutions Manual

a)

In the free trade equilibrium, domestic demand will be 4000, domestic supply will be 1500, and imports will be 2500 units.

1

b) The producer surplus with free trade would be 2 (10 − 0)(1500) = 7,500. With the tariff, 1

domestic supply will increase to 2250 and producer surplus will increase to 2 (15 − 0)(2250) = 16,875. So producer surplus will increase by 9,375. c) With the tariff, domestic demand will fall to 3500 units and domestic demand will increase to 2250 units. Thus, 1250 units will be imported. A tariff of $5 on each of those units will result in government receipts of 6,250. d) The deadweight loss from the tariff will come from two sources. First, the deadweight loss 1 associated the overproduction of domestic suppliers will be 2 (2250 − 1500)5 = 1,875. Second, the deadweight loss associated with the reduction in consumption by consumers due to the tariff 1 is 2 (4000 − 3500)5 = 1,250. Therefore, the total deadweight loss with this tariff is 3,125.

10.26. Suppose that the supply curve in a market is upward sloping and that the demand curve is totally inelastic. In a free market the price is $30 per ton. If an excise tax of $2 per ton is imposed in the market, what will be the resulting deadweight loss?

If demand is perfectly inelastic, the demand curve will be a vertical line. The price will rise by exactly $2 after the tax is imposed and consumers will take on 100% of the tax burden.

Consumer surplus will fall by $2 times the market quantity, which will be the same as the pre-tax quantity given the vertical demand curve. Government tax receipts will increase by $2 times the market quantity completely offsetting the reduction in consumer surplus. Producer surplus will remain the same since consumers have 100% of the burden of the tax. Thus, since government receipts completely offset the reduction in consumer surplus, there is nothing lost to society.

There is no deadweight loss from an excise tax when the demand curve is perfectly inelastic.

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Chapter 10-28

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.27. Suppose that the domestic demand for television sets is described by Q = 40,000 − 180P and that the supply is given by Q = 20P. If televisions can be freely imported at a price of $160, how many televisions would be produced in the domestic market? By how much would domestic producer surplus and deadweight loss change if the government introduces a $20 tariff per television set? What if the tariff was $70?

When televisions can be freely imported at a price of PW = $160, domestic producers will produce 20(160) = 3200 television sets. Domestic demand is 40,000 – 180*160 = 11,200 units.

P

Domestic Supply

230 222 A 200 B

J

PW + $20

180 C

F

G

K

PW

160 E

Demand 3200 3600 7600

11,200

40,000

Q

When the import duty of $20 is introduced, the effective price of importing televisions is $180.

At this price, domestic firms will supply 20(180) = 3600 televisions, and demand will be 40,000 – 180(180) = 7600. Domestic producer surplus will increase by area C = (180 – 160)(3200) + 0.5(180 – 160)(3600 – 3200) = 68,000. The tariff creates a deadweight equal to area F + K = 0.5(180 – 160)(3600 – 3200) + 0.5(180 – 160)(11,200 – 7600) = 40,000.

An import duty of $70 raises the effective import price to $230. You can see from the graph that this is above the equilibrium price of $200 that would prevail in the domestic market without any foreign trade. Thus, imposing such a high import duty is equivalent to banning trade in this industry altogether. The new price will be $200 and the quantity demanded 4000. Relative to the free trade equilibrium, producer surplus would now increase by area B + C = 0.5(200)(4000) – 0.5(160)(3200) = 144,000. The $70 import tariff creates a deadweight loss equal to area F + G + J + K = 0.5(200 – 160)(11,200 – 3200) = 160,000.

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Chapter 10-29

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

10.28. Suppose that the domestic demand for television sets is described by Q = 40,000 − 180P and that the supply is given by Q = 20P. Televisions can currently be freely imported at the world price of $160. Suppose the government bans the import of television sets. How much would domestic producer surplus and deadweight loss change?

When televisions can be freely imported at a price of PW = $160, domestic producers will produce 20(160) = 3200 television sets. Domestic demand is 40,000 – 180*160 = 11,200 units.

Producer surplus is area C in the graph.

With imports banned, equilibrium occurs where domestic supply intersects demand: 40,000 – 180P = 20P which implies P = 200 and Q = 4000. Producer surplus has now increased by area B = (200 – 160)(3200) + 0.5(200 – 160)(4000 – 3200) = 144,000. Banning imports creates a deadweight loss equal to area E + F = 0.5(200 – 160)(11,200 – 3200) = 160,000.

P

Domestic Supply

222 200

160

A B

E

F PW

C Demand 3200

4000

11,200

40,000

Q

10.29. Suppose that demand and supply curves in the market for corn are Qd = 20,000 − 50P and Qs = 30P. Suppose that the government would like to see the price at $300 per unit and is prepared to artificially increase demand by initiating a government purchase program. How much would the government need to spend to achieve this? What is the total deadweight loss if the government is successful in its objective?

Without government intervention, equilibrium occurs where 20,000 – 50P = 30P, or P = 250 and Q = 7500.

If the price were to be pushed up to $300, suppliers would like to produce 30(300) = 9000 units.

However, demand would be just 20,000 – 50(300) = 5,000 units. Therefore the government must buy the difference, which is 4,000 units. At $300 each, total government expenditure is $1.2 million. Relative to no government intervention, area A remains consumer surplus and C remains producer surplus, while area B is transferred from consumers to producers. To find deadweight loss, note that area E + F represents potential benefits no longer captured by anyone, while area G + H + J + K represents production costs that are incurred for units of corn that no one

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Chapter 10-30

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consumes. Thus deadweight loss is equal to area E + F + G + H +J + K. Alternatively, you can think of the deadweight loss as total government expenditures minus area L, or area (E + F + G + H +J + K + L) – L = 300(9000 – 5000) – 0.5(300 – 250)(9000 – 5000) = $1,100,000.

P Supply

400 300 250

A B

E

L

C

F G

K J H Demand

5000

7500 9000

20,000

Q

10.30. Suppose that demand and supply curves in the market for corn are Qd = 20,000 − 50P and Qs = 30P. Suppose that the government would like to see the price at $300 per unit and would like to do so with an acreage limitation program. How much would the government need to spend to achieve this? What is the total deadweight loss at the point where the government is successful in its objective?

Without government intervention, equilibrium occurs where 20,000 – 50P = 30P, or P = 250 and Q = 7500. As shown in the graph below, producer surplus is area C + F = 0.5(250)(7500) = 937,500.

To fix the price at $300, the government needs to ensure that only 20,000 – 50(300) = 5000 units will be supplied. At this price, producers would like to supply a total of 30(300) = 9000 units and earn a surplus equal to areas B + C + E + F + L. To compensate the producers for limiting production to 5000 units, the government must therefore transfer to producers a sum equal to area E + F + L ≈ 0.5(300 – 167)(9000 – 5000) = $266,000. (To see that the lower corner of triangle F occurs near P = 167, note that at Q = 5000 along the supply curve, 5000 = 30P or P = 500/3 ≈167.) Since production is restricted to 5000 units, deadweight loss is simply equal to the potential benefits that no one captures, or area E + F ≈ 0.5(300 – 167)(7500 – 5000) = 332,500.

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Chapter 10-31

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

P Supply

400 300 250

A B

E

C

F

167

L

G

K J H Demand

5000

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7500 9000

20,000

Q

Chapter 10-32

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Microeconomics, 6th edition Solutions Manual

Chapter 11 Monopoly and Monopsony Solutions to Review Questions 1. Why is the demand curve facing a monopolist the market demand curve?

A monopoly market consists of a single seller facing many buyers. Because the firm is by definition supplying the entire market, it faces the entire set of buyers making up the market demand curve.

2. The marginal revenue for a perfectly competitive firm is equal to the market price. Why is the marginal revenue for a monopolist less than the market price for positive quantities of output?

Marginal revenue is less than price for a monopolist. This is because as it lowers its price two things happen. First, the firm’s revenue increases from the additional units it sells (these are the marginal units). Second, the firm’s revenue decreases because it loses revenue from selling units at a lower price than it could have had it chosen a lower quantity of output (these are the inframarginal units.) The change in revenue is the sum of the increase from the marginal units and the decrease from the inframarginal units. This change can be summarized as 𝑀𝑅 =

𝛥𝑇𝑅 𝛥𝑃 =𝑃+𝑄 𝛥𝑄 𝛥𝑄

Since demand is downward sloping, the second term will be negative implying marginal revenue will be less than price.

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Microeconomics, 6th edition Solutions Manual

3. Why can a monopolist’s marginal revenue be negative for some levels of output? Why is marginal revenue negative when market demand is price inelastic?

The firm’s marginal revenue could be negative if the increase in revenue the firm gets from selling additional (marginal) units at a lower price is more than offset by the decrease in revenue from selling (inframarginal) units at a lower price than if it had chosen a lower quantity of output.

If demand is price inelastic then %Q  −1 %P Q Q  −1 P P Q  P     −1 P  Q   P  P  −Q    Q   P  P +Q 0  Q  𝛥𝑃

But, 𝑀𝑅 = 𝑃 + 𝑄 𝛥𝑄 Thus, when demand is price inelastic, marginal revenue is negative.

4. Assume that the monopolist’s marginal cost is positive at all levels of output. a) True or false: When the monopolist operates on the inelastic region of the market demand curve, it can always increase profit by producing less output. b) True or false: When the monopolist operates on the elastic region of the market demand curve, it can always increase profit by producing more output. a) True. Because the firm is operating on the inelastic region of the demand curve marginal revenue is negative. Thus, decreasing output will increase total revenue. And, since output is lower, total cost will be lower. Thus, by decreasing output and increasing price the firm can increase profits. b) False. When the firm operates on the elastic portion of the market demand curve, increasing output will increase total revenue. In addition, increasing output will increase total costs. Thus, the effect on profit will depend on how costs increase in relation to revenue.

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Microeconomics, 6th edition Solutions Manual

5. At the quantity of output at which the monopolist maximizes total profit, is the monopolist’s total revenue maximized? Explain.

The firm will not be maximizing total revenue at the point where the firm maximizes total profit.

The firm maximizes revenue at the point where 𝑀𝑅 = 0 and the firm maximizes profit at the point where 𝑀𝑅 = 𝑀𝐶. Thus, unless 𝑀𝐶 = 0, the firm will not maximize both revenue and profit at the same point.

6. What is the IEPR? How does it relate to the monopolist’s profit-maximizing condition, MR = MC? IEPR is the Inverse Elasticity Pricing Rule. This rule states that a profit- maximizing firm that 𝑃 ∗ −𝑀𝐶 ∗ 1 sets 𝑀𝑅 = 𝑀𝐶 will satisfy the condition that 𝑃∗ = − 𝜀 . where the asterisks indicate the 𝑄,𝑃

price and marginal cost at the profit-maximizing level of output.

7. Evaluate the following statement: Toyota faces competition from many other firms in the world market for automobiles; therefore, Toyota cannot have market power.

While perfectly competitive firms do not have market power, it is not true that any firm that faces competition does not have market power. In this particular example, while Toyota clearly has competition, Toyota also sells a differentiated product from the other automobile manufacturers. This will allow Toyota to control its price since no other manufacturer is producing the identical product. Thus, Toyota will have some market power. It is true, however, that this market power may be limited by the prices other manufacturers set for their automobiles. If Toyota’s product is not seen as being much different from other autos, it will not be able to set a price far out of line with the rest of the auto market.

8. What rule does a multiplant monopolist use to allocate output among its plants? Would a multiplant perfect competitor use the same rule?

A multi-plant monopolist will choose a level of output and then allocate output between plants so that marginal costs are equalized across plants. If a perfectly competitive firm had multiple plants it would follow the same rule. To see why, imagine it did not and allowed marginal costs to be different across plants. If marginal costs were different, then it reallocates one unit of output from the high marginal cost plant to the low marginal cost plant. This would reduce total cost without changing revenue. Thus, profit would increase. Therefore, to maximize profit the firm should allocate output between plants to equalize marginal cost.

9. Why does the monopoly equilibrium give rise to a deadweight loss?

A monopolist creates a deadweight loss because it produces a lower level of output and charges a higher price than would occur in perfect competition. This choice allows the monopolist to generate economic profits and increase producer surplus, essentially extracting surplus away from consumers. The firm, however, will not be able to gain as much surplus as consumers lose, lowering total net benefits, and creating a deadweight loss.

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Microeconomics, 6th edition Solutions Manual

10. How does a monopsonist differ from a monopolist? Could a firm be both a monopsonist and a monopolist?

A monopsonist is a firm that is a single buyer that can purchase from many sellers, whereas a monopolist is a firm that is a single seller that can sell to many buyers. It is possible for a firm to be both a monopolist and a monopsonist. Using the text example, suppose some local area had only one hospital. It would be a monopolist in the provision of some hospital services, e.g. emergency room services, and it would be a monopsonist in the purchase of some hospital inputs, e.g. nurses.

11. What is a monopsonist’s marginal expenditure function? Why does a monopsonist’s marginal expenditure exceed the input price at positive quantities of the input?

The monopsonist’s marginal expenditure function is the rate at which the monopsonist’s total cost goes up, per unit of input, as it hires more units of the input. Marginal expenditure will exceed unit cost because as the monopsonist increases the price it pays for units of input it must pay this higher price for the units it could have purchased at lower prices. This marginal 𝛥𝑤 expenditure can be summarized as 𝑀𝐸𝐿 = 𝑤 + 𝐿 𝛥𝐿 .

Since the second term is positive (the monopsonist must pay a higher wage to increase the supply of labor) the marginal expenditure will exceed the wage.

12. Why does the monopsony equilibrium give rise to a deadweight loss?

The monopsonist creates a deadweight loss. This occurs because the monopsonist hires a lower quantity and pays a lower price for its input than would occur in perfect competition. This allows the monopsonist to extract surplus away from suppliers, but the monopsonist is unable to earn as much additional surplus as suppliers lose, lowering net total benefits, and creating a deadweight loss.

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Chapter 11-4

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

Solutions to Problems 11.1. Suppose that the market demand curve is given by Q = 100 - 5P. a) What is the inverse market demand curve? b) What is the average revenue function for a monopolist in this market? c) What is the marginal revenue function that corresponds to this demand curve? a) If demand is given by 𝑄 = 100 − 5𝑃, inverse demand is found by solving for 𝑃. This implies 1 inverse demand is 𝑃 = 20 − 5 𝑄. 𝑇𝑅

𝑃𝑄

b) Average revenue is given by 𝐴𝑅 = 𝑄 = 𝑄 = 𝑃.

1

Therefore, average revenue will be 𝑃 = 20 − 5 𝑄. c) For a linear demand curve 𝑃 = 𝑎 − 𝑏𝑄, marginal revenue is given by 𝑀𝑅 = 𝑎 − 2𝑏𝑄. In this 1 2 instance demand is 𝑃 = 20 − 5 𝑄 implying marginal revenue is 𝑀𝑅 = 20 − 5 𝑄.

11.2. The market demand curve for a monopolist is given by P = 40 - 2Q. a) What is the marginal revenue function for the firm? b) What is the maximum possible revenue that the firm can earn? a) Since the demand curve is written in inverse form and is linear, the MR curve has the same vertical intercept and twice the slop as the demand curve. Thus, MR = 40 – 4Q. b) Total revenue will be maximized when MR = 0, or when Q = 10. At that quantity, the price will be P = 40 – 2Q = 20. Total revenue is PQ = 20(10) = 200.

11.3. Show that the price elasticity of demand is -1 if and only if the marginal revenue is zero. 𝛥𝑃

𝑄 𝛥𝑃

𝑀𝑅 = 𝑃 + 𝑄 𝛥𝑄 = 𝑃 [1 + 𝑃 𝛥𝑄] = 𝑃 [1 + 𝜀

1 𝑄,𝑃

]. Since P > 0, MR = 0 if and only if

1 + (1/𝜀𝑄,𝑃 )= 0, which is equivalent to 1/𝜀𝑄,𝑃 = −1 or 𝜀𝑄,𝑃 = −1.

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Microeconomics, 6th edition Solutions Manual

11.4. Suppose that Intel has a monopoly in the market for microprocessors in Brazil.

During the year 2005, it faces a market demand curve given by P = 9 - Q, where Q is millions of microprocessors sold per year. Suppose you know nothing about Intel’s costs of production. Assuming that Intel acts as a profit-maximizing monopolist, would it ever sell 7 million microprocessors in Brazil in 2005?

If demand is 𝑃 = 9 − 𝑄, then 𝑀𝑅 = 9 − 2𝑄. If the firm sets 𝑄 = 7, then 𝑀𝑅 = −5. At this point, if the firm lowered its output it would increase total revenue, and with the lower level of output total cost would fall. Thus, decreasing output would increase profit. Therefore, a profitmaximizing monopolist facing this demand curve would never choose 𝑄 = 7.

11.5. A monopolist operates in an industry where the demand curve is given by Q = 1000 - 20P. The monopolist’s constant marginal cost is $8. What is the monopolist’s profit-maximizing price?

Recall that the MR curve can easily be derived from the demand curve when the latter is written in the inverse form. The inverse demand curve is P = 50 – (Q/20) so the marginal revenue curve is P = 50 – (Q/10) (using the fact that the slope of the MR curve is twice that of the inverse demand curve, with the same intercept). Using the rule MR=MC, we get 50 – (Q/10) = 8, so Q = 420. Plugging this back into the demand curve (or the inverse demand curve) we can calculate the profit maximizing price, P = 29.

11.6. Suppose that United Airlines has a monopoly on the route between Chicago and Omaha, Nebraska. During the winter (December–March), the monthly demand on this route is given by P = a1 - bQ. During the summer (June–August), the monthly demand is given by P = a2 - bQ, where a2 > a1. Assuming that United’s marginal cost function is the same in both the summer and the winter, and assuming that the marginal cost function is independent of the quantity Q of passengers served, will United charge a higher price in the summer or in the winter?

If marginal cost is independent of Q, then marginal cost is constant. Assume 𝑀𝐶 = 𝑐. Then in the winter the firm will produce where 𝑀𝑅 = 𝑀𝐶.

a1 − 2bQ = c Q=

a1 − c 2b

At this quantity the price charged will be 𝑎 −𝑐

1 𝑃 = 𝑎1 − 𝑏 ( 2𝑏 ) 𝑎1 + 𝑐 𝑃= 2

In the summer the firm will also produce where 𝑀𝑅 = 𝑀𝐶.

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Chapter 11-6

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Microeconomics, 6th edition Solutions Manual

a2 − 2bQ = c Q=

a2 − c 2b

At this quantity the price charged will be 𝑎 −𝒄

2 𝑃 = 𝑎2 − 𝑏 ( 𝟐𝒃 ) 𝒂𝟐 + 𝒄 𝑷= 𝟐

Since we are told that 𝒂𝟐 > 𝒂𝟏 , the price charged during the summer months will be greater than the price charged during the winter months.

11.7. A monopolist operates with the following data on cost and demand. It has a total fixed cost of $1,400 and a total variable cost of Q2, where Q is the number of units of output it produces. The firm’s demand curve is P = $120 - 2Q. The size of its sunk cost is $600. The firm expects the conditions of demand and cost to continue in the foreseeable future. a) What is the firm’s profit if it operates and it maximizes profit? b) Should the firm continue to operate in the short run, or should it shut down? Explain. a) The monopolist chooses Q so that MR = MC: 120 – 4Q = 2Q => Q = 20.

P = 120 – 2(20) = 80.

Profit = PQ – V – F = 80(20) – 202 – 1400 = - 200.

The firm has nonsunk fixed costs: FNonsunk = F - FSunk = 1400 – 600 = 800. b) Producer surplus = PQ – V – FNonsunk = 80(20) – 202 – 800 = 400. So the firm should continue to operate in the short run. If it operates, its profit is -200. But if it shuts down, its profit = - FSunk = -600. So it can lessen its losses by 400 if it continues to operate (and this is why producer surplus is +400 annually.)

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Chapter 11-7

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.8. A monopolist operates with a fixed cost and a variable cost. Part of the fixed cost is sunk, and part nonsunk. How will the sunk and nonsunk fixed costs affect the firm’s decisions as it tries to maximize profit in the short run?

We can answer this question in two stages. In the first stage let’s ask what price and quantity the firm should choose if it remains in production. When the firm chooses the optimal quantity, it sets MR = MC. The optimal quantity choice rule therefore does not depend on the fixed costs, and thus the optimal quantity and price (again, assuming the firm stays in operation) do not depend on the size of the sunk and nonsunk fixed costs. Once the optimal price is determined, the optimal revenues R* and variable costs V* can be determined.

Now we can move to the second decision: should the firm stay in business, at least in the short run?

If it remains in production, the firms producer surplus will be R* - V* – Fnonsunk.

The firm would want to remain in business in the short run if the producer surplus is positive, and it would shut down if the producer surplus is negative.

11.9. Under what conditions will a profit-maximizing monopolist and a revenuemaximizing monopolist set the same price?

A profit-maximizing monopolist would choose the output at which MR = MC. A revenuemaximizing monopolist would choose the output at which MR = 0. The two would therefore choose the same output (and set the same price) when MC = 0.

11.10. Assume that a monopolist sells a product with the cost function C = F + 20Q, where C is total cost, F is a fixed cost, and Q is the level of output. The inverse demand function is P = 60 - Q, where P is the price in the market. The firm will earn zero economic profit when it charges a price of 30 (this is not the price that maximizes profit). How much profit does the firm earn when it charges the price that maximizes profit?

When the P = 30, the demand function shows that Q = 30.

At that price, profit = 0 = PQ – C = (30)(30) – F – 20(30); therefore F = 300.

So total cost is C = 300 – 20Q. Now find the quantity that maximizes profit. Set MR = MC. MR = 60 – 2Q and MC = 20.

60 – 2Q = 20 implies that Q = 20 and P = 40.

So, the profit-maximizing profit will be PQ – C = (40)(20) – 300 – (20)(20) = 100.

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Chapter 11-8

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Microeconomics, 6th edition Solutions Manual

11.11. Assume that a monopolist sells a product with a total cost function TC = 1,200 + 0.5Q2 and a corresponding marginal cost function MC = Q. The market demand curve is given by the equation P = 300 - Q. a) Find the profit-maximizing output and price for this monopolist. Is the monopolist profitable? b) Calculate the price elasticity of demand at the monopolist’s profit-maximizing price.

Also calculate the marginal cost at the monopolist’s profit-maximizing output. Verify that the IEPR holds. a) If demand is given by 𝑷 = 𝟑𝟎𝟎 − 𝑸then 𝑴𝑹 = 𝟑𝟎𝟎 − 𝟐𝑸. To find the optimum set 𝑴𝑹 = 𝑴𝑪.

300 − 2Q = Q Q = 100

At 𝑸 = 𝟏𝟎𝟎 price will be 𝑷 = 𝟑𝟎𝟎 − 𝟏𝟎𝟎 = 𝟐𝟎𝟎. At this price and quantity total revenue will be 𝑻𝑹 = 𝟐𝟎𝟎(𝟏𝟎𝟎) = 𝟐𝟎, 𝟎𝟎𝟎 and total cost will be 𝑻𝑪 = 𝟏𝟐𝟎𝟎+. 𝟓(𝟏𝟎𝟎)𝟐 = 𝟔, 𝟐𝟎𝟎.

Therefore, the firm will earn a profit of 𝝅 = 𝑻𝑹 − 𝑻𝑪 = 𝟏𝟑, 𝟖𝟎𝟎. 𝜟𝑸 𝑷

b) The price elasticity at the profit-maximizing price is 𝜺𝑸,𝑷 = 𝜟𝑷 𝑸. 𝜟𝑸

With the demand curve 𝑸 = 𝟑𝟎𝟎 − 𝑷, 𝜟𝑷 = −𝟏. Therefore, at the profit-maximizing price 𝜺𝑸,𝑷 = −𝟏 ( 𝜺𝑸,𝑷 = −𝟐

𝟐𝟎𝟎 ) 𝟏𝟎𝟎

The marginal cost at the profit-maximizing output is MC = Q = 100. The inverse elasticity pricing rule states that at the profit-maximizing price 𝑷 − 𝑴𝑪 𝟏 =− 𝑷 𝜺𝑸,𝑷 In this case we have 200 − 100 1 =− 200 −2 1 1 = 2 2

Thus, the IEPR holds for this monopolist.

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Chapter 11-9

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.12. A monopolist faces a demand curve P = 210 - 4Q and initially faces a constant marginal cost MC = 10. a) Calculate the profit-maximizing monopoly quantity and compute the monopolist’s total revenue at the optimal price. b) Suppose that the monopolist’s marginal cost increases to MC = 20. Verify that the monopolist’s total revenue goes down. c) Suppose that all firms in a perfectly competitive equilibrium had a constant marginal cost MC = 10. Find the long-run perfectly competitive industry price and quantity. d) Suppose that all firms’ marginal costs increased to MC = 20. Verify that the increase in marginal cost causes total industry revenue to go up. a) With demand 𝑷 = 𝟐𝟏𝟎 − 𝟒𝑸, 𝑴𝑹 = 𝟐𝟏𝟎 − 𝟖𝑸. Setting 𝑴𝑹 = 𝑴𝑪 implies 210 − 8Q = 10 Q = 25

With 𝑸 = 𝟐𝟓, price will be 𝑷 = 𝟐𝟏𝟎 − 𝟒𝑸 = 𝟏𝟏𝟎. At this price and quantity total revenue will be 𝑻𝑹 = 𝟏𝟏𝟎(𝟐𝟓) = 𝟐, 𝟕𝟓𝟎. b) If 𝑴𝑪 = 𝟐𝟎, then setting 𝑴𝑹 = 𝑴𝑪 implies 210 − 8Q = 20 Q = 23.75

At 𝑸 = 𝟐𝟑. 𝟕𝟓, price will be 𝑷 = 𝟏𝟏𝟓. At this price and quantity total revenue will be 𝑻𝑹 = 𝟏𝟏𝟓(𝟐𝟑. 𝟕𝟓) = 𝟐, 𝟕𝟑𝟏. 𝟐𝟓. Therefore, the increase in marginal cost will result in lower total revenue for the firm. c) Competitive firms produce until P = MC, so in this case we know the market price would be P = 10 and the market quantity would be: 210 − 4Q = 10 Q = 50

d) In this case, the market price will be𝑷 = 𝑴𝑪 = 20, implying that the industry quantity is given by 210 − 4Q = 20 Q = 47.50

At this quantity, price will be 𝑷 = 𝟐𝟎. When 𝑴𝑪 = 𝟏𝟎, total industry revenue is𝟏𝟎(𝟓𝟎) = 𝟓𝟎𝟎. With 𝑴𝑪 = 𝟐𝟎, total industry revenue is 𝟐𝟎(𝟒𝟕. 𝟓𝟎) = 𝟗𝟓𝟎. Thus, total industry revenue increases in the perfectly competitive market after the increase in marginal cost.

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Chapter 11-10

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.13. A monopolist serves a market in which the demand is P = 120 - 2Q. It has a fixed cost of 300. Its marginal cost is 10 for the first 15 units (MC = 10 when 0 Q 15). If it wants to produce more than 15 units, it must pay overtime wages to its workers, and its marginal cost is then 20. What is the maximum amount of profit the firm can earn?

The marginal revenue curve is MR = 120 – 4Q. Initially we are not sure whether the optimal quantity will be less than 15 units (in which case MC = 10), or more than 20 units (where MC = 20).

There are two regions of output: Region I: where MC = 10 and 0 < Q < 15 Region II: where MC = 20 and 15 < Q Let’s assume that the MC = 10 and optimal quantity is less than or equal to 15 units. In that case, setting MR = MC, we find that 120 – 4Q = 10, or that Q = 27.5. But when Q = 27.5, MC is not 10, so the assumption that the optimal quantity is in Region I is not correct.

Now let’s assume that the MC = 20 and optimal quantity is greater than 15 units. In that case, setting MR = MC, we find that 120 – 4Q = 20, or that Q = 25. When Q = 25, MC is 20, so that marginal cost we have assumed is correct at the optimal output level we have calculated. The market price is P = 120 – 2(25) = 70.

Revenue = PQ = 70(25) = 1750 Variable cost = 10(15) + 20 (25 – 15) = 350 Fixed Cost = 300 Profit = 1750 – 350 – 300 = 1100.

11.14. A monopolist faces the demand function P = 100 - Q + I, where I is average consumer income in the monopolist’s market. Suppose we know that the monopolist’s marginal cost function is not downward sloping. If consumer income goes up, will the monopolist charge a higher price, a lower price, or the same price?

If demand is initially 𝑷 = 𝟏𝟎𝟎 − 𝑸 + 𝑰, then initially 𝑴𝑹 = 𝟏𝟎𝟎 + 𝑰 − 𝟐𝑸. Setting 𝑴𝑹 = 𝑴𝑪 implies

100 + I − 2Q1 = MC1 Q1 =

100 + I − MC1 2

where 𝑸𝟏 is the profit-maximizing quantity when income equals 𝑰 and 𝑴𝑪𝟏 is the corresponding level of marginal cost.

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Chapter 11-11

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

With this quantity, price will be 𝟏𝟎𝟎 + 𝑰 − 𝑴𝑪𝟏 )+𝑰 𝑷𝟏 = 𝟏𝟎𝟎 − ( 𝟐 𝟏𝟎𝟎 + 𝑰 + 𝑴𝑪𝟏 𝑷𝟏 = 𝟐 Now suppose income increases by a factor 𝑲 where 𝑲 > 𝟏. Then setting 𝑴𝑹 = 𝑴𝑪 implies

100 + KI − 2Q2 = MC2 Q2 =

100 + KI − MC2 2

where 𝑸𝟐 is the profit-maximizing quantity when income equals 𝑲𝑰 and 𝑴𝑪𝟐 is the corresponding level of marginal cost. This quantity must be greater than the quantity when income equals 𝑰, i.e., 𝑸𝟐 > 𝑸𝟏 . If it were not, i.e., if 𝑸𝟐 ≤ 𝑸𝟏 , then the marginal cost 𝑴𝑪𝟐 would be less than or equal to 𝑴𝑪𝟏 (since we know marginal cost is not decreasing). But that would mean that 𝑸𝟐 =

𝟏𝟎𝟎+𝑲𝑰−𝑴𝑪𝟐 𝟐

>

𝟏𝟎𝟎+𝑰−𝑴𝑪𝟐 𝟐

= 𝑸𝟏

contradicting the assumption that 𝑸𝟐 ≤ 𝑸𝟏 At quantity 𝑸𝟐 , price will be 𝟏𝟎𝟎 + 𝑲𝑰 − 𝑴𝑪𝟐 ) 𝟐 𝟏𝟎𝟎 + 𝑲𝑰 + 𝑴𝑪𝟐 𝑷𝟐 = 𝟐 𝑷𝟐 = 𝟏𝟎𝟎 + 𝑲𝑰 − (

In this case the new price charged by the monopolist will be greater than the initial price. Clearly 𝑲𝑰 > 𝑰 since 𝑲 > 𝟏, and because the marginal cost function is assumed to not be downward sloping, the increase in 𝑸 at the higher income level will result in a marginal cost at least as high as the initial marginal cost, i.e., 𝑴𝑪𝟐 ≥ 𝑴𝑪𝟏 . Therefore, the price will increase when consumer income increases.

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Chapter 11-12

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.15. Two monopolists in different markets have identical, constant marginal cost functions. a) Suppose each faces a linear demand curve and the two curves are parallel. Which monopolist will have the higher markup (ratio of P to MC): the one whose demand curve is closer to the origin or the one whose demand curve is farther from the origin? b) Suppose their linear demand curves have identical vertical intercepts but different slopes. Which monopolist will have a higher markup: the one with the flatter demand curve or the one with the steeper demand curve? c) Suppose their linear demand curves have identical horizontal intercepts but different slopes. Which monopolist will have a higher markup: the one with the flatter demand curve or the one with the steeper demand curve? a) If the two demand curves are linear and parallel they differ only by a constant; call this constant 𝒄. Then 𝑷𝟏 = 𝒂 − 𝒃𝑸𝟏 𝑷𝟐 = 𝒂 + 𝒄 − 𝒃𝑸𝟐 In this instance demand for the second firm will be further from the origin assuming 𝒄 > 𝟎. Now assume that both firms have identical constant marginal cost 𝒆. Then the first firm will maximize profit where 𝑴𝑹 = 𝑴𝑪.

a − 2bQ1 = e Q1 =

a−e 2b

At this quantity price will be 𝒂−𝒆 𝑷𝟏 = 𝒂 − 𝒃 ( ) 𝟐𝒃 𝒂+𝒆 𝑷𝟏 = 𝟐 The second firm will also maximize profit where 𝑴𝑹 = 𝑴𝑪.

a + c − 2bQ2 = e Q2 =

a+c−e 2b

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Chapter 11-13

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

At this quantity price will be 𝒂+𝒄−𝒆 ) 𝑷𝟐 = 𝒂 + 𝒄 − 𝒃 ( 𝟐𝒃 𝒂+𝒄+𝒆 𝑷𝟐 = 𝟐 For the first monopolist 𝑷𝟏 = 𝑴𝑪

(𝒂 + 𝒆) 𝟐 𝒆

and for the second monopolist 𝑷𝟐 = 𝑴𝑪

(𝒂 + 𝒄 + 𝒆) 𝟐 𝒆

𝑷

𝑷

Here 𝑴𝑪𝟐 > 𝑴𝑪𝟏 implying the firm with the demand curve further from the 𝑷 axis will have the higher mark-up ratio. b) Suppose the first monopolist faces demand 𝑷𝟏 = 𝒂 − 𝒃𝑸𝟏 and the second monopolist faces demand 𝑷𝟐 = 𝒂 − 𝒌𝒃𝑸𝟐 where 𝒌 > 𝟏. In this case the demand curve for the second monopolist is steeper. As in part a), the first monopolist will maximize profit at a−e 2b a+e P1 = 2

Q1 =

For the second monopolist profit will be maximized where 𝑴𝑹 = 𝑴𝑪.

a − 2kbQ2 = e Q2 =

a−e 2kb

At this quantity price will be 𝒂−𝒆 𝑷𝟐 = 𝒂 − 𝒌𝒃 ( ) 𝟐𝒌𝒃 𝒂+𝒆 𝑷𝟐 = 𝟐 Since both monopolists will charge the same price and since marginal cost is constant, both monopolists will have the same mark-up ratio.

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Chapter 11-14

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

c) Suppose the first monopolist faces demand 𝑷𝟏 = 𝒂 − 𝒃𝑸𝟏 and the second monopolist faces demand 𝑷𝟐 = 𝒌(𝒂 − 𝒃𝑸𝟐 ) where 𝒌 > 𝟏. In this case both firms face linear demand curves with the same horizontal intercept (at Q = a/b) but the demand for monopolist 2 is steeper.

The first firm maximizes profit as in parts A and B at a−e 2b a+e P1 = 2

Q1 =

For the second monopolist, profit will be maximized where 𝑴𝑹 = 𝑴𝑪.

ak − 2kbQ2 = e Q2 =

ak − e 2b

At this quantity price will be 𝑷𝟐 = 𝒂𝒌 − 𝒌𝒃 ( 𝑷𝟐 =

𝒂𝒌 + 𝒆 𝟐

𝒂𝒌 − 𝒆 ) 𝟐𝒌𝒃

Since 𝒌 > 𝟏, 𝑷𝟐 > 𝑷𝟏 . Since marginal cost is constant, the monopolist with the steeper demand function will have the higher mark-up ratio.

11.16. Suppose a monopolist faces the market demand function P = a - bQ. Its marginal cost is given by MC = c + eQ. Assume that a > c and 2b + e > 0. a) Derive an expression for the monopolist’s optimal quantity and price in terms of a, b, c, and e. b) Show that an increase in c (which corresponds to an upward parallel shift in marginal cost) or a decrease in a (which corresponds to a leftward parallel shift in demand) must decrease the equilibrium quantity of output. c) Show that when e  0, an increase in a must increase the equilibrium price.

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Chapter 11-15

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

a) The monopolist will operate where 𝑴𝑹 = 𝑴𝑪. With demand 𝑷 = 𝒂 − 𝒃𝑸, marginal revenue is given by 𝑴𝑹 = 𝒂 − 𝟐𝒃𝑸. Setting this equal to marginal cost implies a − 2bQ = c + eQ Q=

a−c 2b + e

At this quantity price is 𝒂−𝒄 ) 𝟐𝒃 + 𝒆 𝒂𝒃 + 𝒂𝒆 + 𝒃𝒄 𝑷= 𝟐𝒃 + 𝒆 𝑷 = 𝒂−𝒃(

𝒂−𝒄

b) Since 𝑸 = 𝟐𝒃+𝒆, increasing 𝒄 or decreasing 𝒂 will reduce the numerator of the expression, reducing 𝑸. 𝒂𝒃+𝒂𝒆+𝒃𝒄

c) Since e≥ 0 and 𝑷 = 𝟐𝒃+𝒆 , increasing 𝒂 will increase the numerator for this expression.

This will therefore increase the equilibrium price.

11.17. Suppose a monopolist has the demand function Q = 1,000P-3. What is the monopolist’s optimal markup of price above marginal cost?

With demand 𝑸 = 𝟏𝟎𝟎𝟎𝑷−𝟑 , elasticity along the demand curve is constant at −𝟑. Employing 𝑷−𝑴𝑪 𝟏 𝟏 the inverse elasticity pricing rule implies 𝑷 = − −𝟑 = 𝟑. Therefore, the optimal percentage 𝟏

mark-up of price over marginal cost is 𝟑, or 33 percent.

11.18. Suppose a monopolist has an inverse demand function given by P = 100Q-1/2. What is the monopolist’s optimal markup of price above marginal cost?

Remember that the demand elasticity in a constant elasticity demand function is the exponent on P when the demand function is written in the regular form, i.e. 𝑸 = 𝒇(𝑷). We can manipulate the inverse demand function to get the regular demand function, 𝑸 = 𝟏𝟎, 𝟎𝟎𝟎𝑷−𝟐 . This implies 𝑷−𝑴𝑪 𝟏 that the demand elasticity is –2. Therefore, using the IEPR, 𝑷 = 𝟐. So the optimal percentage mark-up of price over marginal cost is ½, or 50 percent.

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Chapter 11-16

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.19. The marginal cost of preparing a large latte in a specialty coffee house is $1. The firm’s market research reveals that the elasticity of demand for its large lattes is constant, with a value of about -1.3. If the firm wants to maximize profit from the sale of large lattes, about what price should the firm charge?

Since the elasticity of demand is constant, we can use the inverse elasticity rule for a monopolist.

[P - MC]/P = -1/eQ,P . The inverse elasticity rule then becomes [P - 1]/P = -1/1.3. Thus we would expect to see the firm charge about $4.33 for a large latté.

11.20. The following diagram shows the average cost curve and the marginal revenue curve for a monopolist in a particular industry. What range of quantities could it be possible to observe this firm producing, assuming that the firm maximizes profit? You can read your answers off the graph, and therefore approximate values are permissible.

The graph is reproduced below. The MES appears to be at about 16 units of output, and the point where the MR curve intersects the AC curve is at about 20 units.

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Chapter 11-17

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

The monopolist’s profit maximizing output must fall between 16 and 20 units. To see this, remember that the firm will produce where MR = MC. This cannot happen at any point less than 16 units because the AC curve is decreasing for Q < 16. Therefore the MC curve lies below the AC curve and clearly MR > MC for Q > 16. Similarly, since the MC curve must lie above the AC curve to the right of 16 units, it must intersect the MR curve before the MR curve intersects the AC curve. That is, the profit maximizing quantity must be less than 20 units.

11.21. Imagine that Gillette has a monopoly in the market for razor blades in Mexico. The market demand curve for blades in Mexico is P = 968 - 20Q, where P is the price of blades in cents and Q is annual demand for blades expressed in millions. Gillette has two plants in which it can produce blades for the Mexican market: one in Los Angeles and one in Mexico City. In its L.A. plant, Gillette can produce any quantity of blades it wants at a marginal cost of 8 cents per blade. Letting Q1 and MC1 denote the output and marginal cost at the L.A. plant, we have MC1(Q1) = 8. The Mexican plant has a marginal cost function given by MC2(Q2) = 1 + 0.5Q2. a) Find Gillette’s profit-maximizing price and quantity of output for the Mexican market overall. How will Gillette allocate production between its Mexican plant and its U.S. plant? b) Suppose Gillette’s L.A. plant had a marginal cost of 10 cents rather than 8 cents per blade. How would your answer to part (a) change? a) Profit-maximizing firms generally allocate output among plants so as to keep marginal costs equal. But notice that MC2 < MC1 whenever 1 + 0.5Q2 < 8, or Q2 < 14. So for small levels of output, specifically Q < 14, Gillette will only use the first plant. For Q > 14, the cost-minimizing approach will set Q2 = 14 and Q1 = Q – 14.

Suppose the monopolist’s profit-maximizing quantity is Q > 14. Then the relevant MC = 8, and with 𝑴𝑹 = 𝟗𝟔𝟖 − 𝟒𝟎𝑸 we have 968 − 40Q = 8 Q = 24

Since we have found that Q > 14, we know this approach is valid. (You should verify that had we supposed the optimal output was Q < 14 and set MR = MC2 = 1 + 0.5Q, we would have found Q > 14. So this approach would be invalid.) The allocation between plants will be Q2 = 14 and Q1 = 10. With a total quantity Q = 24, the firm will charge a price of P = 968 – 20(24) = 488. Therefore the price will be $4.88 per blade.

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Chapter 11-18

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

b) If 𝑴𝑪 = 𝟏𝟎 at plant 1, by the logic in part (a) Gillette will only use plant 2 if Q < 18. It will produce all output above Q = 18 in plant 1 at MC = 10. Assuming Q > 18, setting 𝑴𝑹 = 𝑴𝑪 implies 968 − 40Q = 10 Q = 23.95

(So again, this approach is valid. You can verify that setting MR = MC2 would again lead to Q > 18.) The firm will allocate production so that Q2 = 18 and Q1 = 5.95. At Q = 23.95, price will be $4.89.

11.22. Market demand is P = 64 - (Q/7). A multiplant monopolist operates three plants, with marginal cost functions:

a) Find the monopolist’s profit-maximizing price and output at each plant. b) How would your answer to part (a) change if MC2 (Q2) = 4? a) Equating the marginal costs at MCT, we have Q = Q1 + Q2 + Q3 = 0.25MCT + 0.5MCT – 1 + MCT – 6, which can be rearranged as MCT = (4/7)Q + 4. Setting MR = MC yields 64 – (2/7)*Q = (4/7)*Q + 4 or Q = 70 and P = 54. At this output level, MCT = 44, implying that Q1 = 11, Q2 = 21, and Q3 = 38. b) In this case, using plant 3 is inefficient because its marginal cost is always higher than that of plant 2. Hence, the firm will use only plants 1 and 2. Moreover, the firm will not use plant 1 once its marginal cost rises to MC2 = 4, so we can immediately see that it will only produce 4Q1 = 4 or Q1 = 1 unit at plant 1. Its total production can be found by setting MR = MC2, yielding 64 – (2/7)*Q = 4 or Q = 210 and P = 34. So it produces Q1 = 1 unit in plant 1 and Q2 = 209 units in plant 2, while producing no units in plant 3 (i.e. Q3 = 0).

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Chapter 11-19

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

11.23. A monopolist producing only one product has two plants with the following marginal cost functions: MC1 = 20 + 2Q1 and MC2 = 10 + 5Q2, where MC1 and MC2 are the marginal costs in plants 1 and 2, and Q1 and Q2 are the levels of output in each plant, respectively. If the firm is maximizing profits and is producing Q2 = 4, what is Q1?

The firm will be maximizing its profit when the marginal costs are equal for the two plants.

(Otherwise, the firm could take the last unit produced at the high-cost plant and instead produce that same unit at the low cost plant, not changing revenues and reducing costs.) When Q2 =4, MC2 = 30, plant 1 must be operating with MC1 = 30. This means that Q1 = 5.

11.24. Suppose that you are hired as a consultant to a firm producing a therapeutic drug protected by a patent that gives a firm a monopoly in two markets. The drug can be transported between the two markets at no cost, so the firm must charge the same price in both markets. The demand schedule in the first market is P1 = 200 - 2Q1, where P1 is the price of the product and Q1 is the amount sold in the market. In the second market, the demand is P2 = 140 - Q2, where P2 is the price and Q2 the quantity. The firm’s overall marginal cost is MC = 20 + Q1 + Q2. What price should the firm charge?

Because the firm needs to charge the same price in both markets, it needs to set its marginal cost equal to the marginal revenue associated with the aggregate demand curve.

To get the aggregated demand curve, it must sum the demands “horizontally,” i.e., add the quantities when P1 = P2 (= P).

Q1 = 100 – 0.5P and Q2 = 140 – P. The aggregate quantity demanded is Q = Q1 + Q2.

Then the aggregate demand is Q = 240 – 1.5P. Now find the inverse aggregate demand curve: P = 160 - (2/3)Q. The marginal revenue associated with the aggregate demand curve has the same vertical intercept and twice the slope as the demand curve: MR = 160 - (4/3)Q. The marginal cost is MC = 20 + Q. Set MR = MC. 160 - (4/3)Q = 20 + Q. Thus the profit-maximizing total quantity to produce is Q = 60.

The optimal price is P = 160 - (2/3)(60) = 120.

11.25. A firm has a monopoly in the production of a software application in Europe. The demand schedule in Europe is Q1 = 120 - P, where Q1 is the amount sold in Europe when the price is P. The firm’s marginal cost is 20. a) What price would the firm choose if it wishes to maximize profits? b) Now suppose the firm also receives a patent for the application in the United States. The demand for the application in the United States is Q2 = 240 - 2P, where Q2 is the quantity sold when the price is P. Because it costs essentially nothing to transport software over the Internet, the firm must charge the same price in Europe and the United States. What price would maximize the firm’s profit? c) Use the monopoly midpoint rule (Learning-By-Doing Exercise 11.5) to explain the relationship between your answers to parts (a) and (b).

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Chapter 11-20

Besanko & Braeutigam –

Microeconomics, 6th edition Solutions Manual

a) Set MR = MC in Europe. The inverse demand is P = 120 – Q, so MR = 120 – 2Q. MR = MC implies that 120 – 2Q = 20, or Q = 50. P = 120 – 50 = 70. b) Because the firm needs to charge the same price in both markets, it needs to set its marginal cost equal to the marginal revenue associated with the aggregate demand curve. To get the aggregated demand curve, it must sum the demands “horizontally,” i.e., add the quantities.

Q1 = 120 – P and Q2 = 240 – 2P. The aggregate quantity demanded is Q = Q1 + Q2.

Then the aggregate demand is Q = 360 – 3P. Now find the inverse aggregate demand curve: P = 120 - (1/3)Q. The marginal revenue associated with the aggregate demand curve has the same vertical intercept and twice the slope as the demand curve: MR = 120 - (2/3)Q. The marginal cost is MC = 20. Set MR = MC. 120 - (2/3)Q = 20.

Thus the profit-maximizing total quantity to produce is Q = 150.

The optimal price is P = 120 - (1/3)(150) = 70. c) The demand in Europe is linear and has a choke price of 120. The aggregate demand in part (b) is also linear, with a choke price of 120. The marginal cost is constant at 20. The Monopoly Midpoint Rule states that with a linear demand and a constant marginal cost, the profit maximizing price will be (choke price + marginal cost)/2, or (120 + 20)/2 = 70. This is the same in parts (a) and (b).

11.26. Suppose that a monopolist’s market demand is given by P = 100 - 2Q and that marginal cost is given by MC = Q/2. a) Calculate the profit-maximizing monopoly price and quantity. b) Calculate the price and quantity that arise under perfect competition with a supply curve P = Q/2. c) Compare consumer and producer surplus under monopoly versus marginal cost pricing.

What is the deadweight loss due to monopoly? d) Suppose market demand is given by P = 180 - 4Q. What is the deadweight loss due to monopoly now? Explain why this deadweight loss differs from that in part (c). a) With demand 𝑷 = 𝟏𝟎𝟎 − 𝟐𝑸, 𝑴𝑹 = 𝟏𝟎𝟎 − 𝟒𝑸. Setting 𝑴𝑹 = 𝑴𝑪 implies 100 − 4Q = .5Q Q = 22.2

(All figures are rounded.) At this quantity, price will be 𝑷 = 𝟓𝟓. 𝟔.

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Microeconomics, 6th edition Solutions Manual

b) A perfectly competitive market produces until P = MC, or 100 − 2Q = .5Q Q = 40

At this quantity, price will be P = 20. c) Under monopoly, consumer surplus is 0.5(100 – 55.6)(22.2) = 493. Since MC(22.2) = 11.1, producer surplus is 0.5(11.1)(22.2) + (55.6 – 11.1)(22.2) = 1111. Net benefits are 1604. (All figures are rounded.)

Under perfect competition, consumer surplus is 0.5(100 – 20)(40) = 1600, and producer surplus is 0.5(20)(40) = 400. Net benefits are 2000. Therefore, the deadweight loss due to monopoly is 396. d) Now setting MR = MC gives 180 − 8Q = 0.5Q Q = 21.2

At this quantity, price is 95.2. Consumer surplus is 0.5(100 – 95.2)(21.1) = 51 and producer surplus is 0.5(10.6)(21.2) + (95.2 – 10.6)(21.2) = 1906. Net benefits are 1957.

Setting P = MC as in perfect competition yields 180 − 4Q = .5Q Q = 40

At this quantity, price is 20. Consumer surplus is 0.5(180 – 20)(40) = 3200 and producer surplus is 0.5(20)(40) = 400. Net benefits with perfect competition are 3600. Therefore, the deadweight loss in this case is 1643.

While the competitive solution is identical with both demand curves, the deadweight loss in the first case is far greater. This difference occurs because with the second demand curve demand is less elastic at the perfectly competitive price. If consumers are less willing to change quantity as price increases toward the monopoly level, the firm will be able to extract more surplus from the market.

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Chapter 11-22

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Microeconomics, 6th edition Solutions Manual

11.27. The demand curve for a certain good is P = 100 - Q. The marginal cost for a monopolist is MC(Q) = Q, for Q 30. The maximum that can be supplied in this market is Q = 30, that is, the marginal cost is infinite for Q > 30. a) What price will the profit-maximizing monopolist set? b) What is the deadweight loss due to monopoly in this market? a) See the figure below. The monopolist will produce the quantity that corresponds to MR = MC. However, because the MC curve is vertical at Q = 30, this is also the quantity corresponding to the point where the MC curve intersects the demand curve. The monopolist produces 30 units and sells at a price of 70.

MC

A

B

D

C

MR

Demand

b) The deadweight loss is zero. To see this, notice that the price and quantity are the same in the case of monopoly and the case of a competitive market, if P = MC. Therefore, there is no deadweight loss from monopoly.

11.28. A coal mine operates with a production function Q = L/2, where L is the quantity of labor it employs and Q is total output. The firm is a price taker in the output market, where the price is currently 32. The firm is a monopsonist in the labor market, where the supply curve for labor is w = 4L. a) What is the monopsonist’s marginal expenditure function, MEL? b) Calculate the monopsonist’s optimal quantity of labor. What wage rate must the monopsonist pay to attract this quantity of labor? c) What is the deadweight loss due to monopsony in this market?

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Microeconomics, 6th edition Solutions Manual

a) For this monopsonist 𝜟𝒘 𝜟𝑳 𝑴𝑬𝑳 = 𝟒𝑳 + 𝑳(𝟒) 𝑴𝑬𝑳 = 𝟖𝑳 𝑴𝑬𝑳 = 𝒘 + 𝑳

𝛥𝑄

b) The monopsonist will maximize profit at the point where 𝑴𝑹𝑃𝐿 = 𝑀𝐸𝐿 , where 𝑀𝑅𝑃𝐿 = 𝑃 𝛥𝐿 𝛥𝑄

In this example, 𝛥𝐿 = 0.5, so 𝑀𝑅𝑃𝐿 = 0.5𝑃. Since 𝑃 = 32, 𝑀𝑅𝑃𝐿 = 16. Now setting 𝑀𝑅𝑃𝐿 = 𝑀𝐸𝐿 implies

16 = 8L L=2 At this quantity of labor, 𝑤 = 4𝐿 = 8. c) In a competitive labor market, w = MRPL. So the competitive supply of labor satisfies 4L = 16 or L = 4, with w = 4L = 16. The deadweight loss due to monopsony is equal to area A in the graph below, or 0.5(16 – 8)(4 – 2) = 8. w

MEL = 8L w = 4L

16

MRPL = 16

A 8 2

4

L

11.29. A firm produces output, measured by Q, which is sold in a market in which the price P = 20, regardless of the size of Q. The output is produced using only one input, labor (measured by L); the production function is Q(L) = L. There are many suppliers of labor, and the supply schedule is w = 2L, where w is the wage rate. The firm is a monopsonist in the labor market. a) What wage rate will the monopsonist pay? b) How much extra profit does the firm earn when it pays labor as a monopsonist instead of paying the wage rate that would be observed in a perfectly competitive market?

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Chapter 11-24

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Microeconomics, 6th edition Solutions Manual

a) The monopsonist will choose L so that the marginal expenditure on labor equals the marginal revenue product of labor. Since the supply of labor is linear, the marginal expenditure will have the same vertical intercept (zero in this case) and twice the slope of the supply curve. Thus, MEL = 4L. The marginal revenue product of labor is just the market price (20) times the marginal product of labor (1); so MRPL = 20. The monopsonist would hire labor so that 20 = 4L; thus, L = 5. The monopsonist will pay a wage rate on the supply curve, so w = 2(5) = 10. (b) The monopsonist produces 5 units of output with the 5 units of labor it hires. Revenue = 20(5) = 100.

Costs are wL= 10(5) = 50. So profit = revenue minus cost = 100 – 50 = 50.

By contrast, if it operated as a perfectly competitive firm, it would produce where the marginal revenue product of labor (20) equals the wage rate on the supply curve for labor (2L), with L = 10.

With its 10 units of labor the firm would produce 10 units of out, and receive revenue or 20(10) = 200.

It would pay the wage rate = 2L = 2(10) = 20, so its cost would be wL = 20(10) = 200. Its profits would be zero.

Thus the firm increases its profit by 50 by acting as a monopsonist.

11.30. A firm produces output, measured by Q, which is sold in a market in which the price is 4, regardless of the size of Q. The output is produced using only one input, labor (measured by L); the production function is Q(L) = 10L. Labor is supplied by competitive suppliers, and everywhere along the supply curve the elasticity of supply is 3. The firm is a monopsonist in the labor market. What wage rate will it pay its workers?

Since the elasticity of supply of labor is constant, we can use the inverse elasticity rule for a monopsonist. [MRPL - w]/w = 1/eL,w . Each unit of labor produces 10 units of output (MPL = 10), each of which can be sold at a price of 4. Thus, MRPL = P(MPL) = 40.

The inverse elasticity rule then becomes [40 - w]/w = 1/3. Thus, w = 30.

11.31. National Hospital is the only employer of nurses in the country of Castoria, and it acts as a profit maximizing monopsonist in the market for nursing labor. The marginal revenue product for nurses is w = 50 - 2N, where w is the wage rate and N is the number of nurses employed (measured in hundreds of nurses). Nursing services are provided according to the supply schedule w = 14 + 2N. a) How many nurses does National Hospital employ, and what wage will National pay its nurses? b) What is the deadweight loss arising from monopsony?

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Microeconomics, 6th edition Solutions Manual

a) Since the supply of nurses is linear, the marginal expenditure will have the same vertical intercept (14 in this case) and twice the slope of the supply curve. The marginal expenditure on nurses is represented by MEN = 14 + 4N. The monopsonist would choose N so that 14 + 4N = 50 – 2N. Thus N = 6.

To determine the wage rate paid, find out w on the supply curve when N=6; w = 14 + 2(6) = 26. b) If the market for nurses were competitive, the market clearing quantity of nurses would be determined by the intersection of the marginal revenue product of nurses and the supply of nurses: 50 – 2N = 14 + 2N. Thus, N = 9 and w = 32.

However, the monopsonist hires N = 6. The value of w on the marginal revenue product curve is w = 50 – 2(6) = 38, and the value of w on the supply curve is w= 14 + 2(6) = 26.

DWL = ½(38 – 26)(9 – 6) = 18 (measured in hundreds because N is in hundreds.)

11.32. A hospital is a monopsonist in the market for nursing services in a city. At its profitmaximizing input combination, the elasticity of supply for nursing services is +1. What does this tell you about the magnitude of the marginal revenue product of labor relative to the wage that the firm is currently paying its workers?

We can use the IEPR condition for monopsony:

𝑀𝑅𝑃𝐿 −𝑤 𝑤

=∈

1

𝐿,𝑤

. Since labor supply is unit elastic,

it means that 𝑀𝑅𝑃𝐿 − 𝑤 = 𝑤 or that 𝑀𝑅𝑃𝐿 = 2𝑤. So the marginal revenue product of labor is twice as much as the wage rate.

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Chapter 11-26

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 12 Capturing Surplus Solutions to Review Questions 1. Why must a firm have at least some market power to price discriminate?

If a firm has no market power it will not be able to price discriminate to increase profits. Without market power the firm is a price taker and has no ability to set different prices for different units of output. As the firm attempts to set higher prices for some units, consumers will simply purchase elsewhere if the firm has no market power.

2. Does a firm need to be a monopolist to price discriminate?

No, a firm does not need to be a monopolist to price discriminate. The firm simply needs to have market power and face a downward sloping demand curve.

3. Why must a firm prevent resale if it is to price discriminate successfully?

If the firm cannot prevent resale, then customers who buy at a low price can act as middlemen and resell the goods to customers willing to pay more. In this case the firm won’t earn the additional surplus; the middlemen will capture the surplus instead of the firm.

4. What are the differences among first-degree, second degree, and third-degree price discrimination?

With first-degree price discrimination, the firm charges each consumer a price close to the consumer’s maximum willingness to pay. In this way, the firm is able to extract virtually all the available surplus for itself. With second-degree price discrimination, the firm offers quantity discounts. This induces some customers to purchase more than they would if all units were priced the same. With third-degree price discrimination the firm charges different prices to different market segments. For example, the firm might charge a lower price to students and senior citizens to induce them to purchase when they might not otherwise.

5. With first-degree price discrimination, why is the marginal revenue curve the same as the demand curve?

With perfect first-degree price discrimination, the marginal revenue and demand curves are the same. This is because with perfect first-degree price discrimination the firm charges each consumer their maximum willingness to pay, as measured by the demand curve. Therefore, the demand curve represents the additional revenue the firm will bring in for each additional unit it sells, or marginal revenue. Note, with perfect first-degree price discrimination the firm charges each individual a different price, so as it lowers its price to gain marginal customers it doesn’t lose revenue on the inframarginal customers who would have paid a higher price.

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Chapter 12-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

6. How large will the deadweight loss be if a profit maximizing firm engages in perfect firstdegree price discrimination?

With perfect first-degree price discrimination there is no deadweight loss. The outcome is economically efficient because every consumer who purchases the good has a willingness to pay meeting or exceeding the marginal cost of production and every consumer who does not receive the good has a willingness to pay below marginal cost. Note, however, that while there is no deadweight loss, all of the surplus is captured by the firm leaving no consumer surplus.

7. What is the difference between a uniform price and a nonuniform (nonlinear) price?

Give an example of a nonlinear price.

With a uniform price the firm sells every unit to every consumer at the same price. With nonuniform or nonlinear pricing, the firm charges different prices for different units of output. For example, a telephone company might charge each user a subscription fee of $10 and then a usage fee of $0.05 per call. Nonlinear pricing is one type of second-degree price discrimination. With second-degree price discrimination, the firm charges a lower average price to consumers who are willing to buy large quantities of the good.

8. Suppose a company is currently charging a uniform price for its two products, creamy and crunchy peanut butter. Will third-degree price discrimination necessarily improve its profit? Would the firm ever be worse off with price discrimination?

The firm could never do worse with third-degree price discrimination than without it because the firm could always charge the uniform price and earn the same profit. So long as the firm can reliably identify a difference in willingness to pay among its customers and prevent resale, thirddegree price discrimination should in fact increase its profits.

9. How might screening help a firm price discriminate? Give an example of screening and explain how it works.

Screening is a mechanism that allows a firm to sort consumers according to their willingness to pay or their price elasticity of demand. For example, the firm might screen the consumers based on some observable consumer characteristic such as age. This allows the firm to charge a price closer to the consumer’s willingness to pay given that the characteristic, such as age, is correlated with willingness to pay. In addition, because the characteristic is observable, it is possible for the firm to prevent arbitrage.

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Chapter 12-2

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Solutions Manual

10. Why might a firm try to implement a tying arrangement? What is the difference between tying and bundling?

A firm implementing a tying arrangement tries to force a consumer to purchase another product from the firm in addition to the original product when it is unable to price discriminate on the original product. In this way the firm can earn extranormal profits on the supplemental product that it could not on the original product. An example would be a computer manufacturer that allowed only its software to operate on the computer. In this way, while the firm might not be able to price discriminate and earn extra profit on the sale of the computer, it can force the consumer to only purchase software from the firm and charge a higher price for the software than it would if the consumer were allowed to use any software. In this way the firm increases its profits.

There is a difference between tying and bundling. Tying requires that if you buy product A, you must also buy product B. However, it’s possible to purchase product B (the tied product) individually. Bundling requires that A and B can only be purchased together, in a package.

11. How might bundling increase a firm’s profits? When is bundling not likely to increase profits?

Bundling can increase the firm’s profits when demands for products are negatively correlated.

This means that consumers who are willing to pay more for one good than another consumer are willing to pay less for another good than another consumer. By bundling the products, the consumers are induced to buy both products, allowing the firm to earn profits from both consumers on both goods, increasing total profits. Bundling is not likely to increase profits when demands for products are not negatively correlated. If one consumer is willing to pay more for both goods, the firm is unlikely to be able to increase profits by bundling goods.

12. Even if a monopolist knows that advertising shifts the demand curve for its product to the right, why might it decide not to advertise at all? If it does advertise, what factors determine how much advertising it will do?

While advertising will shift the demand curve to the right, advertising is costly. A firm would not choose to advertise if the increase in total revenue associated with the demand curve shifting to the right did not exceed the advertising costs necessary to induce the shift in demand. The amount of advertising a firm will purchase will depend primarily on the firm’s price elasticity of demand and on the firm’s advertising elasticity of demand. In particular, the advertising-to-sales ratio should be 𝜀𝑄,𝐴 𝐴 =− 𝑃𝑄 𝜀𝑄,𝑃

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Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 12.1. Which of the following are examples of first-degree, second-degree, or third-degree price discrimination? a) The publishers of the Journal of Price Discrimination charge a subscription price of $75 per year to individuals and $300 per year to libraries. b) The U.S. government auctions off leases on tracts of land in the Gulf of Mexico. Oil companies bid for the right to explore each tract of land and to extract oil. c) Ye Olde Country Club charges golfers $12 to play the first 9 holes of golf on a given day, $9 to play an additional 9 holes, and $6 to play 9 more holes. d) The telephone company charges you $0.10 per minute to make a long-distance call from Monday through Saturday and $0.05 per minute on Sunday. e) You can buy one computer disk for $10, a pack of 3 for $27, or a pack of 10 for $75. f) When you fly from New York to Chicago, the airline charges you $250 if you buy your ticket 14 days in advance, but $350 if you buy the ticket on the day of travel. a) Third degree – the firm is charging a different price to different market segments, individuals and libraries. b) First degree – each consumer is paying near their maximum willingness to pay. c) Second degree – the firm is offering quantity discounts. As the number of holes played goes up, the average expenditure per hole falls. d) Third degree – the firm is charging different prices for different segments. Business customers (M-F) are being charged a higher price than those using the phone on Sunday, e.g., family calls. e) Second degree – the firm is offering a quantity discount. f) Third degree – the airline is charging different prices to different segments. Those who can purchase in advance pay one price while those who must purchase with short notice pay a different price.

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12.2. Suppose a profit-maximizing monopolist producing Q units of output faces the demand curve P = 20 - Q. Its total cost when producing Q units of output is TC = 24 + Q2.

The fixed cost is sunk, and the marginal cost curve is MC = 2Q. a) If price discrimination is impossible, how large will the profit be? How large will the producer surplus be? b) Suppose the firm can engage in perfect first-degree price discrimination. How large will the profit be? How large is the producer surplus? c) How much extra surplus does the producer capture when it can engage in first-degree price discrimination instead of charging a uniform price? a) If price discrimination is impossible the firm will set 𝑀𝑅 = 𝑀𝐶.

20 − 2Q = 2Q Q=5

At this quantity, price will be 𝑃 = 15, total revenue will be 𝑇𝑅 = 75, total cost will be 𝑇𝐶 = 49, and profit will be 𝜋 = 26. Producer surplus is total revenue less non-sunk cost, or, in this case, total revenue less variable cost. Thus producer surplus is 75 − 52 = 50. b) With perfect first-degree price discrimination the firm sets 𝑃 = 𝑀𝐶 to determine the level of output.

20 − Q = 2Q Q = 6.67 The price charged each consumer, however, will vary. The price charged will be the consumer’s maximum willingness to pay and will correspond with the demand curve. Total revenue will be the area underneath the demand curve out to Q = 6.67 units, or 0.5(20 – 13.33)(6.67) + 13.33(6.67) = 111.16. Since the firm is producing a total of 6.67 units, total cost will be 𝑇𝐶 = 68.49. Profit is then 𝜋 = 42.67, while producer surplus is revenue less variable cost, or 111.16 − 6.672 = 66.67. c) By being able to employ perfect first-degree price discrimination the firm increases profit and producer surplus by 16.67.

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Chapter 12-5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.3. Suppose a monopolist producing Q units of output faces the demand curve P = 20 - Q. Its total cost when producing Q units of output is TC = F + Q2, where F is a fixed cost. The marginal cost is MC = 2Q. a) For what values of F can a profit-maximizing firm charging a uniform price earn at least zero economic profit? b) For what values of F can a profit-maximizing firm engaging in perfect first-degree price discrimination earn at least zero economic profit? a) With demand 𝑃 = 20 − 𝑄, 𝑀𝑅 = 20 − 2𝑄. A profit-maximizing firm charging a uniform price will set 𝑀𝑅 = 𝑀𝐶.

20 − 2Q = 2Q Q=5

At this quantity, price will be 𝑃 = 15. At this price and quantity profit will be 𝜋 = 15(5) − (𝐹 + 52 ) 𝜋 = 50 − 𝐹 Therefore, the firm will earn positive profit as long as 𝐹 < 50. b) A firm engaging in first-degree price discrimination with this demand will produce where demand intersects marginal cost: 20 – Q = 2Q or Q = 6.67 units. Its total revenue will be the area underneath the demand curve out to Q = 6.67 units; 𝑇𝑅 = .5(20 − 13.33)(6.67) + 13.33(6.67) = 111.16. Profit will be 𝜋 = 111.16 − (𝐹 + 6.672 ) 𝜋 = 66.67 − 𝐹 Therefore, profit will be positive as long as 𝐹 < 66.67. Comparing the solution to parts (a) and (b), for values of F between 50 and 66.67 the firm would be unwilling to operate unless it is able to practice first-degree price discrimination.

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Chapter 12-6

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Solutions Manual

12.4. A firm serving a market operates with total variable cost TVC = Q2. The corresponding marginal cost is MC = 2Q. The firm faces a market demand represented by P = 40 - 3Q. a) Suppose the firm sets the uniform price that maximizes profit. What would that price be? b) Suppose the firm were able to act as a perfect first degree price-discriminating monopolist. How much would the firm’s profit increase compared with the uniform profitmaximizing price you found in (a)? a) The firm would maximize profit by producing until MR = MC, or 40 – 6Q = 2Q. Thus Q = 5 and the profit-maximizing price is P = 25. With MC = 2Q and no fixed costs, its total costs are C = Q2, so  = 25(5) – 52 = 100. b) With perfect first degree price discrimination, the firm will charge a price on the demand curve for all units up to the quantity at which the demand curve intersects the marginal cost curve. The demand curve intersects the marginal cost curve when 40 – 3Q = 2Q, or when Q = 8.

Total revenue will be the area under the demand curve, or 0.5(40 – 16)8 + 16(8) = 224. Total variable cost is the area of the triangle under its marginal cost curve up to the quantity produced, that is, 0.5(16)(8) = 64. Economic profit will be 224 – 64 = 160. So by price discriminating, the firm will be able to earn an extra profit of (160 – 100) = 60.

12.5. A natural monopoly exists in an industry with a demand schedule P = 100 - Q. The marginal revenue schedule is then MR = 100 - 2Q. The monopolist operates with a fixed cost F, and a total variable cost TVC = 20Q. The corresponding marginal cost is thus constant and equal to 20. a) Suppose the firm sets a uniform price to maximize profit. What is the largest value of F for which the firm could earn zero profit? b) Suppose the firm is able to engage in perfect first degree price discrimination. What is the largest value of F for which the firm could earn zero profit? a) When the firm sets a uniform price, it sets MR = MC: 100 – 2Q = 20. The quantity that maximizes profit is therefore Q = 40. The profit maximizing uniform price is P = 100 – Q = 100 – 40 = 60. Profit is PQ – F -20Q = (60)(40) – F – (20)(40) = 1600 – F. So the firm could earn at least zero economic profit as long as F < 1600. b) With perfect first degree price discrimination, the firm will charge a price on the demand curve for all units up to the quantity at which the demand curve intersects the marginal cost curve. The demand curve intersects the marginal cost curve when 100 – Q = 20, or when Q = 80.

Total revenue will be the area under the demand curve, or 0.5(100 – 20)80 + 20(80) = 4800.

Total cost will be F + 20(80) = F + 1600. Economic profit will be 4800 – F – 1600 = 3200 – F. So the firm will be able to earn at least zero economic profit as long as F < 3200.

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Chapter 12-7

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.6. Suppose a monopolist is able to engage in perfect first-degree price discrimination in a market. It can sell the first unit at a price of 10 Euros, the second at a price of 9 Euros, the third at a price of 8 Euros, the fourth at a price of 7 Euros, the fifth at a price of 6 Euros, and the sixth at a price of 5 Euros. It must sell whole units, not fractions of units. a) What is the firm’s total revenue when it produces two units? b) What is the total revenue when it produces three units? c) What is the relationship between the price of the third unit and the marginal revenue of the third unit? d) What is the relationship between the price and the marginal revenue of the fourth unit? a) The firm’s total revenue when it produces 2 units is 19 Euros (10 from the first unit and 9 from the second). b) The firm’s total revenue when it produces 3 units is 27 Euros (10 from the first unit, 9 from the second, and 8 from the third). c) They are equal, as we would expect with perfect first-degree price discrimination. The price of the third unit is 8 Euros. The marginal revenue of the third unit is also 8 Euros (27 Euros – 19 Euros). d) By similar reasoning, the price and the marginal revenue of the fourth unit will also be equal to each other (in this case 7 Euros).

12.7. Suppose the monopolist in Problem 12.6 incurs a marginal cost of 5.50 Euros for every unit it produces. The firm has no fixed costs. a) How many units will it produce if it wants to maximize its profit? (Remember, it must produce whole units.) b) What will its profit be when it maximizes profit? c) What will the deadweight loss be when it maximizes profit? Explain. a) The firm will want to produce a unit when the marginal revenue of the next unit is greater than the marginal cost. Since the firm can implement perfect first-degree price discrimination, the marginal revenue of the next unit equals the price at which it can sell the next unit. It will thus sell 5 units; the marginal revenue of the fifth unit is six euros, which exceeds the marginal cost.

Since the marginal revenue of the sixth (5 euros) does not cover the marginal cost, the firm will not want to produce the sixth unit. b) The profit from each unit will be the difference between the price of that unit and the marginal cost. Total profit will be the sum of the profits from all 5 units it produces.

Total profit = (10 – 5.50) from unit 1 + (9 – 5.50) from unit 2+ (8 – 5.50) from unit 3 + (7 – 5.50) from unit 4 + (6 – 5.50) from unit 5 = (4.50 + 3.50 + 2.50 + 1.50 + 0.50) euros = 12.50 euros.

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Chapter 12-8

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

c) There will be no deadweight loss. The first five units will be produced and sold; this is economically efficient because these are the units for which price (the value the consumer places on that unit) exceeds marginal cost. The additional units will not be sold, and that is also efficient because the price is less than the marginal cost of production.

12.8. Fore! is a seller of golf balls that wants to increase its revenues by offering a quantity discount. For simplicity, assume that the firm sells to only one customer and that the demand for Fore!’s golf balls is P = 100 - Q. Its marginal cost is MC = 10. Suppose that Fore! sells the first block of Q1 golf balls at a price of P1 per unit. a) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 20 and P1 = 80. b) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 30 and P1 = 70. c) Find the profit-maximizing quantity and price per unit for the second block if Q1 = 40 and P1 = 60. d) Of the three options in parts (a) through (c), which block tariff maximizes Fore!’s total profits? a) We can represent the marginal willingness to pay for each unit beyond Q1 = 20 as P = 100 – (20 + Q2) = 80 – Q2. The associated marginal revenue is then MR = 80 – 2Q2, so the profit maximizing second block is MR = MC: 80 – 2Q2 = 10. Thus Q2 = 35 and P2 = 80 – 35 = 45. So the firm sells the first 20 units at a price of $80 apiece, while the firm sells any quantity above 20 at $45 apiece. The firm’s total profit will be (80 – 10)*20 + (45 – 10)*35 = $2625. b) The marginal willingness to pay for each unit beyond Q1 = 30 is P = 70 – Q2. So MR = 70 – 2Q2 and we have MR = MC: 70 – 2Q2 = 10. Thus Q2 = 30 and P2 = 40. The firm’s total profit will be (70 – 10)*30 + (40 – 10)*30 = $2700. c) The marginal willingness to pay for each unit beyond Q1 = 40 is P = 60 – Q2. So MR = 60 – 2Q2 and we have MR = MC: 60 – 2Q2 = 10. Thus Q2 = 25 and P2 = 35. The firm’s total profit will be (60 – 10)*40 + (35 – 10)*25 = $2625. d) The option in part (b) yields the highest profits, of $2700.

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Chapter 12-9

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.9. Consider the manufacturer of golf balls in Problem 12.8. The firm faces the demand curve P = 100 - Q, and operates with a marginal cost of 10 for all units produced. Among all the possible block tariffs (with two blocks), what block tariff structure will maximize profit? In other words, what choices of P1, Q1 for the first block and P2, Q2 for the second block will maximize profit?

To answer this question, we follow the procedure outlined in the text’s discussion of Figure 12.5.

In this problem producer surplus will equal profit because there are no sunk fixed costs – in fact, no fixed costs at all.

Using the monopoly midpoint rule, we know that the optimal quantity in the second block will be half way between Q1 and 90; thus Q2= (Q1 + 90)/2.

Total profits will be P1Q1 + P2 (Q2 – Q1 ) – 10Q2 = (100 – Q1)Q1 + (100-Q2)(Q2 – Q1 ) – 10Q2 –Q12 +90Q2 +Q1Q2 – Q22 = -Q12+Q2(90 +Q1 - Q2) Substituting Q2= (Q1 + 90)/2 into the expression for profit leads to: Total profits = -Q12+[(Q1 + 90)/2]2 = [-3Q12 + 180Q1 + 8100]/4 = -3(Q1- 30)2/4 + 2700.

Thus, profits are maximized when Q1 =30 , with P1 = 70 in the first block, and when Q2 =60 , with P2 = 40 in the second block. Total profit will be 2700, the shaded area in the graph below.

P 100

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P1 =70

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An individual consumer's demand for golf balls B

F

P2 =40

K

Z

L

O

G

Y

E

60 Q2

90

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10

30 Q1

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Marginal Cost Q

N

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Chapter 12-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.10. Suppose that you are a monopolist who produces gizmos, Z, with the total cost function C(Z) = F + 50Z, where F represents the firm’s fixed cost. Your marginal cost is MC = 50. Suppose also that there is only one consumer in the market for gizmos, and she has the demand function P = 60 - Z. a) If you use a constant per-unit price for gizmos, what price maximizes your profits? What is the smallest value of F such that you could earn positive profits at this price? b) Suppose instead that you charge a per-unit price equal to marginal cost, that is, P = MC = 50. How many units would the customer purchase at this price? Illustrate your answer in a graph (featuring the individual demand curve and marginal cost). c) Now consider charging the customer a “subscription fee” of S in addition to a usage fee.

If you set the usage fee as in part (b), what is the largest fixed fee you could charge the consumer, while ensuring that she is willing to participate in this market? d) For what values of F will you be able to earn positive profits if you follow the pricing strategy you outlined in part (c)? How does this relate to your answer in part (a)? e) Suppose now that there are N consumers in the market for gizmos, each with the individual demand function P = 60 - Z. Expressing your answer in terms of N, how large can the fixed costs F be for you to still earn positive profits if you use the above nonlinear pricing strategy. a) Setting MR =MC, we have 60 – 2Z = 50 or Z = 5, with P = 55. You earn π = 55*5 – (K + 50*5) = 25 – K, so profits are positive only if K < 25. b) P = MC = 50 implies the customer purchases Z = 10 units. See the graph below.

P 60

Demand

50

MC

10

Q

c) At P = 10, the customer gets CS = ½*(60 – 50)*10 = 50. Thus, the largest fixed-fee you could charge her, while ensuring that she is willing to participate in this market, is F = CS = 50. d) Now, your revenues are R = 50 + 50*5 = 300, so profits are π = 300 – (K + 50*5) = 50 – K. Now the firm can operate profitably so long as K < 50. By enabling the firm to extract more surplus, (here, second-degree) price discrimination allows you to operate in a market where sunk fixed costs range as high as K = 50, whereas using standard monopoly pricing the firm wouldn’t participate unless K < 25.

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Chapter 12-11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

e) For N customers, your profits are π = N*300 – (K + 50*5*N) = 50*N – K, so profits are positive only when K < 50*N. 12.11. In part (c) of Learning-By-Doing Exercise 12.3, we suggested that the profitmaximizing structure for the first and second blocks for Softco is something other than the pricing structure we determined in part (b), selling the first 60 units at a price of $40 apiece, and selling any quantity above 60 at $25 apiece. Find the structure that maximizes profit.

Suppose the optimal structure is to sell the first Q1 units at the price P1, and any additional units at P2. First, we know that P1 = 70 – 0.5Q1. Second, if price is equal to marginal cost, then consumers would demand Q = 120 units. So in the optimal block tariff, the quantity sold in the second block will be halfway between Q1 and 120; that is, Q2 = 0.5(Q1 + 120). Third, we can then say that the optimal price P2 must satisfy P2 = 70 – 0.5Q2. Expressing in terms of Q1, that implies P2 = 40 – 0.25Q1. With a marginal cost of 10, the firm’s producer surplus is thus

PS = P1Q1 + P2 (Q2 − Q1 ) − 10Q2

= (70 − 0.5Q1 )Q1 + (40 − 0.25Q1 )0.5(Q1 + 120) − Q1  − 10 * 0.5(Q1 + 120) =−

3 (Q1 − 40)2 + 2400 8

Since the first term is negative, PS is maximized (at 2400) when Q1 = 40. Therefore, these units should be priced at P1 = 50. The optimal second block involves P2 = 30 and Q2 = 80. That is, the firm will sell the first 40 units for $50 apiece and a second 40 units (because Q2 – Q1 = 40) at $30 apiece.

12.12. Consider a market with 100 identical individuals, each with the demand schedule for electricity of P = 10 - Q. They are served by an electric utility that operates with a fixed cost 1,200 and a constant marginal cost of 2. A regulator would like to introduce a two-part tariff, where S is a fixed subscription charge and m is a usage charge per unit of electricity consumed. How should the regulator set S and m to maximize the sum of consumer and producer surplus while allowing the firm to earn exactly zero economic profit?

To maximize the sum of consumer and producer surplus, the regulator must set the usage charge m = 2; this will induce consumers to buy units of electricity as long as their willingness to pay is at least as high as the marginal cost of providing electricity service. This means that each consumer will buy 8 units of electricity. There will be zero deadweight loss in the market.

If there were no subscription charge, each consumer would realize a consumer surplus of 0.5*(10 – 2)*8 = 32. This means that each consumer will be willing to buy electricity as long as the subscription charge is less than 32. With 100 consumers, the electric utility can then charge each customer a subscription fee of $12 to cover its fixed costs of $1200, leaving each consumer with a consumer surplus of 32 – 12 = 20. So the total revenue for the firm will be the sum of the revenue from the subscription charge (1200) and the revenue from the usage charge 100*8*2 = 200. Total revenue will just cover total cost, and the firm will earn zero economic profit.

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Chapter 12-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.13. A monopolist faces two market segments. In each market segment, the demand curve is of the constant elasticity form. In market segment 1, the price elasticity of demand is -3, while in market segment 2, the price elasticity of demand is -1.5. The monopolist has a constant marginal cost of $5 per unit, which is the same in each market segment. What is the monopolist’s profit maximizing price in each segment?

We use the inverse elasticity rule to determine the profit-maximizing prices: P1 = MC[1/(1 + 1/1)] = 5[1/(1+1/(-3))] = 5(3/2) = 7.5 P2 = MC[1/(1 + 1/2)] = 5[1/(1+1/(-1.5))] = 5(3) = 15 12.14. Suppose that Acme Pharmaceutical Company discovers a drug that cures the common cold. Acme has plants in both the United States and Europe and can manufacture the drug on either continent at a marginal cost of 10. Assume there are no fixed costs. In Europe, the demand for the drug is QE = 70 - PE, where QE is the quantity demanded when the price in Europe is PE. In the United States, the demand for the drug is QU = 110 PU, where QU is the quantity demanded when the price in the United States is PU. a) If the firm can engage in third-degree price discrimination, what price should it set on each continent to maximize its profit? b) Assume now that it is illegal for the firm to price discriminate, so that it can charge only a single price P on both continents. What price will it charge, and what profits will it earn? c) Will the total consumer and producer surplus in the world be higher with price discrimination or without price discrimination? Will the firm sell the drug on both continents? a) With third-degree price discrimination the firm should set 𝑀𝑅 = 𝑀𝐶 in each market to determine price and quantity. Thus, in Europe setting 𝑀𝑅 = 𝑀𝐶 70 − 2QE = 10 QE = 30

At this quantity, price will be 𝑃𝐸 = 40. Profit in Europe is then 𝜋𝐸 = (𝑃𝐸 − 10)𝑄𝐸 = (40 − 10)30 = 900. Setting 𝑀𝑅 = 𝑀𝐶 in the US implies 110 − 2QU = 10 QU = 50

At this quantity price will be 𝑃𝑈 = 60. Profit in the US will then be 𝜋𝑈 = (𝑃𝑈 − 10)𝑄𝑈 = (60 − 10)50 = 2500. Total profit will be 𝜋 = 3400.

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Chapter 12-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) If the firm can only sell the drug at one price, it will set the price to maximize total profit. The total demand the firm will face is 𝑄 = 𝑄𝐸 + 𝑄𝑈 . In this case 𝑄 = 70 − 𝑃 + 110 − 𝑃 𝑄 = 180 − 2𝑃 The inverse demand is then 𝑃 = 90 − 0.5𝑄. Since 𝑀𝐶 = 10, setting 𝑀𝑅 = 𝑀𝐶 implies 90 − Q = 10 Q = 80

At this quantity price will be 𝑃 = 50. If the firm sets price at 50, the firm will sell 𝑄𝐸 = 20 and 𝑄𝑈 = 60. Profit will be 𝜋 = 50(80) − 10(80) = 3200. c) The firm will sell the drug on both continents under either scenario. If the firm can price discriminate, total consumer surplus will be 0.5(70 – 40)30 + 0.5(110 – 60)50 = 1700 and producer surplus (equal to profit) will be 3400. Thus, total surplus will be 5100. If the firm cannot price discriminate, consumer surplus will be 0.5(70 – 50)20 + 0.5(110 – 50)60 = 2000 and producer surplus will be equal to profit of 3200. Thus, total surplus will be 5200.

12.15. Consider Problem 12.14 with the following change. Suppose the demand for the drug in Europe declines to QE = 30 - PE. If the firm cannot price discriminate, will it be in the firm’s interest to sell on both continents?

Let’s start by assuming that the optimal uniform price (i.e., no price discrimination) is one at which the firm would sell in both markets. If the firm cannot price discriminate then 𝑄 = 𝑄𝐸 + 𝑄𝑈 𝑄 = 30 − 𝑃 + 110 − 𝑃 𝑄 = 140 − 2𝑃 Inverse demand is P = 70 – 0.5Q. Setting 𝑀𝑅 = 𝑀𝐶 implies 70 − Q = 10 Q = 60

At this quantity, price will be 𝑃 = 40. This price exceeds the choke price in Europe, so the firm will not be able to sell any units in Europe. Since the firm will not sell any units in Europe, the firm should set its marginal cost equal to the marginal revenue in the US market: MRU = 110 – 2QU = MC = 10, implying QU = 50 and PU = 60.

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Chapter 12-14

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.16. Consider Problem 12.14 with the following change. Suppose the demand for the drug in Europe becomes QE = 55 - 0.5PE. Will third-degree price discrimination increase the firm’s profits?

Without price discrimination, if 𝑄𝐸 = 55 − .5𝑃𝐸 , then 𝑄 = 𝑄𝐸 + 𝑄𝑈 is 𝑄 = 55 − .5𝑃 + 110 − 𝑃 𝑄 = 165 − 1.5𝑃 2

Inverse demand is P = 110 – 3Q. Setting 𝑀𝑅 = 𝑀𝐶 implies 110 − 4 3 Q = 10 Q = 75

At this quantity the firm will charge a price of 𝑃 = 60. At this price the firm will sell 25 units in Europe and 50 units in the US and earn a total profit of 𝜋 = 3750.

To maximize profits with third-degree price discrimination the firm should set 𝑀𝑅 = 𝑀𝐶 in each market. In the US 110 − 2QU = 10 Q = 50

At this quantity the firm will charge a price 𝑃𝑈 = 60 and profits in the US will be 𝜋 = 2500.

Setting 𝑀𝑅 = 𝑀𝐶 in Europe implies 110 − 4Q = 10 Q = 25

At this quantity the firm will charge a price 𝑃𝐸 = 60 and profits in Europe will be 𝜋 = 1250.

Total profits with price discrimination will equal the sum of the profits from each market 𝜋 = 𝜋𝐸 + 𝜋𝑈 = 1250 + 2500 = 3750. This is equal to the profits without price discrimination, so the firm gains no advantage by being able to price discriminate. With these demands, the firm would charge the same price in each market regardless of whether it could price discriminate or not.

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Chapter 12-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.17. Think about the problem that Acme faces in Problem 12.14. Consider any demand curves for the drug in Europe and in the United States. Will its profits ever be lower with third-degree price discrimination than they would be if price discrimination were impossible?

A firm could never do worse with third-degree price discrimination than without. Why? Because with third-degree price discrimination the firm is trying to find a price to charge each market to increase profits above the profits the firm would earn if it charged each market segment the same price. Profits can never be worse because the firm could always choose the solution where it charges all segments the same price and earn profit equal to the non-discriminating solution. If the firm varies the solution from this point, it will only do so if the profits will increase.

Therefore, profits for the discriminating firm could never be worse than the profits for the nondiscriminating firm.

12.18. There is another way to solve Learning-By-Doing Exercise 12.5. Recall that marginal revenue can be written as MR = P + (ΔP/ΔQ)Q. By factoring out P, we can write Since third-degree price discrimination means that marginal cost equals marginal revenue in each market segment, the profit-maximizing regular and vacation fares will be determined

by MRR = MRV = MC. (Remember the marginal cost of both classes of service is assumed to be the same in the exercise.) Thus use this relationship to verify the answer given in the exercise.

For this problem, Learning-By-Doing Exercise 12.5 gives 𝜀𝑄𝑅,𝑃𝑅 = −1.15 and 𝜀𝑄𝑉,𝑃𝑉 = −1.52.

Verifying the relationship in this problem implies 1  1    PR 1 +  = PV 1 +   −1.15   −1.52  0.130 PR = 0.342 PV PR 0.342 = PV 0.130

The same solution as that given in Learning-By-Doing Exercise 12.5.

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Chapter 12-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.19. J. Cigliano (“Price and Income Elasticities for Airline Travel: The North Atlantic Market,” Business Economics, September 1980) estimated the price elasticity of demand for regular (full-fare) travel in coach class in the North Atlantic market to be ϵB = -1.3. He also found the price elasticity of demand for excursion (vacation) travel to be about ϵV = 1.8. Suppose Transatlantic Airlines faces these price elasticities of demand, and that the elasticities are constant; that is, they do not vary with price. Since both are coach fares, you may also assume that the marginal cost of service is about the same for business and vacation travelers. Suppose an airline facing these demand elasticities wants to set PR (the price of a round-trip ticket to regular business travelers) and PV (the price of a round-trip ticket to vacation travelers) to maximize profit. What prices should the firm charge if the marginal cost of a round trip is 200?

Use the inverse elasticity pricing rule to find the profit maximizing level of each price. For business travelers 𝑃𝐵 − 200 1 =− which implies 𝑃𝐵 ≈ 867. 𝑃𝐵 −1.3 For vacation travelers 𝑃𝑉 − 200 1 =− which implies 𝑃𝑉 = 450. 𝑃𝑉 −1.8 12.20. La Durazno is the only resort hotel on a small desert island off the coast of South America. It faces two market segments: bargain travelers and high-end travelers. The demand curve for bargain travelers is given by Q1 = 400 - 2P1. The demand curve for highend travelers is given by Q2 = 500 - P2. In each equation, Q denotes the number of travelers of each type who stay at the hotel each day, and P denotes the price of one room per day.

The marginal cost of serving an additional traveler of either type is $20 per traveler per day. a) Under the assumption that there is a positive demand from each type of traveler, what is the equation of the overall market demand curve facing the resort? b) What is the profit-maximizing price under the assumption that the resort must set a uniform price for all travelers? For the purpose of this problem, you may assume that at the profit-maximizing price, both types of travelers are served. Under the uniform price, what fraction of customers are bargain travelers, and what fraction are high end? c) Suppose that the resort can engage in third-degree price discrimination based on whether a traveler is a high-end traveler or a bargain traveler. What is the profitmaximizing price in each segment? Under price discrimination, what fraction of customers are bargain travelers and what fraction are high end? d) The management of La Durazno is probably unable to determine, just from looking at a customer, whether he or she is a high-end or bargain traveler. How might La Durazno

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Chapter 12-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

screen its customers so that it can charge the profit-maximizing discriminatory prices you derived in part (c)? a) In the range of prices in which consumers in both market segments purchase at a common price P, the total market demand curve is Q1 + Q2 = 400 – 2P + 500 – P, or Q = 900 – 3P. (b) If Q = 900 – 3P, then the inverse market demand curve is P = 300 – (1/3)Q. Marginal revenue is thus MR = 300 – (2/3)Q, and so the profit-maximizing quantity is found by solving 300 – (2/3)Q = 20, or Q = 420. The profit-maximizing price is thus: P = 300 – (1/3)(420) = 160.

At this price, Q1 = 400 – 2(160) = 80 bargain travelers stay at the resort, and Q2 = 500 – 160 = 340 high-end travelers stay at the resort. Thus, about 19 percent of the resort’s guests are bargain travelers. c) We find the profit maximizing quantity and price in each market segment as follows: Bargain travelers: Q1 = 400 – 2P1  P1 = 200 – ½ Q1.

This implies that marginal revenue is: MR1 = 200 – Q1.

Equating marginal revenue to marginal cost gives us: 200 – Q1 = 20, or Q1 = 180.

This implies P1 = 200 – ½ (180) = 110.

High-end travelers: Q1 = 500 – P1  P1 = 500 – Q1.

This implies that marginal revenue is: MR1 = 500 – 2Q1.

Equating marginal revenue to marginal cost gives us: 500 – 2Q1 = 20, or Q1 = 240.

This implies P1 = 500 – (240) = 260.

The percentage of bargain travelers is now 180/(180+240) = 0.428 or about 43 percent. d) There are a number of ways the resort can screen passengers.

If there is a correlation between age and membership in the bargain segment (perhaps elderly individuals are more price sensitive), then the resort could screen on the basis of age.

If there is a correlation between a traveler’s willingness to pay for a room in the resort and his/her propensity to purchase complementary resort services such as spa treatments or workouts with a personal trainer. If so, the resort may set a uniform price for rooms but offer the complementary services for sale at a high mark-up to allow the resort to, in effect, collect a higher “overall” price (room charge plus other services) from high-end travelers.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 12-18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

If there is a correlation between the propensity of a traveler to mail in rebate cards or coupons and the traveler’s membership in the bargain segment, the resort could screen by offering discounts to those travelers who, upon returning home, mail in a rebate card or coupon.

Finally, the resort could offer a few rooms on Priceline.com. This is a selling channel that appeals disproportionately to price sensitive travelers.

12.21. A pipeline transports gasoline from a refinery at point A to destinations at R and T. The marginal cost of transporting gasoline to each destination is MC = 2. The pipeline has a fixed cost of 160. The demand curve for the transportation of gasoline from A to R is QR = 100 - 10PR, where QR is the number of units transported when PR is the transport price per unit. The demand for pipeline movements from A to T will be 20 units as long as if the customers at T will purchase gasoline from another source, buying no gasoline shipped through the pipeline. These demand curves are shown below. a) If this firm was unable to engage in price discrimination (so that it can only choose a single P for the two markets), what would the profit-maximizing tariff be? What level of profit would the firm realize? b) If this firm were able to implement third-degree price discrimination to maximize profits, what would the profit-maximizing prices be? What level of profits would the firm realize?

a) One way to solve the problem is to form the aggregate demand: Y = YR + YT = 100 – 10P + 20 = 120 – 10P. The inverse form of this demand is P = 12 – 0.1Y. Then set MR = MC: 12 – 0.2Y = 2. Thus, Y = 50, and P = 12 – 0.1(50) = 7. π = PY – 160 – 2Y = 7(50) – 160 – 2(50) = 90. b) In T, the firm should extract all consumer surplus by setting PT = 12 for all YT = 20 units. In R, inverse demand is PR = 10 – 0.1YR. Setting MR = MC, we have 10 – 0.2YR = 2, which implies YR = 40 and PR = 6. Then π = PRYR + PTYT – 160 – 2(YR + YT) = 6(40) + 12(20) – 160 – 2(60) = 200.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 12-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12.22. A seller produces output with a constant marginal cost MC = 2. Suppose there is one group of consumers with the demand curve P1 = 16 - Q1, and another with the demand curve P2 = 10 - (1/2)Q2. a) If the seller can discriminate between the two markets, what prices would she charge to each group of consumers? (You may want to exploit the monopoly midpoint rule from Learning-By-Doing Exercise 11.5.) b) If the seller cannot discriminate, but instead must charge the same price P1 = P2 = P to each consumer group, what will be her profit-maximizing price? c) Which, if any, consumer group benefits from price discrimination? d) If instead P1 = 10 - Q1, does either group benefit from price discrimination? a) With linear demand and constant marginal costs, we can use the monopoly midpoint rule to quickly determine that the profit-maximizing prices are P1 = 0.5(14 + 2) = 8 and P2 = 0.5(10 + 2) = 6. b) Market demand will be Q = Q1 + Q2 = 16 – P + 20 – 2P. Inverse market demand is then P = 12 – ⅓*Q. So the profit-maximizing non-discriminatory price will be P = 0.5(12 + 2) = 7. c) For any two general demand curves P1= a1 – Q1 and P2 = a2 – 0.5Q2 and constant marginal cost c, the profit-maximizing discriminatory prices will be P1 = 0.5(a1 + c) and P2 = 0.5(a2 + c).

In the case where the seller cannot discriminate, market demand will be Q = Q1 + Q2 = a1 + 2a2 – 3P. Inverse demand is P = ⅓*(a1 + 2a2) – ⅓*Q. So the profit-maximizing non-discriminatory price will be P = ½*[⅓*(a1 + 2a2) + c].

Note that P – P1 = ⅓*(a2 – a1) while P – P2 = (1/6)*(a1 – a2). If a1 > a2, then group 2 (i.e., consumers with the lower choke price) benefits from relatively lower prices under price discrimination while group 1 is hurt by relatively higher prices. d) If a1 = a2, then all three prices are the same: P1 = P2 = P. That is, even though the demand curves have different slopes, because each has the same intercept (choke price), the monopoly midpoint rule implies that the profit-maximizing monopoly price is the same regardless of whether the seller uses price discrimination or not. Thus, neither group benefits (nor is harmed by) price discrimination.

12.23. A cruise line has space for 500 passengers on each voyage. There are two market segments: elderly passengers and younger passengers. The demand curve for the elderly market segment is Q1 = 750 - 4P1. The demand curve for the younger market segment is Q2 = 850 - 2P2. In each equation, Q denotes the number of passengers on a cruise of a given length and P denotes the price per day. The marginal cost of serving a passenger of either type is $40 per person per day. Assuming the cruise line can price discriminate, what is the profit-maximizing number of passengers of each type? What is the profit-maximizing price for each type of passenger?

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 12-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Q1 = 750 – 4P1  P1 = 187.5 – (1/4)Q1. This implies MR1 = 187.5 – (1/2)Q1.

Q2 = 850 – 2P2  P2 = 425 – (1/2)Q2. This implies MR2 = 425 – Q2.

When we have limited capacity we solve the following two equations: MR1 = MR2  187.5 – (1/2)Q1 = 425 – Q2. (Equate the MRs) Q1 + Q2 = 500. (Quantities sold must add up to capacity) We thus have two equations in two unknowns: 187.5 – (1/2)Q1 = 425 – Q2 Q1 + Q2 = 500 Solving these equations yields: Q1 = 175.

Q2 = 325.

Plugging these back into the inverse demand curves gives us the profit-maximizing prices: P1 = 187.5 – (1/4)(175) = 143.75.

P2 = 425 – (1/2)(325) = 262.5.

12.24. An airline has 200 seats in the coach portion of the cabin of an Airbus A340. It is attempting to determine how many seats it should sell to business travelers and how many to vacation travelers on a flight between Chicago and Dubai that departs on Monday morning, January 25. It has tentatively decided to sell 150 seats to business travelers and 50 seats to vacation travelers at $4,000 and $1,000, respectively. It also knows: a) To sell an additional seat to business travelers, it would need to reduce price by $25. To reduce demand by business travelers by one seat, it would need to increase price by $25. b) To reduce demand by one unit among vacation travelers, it would need to increase price by $5. To sell an additional seat to vacation travelers, it would need to reduce price by $5.

Assuming that the marginal cost of carrying either type of passenger is zero, is the current allocation of seats profit maximizing? If not, would you sell more seats to business travelers or vacation travelers?

The easiest way to check whether the current allocation of seats is optimal is to compare the marginal revenues of the two market segments. In the equations below, let’s let “1” denote the business traveler segment and “2” denote the vacation traveler segment.

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Note that the marginal revenue in each segment is given by: MR1 = P1 + (P1/Q1)Q1 MR2 = P2 + (P2/Q2)Q2 We know: P1 = $4,000, Q1 = 150, and P1/Q1 = - $25. Hence, MR1 = $4,000 - $25(150) = $250.

P2 = $1,000, Q2 = 50, and P2/Q2 = - $5. Hence, MR2 = $1,000 - $5(50) = $750.

Since MR2 > MR1, the airline can increase its profits by selling more seats to vacation traveler and fewer seats to business travelers.

12.25. A summer theater has a capacity of 200 seats for its Saturday evening concerts. The marginal cost of admitting a spectator is zero up to that capacity. The theater wants to maximize profits and recognizes that there are two kinds of customers. It offers discounts to senior citizens and students, who generally are more price sensitive than other customers. The demand curve for tickets by seniors and students is described by P1 = 16 0.04Q1, where Q1 is the number of discount tickets sold at a price of P1. The demand schedule for tickets by customers who do not qualify for a discount is represented by P2 = 28 - 0.1Q2, where Q2 is the number of nondiscount tickets sold at a price of P2. What are the two prices that would maximize profit for the Saturday evening concerts?

The theater would want to set the prices to equate the marginal revenue for the two types of customers. Thus, it would choose the quantities so that: MR1 = 16 – 0.08Q1 = MR2 = 28 – 0.2Q2.

In addition, the capacity constraint must be satisfied, so that Q1 + Q2 = 200.

Thus, 16 – 0.08Q1 = 28 – 0.2(200 – Q1), which tells us that Q1 = 100 and Q2 = 100.

Note that this implies that the marginal revenue from each type of ticket is 8. Since this exceeds that marginal cost (which is zero), the firm does want sell all its capacity.

The profit maximizing prices are as follows: P1 = 16 – 0.04Q1 = 16 – 0.04(100) = 12. P2 = 28 – 0.1Q2 = 28 – 0.1(100) = 18.

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Solutions Manual

12.26. A small island near a major city has a beautiful beach. The company that owns the island sells day passes for the beach, including travel by ferry to and from the beach.

Because the beach is small, the company does not want to sell more than 200 excursion tickets per day. The company knows there are two kinds of visitors: those who are willing to buy tickets a month in advance and those who want to buy on the day of the trip. Those willing to buy in advance are typically more price sensitive. The demand curve for advance purchase excursion tickets is described by P1 = 100 - 0.2Q1, where Q1 is the number of advance purchase tickets sold at a price of P1. The demand schedule for tickets by day-oftravel excursions is represented by P2 = 200 - 0.8Q2, where Q2 is the number of tickets sold at a price of P2. a) Suppose the marginal cost of the ferry trip and use of beach is 50 per customer. What prices should the firm charge for its excursion tickets? b) If the marginal cost were high enough, the firm would want to sell fewer than 200 tickets. Suppose the marginal cost of the ferry trip and use of beach is 80 per customer.

What prices should the firm charge for its beach excursion tickets? a) The company would want to set the prices to equate the marginal revenue for the two types of customers. Let’s assume that the company wants to sell its capacity of 200 tickets. (We will verify that it does in a moment.) It would choose the quantities so that: MR1 = 100 – 0.4Q1 = MR2 = 200 – 1.6Q2.

In addition, the capacity constraint would be satisfied, so that Q1 + Q2 = 200.

Thus, 100 – 0.4Q1 = MR2 = 200 – 1.6(200 – Q1), which tells us that Q1 = 110 and Q2 = 90.

Note that this implies that the marginal revenue from each type of ticket is 56.

Since the marginal revenue exceeds that marginal cost (which is 50), the firm does want sell all its capacity. Thus, our assumption that the firm would want to sell to capacity when the marginal cost is 50 is correct.

The profit maximizing prices are as follows: P1 = 100 – 0.2Q1= 100 – 0.2(110) = 78. P2 = 200 – 0.4Q2 = 200 – 0.8(90) = 128. b) As we showed in part a), if the firm sells tickets to exhaust its capacity, the marginal revenue from sales to each type of customer is 56. So it will not want to sell to capacity if the marginal cost is 80. Instead, it will just set the price so that MR = MC in each market.

MR1 = 100 – 0.4Q1 = 80. This implies that Q1 = 50, with P1 = 100 – 0.2(50) = 90.

MR2 = 200 – 1.6Q2 = 80. This implies that Q2 = 75, with P2 = 200 – 0.8(75) = 140.

Note that the firm now sells only Q1 + Q2 = 50 + 75 = 125 excursion tickets – less than the capacity of 200.

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12.27. You are the only European firm selling vacation trips to the North Pole. You know only three customers are in the market. You offer two services, round trip airfare and a stay at the Polar Bear Hotel. It costs you 300 euros to host a traveler at the Polar Bear and 300 euros for the airfare. If you do not bundle the services, a customer might buy your airfare but not stay at the hotel. A customer could also travel to the North Pole in some other way (by private plane), but still stay at the Polar Bear. The customers have the following reservation prices for these services:

a) If you do not bundle the hotel and airfare, what are the optimal prices PA and PH, and what profits do you earn? b) If you only sell the hotel and airfare in a bundle, what is the optimal price of the bundle PB, and what profits do you earn? c) If you follow a strategy of mixed bundling, what are the optimal prices of the separate hotel, the separate airfare, and the bundle (PA, PH, and PB, respectively) and what profits do you earn? a) Without bundling, the best the firm can do is set the price of airfare at $800 and the price of hotel at $800. In each case the firm attracts a single customer and earns profit of $500 from each for a total profit of $1000. The firm could attract two customers for each service at a price of $500, but it would earn profit of $200 on each customer for a total of $800 profit, less profit than the $800 price. b) With bundling, the best the firm can do is charge a price of $900 for the airfare and hotel. At this price the firm will attract all three customers and earn $300 profit on each for a total profit of $900. The firm could raise its price to $1000, but then it would only attract one customer and total profit would be $400. Notice that with bundling the firm cannot do as well as it could with mixed bundling. This is because while a) the demands are negatively correlated, a key to increasing profit through bundling, b) customer 1 has a willingness-to-pay for airfare below marginal cost and customer 3 has a willingness-to-pay for hotel below marginal cost. The firm should be able to do better with mixed bundling c) Because customer 1 has a willingness-to-pay for airfare below marginal cost and customer 3 has willingness-to-pay for hotel below marginal cost, the firm can potentially earn greater profits through mixed bundling. In this problem, if the firm charges $800 for airfare only, $800 for hotel only, and $1000 for the bundle, then customer 1 will purchase hotel only, customer 2 will purchase the bundle, and customer 3 will purchase airfare only. This will earn the firm $1400 profit, implying that mixed bundling is the best option in this problem.

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Solutions Manual

12.28. You operate the only fast-food restaurant in town, selling burgers and fries. There are only two customers, one of whom is on the Atkins diet and the other on the Zone diet, whose willingness to pay for each item is displayed in the following table. For simplicity, assume you have zero fixed and marginal costs for each item.

a) If x = 1 and you do not bundle the two products, what are your profit-maximizing prices PB and PF? Calculate total surplus under this outcome. b) Now assume only that x > 0. Instead, suppose that you hired an economist who tells you that the profit-maximizing bundle price (for a burger and fries) is $8, while if you sold the items individually (and did not offer a bundle) your profit-maximizing price for fries would be greater than $3. Using this information, what is the range of possible values for x? a) You should sell two burgers for PB = 5, and one order of fries for PF = 3. Total surplus is then PS + CS = (10 + 3) + (3 + 0) = 16. b) In order for the profit-maximizing bundle price to be $8, it must be true that 8 + x < 2*8, i.e. that x < 8. In order for the profit-maximizing price of fries to be greater than $3, it must be true that x > 2*3, or x > 6. Thus, we know that 6 ≤ x < 8.

12.29. Suppose your company produces athletic footwear. Marketing studies indicate that your own price elasticity of demand is -3 and that your advertising elasticity of demand is 0.5. You may assume these elasticities to be approximately constant over a wide range of prices and advertising expenses. a) By how much should the company mark up price over marginal cost for its footwear? b) What should the company’s advertising-to-sales ratio be? a) Using the inverse elasticity price rule, P − MC 1 =− P  Q,P P − MC 1 =− P −3 P = 1.5 MC

The firm should set price at about 1.5 times marginal cost.

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Solutions Manual

b) The optimal advertising-to-sales ratio can be found be equating

 Q, A A =− PQ  Q,P A 0.5 =− PQ −3 A = 0.167 PQ

Thus, advertising expense should be about 16 or 17 percent of sales revenue.

12.30. The motor home industry consists of a small number of large firms. In 2003, producers of motor homes had an average advertising sales ratio of 1.8 percent. Assuming that the price elasticity of demand facing a typical motor home producer is -4, what is the advertising elasticity of demand facing a typical producer, under the assumption that each producer has chosen its price and advertising level to maximize profits?

The condition for the ratio profit-maximizing advertising-to-sales ratio is: 𝜀

Advertising-to-sales ratio= − 𝜀𝑄,𝐴, where Q,A is the advertising elasticity of demand and Q,P is 𝑄,𝑃

the price elasticity of demand. We know that the advertising-to-sales ratio of a typical producer of motor homes is 1.8 percent or 0.018. We also know that the price elasticity of demand of a typical firm is -4. We thus have: 𝜀

𝑄,𝐴 0.018 = = − −4 , which implies Q,A = (4)(0.018) = 0.072.

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Solutions Manual

Chapter 13 Market Structure and Competition Solutions to Review Questions 1. Explain why, at a Cournot equilibrium with two firms, neither firm would have any regret about its output choice after it observes the output choice of its rival.

In a Cournot setting, each firm chooses a level of output that maximizes its own profit given the output choice of the other firm. In equilibrium each firm is choosing the profit-maximizing level of output given the other firm’s output choice. Thus, neither firm will have any regret since it is doing the best it can given the other firm’s choice.

2. What is a reaction function? Why does the Cournot equilibrium occur at the point at which the reaction functions intersect?

A reaction function represents a firm’s best response to each possible choice of another firm. For example, in a Cournot model a reaction function represents the firm’s profit-maximizing level of output given another firm’s choice of output.

At the point where the reaction functions intersect, both firms are choosing a level of output that maximizes its profit given the output choice of the other firm. Thus, neither firm would choose to deviate from this point since doing so would reduce its profit. This point, therefore, represents an equilibrium since neither firm would choose to change what it is doing given the other firm’s choice.

3. Why is the Cournot equilibrium price less than the monopoly price? Why is the Cournot equilibrium price greater than the perfectly competitive price?

The Cournot equilibrium price is less than the monopoly price because the firms in the Cournot oligopoly are not trying to maximize industry profits. By acting in their own self-interest, the firms will expand their output to achieve greater profits. By expanding output, market price falls.

The Cournot equilibrium price is greater than the perfectly competitive price because the firms in the Cournot oligopoly exhibit market power. This market power allows the firms to set a price above the perfectly competitive price.

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Solutions Manual

4. Explain the difference between the Bertrand model of oligopoly and the Cournot model of oligopoly. In a homogeneous products oligopoly, what predictions do these models make about the equilibrium price relative to marginal cost?

In the Cournot model of oligopoly, firms choose a level of output given the output choices of rival firms. In the Bertrand model of oligopoly, firms choose a price given the prices set by rival firms. In the Bertrand model it is assumed that the firm with the lowest price will achieve 100% market share. Therefore, firms will undercut the prices of rival firms until price is driven down to the firm’s marginal cost.

In a homogenous products oligopoly, Cournot firms exhibit market power and set a price above the perfectly competitive price and provide a level of output below the perfectly competitive level. Bertrand firms, by undercutting prices of rival firms, drive the price down to the level of marginal cost, achieving the perfectly competitive solution. Thus, in the Cournot model price is above the perfectly competitive price and in the Bertrand model price is equal to the perfectly competitive price.

5. What is the role played by the competitive fringe in the dominant firm model of oligopoly? Why does an increase in the size of the fringe result in a reduction in the dominant firm’s profit-maximizing price?

In a dominant firm market, one dominant firm splits the market with a competitive fringe. This competitive fringe is some number of producers that act as perfect competitors: each chooses a quantity of output taking the market price as given. The market price is set by the dominant firm who chooses a price to maximize its own profit.

As more firms enter the competitive fringe, the supply of the competitive fringe increases. As this supply increases, the dominant firm’s residual demand – market demand less fringe supply – falls. As this residual demand falls, marginal revenue declines and the firm will maximize its own profits at a lower price.

6. What is the difference between vertical product differentiation and horizontal product differentiation?

With vertical product differentiation, consumers view one product as unambiguously better or worse than another product. With horizontal product differentiation, some consumers regard product A as a poor substitute for product B, while other consumers regard product B as a poor substitute for product A. In other words, with horizontal differentiation some consumers prefer A over B at equal prices while others prefer B over A at equal prices.

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Solutions Manual

7. Explain why, in the Bertrand model of oligopoly with differentiated products, a greater degree of product differentiation is likely to increase the markup between price and marginal cost.

As the degree of product differentiation increases, the firm gains more market power for its product. This occurs because with greater product differentiation there are fewer close substitutes for the product to compete with. As the firm gains market power for its product, it can charge a higher price for the product.

Viewing this from the opposite standpoint, if there was less product differentiation, there would be more close substitutes and price increases would send consumers to other products. In the limit, if there were no product differentiation in the Bertrand model, prices would fall to marginal cost and all producers would charge the same price.

8. What are the characteristics of a monopolistically competitive industry? Provide an example of a monopolistically competitive industry.

A monopolistically competitive market has three key features. First, the market is fragmented, meaning it consists of many buyers and many sellers. Second, there is free entry and exit into the market. Third, firms produce horizontally differentiated products.

The restaurants in a city might represent a monopolistically competitive market. There are likely many restaurants, and many consumers who want to eat at a restaurant; entrepreneurs can easily enter the restaurant market; and consumers likely view the restaurants as imperfect substitutes.

Clothing retailers might represent another monopolistically competitive market.

9. Why is it the case in a long-run monopolistically competitive equilibrium that the firm’s demand curve is tangent to its average cost curve? Why could it not be a long-run equilibrium if the demand curve “cut through” the average cost curve?

In a monopolistically competitive market, each firm’s demand curve will be tangent to its average cost curve because there is free entry and exit of firms, so firms will enter if there are any economic profits to be earned. This entry will occur until economic profits are driven to zero, much like the perfectly competitive environment. When economic profits are zero, it must be the case that price is equal to average cost since profit can be written as 𝜋 = (𝑃 − 𝐴𝐶)𝑄. If price is equal to average cost, then the demand curve must touch the average cost curve at the optimal quantity. This is seen as a tangency between demand and average cost in the long run.

If demand ‘cut through’ the average cost curve, then there would be a range of outputs over which economic profit would be positive. This would entice new firms to enter, driving profits down. Therefore, the demand curve ‘cutting through’ the average cost curve cannot be an equilibrium in the long run since new firms will enter.

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Solutions Manual

Solutions to Problems 13.1 Beryllium oxide is a chemical compound used in pharmaceutical applications.

Beryllium oxide can only be made in one particular way, and all firms produce their version of beryllium oxide to the exact same standards of purity and safety. The largest firms have market shares given in the table below:

a. What is the four-firm (4CR) concentration ratio for this industry? b. What is the Herfindahl-Hirschman Index (HHI) for this industry? c. Of the market structures described in Table 13.1, which one best describes the Beryllium oxide industry? a. 4CR = 80+1+1+1 = 84 b. HHI  802 + 12 +12 +12 = 6,403 (ignoring the effect of smaller firms in the industry). c. This is an example of a dominant firm industry. The firms sell homogeneous products, and the industry is dominated by a single large firm.

13.2 The cola industry in the country of Inner Baldonia consists of five sellers: two global brands, Coke and Pepsi, and three local competitors, Bright, Quite, and Zight. Consumers view these products as similar, but not identical. The market shares of the five sellers are as follows:

a. What is the four-firm (4CR) concentration ratio for this industry? b. What is the Herfindahl-Hirschman Index (HHI) for this industry? c. Of the market structures described in Table 13.1, which one best describes the cola industry in Inner Baldonia?

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Solutions Manual

a. 4CR = 25+24+23+20 = 92 b. HHI =252 + 242 +232 +202 + 82 = 2,194 c. This is an example of a differentiated products oligopoly. The firms produce similar but not identical products. And the fact that some consumers are loyal (even though others switch back and forth among brands based on price) suggests that there is some measure of horizontal differentiation in this market.

13.3 Outer Baldonia is a largely rural country with many small towns. Each town typically contains a retail store selling livestock feed. In virtually all towns, there is only one such store. The farmers who purchase feed from these stores typically live outside of town, sometimes at a significant distance. Often, they will purchase from a store in the town closest to them, but if farmers learn through word of mouth that a feed retailer in a more distant town is selling feed at a significantly lower price, they will sometimes go to that store to obtain feed.

The country-wide market shares of the six largest feed stores in Outer Baldonia are shown in the table below:

a. What is the four-firm (4CR) concentration ratio for the livestock feed store market in Outer Baldonia? b. What is the (approximate) Herfindahl-Hirschman Index (HHI) for this industry? c. Of the market structures described in Table 13.1, which one best describes the livestock feed market in Outer Baldonia? a. 4CR = 2+1+1+0.5 = 4.5. b. HHI =22 + 12 +12 +(0.5)2 + (0.25)2 + (0.25)2 = 6.375. This is only an approximation to the industry’s HHI because to compute the HHI with complete accuracy, we would need to know the market shares of all livestock stores in Outer Baldonia. However, our approximation is likely to be very good because the remaining feed stores have market shares less than or equal to 0.25%.

When we square such small market shares, the result is a number very close to zero. Adding additional stores would thus not change our computed HHI very much. c. This is an example of a monopolistically competitive industry. The reason that the industry is monopolistically competitive rather than perfectly competitive is that feed stores are most likely differentiated geographically. Farmers are willing to travel longer distances if they can get a

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significantly better price, but if prices are about the same, farmers will go to the store in the closet town.

13.4. In the following, let the market demand curve be P = 70 - 2Q, and assume all sellers can produce at a constant marginal cost of c = 10, with zero fixed costs. a) If the market is perfectly competitive, what is the equilibrium price and quantity? b) If the market is controlled by a monopolist, what is the equilibrium price and quantity?

How much profit does the monopolist earn? c) Now suppose that Amy and Beau compete as Cournot duopolists. What is the Cournot equilibrium price? What is total market output, and how much profit does each seller earn? a)

P = MC implies 70 – 2Q = 10, or Q = 30 and P = 10.

b) A monopolist produces until MR = MC yielding 70 – 4Q = 10 so Qm = 15 and Pm = 40. m Thus π = (40 – 10)*15 = 450. c) For Amy, MRA = MC implies 70 – 4qA – 2qB = 10. We could either calculate Beau’s profit-maximization condition (and solve two equations in two unknowns), or, inferring that the equilibrium will be symmetric since each seller has identical costs, we can exploit the fact that qA = qB in equilibrium. (Note: You can only do this after calculating marginal revenue for one Cournot firm, not before.) Thus 70 – 6qA = 10 or qA = 10. Similarly, qB = 10. Total market output under Cournot duopoly is Qd = qA + qB = 20, and the market price is Pd = 70 – 2*20 = 30.

Each duopolist earns πd = (30 – 10)*10 = 200.

13.5. A homogeneous products duopoly faces a market demand function given by P = 300 3Q, where Q = Q1 + Q2. Both firms have a constant marginal cost MC = 100. a) What is Firm 1’s profit-maximizing quantity, given that Firm 2 produces an output of 50 units per year? What is Firm 1’s profit-maximizing quantity when Firm 2 produces 20 units per year? b) Derive the equation of each firm’s reaction curve and then graph these curves. c) What is the Cournot equilibrium quantity per firm and price in this market? d) What would the equilibrium price in this market be if it were perfectly competitive? e) What would the equilibrium price in this market be if the two firms colluded to set the monopoly price? f ) What is the Bertrand equilibrium price in this market? g) What are the Cournot equilibrium quantities and industry price when one firm has a marginal cost of 100 but the other firm has a marginal cost of 90?

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Solutions Manual

a) With two firms, demand is given by 𝑃 = 300 − 3𝑄1 − 3𝑄2. If 𝑄2 = 50, then 𝑃 = 300 − 3𝑄1 − 150 or 𝑃 = 150 − 3𝑄1. Setting 𝑀𝑅 = 𝑀𝐶 implies 150 − 6Q1 = 100 Q1 = 8.33

If 𝑄2 = 20, then 𝑃 = 240 − 3𝑄1. Setting 𝑀𝑅 = 𝑀𝐶 implies 240 − 6Q1 = 100 Q1 = 23.33

b)

For Firm 1, 𝑃 = (300 − 3𝑄2 ) − 3𝑄1. Setting 𝑀𝑅 = 𝑀𝐶 implies

(300 − 3Q2 ) − 6Q1 = 100 Q1 = 33.33 − 0.5Q2

Since the marginal costs are the same for both firms, symmetry implies 𝑄2 = 33.33 − 0.5𝑄1.

Graphically, these reaction functions appear as

c) Because of symmetry, in equilibrium both firms will choose the same level of output.

Thus, we can set 𝑄1 = 𝑄2 and solve 𝑄2 = 33.33 − 0.5𝑄2 𝑄2 = 22.22 Since both firms will choose the same level of output, both firms will produce 22.22 units. Price can be found by substituting the quantity for each firm into market demand. This implies price will be 𝑃 = 300 − 3(44.44) = 166.67. d) If this market were perfectly competitive, then equilibrium would occur at the point where 𝑃 = 𝑀𝐶 = 100.

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e)

Solutions Manual

If the firms colluded to set the monopoly price, then

300 − 6Q = 100 Q = 33.33 200

At this quantity, market price will be 𝑃 = 300 − 3( 6 ) = 200. f) If the firms acted as Bertrand oligopolists, the equilibrium would coincide with the perfectly competitive equilibrium of 𝑃 = 100. g) Suppose Firm 1 has 𝑀𝐶 = 100 and Firm 2 has 𝑀𝐶 = 90. For Firm 1, 𝑃 = (300 − 3𝑄2 ) − 3𝑄1. Setting 𝑀𝑅 = 𝑀𝐶 implies (300 − 3Q2 ) − 6Q1 = 100 Q1 = 33.33 − 0.5Q2

For Firm 2, 𝑃 = (300 − 3𝑄1 ) − 3𝑄2. Setting 𝑀𝑅 = 𝑀𝐶 implies (300 − 3Q1 ) − 6Q2 = 90 Q2 = 35 − 0.5Q1

Solving these two reaction functions simultaneously yields 𝑄1 = 21.11 and 𝑄2 = 24.44. With these quantities, market price will be 𝑃 = 163.36.

13.6. Zack and Andon compete in the peanut market. Zack is very efficient at producing nuts, with a low marginal cost cZ = 1; Andon, however, has a constant marginal cost cA = 10. If the market demand for nuts is P = 100 - Q, find the Cournot equilibrium price and the quantity and profit level for each competitor.

For Zack, MRZ = MCZ implies 100 – 2qZ – qA = 1, so Zack’s reaction function is qZ = ½*(99 – qA). Similarly, MRA = MCA implies 100 – 2qA – qZ = 10 so Andon’s reaction function is qA = ½*(90 – qZ). Solving these two equations in two unknowns yields qZ = 36 and qA = 27. The market price is P = 100 – (36 + 27) = 37. Zack earns πZ = (37 – 1)*36 = 1296 and Andon earns πA = (36 – 10)*27 = 702.

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13.7. Let’s consider a market in which two firms compete as quantity setters, and the market demand curve is given by Q = 4000 - 40P. Firm 1 has a constant marginal cost equal to MC1 = 20, while Firm 2 has a constant marginal cost equal to MC2 = 40. a) Find each firm’s reaction function. b) Find the Cournot equilibrium quantities and the Cournot equilibrium price. a) The inverse market demand curve is P = 100 – (Q/40) = 100 – (Q1 + Q2)/40. Firm 1’s reaction function is found by equating MR1 = MC1: MR1 = [100 – Q2/40] – Q1/20 R1 = MC1  [100 – Q2/40] – Q1/20 = 20 Solving this for Q1 in terms of Q2 gives us: Q1 = 1,600 – 0.5Q2 Similarly, Firm 2’s reaction function is found by equating MR2 = MC2: MR2 = [100 – Q1/40] – Q2/20 MR2 = MC2  [100 – Q1/40] – Q2/20 = 40 Solving this for Q2 in terms of Q1 gives us: Q2 = 1,200 – 0.5Q1 b) The two reaction functions give us two equations in two unknowns. Using algebra we can solve these to get: Q1 = 1,333.33 and Q2 = 533.33.

We find the Cournot equilibrium price by plugging these quantities back into the inverse market demand curve: P = 100 – (1333.33 + 533.33)/40 = 53.33

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Chapter 13-9

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.8. In a homogeneous products duopoly, each firm has a marginal cost curve MC = 10 + Qi, i = 1, 2. The market demand curve is P = 50 - Q, where Q = Q1 + Q2. a) What are the Cournot equilibrium quantities and price in this market? b) What would be the equilibrium price in this market if the two firms acted as a profitmaximizing cartel? c) What would be the equilibrium price in this market if firms acted as price-taking firms? a) Market demand is given by 𝑃 = 50 − 𝑄1 − 𝑄2 . Firm 1’s reaction function can be found by equating its marginal revenue to its marginal cost.

(50 − Q2 ) − 2Q1 = 10 + Q1 Q1 =

40 1 − Q2 3 3 40

1

Since both firms have the same marginal cost, symmetry implies 𝑄2 = 3 − 3 𝑄1 . Solving these two reaction functions simultaneously yields 𝑄1 = 10 and 𝑄2 = 10. Industry price is found by substituting these quantities into the market demand function. This implies 𝑃 = 30. b) As discussed in Chapter 11, a multiplant monopolist will equate the marginal cost of production across all its plants. At any level of marginal cost MCT, each plant would operate so that MCT = Qi + 10, or Qi = MCT – 10. Thus, total output Q = Q1 + Q2 = 2MCT – 20. So, the multiplant marginal cost curve is given by MCT = 0.5Q + 10. Equating MR with MCT yields 50 – 2Q = 0.5Q + 10 Q = 16 Thus, both plants will produce Qi = 8 units. Industry price is found by substituting these quantities into the market demand function. This implies P = 34. c) If the firms acted as price takers, the market would see the perfectly competitive solution.

Setting 𝑃 = 𝑀𝐶 for both firms implies 50 − 𝑄1 − 𝑄2 = 10 + 𝑄1 50 − 𝑄1 − 𝑄2 = 10 + 𝑄2 Since 𝑄1 = 𝑄2 in equilibrium we have

50 − 2Q1 = 10 + Q1 40 3 40 Both firms will produce 3 = 13.3 units. Industry price is found by substituting these quantities Q1 =

70

into the market demand function. This implies 𝑃 = 3 = 23.3.

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Chapter 13-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.9. Suppose that demand for cruise ship vacations is given by P = 1200 - 5Q, where Q is the total number of passengers when the market price is P. a) The market initially consists of only three sellers, Alpha Travel, Beta Worldwide, and Chi Cruiseline. Each seller has the same marginal cost of $300 per passenger. Find the symmetric Cournot equilibrium price and output for each seller. b) Now suppose that Beta Worldwide and Chi Cruiseline announce their intention to merge into a single firm. They claim that their merger will allow them to achieve cost savings so that their marginal cost is less than $300 per passenger. Supposing that the merged firm, BetaChi, has a marginal cost of c < $300, while Alpha Travel’s marginal cost remains at $300, for what values of c would the merger raise consumer surplus relative to part (a)? a) Consider first the problem of Alpha Travel. It produces until MRA = MCA or 1200 – 5(QB + QC) – 10QA = 300. Thus its reaction function is QA = 90 – 0.5(QB + QC).

Symmetry implies that in equilibrium QA = QB = QC, so we can solve to find that Qi = 45 for each firm. Thus, the equilibrium price is P = 525. b) Reconsidering Alpha’s profit-maximization problem, we now have that MRA = MCA or 1200 – 5QBC – 10QA = 300. Thus, its reaction function is QA = 90 – 0.5QBC. The merged firm will produce until MRBC = MCBC or 1200 – 5QA – 10QBC = c, so its reaction function is QBC = 120 – 0.1c – 0.5QA Solving these two equations as a function of c yields QBC = 100 – (2/15)*c and QA = 40 + (1/15)*c. Total output is then Q = 140 – (1/15)*c, and so the market price is P = 500 + ⅓*c. Put simply, the merger raises consumer surplus only if the price falls; thus consumer surplus rises only when 500 + ⅓*c < 525, or c < 75.

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Chapter 13-11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.10. A homogeneous products oligopoly consists of four firms, each of which has a constant marginal cost MC = 5. The market demand curve is given by P = 15 - Q. a) What are the Cournot equilibrium quantities and price? Assuming that each firm has zero fixed costs, what is the profit earned by each firm in equilibrium? b) Suppose Firms 1 and 2 merge, but their marginal cost remains at 5. What are the new Cournot equilibrium quantities and price? Is the profit of the merged firm bigger or smaller than the combined profits of Firms 1 and 2 in the initial equilibrium in part (a)?

Provide an explanation for the effect of the merger on profit in this market. a) With four firms, demand is given by 𝑃 = 15 − 𝑄1 − 𝑄2 − 𝑄3 − 𝑄4. Let 𝑋 represent total output for Firms 2, 3, and 4. Then demand faced by Firm 1 is 𝑃 = (15 − 𝑋) − 𝑄1 . Setting 𝑀𝑅 = 𝑀𝐶 implies (15 − X ) − 2Q1 = 5 Q1 = 5 − 0.5 X

Since all firms have the same marginal cost, the solution will be symmetric. Letting 𝑄 ∗ represent the optimal output for each firm,

Q* = 5 − 0.5(3Q* ) Q* = 2 Thus, total industry output will be 8 with each firm producing 2 units of output. At this quantity price will be 𝑃 = 15 − 8 = 7. Profits for each firm will be 𝜋 = 𝑇𝑅 − 𝑇𝐶 = 7(2) − 5(2) = 4. b) If two firms merge, then the number of firms in the market will fall to three. The new quantity for each firm will be 𝑄 ∗ = 5 − 0.5(2𝑄 ∗ ) 𝑄 ∗ = 2.5 Now total industry output will be 7.5 with each of the three firms producing 2.5 units. At this quantity, price will be 𝑃 = 15 − 7.5 = 7.5. Profit per firm will be 𝜋 = 𝑇𝑅 − 𝑇𝐶 = 7.5(2.5) − 5(2.5) = 6.25.

Thus, while profit per firm does increase after the merger, profits do not double, and the merger nets the two firms a smaller total profit. Profit per firm increases after the merger because as the total number of firms falls, each individual firm has greater market power. This greater market power allows the firms to charge a higher price, produce less in aggregate, and earn greater profit per firm.

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Chapter 13-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.11. An industry is known to face market price elasticity of demand €Q, P = -3 (Assume this elasticity as constant as the industry moves along its demand curve.) The marginal cost of each firm in this industry is $10 per unit, and there are five firms in the industry. What would the Lerner Index be at the Cournot equilibrium in this industry?

In a Cournot equilibrium, the percentage contribution margin is given by 𝑃 − 𝑀𝐶 1 1 =− ( ) 𝑃 𝑁 𝜀𝑄,𝑃 where 𝑁 is the number of firms in the market. For this example, 𝑃 − 𝑀𝐶 1 1 =− ( ) 𝑃 5 −3 𝑃 − 𝑀𝐶 1 = 𝑃 15 Thus, the Lerner Index in this industry would be 0.067, or 6.7 percent.

13.12. Besanko, Inc., is one of two Cournot duopolists in the market for gizmos. It and its main competitor Schmedders Ltd. face a downward-sloping market demand curve. Each firm has an identical marginal cost that is independent of output. Please indicate how the following will affect Besanko’s and Schmedders’s reaction functions, and the Cournot equilibrium quantities produced by Besanko and Schmedders. a) Leading safety experts begin to recommend that all home owners should replace their smoke detectors with gizmos. b) Besanko and Schmedders’s gizmos are made out of platinum, with each gizmo requiring 1 kg of platinum. The price of platinum goes up. c) Besanko, Inc.’s total fixed cost increases. d) The government imposes an excise tax on gizmos produced by Schmedders, but not on those produced by Besanko.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 13-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

As you read the answer to this, think of the reaction functions being graphed in coordinate system with Besanko’s Q on the horizontal axis and Schmedders’ Q on the vertical axis.

Shifts Besanko’s Reaction Function?

Yes – shifts it rightward (more quantity for any given quantity of rival)

Shifts Schmedders’s Reaction Function?

Effect on Cournot equilibrium quantities?

Yes – shifts it upward (more quantity for any given quantity of rival)

b) Increase in MC of both firms

Yes – shifts it leftward (less quantity for any given quantity of rival)

Yes – shifts it downward (less quantity for any given quantity of rival)

c) Increase in Besanko’s total fixed cost d) Excise tax on Schmedders

No change – fixed costs don’t affect Besanko’s reaction function No – Besanko’s fundamentals (demand, MC) haven’t changed

No change

Both Besanko and Schmedders produce more in the new equilibrium (intersection of reaction curves occurs further to northeast) Both Besanko and Schmedders produce less in the new equilibrium (intersection of reaction curves occurs further to southwest) No change

a) Increase in market demand

Copyright © 2020 John Wiley & Sons, Inc.

Yes – shifts it downward (less quantity for any given quantity of rival)

Schmedders produces less in new equilibrium, while Besanko produces more (intersection of reaction function occurs further to the southeast along Besanko’s reaction function)

Chapter 13-14

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.13. Suppose that firms in a two-firm industry choose quantities every month, and each month the firms sell at the market-clearing price determined by the quantities they choose.

Each firm has a constant marginal cost, and the market demand curve is linear of the form P = a - bQ, where Q is total industry quantity and P is the market price. Suppose that initially each firm has the same constant marginal cost. Further suppose that each month the firms attain the Cournot equilibrium in quantities. a) Suppose that it is observed that from one month to the next Firm 1’s quantity goes down, Firm 2’s quantity goes up, and the market price goes up. A change in the demand and/or cost conditions consistent with what we observe is: i) The market demand curve shifted leftward in a parallel fashion. ii) The market demand curve shifted rightward in a parallel fashion. iii) Firm 1’s marginal cost went up, while Firm 2’s marginal cost stayed the same. iv) Firm 2’s marginal cost went up, while Firm 1’s marginal cost stayed the same. v) All of the above are possible. b) Suppose that it is observed that from one month to the next, Firm 1’s quantity goes down, Firm 2’s quantity goes down, and the market price goes down. A change in the demand and/or cost conditions consistent with what we observe is: i) The market demand curve shifted leftward in a parallel fashion. ii) The market demand curve shifted rightward in a parallel fashion. iii) Firm 1’s marginal cost went up, while Firm 2’s marginal cost stayed the same. iv) Firm 2’s marginal cost went down, while Firm 1’s marginal cost stayed the same. v) All of the above are possible. c) Suppose that it is observed that from one month to the next, Firm 1’s quantity goes up, Firm 2’s quantity goes up, and the market price goes up. A change in the demand and/or cost conditions consistent with what we observe is: i) The market demand curve shifted leftward in a parallel fashion. ii) The market demand curve shifted rightward in a parallel fashion. iii) Both firms’ marginal costs went up by the same amount. iv) Both firms’ marginal costs went down by the same amount. v) All of the above are possible. d) Suppose that it is observed that from one month to the next, Firm 1’s quantity goes up, Firm 2’s quantity goes up, and the market price goes down. A change in the demand and/or cost conditions consistent with what we observe is: i) The market demand curve shifted leftward in a parallel fashion. ii) The market demand curve shifted rightward in a parallel fashion. iii) Both firms’ marginal costs went up by the same amount.

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Chapter 13-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

iv) Both firms’ marginal costs went down by the same amount. v) All of the above are possible. a) Correct answer: C. Why? We can rule out A and B because under these two scenarios, the quantities of firms 1 and 2 would have moved in the same direction. This then eliminates E. D is incorrect because if firm 2’s marginal cost had gone up; its equilibrium quantity would have gone down. Thus, the only possible answer is the correct answer C. b) Correct answer: A. Why? We can rule out C and D. Here’s why. Under these scenarios, the reaction function of the firm whose marginal went up would shift leftward (if the firm’s quantity is being measured along the horizontal axis) or downward if the firm’s quantity is being measured along the vertical axis). The reaction function of the firm whose marginal cost did not change would remain unchanged. As a result, the equilibrium would involve a lower quantity for the firm experiencing the unfavorable marginal cost change and a higher quantity for the firm whose marginal cost remained the same. By ruling out C and D, we also rule out E. The only two possibilities are A and B. But we can rule out B because a rightward parallel shift in the market demand curve would result in each firm producing more in equilibrium not less (See Table 13.4 in the text). Hence, the correct answer must be A. c) Correct answer: B. We can rule out C. If both firm’s marginal costs went up by the same amount, then each firm’s quantity would fall, not go up (see Table 13.4). We also can rule out D. If both firm’s marginal costs went down by the same amount, then the market price would fall, not go up (see Table 13.4 again). Because C and D are incorrect, can therefore rule out E. We also can rule out A. If the market demand curve shifted leftward, then that would mean that the “a” term in the market demand curve P = a – bQ has gone down. In this case, Table 13.4 would tell us that each firm’s quantity would down and the market price would go down. Gathering together all of the above tells us that the correct answer must be B. d) Correct answer: D. We can rule out both A and B. If the market demand curve had shifted leftward or rightward in a parallel fashion, then the “a” term in the market demand curve would either increase or decrease. Table 13.4 would then imply that the change in the firms’ quantities and market price would be in the same direction (the quantities and the market price would each increase or would each decrease). Because A and B cannot be correct, E cannot be correct either. This leaves us with C and D. We can rule out C because if each firm’s marginal cost went up then Table 13.4 would tell us that each firm would produce less quantity and the market price would go up. Pulling this all together implies that the correct answer must be D.

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Chapter 13-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

13.14. An industry consists of two Cournot firms selling a homogeneous product with a market demand curve given by P = 100 - Q1 - Q2. Each firm has a marginal cost of $10 per unit. a) Find the Cournot equilibrium quantities and price. b) Find the quantities and price that would prevail if the firms acted “as if” they were a monopolist (i.e., find the collusive outcome). c) Suppose Firms 1 and 2 sign the following contract. Firm 1 agrees to pay Firm 2 an amount equal to T dollars for every unit of output it (Firm 1) produces. Symmetrically, Firm 2 agrees to pay Firm 1 an amount T dollars for every unit of output it (Firm 2) produces. The payments are justified to the government as a cross licensing agreement whereby Firm 1 pays a royalty for the use of a patent developed by Firm 2, and similarly, Firm 2 pays a royalty for the use of a patent developed by Firm 1. What value of T results in the firms achieving the collusive outcome as a Cournot equilibrium? d) Draw a picture involving reaction functions that shows what is going on in this situation. a) Equating MR to MC for each firm yields the reaction function Q1 = 45 – 0.5Q2 for firm 1 and Q2 = 45 – 0.5Q1 for firm 2. Solving these two equations in two unknowns yields Q1 = Q2 = 30 with P = $40. b) A monopolist produces until MR = MC or 100 – 2Q = 10, implying Q = 45 and P = $55/unit. Thus, each firm produces Q1 = Q2 = 22.5 in the collusive outcome. c) The cross-licensing agreement implies that each firm has a marginal cost of MC = 10 + T. Setting MR = MC for each firm generates the reaction functions: Q1 = 45 – 0.5T – 0.5Q2 and Q2 = 45 – 0.5T – 0.5Q1. Solving these reaction functions simultaneously for Q1 and Q2, we get the reduced form equations for each firm’s output as a function of T: Q1 = Q2 = 30 – (1/3)T. We want to find a value of T so that Q1 = Q2 = 22.5. Thus we set 30 – (1/3)T = 22.5 which yields T = 22.5. d) The figure below shows us what is going on here. The cross-licensing agreement has, in effect, shifted each firm’s reaction function inward (from the solid to the dotted lines) so that they now cross at the “as if” monopoly outcome.

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Chapter 13-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Firm 2’s Quantity

R1

Cournot equilibrium

“As if” monopoly outcome

R2

Firm 1’s Quantity

13.15. Consider an oligopoly in which firms choose quantities. The inverse market demand curve is given by P = 280 - 2(X + Y ), where X is the quantity of Firm 1, and Y is the quantity of Firm 2. Each firm has a marginal cost equal to 40. a) What are the Cournot equilibrium outputs for each firm? What is the market price at the Cournot equilibrium? What is the profit of each firm? b) What is the Stackelberg equilibrium, when Firm 1 acts as the leader? What is the market price at the Stackelberg equilibrium? What is the profit of each firm?

The table below summarizes the answer to this problem. The solution details follow.

Cournot Stackelberg with Firm 1 as leader a)

Firm 1 output 40 60

Firm 2 output 40 30

Market Price 120 100

Firm 1 Profit 3,200 3,600

Firm 2 Profit 3,200 1,800

Firm 1’s marginal revenue is MR = 280 – 2Y – 4X. Equating MR to MC gives us:

280 – 2Y – 4X = 40, 240 – 2Y = 4X X = 60 – 0.5Y This is Firm 1’s reaction function. Firm 2 is identical, and so Firm 2’s reaction function will be a mirror image of Firm 1’s Y = 60 – 0.5X

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 13-18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

If we solve these reaction functions simultaneously, we find X = Y = 40. At this output, the corresponding market price is P = 280 – 2(40 + 40) = 120. Each firm’s profit is thus: 120*40 – 40*40 = 3,200. b) To find the Stackelberg equilibrium in which Firm 1 is the leader, we start by writing the expression for Firm 1’s total revenue: TR = (280 – 2Y – 2X)X In place of Y, we substitute in Firm 2’s reaction function: Y = 60 – 0.5X TR = [280 – 2(60 – 0.5X) – 2X]X = (160 – X)X Firm 1’s marginal revenue is therefore MR = 160 – 2X. Equating marginal revenue to marginal cost gives us: 160 – 2X = 40, or X = 60.

To find Firm 2’s output, we plug X = 60 back into Firm 2’s reaction function: Y = 60 – 0.5(60) = 30.

The market price is found by plugging X = 60 and Y = 30 back into the demand curve: P = 280 – 2(60 + 30) = 100.

Thus, at the Stackelberg equilibrium, Firm 1’s profit is: 100*60 – 40*60 = 3,600. Firm 2’s profit is 100*30 – 40*30 = 1,800.

13.16. The market demand curve in a commodity chemical industry is given by Q = 600 3P, where Q is the quantity demanded per month and P is the market price in dollars.

Firms in this industry supply quantities every month, and the resulting market price occurs at the point at which the quantity demanded equals the total quantity supplied.

Suppose there are two firms in this industry, Firm 1 and Firm 2. Each firm has an identical constant marginal cost of $80 per unit. a) Find the Cournot equilibrium quantities for each firm. What is the Cournot equilibrium market price? b) Assuming that Firm 1 is the Stackelberg leader, find the Stackelberg equilibrium quantities for each firm. What is the Stackelberg equilibrium price? c) Calculate and compare the profit of each firm under the Cournot and Stackelberg equilibria. Under which equilibrium is overall industry profit the greatest, and why?

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 13-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) Begin by inverting the market demand curve: Q = 600 – 3P  P = 200 – (1/3)Q. The marketing-clearing price if firm 1 produces Q1 and firm 2 produces Q2 is: P = 200 – (1/3)(Q1 + Q2). Let’s focus on firm 1 first. Firm 1’s residual demand curve has the equation P = [200 – (1/3)Q2] – (1/3)Q1. The corresponding marginal revenue curve is thus: MR1 = [200 – (1/3)Q2] – (2/3)Q1. Equating firm 1’s marginal revenue to marginal cost and solving for Q1 gives us: [200 – (1/3)Q2] – (2/3)Q1 = 80, or 120 – (1/3)Q2 = (2/3)Q1  Q1 = 180 – ½ Q2. This is Firm 1’s reaction function. Similar logic gives us Firm 2’s reaction function: Q2 = 180 – ½ Q1. Now, we have two equations (the two reaction functions) in two unknowns (Q1 and Q2). Solving this system of linear equations gives us: Q1 = Q2 = 120. The resulting market price is: P = 200 – (1/3)(120+120) = 120. Each firm’s profit is (P – M)Qi, for i = 1,2, or (120 – 80)(120) = $4,800 per month.

Industry profit is thus: $9,600 per month. b) To find the Stackelberg equilibrium, we begin by substituting Firm 2’s reaction function into the expression for the market-clearing price to get Firm 1’s residual demand curve. This gives us: P = 200 – (1/3)(Q1 + 180 – ½ Q1)  P = 140 – (1/6)Q1.The corresponding marginal revenue curve is: MR1 = 140 – (1/3)Q1. Equating marginal revenue to marginal cost gives us: 140 – (1/3)Q1 = 80, or Q1 = 180. This is the Stackelberg leader’s quantity. The Stackelberg follower’s quantity is found by substituting the leader’s quantity into the follower’s reaction function: Q2 = 180 – ½ (180) = 90. The resulting market price is: P = 200 – (1/3)(180 + 90) = 110. Notice that this is lower than the Cournot equilibrium price. c) Let’s now compute the profit of each firm under Stackelberg leadership and compare to the profit under Cournot. The leader’s profit is: (110 – 80)(180) = $5,400 per month. The follower’s profit is: (110 – 80)(90) = $2,700 per month. Notice that the leader earns higher profit than under the Cournot equilibrium, while the follower earns lower profit. Overall industry profit under Stackelberg leadership --- $5,400 + $2,700 = $8,100 --- is less than it is in the Cournot model.

13.17. Consider a market in which the market demand curve is given by P = 18 - X - Y, where X is Firm 1’s output, and Y is Firm 2’s output. Firm 1 has a marginal cost of 3, while Firm 2 has a marginal cost of 6. a) Find the Cournot equilibrium outputs in this market. How much profit does each firm make? b) Find the Stackelberg equilibrium in which Firm 1 acts as the leader. How much profit does each firm make?

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 13-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

The table below summarizes the solution. The details follow.

Cournot Stackelberg with Firm 1 as leader

Firm 1 output 6 9

Firm 2 output 3 1.5

Market Price 9 7.5

Firm 1 Profit 36 40.5

Firm 2 Profit 9 2.25

a) For Firm 1, equating MR to MC yields 18 – Y – 2X = 3, or X = 7.5 – 0.5Y. For Firm 2, equating MR to MC yields 18 – X – 2Y = 6, or Y = 6 – 0.5X. If we solve these reaction functions simultaneously, we have X = 6, Y = 3.

The resulting market price is thus: 18 – 6 – 3 = 9. Firm 1’s profit is: (9 – 3)*6 = 36. Firm 2’s profit is: (9 – 6)*3 = 9 b) If Firm 1 is the Stackelberg leader, we plug Firm 2’s reaction function into the expression for Firm 1’s total revenue: TR = [18 – (6 – 0.5X) – X]X = (12 – 0.5X)X. Firm 1’s marginal revenue is thus: 12 – X. Equating this to Firm 1’s marginal cost gives us: 12 – X = 3, or X = 9.

Given this, Firm 2’s output is Y = 6 – 0.5*9 = 1.5. The resulting market price is (18 – 9 – 1.5) = 7.5.

Firm 1’s profit is thus: (7.5 – 3)*9 = 40.5, while Firm 2’s profit is thus: (7.5 – 6)*1.5 = 2.25.

The problem demonstrates that Stackelberg leadership exaggerates the equilibrium differences (quantities and profits) between firms. Put another way, Firm 1’s cost advantage is even more powerful when Firm 1 acts as a Stackelberg leader.

13.18. Consider a market in which we have two firms, one of which will act as the Stackelberg leader and the other as the follower. As we know, this means that each firm will choose a quantity, X (for the leader) and Y (for the follower). Imagine that you have determined the Stackelberg equilibrium for a particular linear demand curve and set of marginal costs. Please indicate how X and Y would change if we then “perturbed” the initial situation in the following way: a) The leader’s marginal cost goes down, but the follower’s marginal cost stays the same. b) The follower’s marginal cost goes down, but the leader’s marginal cost stays the same. a)

The leader would produce more, and the follower would produce less.

Intuitively, with a lower marginal cost, the leader is more “eager” to supply output than before.

(Technically, with lower marginal cost, the intersection of the leader’s MR curve with its new MC must occur at a higher quantity than before). But if the leader produces more, the follower

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Chapter 13-21

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

must produce less. (Technically speaking, this is because by committing to a higher quantity, the leader “moves” the follower further to southeast along the follower’s reaction curve.) b)

The leader would produce less, and the follower would produce more.

Intuitively, with a lower marginal cost, the follower is now more eager to supply output than before. Therefore, it is “harder” for the leader (by producing lots of output) to get the follower to scale back its output. Thus, from the leader’s perspective, the value of extra output as a “strategic move” has been somewhat diminished. Hence, the leader is not quite as enthusiastic about producing lots of output, and so it will cut back a little.

13.19. Suppose that the market demand for cobalt is given by Q = 200 - P. Suppose that the industry consists of 10 firms, each with a marginal cost of $40 per unit. What is the Cournot equilibrium quantity for each firm? What is the equilibrium market price?

Market demand is given by 𝑃 = 200 − 𝑄. With ten firms, 𝑄 = 𝑄1 + ⋯ + 𝑄10 . Letting 𝑋 = 𝑄2 + ⋯ + 𝑄10 , Firm 1 faces demand 𝑃 = (200 − 𝑋) − 𝑄1 . Setting 𝑀𝑅 = 𝑀𝐶 implies (200 − X ) − 2Q1 = 40 Q1 = 80 − 0.5 X

In equilibrium, each firm will produce the same quantity Q* = Q1 = Q2 = … = Q10 𝑄 ∗ = 80 − 0.5(9𝑄 ∗ ) 𝑄 ∗ = 14.55 Total industry output will be 145.50 with each of the ten firms producing 14.55 units. At this quantity, market price will be 𝑃 = 200 − 145.50 = 54.50.

13.20. Consider the same setting as in the previous problem, but now suppose that the industry consists of a dominant firm, Braeutigam Cobalt (BC), which has a constant marginal cost equal to $40 per unit. There are nine other fringe producers, each of whom has a marginal cost curve MC = 40 + 10q, where q is the output of a typical fringe producer.

Assume there are no fixed costs for any producer. a) What is the supply curve of the competitive fringe? b) What is BC’s residual demand curve? c) Find BC’s profit-maximizing output and price. At this price, what is BC’s market share? d) Repeat parts (a) to (c) under the assumption that the competitive fringe consists of 18 firms.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 13-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) The supply curve of the competitive fringe is the horizontal summation of the marginal cost curves (supply curves) for the individual firms. Since 𝑀𝐶 = 40 + 10𝑞 for each firm, q = 0.1P – 4 is the supply curve for an individual firm, so long as P > 40. For P ≤ 40, fringe supply is zero. Summing these for the 9 fringe producers in this market implies 0 𝑆𝐹 = { 0.9𝑃 − 36

𝑃 ≤ 40 𝑃 > 40

b) For P ≤ 40, fringe supply is zero so residual demand is equal to market demand. For P > 40, residual demand is the horizontal difference between the fringe supply and market demand.

Thus, residual demand is 200 − 𝑃 𝑄𝑅 = { 236 − 1.9𝑃

𝑃 ≤ 40 𝑃 > 40

c) For P > 40, the inverse residual demand curve is P = (10/19)(236 – QR), so the associated marginal revenue curve is MR = (10/19)(236 – 2QR). BC maximizes profit by equating MR to its MC = 40:

(10 / 19)(236 − 2QR ) = 40 QR = 80

Using the residual demand curve, BC’s profit-maximizing price is P = (10/19)(236 – 80) = 82.11 and total market demand is Q = 200 – 82.11 = 117.89. Thus, BC’s market share is 80/117.89 = 0.68, or 68 percent. d)

If the competitive fringe consists of 18 firms, then

0 𝑆𝐹 = { 1.8𝑃 − 72

𝑃 ≤ 40 𝑃 > 40

Residual demand is: 200 − 𝑃 𝑄𝑅 = { 272 − 2.8𝑃

𝑃 ≤ 40 𝑃 > 40

For P > 40, the inverse residual demand is P = (10/28)(272 – QR). BC maximizes profit by setting MR = MC, or

(10 / 28)(272 − 2QR ) = 40 QR = 80

Using the residual demand curve, BC will set P = (10/28)(272 – 80) = 68.58. Total market demand will be 𝑄 = 200 − 68.58 = 131.42. Thus, BC’s market share will be 80/131.43 = 0.61, or 61 percent. As more firms enter the competitive fringe, BC’s market share falls.

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Solutions Manual

13.21. Apple’s iPod has been the portable MP3-player of choice among many gadget enthusiasts. Suppose that Apple has a constant marginal cost of 4 and that market demand is given by Q = 200 - 2P. a) If Apple is a monopolist, find its optimal price and output. What are its profits? b) Now suppose there is a competitive fringe of 12 price-taking firms, each of which has a total cost function TC(q) = 3q2 + 20q with corresponding marginal cost curve MC = 6q + 20.

Find the supply function of the fringe (Hint: A competitive firm supplies along its marginal cost curve above its shutdown point). c) If Apple operates as the dominant firm facing competition from the fringe in this market, now what is its optimal output? How many units will fringe providers sell? What is the market price, and how much profit does Apple earn? d) Graph your answer to part (c). a) A monopolist sets MR = MC (don’t forget to invert the demand curve first!) so 100 – Qm = 4. Thus, Qm = 96 and Pm = 52. As a monopolist, Apple’s profits are  = (52 – 4)*96 = 4608. b) Each fringe firm maximizes profits by setting P = MC = 6q + 20, so we can derive a single firm’s supply curve as q = (P – 20)/6, so long as P > 20. With the fringe comprising 12 firms, total supply is Qfringe = 12q, or 0 𝑄𝑓𝑟𝑖𝑛𝑔𝑒 = { 2𝑃 − 40

𝑃 ≤ 20 𝑃 > 20

c) First, find Apple’s (denoted DF for “dominant firm”) residual demand, for P > 20: QDF = Qmkt – Qfringe = 200 – 2P – (2P – 40) = 240 – 4P. Inverting, this is P = 60 – 0.25Q. So Apple sets MR = MC or 60 – 0.5QDF = 4 QDF = 112 From the residual demand, Apple’s price is P = 60 – 0.25*112 = 32. At this price, the fringe supplies Qfringe = 2P – 40 = 2*32 – 40 = 24. Apple’s profits are  = (32 – 4)*112 = 3136. d) On the graph, note that (i) residual demand begins at P = 60, precisely where fringe supply intersects market demand; (ii) residual demand connects back with market demand at P = 20, where the fringe supply curve intersects the P-axis; and (iii) the dominant firm’s equilibrium output occurs where its MR = MC.

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Solutions Manual

Market demand Fringe supply

Residual demand

MR

Qfringe = 24

MC

QDF = 112 Qmkt = 136

13.22. Britney produces pop music albums with the total cost function TC(Q) = 8Q. Market demand for pop music albums is P = 56 - Q. Suppose there is a competitive fringe of pricetaking pop music artists, with total supply function Qfringe = 2P - y, where y > 0 is some positive integer. If Britney behaves like a dominant firm and maximizes her profit by selling at a price of P = 16, find (i) the value of y, (ii) Britney’s output level, and (iii) the output level of the competitive fringe.

Britney’s residual demand is QB = Q – Qfringe = 56 – P – (2P – y) = 56 + y – 3P. Or in inverse form, P = (56 + y – QB)/3. We’ll use this in two ways.

First, Britney maximizes profits by setting MR = MC or (56 + y – 2QB)/3 = 8 yielding QB = 0.5(32 + y).

Second, knowing that Britney charges a price of P = 16, using her residual demand curve implies 16 = (56 + y – QB)/3, or QB = 8 + y.

Solving these two equations for QB and y implies 8 + y = 0.5(32 + y) yielding y = 16 and QB = 24. Lastly, knowing the value of y and the market price P we can then see that the fringe supplies Qfringe = 2*16 – 16 = 16.

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Solutions Manual

13.23. The market demand curve in the nickel industry in Australia is given by Qd = 400 8P. The industry is dominated by a large firm with a constant marginal cost of $10 per unit. There also exists a competitive fringe of 100 firms, each of which has a marginal cost given by MC = 10 + 50q, where q is the output of a typical fringe firm. a) What is the equation of the supply curve for the competitive fringe? b) Restricting your attention to the range of prices that exceed the dominant firm’s marginal cost, what is the equation of the residual demand curve? c) What is the profit-maximizing quantity of the dominant firm? What is the resulting market price? At this price, how much does the competitive fringe produce, and what is the fringe’s market share (i.e., the fringe quantity divided by total industry quantity)? What is the dominant firm’s market share? d) Let’s consider a twist on the basic dominant firm model. Suppose the Australia government, concerned about the amount of dominance in the nickel industry decides to break the dominant firm into two identical firms, each with a constant marginal cost of $10 per unit. Suppose further that these two firms act as Cournot quantity setters, taking into account the supply curve of the competitive fringe. What is the Cournot equilibrium quantity produced by each dominant firm? What is the equilibrium market price? At this price, how much does the competitive fringe produce, and what is the fringe’s market share? a) To find the supply curve of the competitive fringe, we proceed as follows. Each pricetaking fringe firm produces to the point at which the market price equals marginal cost: P = 10 + 50q, or q = (P – 10)/50. This equation is valid only if the price is greater than or equal to 10. If the price is less than 10, a fringe firm produces nothing. In this problem, there is no loss of generality in assuming that the market price will exceed 10, and that a fringe firm’s supply curve is given by q = (P – 10)/50. This is because the marginal cost of the dominant firm is 10, and the dominant will not operate at a point at which its price is less than its marginal cost. Given this, the fringe’s overall supply curve is found by multiplying the individual fringe supply curve by the number of fringe firms (100): Qs = (100)(P – 10)/50 = 2P – 20. Thus, the overall fringe supply curve is Qs = 2P – 20. b) We find the residual demand curve by subtracting the overall fringe supply from the market demand curve. Letting Qr denote residual demand, we have: Qr = Qd – Qs = (400 – 8P)(2P – 20)  Qr = 420 – 10P. c) To find the profit-maximizing quantity of the dominant firm, we first invert the residual demand (dropping the superscript r) to get: P = 42 – (1/10)Q. The corresponding marginal revenue curve is: MR = 42 – (1/5)Q. Equating marginal revenue to marginal cost gives us: 42 – (1/5)Q = 10, which implies: Q = 160. The resulting market price is P = 42 – (1/10)(160) = 26. At this price, the fringe’s overall supply is: 2(26) – 20 = 32. Hence, the fringe’s market share is 32/(32 + 160) = (1/6) or 16.67 percent. The dominant firm’s market share is (5/6) or 83.33 percent. d) We solve this problem as we would solve any Cournot except that we use the (inverse) residual demand curve. Recall that the inverse residual demand curve is P = 42 – (1/10)(Q1 + Copyright © 2020 John Wiley & Sons, Inc.

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Solutions Manual

Q2), where Q1 is the output of dominant Firm 1 and Q2 is the output of dominant Firm 2. Firm 1’s marginal revenue curve is MR1 = [42 – (1/10)Q2] – (1/5)Q1. Equating this to marginal cost and solving for Q1 in terms of Q2 gives us Firm 1’s reaction function: [42 – (1/10)Q2] – (1/5)Q1 = 10  Q1 = 160 – ½ Q2. Similar logic is used to derive Firm 2’s reaction function: Q2 = 160 – ½Q1. Solving these reaction functions simultaneously gives us the equilibrium quantities of each firm: Q1 = Q2 = 106.67. The two dominant firms together produce 213.33. The corresponding market price is: 42 – (1/10)(216.33) = 20.367. At this market price, the competitive fringe produces: 2(20.367) – 20 = 20.733. The market share of the competitive fringe is thus: 20.733/(20.733+213.33) = 8.86 percent, which means than the two dominant firms together have a market share of 100 – 8.86 = 91.14 percent. Comparing this solution to the one in part (c), we can see that adding a second dominant firm reduces the equilibrium market share attained by the competitive fringe.

13.24. Consider the Coke and Pepsi example discussed in the chapter. a) Explain why each firm’s reaction function slopes upward. That is, why does Coke’s profit-maximizing price go up the higher is Pepsi’s price? Why does Pepsi’s profitmaximizing price go up the higher Coke’s price is? b) Explain why Pepsi’s profit-maximizing price seems to be relatively insensitive to Coke’s price. That is, why is Pepsi’s reaction function so flat? a) The profit-maximizing price for one firm given the price of the other firm is positively related to the price of the other firm’s product. This occurs because as one firm lowers its price (say Coke), demand will move toward Coke and away from Pepsi. In order for Pepsi to maintain demand and profits, Pepsi must also lower its price. By the same reasoning, if Coke raises its price, demand will begin to move toward Pepsi, allowing Pepsi the opportunity to raise its price and profits without hurting demand. b) Pepsi’s demand is less sensitive to changes in Coke’s price and more sensitive to changes in its own price. This implies that as Coke’s price increases, relatively little demand will move to Pepsi, but as Pepsi raises its price, relatively more demand will move to Coke. This implies brand loyalty for Pepsi is low when compared with Coke meaning that Pepsi has less room to move its price. This is seen graphically by a reaction function that is flatter than Coke’s.

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13.25. Again consider the Coke and Pepsi example discussed in the chapter. Use graphs of reaction functions to illustrate what would happen to equilibrium prices if: a) Coca-Cola’s marginal cost increased. b) For any pair of prices for Coke and Pepsi, Pepsi’s demand went up. a) When Coca Cola’s marginal cost increases, Coke’s reaction function will shift away from the origin, from R1 to R2. This will raise both Coke’s price and Pepsi’s price.

b) When Pepsi’s demand increases, Pepsi’s reaction function shifts upward, from R1 to R2.

This will increase both Coke’s price and Pepsi’s price.

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Solutions Manual

13.26. Two firms, Alpha and Bravo, compete in the European chewing gum industry. The products of the two firms are differentiated, and each month the two firms set their prices.

The demand functions facing each firm are:

where the subscript A denotes the firm Alpha and the subscript B denotes the firm Bravo.

Each firm has a constant marginal cost of $7 per unit. a) Find the equation of the reaction function of each firm. b) Find the Bertrand equilibrium price of each firm. c) Sketch how each firm’s reaction function is affected by each of the following changes: i) Alpha’s marginal cost goes down (with Bravo’s marginal cost remaining the same). ii) Alpha and Bravo’s marginal cost goes down by the same amount. iii) Demand conditions change so that the “150” term in the demand function now becomes larger than 150. iv) The “10” and “9” terms in each demand function now become larger (e.g., they become “50” and “49,” respectively). d) Explain in words how the Bertrand equilibrium price of each firm is affected by each of the following changes: i) Alpha’s marginal cost goes down (with Bravo’s marginal cost remaining the same). ii) Alpha and Bravo’s marginal cost goes down by the same amount. iii) Demand conditions change so that the “150” term in the demand function for each firm now becomes larger than 150. iv) The “10” and “9” terms in each demand function now become larger (e.g., they become “50” and “49,” respectively). a) We will first solve for Alpha’s reaction function. We begin by solving Alpha’s demand function for PA in terms of QA and PB: PA = 15 – (1/10)QA + (9/10)PB. The corresponding marginal revenue equation is: MRA = 14 – (2/10)QA + (9/10)PB. Equating marginal revenue to marginal cost and solving for QA gives us Alpha’s profit-maximizing quantity as a function of Bravo’s price: MRA = MCA  15 – (2/10)QA + (9/10)PB = 7, which gives us: QA = 40 + (9/2)PB. Now, substitute this expression for QA back into the expression for the demand curve with PA on the left-hand side and QA on the right-hand side: PA = 15 – (1/10)[40 + (9/2)PB] + (9/10)PB  PA = 11 + (9/20)PB. This is Alpha’s reaction function. b) We can find Bravo’s reaction function by following steps identical to those followed to derive Alpha’s reaction function. Following these steps gives us: PB = 11 + (9/20)PA. We now have two equations (the two reaction functions) in two unknowns, PA and PB. Solving these equations gives us the Bertrand equilibrium prices: PA = PB = 20.

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c)

Solutions Manual

The following diagrams show how each change affects the reaction functions.

Alpha’s Marginal Cost Goes Down Firm Bravo’s price

Alpha’s reaction function new initial

30

Bravo’s reaction function 25

E 20

F

15

10

5

0 0

5

10

15

20

25

30

Firm Alpha’s price 2

Alpha and Bravo’s Marginal Cost Go Down by the Same Amount Firm Bravo’s price

Alpha’s reaction function new initial

30

Bravo’s reaction function new

25

E

initial

20

15

F

10

5

0 0

5

10

15

20

25

30

Firm Alpha’s price 3

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Demand Curve Intercept Goes Up for Each Firm Alpha’s reaction function initial new initial F

Firm Bravo’s price 30

new

25

Bravo’s reaction function

E 20

15

10

5

0 0

5

10

15

20

25

30

Firm Alpha’s price 4

Demand Curve Price Coefficients Become Larger in Absolute Value Alpha’s reaction function new initial Bravo’s reaction function initial

Firm Bravo’s price 30

25

new

E 20

15

F 10

5

0 0

5

10

15

20

25

30

Firm Alpha’s price 5

d) The above diagrams can used to verify how each of the changes affects the Bertrand equilibrium: Each firm’s equilibrium price goes down.

Each firm’s equilibrium price goes down.

Each firm’s equilibrium price goes up.

Each firm’s equilibrium price goes down.

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Solutions Manual

13.27. When firms choose outputs, as in the Cournot model, reaction functions slope downward. But when firms choose prices, as in the Bertrand model with differentiated products, reaction functions slope upward. Why do output reaction functions differ from price reaction functions in this way?

When firms choose outputs, the reaction functions are downward sloping. Intuitively, if one firm reduces its quantity, the residual demand curve of the second firm shifts outward, as will the associated marginal revenue curve. This causes the second firm’s profit-maximizing quantity to go up. Thus, if one firm lowers its quantity, the other firm should raise its quantity. This implies a negative relationship between the quantities.

When firms choose prices, the reaction functions are upward sloping. Intuitively, if one firm lowers its price, the residual demand curve of the second firm shifts inward, as will the associated marginal revenue curve. This causes the second firm’s profit-maximizing quantity and price to fall. Thus, if one firm lowers its price, the other firm should lower its price too. This implies a positive relationship between firm’s prices.

13.28. Suppose that Jerry and Teddy are the only two sellers of designer umbrellas, which consumers view as differentiated products. For simplicity, assume each seller has a constant marginal cost equal to zero. When Jerry charges a price pJ and Teddy charges pT, consumers would buy a total of

umbrellas from Jerry. In similar fashion, Teddy faces a demand curve of

Illustrate each seller’s best-response function on a graph. What are the equilibrium prices?

How much profit does each seller earn?

Jerry’s demand curve can be written as pJ = (1/3)(100 + pT) – (1/3)qJ. Hence, MRJ = MCJ implies (1/3)(100 + pT) – (2/3)*qJ = 0 yielding Jerry’s reaction function: qJ = 50 + 0.5pT. At this quantity, Jerry will charge a price of pJ = (100 + pT)/6. Similarly, we can find that Teddy’s best response function is pT = (100 + pJ)/6. Solving the two equations yields pT = pJ = 20 in equilibrium. We can also see this graphically:

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pJ

Solutions Manual

Teddy’s best response function pT = (100 + pJ)/6 Jerry’s best response function pJ = (100 + pT)/6

20

20

pT

13.29. United Airlines and American Airlines both fly between Chicago and San Francisco.

Their demand curves are given by QA = 1000 - 2PA + PU and QU = 1000 - 2PU + PA. QA and QU stand for the number of passengers per day for American and United, respectively.

The marginal cost of each carrier is $10 per passenger. a) If American sets a price of $200, what is the equation of United’s demand curve and marginal revenue curve? What is United’s profit-maximizing price when American sets a price of $200? b) Redo part (a) under the assumption that American sets a price of $400. c) Derive the equations for American’s and United’s price reaction curves. d) What is the Bertrand equilibrium in this market? a) If American sets a price of $200, we can plug this price into United’s demand curve to get United’s perceived demand curve.

QU = 1000 − 2 PU + 200 PU = 600 − 0.5QU

To find United’s profit-maximizing price set 𝑀𝑅 = 𝑀𝐶.

600 − QU = 10 QU = 590

At this quantity United will charge a price 𝑃𝑈 = 600 − 0.5(590) = 305. b)

If American sets a price of $400, then United’s perceived demand curve is

QU = 1000 − 2 PU + 400 PU = 700 − 0.5QU

Equating 𝑀𝑅 to 𝑀𝐶 yields

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Solutions Manual

700 − QU = 10 QU = 690

At this quantity, United will charge a price 𝑃𝑈 = 700 − 0.5(690) = 355. c)

American’s demand can be rewritten as

2 PA = (1000 + PU ) − QA PA = (500 + 0.5PU ) − 0.5QA

Setting 𝑀𝑅 = 𝑀𝐶 implies (500 + 0.5 PU ) − QA = 10 QA = 490 + 0.5 PU

At this quantity American will charge a price PA = (500 + 0.5 PU ) − 0.5(490 + 0.5 PU ) PA = 255 + 0.25 PU

Since the firms have identical marginal cost and symmetric demand curves, United’s price reaction function will be 𝑃𝑈 = 255 + 0.25𝑃𝐴 . d) The Bertrand equilibrium will occur where these price reaction functions intersect.

Substituting the expression for 𝑃𝑈 into the expression for 𝑃𝐴 implies 𝑃𝐴 = 255 + 0.25(255 + 0.25𝑃𝐴 ) 𝑃𝐴 = 340 Substituting into the expression for 𝑃𝑈 implies 𝑃𝑈 = 340. So, in the Bertrand equilibrium each firm charges a price of 340 and attracts a quantity of 660.

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Solutions Manual

13.30. Three firms compete as Bertrand price competitors in a differentiated products market. Each of the three firms has a marginal cost of 0. The demand curves of each firm are as follows:

where P23 is the average of the prices charged by Firms 2 and 3, P13 is the average of the prices charged by Firms 1 and 3, and P12 is the average of the prices charged by Firms 1 and 2 [e.g., P12 = 0.5(P1 + P2)]. What is the Bertrand equilibrium price charged by each firm?

Let’s start by deriving Firm 1’s reaction function: Step 1: We can write Firm 1’s demand curve as: P1 = [40 + 0.5P23] – 0.5Q1 The corresponding MR curve is thus: MR1 = [40 + 0.5P23] – Q1 Step 2: Set MR1 equal to MC and solve for Q1 in terms of P23 Since MC = 0, this step gives us: Q1 = 40 + 0.5P23 Step 3: Substitute back into the inverse demand curve to give us P1 in terms of P23.

P1 = [40 + 0.5P23] – 0.5[40 + 0.5P23] = 20 + 0.25P23 = 20 + 0.125(P2 + P3) This is Firm 1’s reaction function.

We could proceed to derive the reaction functions of all of the firms. But notice that in this problem, all firms are symmetric. Thus, their reaction functions will be mirror images of each other.

P2 = 20 + 0.125(P1 + P3) P3 = 20 + 0.125(P1 + P2) We now have three equations in three unknowns, which we can solve for the equilibrium prices.

Doing so, we get P1 = P2 = P3 = 26.67.

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13.31. The Baldonian shoe market is served by a monopoly firm. The demand for shoes in Baldonia is given by Q = 10 - P, where Q is millions of pairs of shoes (a right shoe and left shoe) per year, and P is the price of a pair of shoes. The marginal cost of making shoes is constant and equal to $2 per pair. a) At what price would the Baldonian monopolist sell shoes? How many shoes are purchased? b) Baldonian authorities have concluded that the shoe seller’s monopoly power is not a good thing. Inspired by the U.S. government’s attempt several years ago to break Microsoft into two pieces, Baldonia creates two firms: one that sells right shoes and the other that sells left shoes. Let P1 be the price charged by the right-shoe producer and P2 be the price charged by the left-shoe producer. Of course, consumers still want to buy a pair of shoes (a right one and a left one), so the demand for pairs of shoes continues to be 10 - P1 - P2. If you think about it, this means that the right-shoe producer sells 10 - P1 - P2 right shoes, while the left-shoe producer sells 10 - P1 - P2 left shoes. Since the marginal cost of a pair of shoes is $2 per pair, the marginal cost of the right-shoe producer is $1 per shoe, and the marginal cost of the left-shoe producer is $1 per shoe. i) Derive the reaction function of the right-shoe producer (P1 in terms of P2). Do the same for the left-shoe producer. ii) What is the Bertrand equilibrium price of shoes? How many pairs of shoes are purchased? iii) Has the breakup of the shoe monopolist improved consumer welfare?

Note: To see the potential relevance of this problem to the Microsoft antitrust case, you might be interested in reading Paul Krugman, “The Parable of Baron von Gates,” New York Times (April 26, 2000). a)

Monopoly price = $6 per pair of shoes, Q = 4 million shoes are sold.

b)

Let’s begin by deriving Firm 1’s reaction function.

Step 1: Firm 1’s demand curve can be written as: P1 = [10 – P2] – Q1 The corresponding MR curve is thus: MR1 = [10 – P2] – 2Q1 Step 2: Set MR1 equal to MC and solve for Q1 in terms of P2 [10 – P2] – 2Q1 = 1 Q1 = 4.5 – 0.5P2

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Step 3: Substitute back into the inverse demand curve to give us P1 in terms of P2.

P1 = [10 – P2] – [4.5 – 0.5P2] = 5.5 – 0.5P2 Step 4: Firm 2’s reaction function is easy. The firms are symmetric so it takes the same form as firm 1’s reaction function: P2 = 5.5 – 0.5P1 Step 5: Solve for the equilibrium We have two equations in two unknowns. If we solve these reaction functions, we get P1 = P2 = 3.67.

Since P1 + P2 = the total price of a pair of shoes = 3.67 + 3.67 = 7.33 > 6, a pair of shoes is now more expensive than when a single shoe monopolist controlled the market. Consumers are worse off after the breakup!

In effect what is going on here is this: after the breakup of the shoe monopoly, the right-shoe producer does not consider the negative impact of his raising price on the demand for left-shoes.

Ditto for the left-shoe producer. A monopolist seller of both right and left shoes, by contrast, would internalize this effect. The result: the independent firms raise price “too much” since they are “ignoring” part of the “cost” of raising price.

13.32. Reconsider Problem 13.29, except suppose American and United take each other’s quantity as given rather than taking each other’s price as given. That is, assume that American and United act as Cournot competitors rather than Bertrand competitors. The inverse demand curves corresponding to the demand curves in Problem 13.29 are

a) Suppose that American chooses to carry 660 passengers per day (i.e., QA = 660). What is United’s profit maximizing quantity of passengers? Suppose American carries 500 passengers per day. What is United’s profit maximizing quantity of passengers? b) Derive the quantity reaction function for each firm. c) What is the Cournot equilibrium in quantities for both firms? What are the corresponding equilibrium prices for both firms? d) Why does the Cournot equilibrium in this problem differ from the Bertrand equilibrium in Problem 13.29?

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a) If American chooses to carry 660 passengers, the Bertrand equilibrium, then United’s perceived inverse demand is 2 1 𝑃𝑈 = 1000 − 𝑄𝑈 − (660) 3 3 2 𝑃𝑈 = 780 − 𝑄𝑈 3 Setting 𝑀𝑅 = 𝑀𝐶 implies

4 780 − QU = 10 3 QU = 577.50 If American chooses to carry 500 passengers per day, United’s perceived inverse demand is 2 1 𝑃𝑈 = 1000 − 𝑄𝑈 − (500) 3 3 2 𝑃𝑈 = 833.33 − 𝑄𝑈 3 Setting 𝑀𝑅 = 𝑀𝐶 implies

4 833.33 − QU = 10 3 QU = 617.50 b)

Setting 𝑀𝑅 = 𝑀𝐶 for American implies

1 4 (1000 − QU ) − QA = 10 3 3 QA = 742.50 − 0.25QU Since the firms are symmetric, we will have 𝑄𝑈 = 742.50 − 0.25𝑄𝐴 . c) The Cournot equilibrium will occur where these reaction functions intersect. Substituting the expression for 𝑄𝑈 into the expression for 𝑄𝐴 implies 𝑄𝐴 = 742.50 − 0.25(742.50 − 0.25𝑄𝐴 ) 𝑄𝐴 = 594 Substituting 𝑄𝐴 = 594 into United’s reaction function implies 𝑄𝑈 = 594. At this quantity, each firm will charge a price of 406.

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d) A Bertrand firm chooses its profit-maximizing price assuming the other firm does not change its price. A Cournot firm chooses its profit-maximizing quantity assuming the other firm does not change its quantity. Because the assumptions about its rival’s behavior are different in the two models, it is not surprising that the equilibrium prices will be different.

13.33. Let’s imagine that a local retail market is monopolistically competitive. Each firm (and potential entrant) is identical and faces a marginal cost that is independent of output and is equal to $100 per unit. Each firm has an annual fixed cost of $300,000 per month.

Because each active firm perceives itself facing a price elasticity of demand equal to -2, the inverse elasticity pricing condition implies that the profit-maximizing price for each firm is (P - 100)/P = 1/2 or P = 200. If each firm charges an equal price, they will evenly split the overall market demand of 96,000 units per month. a) How many firms will operate in this market at a long run equilibrium? b) How would your answer change if each firm faced a price elasticity of demand of -4/3 and charged a profit maximizing price of $400 per unit? a) The profit of an active firm is: (200 – 100)*96,000/N – 300,000, where N is the number of active firms. At a long-run equilibrium, each firm earns zero profit. Thus, the equilibrium number of firms makes this equal to zero. Thus: 9,600,000/N = 300,000, or N = 32. b) The profit of an active firm is: (400 – 100)*96,000/N – 300,000, where N is the number of active firms. At a long-run equilibrium, each firm earns zero profit. Thus, the equilibrium number of firms makes this equal to zero. Thus: 28,800,000/N = 300,000, or N = 96.

13.34. The Thai food restaurant business in Evanston, Illinois, is monopolistically competitive. Suppose that each existing and potential restaurant has a total cost function given by TC = 10Q + 40,000, where Q is the number of patrons per month and TC is total cost per month. The fixed cost of $40,000 includes fixed operating expenses (such as the salary of the chef), the lease on the building space where the restaurant is located, and interest expenses on the bank loan needed to start the business in the first place. Currently, there are 10 Thai restaurants in Evanston. Each restaurant faces a demand function given 𝟒,𝟎𝟎𝟎,𝟎𝟎𝟎 −𝟏 𝟒 ̅ where P is the price of a typical entrée at the restaurant, by 𝑸 = 𝑷 𝑷 is the 𝑵 price of a typical entrée averaged over all the other Thai restaurants in Evanston, and N is the total number of restaurants. Each restaurant takes the prices of other Thai restaurants as given when choosing its own price. a) What is the own-price elasticity of demand facing a typical restaurant? b) For a typical restaurant, what is the profit-maximizing price of a typical entrée? c) At the profit-maximizing price, how many patrons does a typical restaurant serve per month? Given this number of patrons, what is the average total cost of a typical restaurant?

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d) What is the long-run equilibrium number of Thai restaurants in the Evanston market? a)

Each firm faces an own price elasticity of demand of -5.

b) Because the demand function exhibits constant elasticity, we can directly solve for a firm’s profit-maximizing price by using the inverse elasticity pricing rule: (P- MC)/P = -1/Q,P From the equation of the total cost function, we see that each firm faces a marginal cost of $10 per patron. This implies (P – 10)/P = -1/(-5), or P = $12.50. c) Each firm will charge a price of $12.50, so therefore each restaurant attracts (4,000,000/10)(12.50)-5(12.50)4 = 32,000 patrons per month. Given this number of patrons each restaurant has an average total cost given by 10 + 40,000/32,000 = $11.25. d) To find the long-run equilibrium number of firms, we note that when there is an arbitrary number of firms N, the number of patrons going to each restaurant is (4,000,000/N)(12.50)5 (12.50)4 = 320,000/N. Given this number of patrons, each firm’s average total cost is 10 + 40,000/(320,000/N) = 10 + 0.125N. Because the demand function is constant elasticity, each firm will charge a price of $12.50 no matter how many firms are in the market. In a long-run equilibrium, the price of 12.50 must equal each firm’s average total cost. Thus: 10 + 0.125N = 12.50, or 20.

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Chapter 14 Game Theory and Strategic Behavior Solutions to Review Questions 1. What is a Nash equilibrium? Why would strategies that do not constitute a Nash equilibrium be an unlikely outcome of a game?

A Nash equilibrium in a game occurs when each player chooses a strategy that gives it the highest payoff, given the strategies chosen by the other players in the game.

If players chose strategies that did not constitute a Nash equilibrium, then the players could choose another strategy that increased their payoff given the strategies chosen by the other players. Since players could increase their payoffs by choosing other strategies, strategies that do not constitute a Nash equilibrium are an unlikely outcome in a game.

2. What is special about the prisoners’ dilemma game? Is every game presented in this chapter a prisoners’ dilemma?

A prisoners’ dilemma game illustrates the conflict between self-interest and collective interest. In the Nash equilibrium of a prisoners’ dilemma game, each player chooses a non-cooperative action even though it is in the players’ collective interest to pursue a cooperative action.

No, not every game in the chapter is a prisoners’ dilemma. For example, the game of Chicken presented in Table 14.7 is not a prisoners’ dilemma.

3. What is the difference between a dominant strategy and a dominated strategy? Why would a player in a game be unlikely to choose a dominated strategy?

A dominant strategy is a strategy that is better than any other strategy the player might follow no matter what the other player does. A player has a dominated strategy when it has other strategies that give it a higher payoff no matter what the other player does. A player would be unlikely to choose a dominated strategy because the player could always improve his payoff by choosing another strategy regardless of the strategies chosen by the other players.

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4. What is special about the game of Chicken? How does the game of Chicken differ from the prisoners’ dilemma game?

The Chicken game is special because it has multiple Nash equilibria. In each Nash equilibria, one player chooses a cooperative strategy (Swerve) while the other player chooses a noncooperative strategy (Stay). There are multiple equilibria in this game because it is uncertain at the outset which player will Swerve and which will Stay.

This game differs from the prisoners’ dilemma. In the prisoners’ dilemma, both players have a dominant strategy to confess, and by choosing this dominant strategy, both players receive a payoff worse than if they both chose to not confess.

5. Can a game have a Nash equilibrium even though neither player has a dominant strategy? Can a game have a Nash equilibrium even though neither player has a dominated strategy?

Yes, a game can have a Nash equilibrium even though neither player has a dominant or dominated strategy. In fact, every game has a Nash equilibrium, possibly in mixed strategies.

The game of Chicken is an example of a game with no dominant or dominated strategies but which has a Nash equilibrium.

6. What is the difference between a pure strategy and a mixed strategy?

A pure strategy is a specific choice of strategy among the possible moves a player may choose in a game. With a mixed strategy, a player chooses among two or more pure strategies according to pre-specified probabilities.

7. How can cooperation emerge in the infinitely repeated prisoners’ dilemma game even though in a single shot prisoners’ dilemma, noncooperation is a dominant strategy?

In the repeated prisoners’ dilemma game, the players might, in equilibrium, play cooperatively.

This could occur if one player chose to cooperate with the other player as long as the other player chose to cooperate and to resort to non-cooperation when the other player cheated. For this to work, the short-term benefit of cheating must be lower than the long-term benefit of not cheating. This would require the player to place a sufficiently strong weight on future payoffs relative to current payoffs.

8. What are the conditions that enhance the likelihood of a cooperative outcome in a repeated prisoners’ dilemma game?

The likelihood of a cooperative outcome is improved when the players are patient, their interactions are frequent, cheating is easy to detect, and the one-shot gain from cheating is small.

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9. What is the difference between a simultaneous move game and a sequential-move game?

In a simultaneous move game, both players make their strategy choices at the same time. In a sequential move game, one player chooses his strategy first, and then seeing the move that the first player made, the second player chooses her strategy.

10. What is a strategic move? Why must strategic moves be hard to reverse in order to have strategic value?

In a sequential move game, a strategic move is an action a player takes in an early stage of a game that alters the player’s behavior and the competitor’s behavior later in the game in a way that is favorable to the player. Strategic moves can limit a player’s flexibility and in so doing can have strategic value. To have strategic value, however, the strategic move must be hard to reverse. If the move is easily reversed, then the competitor will ignore the strategic move when making its choice. If the move is not easily reversed, then the competitor must assume the player is not ‘bluffing’ and take the strategic move into consideration when making its choice.

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Solutions to Problems 14.1. What is the Nash equilibrium in the following game?

The Nash Equilibrium is: Player 1 chooses Up while Player 2 chooses Left.

14.2. Ignoring mixed strategies, does the following game have a Nash equilibrium? Does it have more than one Nash equilibrium? If so, what are they?

There are two Nash equilibria. Nash equilibrium #1: Player 1 chooses SOUTH, Player 2 chooses WEST. Nash equilibrium #2: Player 1 chooses NORTH, Player 2 chooses EAST 14.3. Does either player in the following game have a dominant strategy? If so, identify it.

Does either player have a dominated strategy? If so, identify it. What is the Nash equilibrium in this game?

Player 1 has a dominant strategy (UP), which means that DOWN is dominated.

Player 2 does not have a dominant strategy, but it does have a dominated strategy (MIDDLE).

The Nash equilibrium in this game is for Player 1 to choose UP and Player 2 to choose LEFT

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14.4. Coca-Cola and Pepsi are competing in the Brazilian soft-drink market. Each firm is deciding whether to follow an aggressive advertising strategy, in which the firm significantly increases its spending on media and billboard advertising over last year’s level, or a restrained strategy, in which the firm keeps its advertising spending equal to last year’s level. The profits associated with each strategy are as follows:

What is the Nash equilibrium in this game? Is this game an example of the prisoners’ dilemma?

In this game, “Aggressive” is a dominant strategy for both firms. Thus, the Nash equilibrium strategy for both firms is to choose “Aggressive.”

This game is an example of the prisoners’ dilemma. In this game both players have a dominant strategy that leads to an outcome that does not maximize the collective payoffs of the players in the game. If both players chose the “Restrained” strategy, then both players would increase their profits and the collective payoff would be maximized.

14.5. In the Castorian Airline market there are only two firms. Each firm is deciding whether to offer a frequent flyer program. The annual profits (in millions of dollars) associated with each strategy are summarized in the following table (where the first number is the payoff to Airline A and the second to Airline B):

a) Does either player have a dominant strategy? Explain. b) Is there a Nash equilibrium in this game? If so what is it? c) Is this game an example of the prisoners’ dilemma? Explain.

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a) Each firm finds having a frequent flyer program (FFP) to be a dominant strategy.

If B has an FFP, A prefers a payoff of 200 (with its own FFP) to 160 (no FFP).

If B has no FFP, A prefers a payoff of 340 (with its own FFP) to 240 (no FFP).

Similarly: If A has an FFP, B prefers a payoff of 160 (with its own FFP) to 80 (no FFP).

If A has no FFP, B prefers a payoff of 280 (with its own FFP) to 200 (no FFP). b) Since both players have a dominant strategy with an FFP, at the Nash equilibrium both players will have an FFP. c) Yes. Both players would be better off with no FFP (A earns 240, B earns 200) than they are at the Nash equilibrium (in which A earns 200 and B earns 160). But, as noted above, “No FFP” is not a dominant strategy for either player.

14.6. Asahi and Kirin are the two largest sellers of beer in Japan. These two firms compete head to head in the dry beer category in Japan. The following table shows the profit (in millions of yen) that each firm earns when it charges different prices for its beer:

a) Does Asahi have a dominant strategy? Does Kirin? b) Both Asahi and Kirin have a dominated strategy: Find and identify it. c) Assume that Asahi and Kirin will not play the dominated strategy you identified in part (b) (i.e., cross out the dominated strategy for each firm in the table). Having eliminated the dominated strategy, show that Asahi and Kirin now have another dominated strategy. d) Assume that Asahi and Kirin will not play the dominated strategy you identified in part (c). Having eliminated this dominated strategy, determine whether Asahi and Kirin now have a dominant strategy. e) What is the Nash equilibrium in this game?

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Neither player has a dominant strategy in this game.

b) In this game, Asahi has a dominated strategy, ¥720 is dominated by ¥690, and Kirin has a dominated strategy, ¥720 is dominated by ¥690. Assuming neither player will play these dominated strategies we can remove them from the game. The reduced game is Kirin ¥630 180, 180 178, 184 175, 185

¥630 ¥660 ¥690

Asahi

¥660 184, 178 183, 183 182, 192

¥690 185, 175 192, 182 191, 191

c) Now that we have eliminated a dominated strategy from the original game, both players now have a dominated strategy in the reduced game. Asahi has a dominated strategy, ¥690 is dominated by ¥660, and Kirin has a dominated strategy, ¥690 is dominated by ¥660. Assuming neither player will play these dominated strategies we can remove them from the game. The reduced game is

Asahi

¥630 ¥660

Kirin ¥630 180, 180 178, 184

¥660 184, 178 183, 183

d) Now that we have eliminated another dominated strategy from the original game, both players have a dominant strategy to choose ¥630. e) Based on the analysis above, the Nash equilibrium in this game has both players choosing ¥630.

14.7. Consider the following game:

a) What is the Nash equilibrium in this game? b) If you were Player 1, how would you play this game? a) Player 1 choosing “Up” and Player 2 choosing “Left” is the Nash equilibrium. Player 2 has a dominant strategy to choose “Left,” and Player 1 seeing this will make the best choice assuming Player 2 will play its dominant strategy which is “Up.”

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b) Some students will argue that Player 1 should choose “Down.” By choosing “Down” Player 1 will receive/lose nothing. By selecting “Up” Player 1 earns at most 1 and may lose 100.

To avoid the possibility of losing 100, Player 1 may select “Down.” This reasoning is based on the unstated assumption that Player 2 will not play its dominant strategy of “Left.” Such doubt might be justified if Player 1 was not sure what Player 2’s objective function was or if Player 1 was not certain that the payoffs in the game matrix represented Player 2’s true payoffs. These possibilities are often explored in advanced game theory courses that study what happens when there are uncertainties in the game.

14.8 It is the year 2099, and the moon has been colonized by humans. Huawei (the Chinese mobile phone company) and Nokia (the Finnish mobile phone company) are trying to decide whether to invest in the first cellular telecommunications system on the moon. The market is big enough to support just one firm profitably. Both companies must make huge expenditures in order to construct a cellular network on the moon. The payoffs that each firm gets when it enters or does not enter the moon market are as follows

Ignoring mixed strategies, find all of the Nash equilibria in this game.

This game has two Nash equilibria corresponding to the outcomes where one firm chooses “Enter” and the other firm chooses “Do Not Enter.” Thus, Alcatel choosing “Enter” and Nokia choosing “Do Not Enter” is one Nash equilibrium; and Alcatel choosing “Do Not Enter” and Nokia choosing “Enter” is the other.

14.9 ABC and XYZ are the only two firms selling gizmos in Europe. The following table shows the profit (in millions of euros) that each firm earns at different prices (in euros per unit). ABC’s profit is the left number in each cell; XYZ’s profit is the right number.

Is there a unique Nash equilibrium in this game? If so, what is it? If not, why not? Explain clearly how you arrive at your answer.

Note that a price of 32 is dominated by a price of 28 for both carriers. So we know that 32 will not be a Nash equilibrium for either carrier, and we can eliminate the bottom row and right column.

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Next, note that for the remaining strategies (20, 24, or 28), 28 is dominated by 24 for both carriers. So we know that 28 will not be a Nash equilibrium for either carrier, and we can focus on the 2X2 game with prices of 20 and 24.

Consider the best response functions: If ABC chooses 20. XYZ chooses 20 (preferring a payoff of 60 to 56).

If ABC chooses 24. XYZ chooses 20 (preferring a payoff of 68 to 66).

So XYZ has a dominant strategy of 20. The game is symmetric, so ABC also has a dominant strategy of 20. So the Nash equilibrium is at (20,20).

14.10 Two pipeline firms are contemplating entry into a market delivering crude oil from a port to a refinery. Pipeline 1, the larger of the two firms, is contemplating its capacity strategy, which we might broadly characterize as “aggressive” and “passive.” The “aggressive” strategy involves a large increase in capacity aimed at increasing the firm’s market share, while the passive strategy involves no change in the firm’s capacity. Pipeline 2, the smaller competitor, is also pondering its capacity expansion strategy; it will also choose between an “aggressive strategy” or a “passive strategy.” The following table shows the present value of the profits associated with each pair of choices made by the two firms:

a) If both firms decide their strategies simultaneously, what is the Nash equilibrium? b) If Pipeline 1 could move first and credibly commit to its capacity expansion strategy, what is its optimal strategy? What will Pipeline 2 do? a) “Passive” is a dominant strategy for pipeline 1. Since Pipeline 2 knows that Pipeline 1 will choose to be passive, Pipeline 2 will be aggressive. So the Nash equilibrium is: Pipeline 1-Passive, Pipeline 2 – Aggressive. Pipeline 1 will receive a payoff of 90, and Pipeline 2 will receive a payoff of 45. b) If Pipeline 1 credibly precommits “Aggressive”, Pipeline 2 will choose “Passive.” So Pipeline 1 can receive a payoff of 100 (instead of the 90 it received in the simultaneous-play Nash equilibrium in part (a)).

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14.11 Lucy and Ricky are making plans for Saturday night. They can go to either a ballet or a boxing match. Each will make the choice independently, although as you can see from the following table, there are some benefits if they end up doing the same thing. Ignoring mixed strategies, is there a Nash equilibrium in this game? If so, what is it?

There are two Nash equilibria in this game. In the first one, Lucy and Ricky both choose BALLET. Why? If Ricky chooses BALLET, Lucy’s best response is BALLET (Lucy’s payoff is 100 versus –90).

If Lucy chooses BALLET, Ricky’s best response is BALLET (Ricky’s payoff is 30 versus –90) Thus: BALLET,BALLET is a point of mutual best response.

In the second Nash Equilibrium, Lucy and Ricky both choose BOXING MATCH If Ricky chooses BOXING MATCH, Lucy’s best response is BOXING MATCH (Lucy’s payoff is 30 versus –90).

If Lucy chooses BOXING MATCH, Ricky’s best response is BOXING MATCH (Ricky’s payoff is 100 versus –90) Thus: BOXING MATCH, BOXING MATCH is also a point of mutual best response.

14.12 Suppose market demand is P = 130 - Q. a) If two firms compete in this market with marginal cost c = 10, find the Cournot equilibrium output and profit per firm. b) Find the monopoly output and profit if there is only one firm with marginal cost c = 10. c) Using the information from parts (a) and (b), construct a 2 x 2 payoff matrix where the strategies available to each of two players are to produce the Cournot equilibrium quantity or half the monopoly quantity. d) What is the Nash equilibrium (or equilibria) of the game you constructed in part (c)? a) Each Cournot firms produces until MR = MC, or 130 – 2Qi – Qj = 10. The Cournot equilibrium output per firm is then Qi = 40. The market price is P = 50, and profit per firm is then πi = (50 – 10)*40 = 1600. b)

A monopolist produces until 130 – 2Q = 10, implying Q = 60, P = 70, and π = 3600.

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c) When both firms choose the Cournot quantity, each earns the Cournot profit. Similarly, producing half the monopoly output garners each firm half the monopoly profit. When, for instance, firm 1 produces the Cournot output Q1 = 40 while firm 2 produces half the monopoly output Q2 = 30, the market price would be P = 60. Firm 1 earns π1 = 2000 and firm 2 earns π2 = 1500. When the roles are reversed, so are profits.

Firm 1

Q1 = 30 Q1 = 40

Firm 2 Q2 = 30 1800, 1800 2000, 1500

Q2 = 40 1500, 2000 1600, 1600

d) The game in part (c) is a typical Prisoners’ Dilemma game, where the unique Nash equilibrium is for each firm to produce the Cournot output.

14.13 Consider the following game, where x > 0:

a) For what values of x do both firms have a dominant strategy? What is the Nash equilibrium (or equilibria) in these cases? b) For what values of x does only one firm have a dominant strategy? What is the Nash equilibrium (or equilibria) in these cases? c) Are there any values of x such that neither firm has a dominant strategy? Ignoring mixed strategies, is there a Nash equilibrium in such cases? a)

For x > 50, both firms have a dominant strategy. The unique NE is (Low, Low).

b) For 40 < x < 50 only Firm 2 has a dominant strategy. The unique NE in this case is still (Low, Low). c)

For x < 40, there are zero dominant strategies, and no NE exists.

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14.14 Professor Nash announces that he will auction off a $20 bill in a competition between Jack and Jill, two students chosen randomly at the beginning of class. Each student is to privately submit a bid on a piece of paper; whoever places the highest bid wins the $20 bill.

(In the event of a tie, each student gets $10.) The catch, however, is that each student must pay whatever he or she bid, regardless of who wins the auction. Suppose that each student has only two $1 bills in his or her wallet that day, so the available strategies to each student are to bid $0, $1, or $2. a) Write down a 3 x 3 payoff matrix describing this game. b) Does either student have any dominated strategies? c) What is the Nash equilibrium in this game? d) Suppose that Jack and Jill each could borrow money from the other students in the class, so that each of them had a total of $11 to bid. Would ($11, $11) be a Nash equilibrium? a)

b)

For both players, bidding $0 is always dominated by bidding $2.

c)

The Nash equilibrium is ($2, $2).

d) At ($11, $11), each player would earn $10 – $11 = –$1. This would not be a Nash equilibrium because either player would prefer to bid $0 and lose the auction (earning $0) rather than bidding $11 and losing money in the end.

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14.15 Consider the following game between Sony, a manufacturer of video cassette players, and Columbia Pictures, a movie studio. Each firm must decide whether to use the VHS or Beta format—Sony to make video players, Columbia to release its movies for rental or purchase.

a) Restrict attention to pure strategies. Does either firm have a dominant strategy? What is (are) the Nash equilibrium (equilibria) of this game? b) Is there a mixed strategy Nash equilibrium in this game? If so, what is it? c) Restrict attention again to pure strategies, but now focus on a sequential-move game in which Sony chooses its strategy first. What is (are) the Nash equilibrium (equilibria) of this game? a) Neither firm has a dominant strategy. There are two Nash equilibria: (Beta, Beta) and (VHS, VHS). b) In any mixed-strategy Nash equilibrium, Sony plays Beta with some probability pS and VHS with probability (1 – pS) while Columbia plays Beta with some probability pC and VHS with probability (1 – pC). By playing a mixed strategy, a player ensures that its rival is indifferent between its available strategies. Thus, Sony chooses pS such that Columbia’s expected π from Beta = Columbia’s expected π from VHS pS*10 + (1 – pS)*0 = pS*0 + (1 – pS)*20 This equation can be solved to show pS = 2/3 (intuitively, Sony puts more probability on the alternative (Beta) that Columbia doesn’t like.) Similarly, in any mixed strategy equilibrium we can show that Columbia plays Beta with probability pC = 1/3 (intuitively, Columbia puts less probability on the alternative (Beta) that Sony prefers.) c) In moving first, Sony realizes that it will always be optimal for Columbia to choose the same strategy, as it never wants to end up on different standards (though Columbia prefers the VHS standard and Sony prefers Beta.) Therefore in the Nash equilibrium Sony will choose Beta, followed by Columbia choosing Beta as well.

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14.16 In a World Series game, Tim Lincecum is pitching and Joe Mauer is batting. The count on Mauer is 3 balls and 2 strikes. Lincecum has to decide whether to throw a fastball or a curveball. Mauer has to decide whether to swing or not swing. If Lincecum throws a fastball and Mauer doesn’t swing, the pitch will almost certainly be a strike, and Mauer will be out. If Mauer does swing, however, there is a strong likelihood that he will get a hit.

If Lincecum throws a curve and Mauer swings, there is a strong likelihood that Mauer will strike out. But if Lincecum throws a curve and Mauer doesn’t swing, there is a good chance that it will be ball four and Mauer will walk (assume that a walk is as good as a hit in this instance). The following table shows the payoffs from each pair of choices that the two players can make:

a) Is there a Nash equilibrium in pure strategies in this game? b) Is there a mixed strategy Nash equilibrium in this game? If so, what is it? a)

There are no Nash equilibria in pure strategies in this game.

b) This game does have a Nash equilibrium in mixed strategies. To find the mixed strategy equilibrium, we need to find the probabilities with which, for example, Rodriguez plays “Swing” and “Do Not Swing” so that Wood’s payoff from playing “Fast Ball” and “Curve Ball” are the same. Letting the probability of “Swing” be 𝑃 and “Do Not Swing” be 1 − 𝑃, Wood’s expected payoff if he chooses “Fast Ball” is −100𝑃 + 100(1 − 𝑃). His expected payoff if he chooses “Curve Ball” is 100𝑃 − 100(1 − 𝑃). Equating these yields:

−100 P + 100(1 − P) = 100 P − 100(1 − P) 100 − 200 P = 200 P − 100 P = 0.50 Thus, in equilibrium Rodriguez should play “Swing” with a 50% probability and “Do Not Swing” with a 50% probability. Since the problem is symmetric, Wood should play “Fast Ball” with a 50% probability and “Curve Ball” with a 50% probability.

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14.17 In the mid-1990s, Value Jet wanted to enter the market serving routes that would compete head to head with Delta Airlines in Atlanta. Value Jet knew that Delta might respond in one of two ways: Delta could start a price war or it could be “accommodating,” keeping the price at a high level. Value Jet had to decide whether it would enter on a small scale or on a large scale. The annual profits (in zillions of dollars) associated with each strategy are summarized in the following table (where the first number is the payoff to Value Jet and the second the payoff to Delta):

a) If Value Jet and Delta choose their strategies simultaneously, what strategies would the two firms choose at the Nash equilibrium, and what would be the payoff for Value Jet? Explain. b) As it turned out, Value Jet decided to move first, entering on a small scale. It communicated this information by issuing a public statement announcing that it had limited aspirations in this marketplace and had no plans to grow beyond its initial small size. Analyze the sequential game in which Value Jet chooses “small” or “large” in the first stage and then Delta accommodates or starts a price war in the second stage. Did Value Jet enhance its profit by moving first and entering on a small scale? If so, how much more did it earn with this strategy? If not, explain why not? (Hint: Draw the game tree.) a) Value Jet has a dominant strategy to enter large. Given that, Delta would respond by launching a price war. Thus there is a unique pure strategy Nash equilibrium in which Value Jet enters large and Delta starts a price war. Value Jet’s payoff is $4 zillion. b) VJ’s Payoff

Delta’s Payoff

Accommodate

8

40

Price War

2

32

Accommodate

16

20

Price War

4

24

D

Enter Small VJ

Enter Large D

The game tree above models the two stages of the game. The payoffs are the same as in the matrix in (a).

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If VJ (Value Jet) builds low, D (Delta) will accommodate (preferring 40 over 32). Thus, D will get 8 if it enters small.

If VJ (Value Jet) builds large, D (Delta) will start a price war (preferring 24 over 20). Thus, D will get 4 if it enters small.

Value Jet’s optimal strategy is to build small and for Delta to accommodate. Value Jet will then receive $8 zillion. Value Jet has increased its profit from $4 zillion in (a) to $8 zillion in (b), so by moving first, gains an extra $4 zillion in profit.

14.18 Besanko, Inc. and Braeutigam, Ltd. compete in the high-grade carbon fiber market.

Both firms sell identical grades of carbon fiber, a commodity product that will sell at a common market price. The challenge for each firm is to decide upon a capacity expansion strategy. The following problem pertains to this choice. a) Suppose it is well known that long-run market demand in this industry will be robust. In light of that, the payoffs associated with various capacity expansion strategies that Besanko and Braeutigam might pursue are shown in the following table. What are the Nash equilibrium capacity choices for each firm if both firms make their capacity choices simultaneously? b) Again, suppose that the table gives the payoffs to each firm under various capacity scenarios, but now suppose that Besanko can commit in advance to a capacity strategy.

That is, it can choose no expansion, modest expansion, or major expansion. Braeutigam observes this choice and makes a choice of its own (no expansion or modest expansion).

What is the equilibrium in this sequential-move capacity game?

a) As we see below (where squares represent Besanko’s best response and circles represent Braeutigam’s), the Nash equilibrium is for each player to choose MODEST EXPANSION.

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b) If Besanko moves first, then he will commit to a MAJOR EXPANSION, and Braeutigam will choose NO EXPANSION. Here’s why: If Besanko chooses NO EXPANSION, Braeutigam’s best response is MODEST EXPANSION. Besanko’s payoff is $844.

If Besanko chooses MODEST EXPANSION, Braeutigam’s best response is MODEST EXPANSION. Besanko’s payoff is $900.

If Besanko chooses MAJOR EXPANSION, Braeutigam’s best response is NO EXPANSION. Besanko’s payoff is $1,013.

Besanko does best when he chooses MAJOR EXPANSION, putting Braeutigam in a position in which it is optimal for him to choose NO EXPANSION. 14.19 Boeing and Airbus are competing to fill an order of jets for Singapore Airlines. Each firm can offer a price of $10 million per jet or $5 million per jet. If both firms offer the same price, the airline will split the order between the two firms, 50–50. If one firm offers a higher price than the other, the lower-price competitor wins the entire order. Here is the profit that Boeing and Airbus expect they could earn from this transaction:

a) What is the Nash equilibrium in this game? b) Suppose that Boeing and Airbus anticipate that they will be competing for orders like the one from Singapore Airlines every quarter, from now to the foreseeable future. Each quarter, each firm offers a price, and the payoffs are determined according to the table above. The prices offered by each airline are public information. Suppose that Airbus has made the following public statement: To shore up profit margins, in the upcoming quarter we intend to be statesmanlike in the pricing of our aircraft and will not cut price simply to win an order.

However, if the competition takes advantage of our statesmanlike policy, we intend to abandon this policy and will compete all out for orders in every subsequent quarter.

Boeing is considering its pricing strategy for the upcoming quarter. What price would you recommend that Boeing charge? Important note: To evaluate payoffs, imagine that each quarter, Boeing and Airbus receive their payoff right away. (Thus, if in the upcoming quarter, Boeing chooses $5 million and Airbus chooses $10 million, Boeing will immediately receive its profit of $270 million.) Furthermore, assume that Boeing and Airbus evaluate future payoffs in the following way: a stream of payoffs of $1 starting next

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quarter and received in every quarter thereafter has exactly the same value as a one-time payoff of $40 received immediately this quarter. c) Suppose that aircraft orders are received once a year rather than once a quarter. That is, Boeing and Airbus will compete with each other for an order this year (with payoffs given in the table above), but their next competitive encounter will not occur for another year. In terms of evaluating present and future payoffs, suppose that each firm views a stream of payoffs of $1 starting next year and received every year thereafter as equivalent to$10 received immediately this year. Again assuming that Airbus will follow the policy in its public statement above, what price would you recommend that Boeing charge in this year and beyond? a) In this game both players have a dominant strategy to choose “P = $5m.” Thus, the Nash equilibrium outcome occurs when Airbus chooses “P = $5m” and Boeing chooses “P = $5m.” b) Airbus’ statement implies that they will play “P = $10m” in this quarter and all subsequent quarters as long as Boeing also plays “P = $10m.” However, if Boeing ever plays “P = $5m,” Airbus will play “P = $5m” in all future quarters.

From Boeing’s perspective, if they choose to continue to play “P = $10m” then in every quarter they will receive a payoff of 50. If they choose to lower their price to “P = $5m,” then in the first quarter they will receive 270. In all subsequent quarters the best they will be able to do is play “P = $5m,” as will Airbus, and Boeing will receive 30. Thus, Boeing’s two possible payoff streams look like: Boeing P = $5m P = $10m

270 50

30 50

30 50

…. ….

30 50

Boeing values a stream of payoffs of $1 starting next quarter as a payoff of $40 in the first quarter. Therefore, Boeing values the two payoff streams as P = $5m 270 + 40(30) = 1,470 P = $10m 50 + 40(50) = 2,050 Therefore, the value of “P = $10m” in current dollars is greater, so Boeing should select “P = $10m” in this quarter and all subsequent quarters. c) Now Boeing values the payoff stream differently. In the current situation, Boeing values a stream of payoffs of $1 starting next year as equivalent to $10 received immediately. Thus, Boeing will now value this payoff stream as P = $5m 270 + 10(30) = 570 P = $10m 50 + 10(50) = 550 Now “P = $5m” has a higher value in current dollars than “P = $10m.” Thus, Boeing should select “P = $5m” this year, receive the high payoff in the current year, and select “P = $5m” thereafter receiving a stream of payoffs of 30 each year.

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14.20 Consider a buyer who, in the upcoming month, will make a decision about whether to purchase a good from a monopoly seller. The seller “advertises” that it offers a highquality product (and the price that it has set is based on that claim.) However, by substituting low-quality components for higher-quality ones, the seller can reduce the quality of the product it sells to the buyer, and in so doing, the seller can lower the variable and fixed costs of making the product. The product quality is not observable to the buyer at the time of purchase, and so the buyer cannot tell, at that point, whether he is getting a high-quality or a low-quality good. Only after he begins to use the product does the buyer learn the quality of the good he has purchased. The payoffs that accrue to the buyer and seller from this encounter are as follows:

The buyer’s payoff (consumer surplus) is listed first; the seller’s payoff (profit) is listed second. Answer each of the following questions, using the preceding table. a) What are the Nash equilibrium strategies for the buyer and seller in this game under the assumption that it is played just once? b) Let’s again suppose that the game is played just once (i.e., the buyer makes at most one purchase). But suppose that before the game is played, the seller can commit to offering a warranty that gives the buyer a monetary payment W in the event that he buys the product and is unhappy with the product he purchases. What is the smallest value of W such that the seller chooses to offer a high-quality product and the buyer chooses to purchase? c) Instead of the warranty, let’s now allow for the possibility of repeat purchases by the buyer. In particular, suppose that if the buyer purchases the product and learns that he has bought a high-quality good, he will return the next month and buy again. Indeed, he will continue to purchase, month after month (potentially forever!), as long as the quality of the product he purchased in the previous month is high. However, if the buyer is ever unpleasantly surprised—that is, if the seller sells him a low-quality good in a particular month—he will refuse to purchase from the seller forever after. Suppose that the seller knows that the buyer is going to behave in this fashion. Further, let’s imagine that the seller evaluates profits in the following way: a stream of payoffs of $1 starting next month and received in every month thereafter has exactly the same value as a one-time payoff of $50 received immediately this month. Will the seller offer a low-quality good or a high-quality good? a) The Nash equilibrium is for the Buyer to choose “Do Not Purchase” and the Seller “Sell Low Quality.” (In fact, “Low-Quality” is a dominant strategy for the Seller.)

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The warranty amount W needs to satisfy two conditions:

(1) The Buyer has an incentive to purchase, even if seller offers a low-quality product.

This will hold provided: “Buyer’s payoff from purchasing a low-quality product” > “Buyer’s payoff if he doesn’t purchase.” In payoff terms this requires: –4 + W > 0, or W > 4.

(2) Seller has an incentive to offer a high-quality product when buyer purchases.

This will hold provided: “Seller’s payoff from offering a high-quality product” > “Seller’s payoff from offering a low-quality product.” In payoff terms this requires: 6 > 12 – W, or W > 6.

Thus, if W is greater than 6, both conditions hold, so the lowest possible value of W is W = 6 (or maybe more precisely, one penny above, or 6.01). c) If the Seller offers a high-quality product this month and forever after, the buyer will continue to purchase, and the Seller’s value of payoffs is: 6 + a stream of 6 starting next month forever after: = 6 + 6*50 = 306 If seller offers a low-quality product this month, it gets a one-time payoff of $14, but it loses the buyer’s business forever more. The seller’s payoff is thus $14. Clearly, the seller’s payoff is much higher if it offers a high-quality product.

The lesson of this “fable” is this: The possible loss of future repeat business is a powerful disciplining device to keep a seller from substituting a shoddy good for a high-quality good. A similar logic explains why powerful brands (e.g., Coca Cola) do not exploit loyal, trusting customers by using inferior ingredients that might be cheaper but would compromise quality.

The loss of future business connected to that “brand franchise” would be enormous and the longterm consequences would be devastating in comparison to the short-term reduction in cost.

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14.21 Two firms are competing in an oligopolistic industry. Firm 1, the larger of the two firms, is contemplating its capacity strategy, which could be either “aggressive” or “passive.” The aggressive strategy involves a large increase in capacity aimed at increasing the firm’s market share, while the passive strategy involves no change in the firm’s capacity. Firm 2, the smaller competitor, is also pondering its capacity expansion strategy; it will also choose between an aggressive strategy and a passive strategy. The following table shows the profits associated with each pair of choices:

a) If both firms decide their strategies simultaneously, what is the Nash equilibrium? b) If Firm 1 could move first and credibly commit to its capacity expansion strategy, what is its optimal strategy? What will Firm 2 do? a) Firm 1 will choose its dominant strategy “Passive.” Firm 2, knowing Firm 1 has a dominant strategy, will play its best response, “Aggressive.” This is the only Nash equilibrium in the simultaneous-move game. b) As shown in the diagram below, if Firm 1 can choose first, then if it chooses “Aggressive” Firm 2 will choose “Passive” and Firm 1 will receive 33. If Firm 1 instead chooses “Passive,” then Firm 2 will select “Aggressive” and Firm 1 will receive a payoff of 30.

Therefore, if Firm 1 can move first, it does best to select “Aggressive” in which case Firm 2 will select its best response “Passive” earning Firm 1 a payoff of 33 and Firm 2 a payoff of 10.

Firm 1

Firm 2

25

9

Passive

33

10

Aggressive

30

13

Passive

36

12

Aggressive Aggressive

2

1 Passive

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14.22 The only two firms moving crude oil from an oil-producing region to a port in Atlantis are pipelines: Starline and Pipetran. The following table shows the annual profit (in millions of euros) that each firm would earn at different capacities. Starline’s profit is the left number in each cell; Pipetran’s profit is the right number. At the current capacities (with no expansion) Starline is earning 40 million euros, and Pipetran is earning 18 million euros annually. Each company is considering an expansion of its capacity. Since Pipetran is a fairly small company, it can consider only a small expansion to its capacity. Starline has the ability to consider both a small and a large expansion.

a) If the two firms make their decisions about expansion simultaneously, is there a unique Nash equilibrium? If so, what is it? If not, why not? Explain whether this game is an example of a prisoners’ dilemma. b) Would Starline have a first-mover advantage if capacities were chosen sequentially? If so, briefly explain how it might credibly implement this strategy. c) Suppose you were hired to advise Pipetran about its choice of capacity. If Pipetran has the option of moving first, should it do so? Explain. a) Best responses for Starline are indicated with *; best responses for Pipetran are indicated with **.

Pipetran Starline

No Expansion Small Large

No Expansion 40, 18 48*, 14 38, 10**

Small 28, 22** 32*, 16** 24, 5

There is a unique Nash equilibrium with “Small, Small,” with Starline earning 32 and Pipetran earning 16.

It is an example of a prisoner’s dilemma. Both pipelines would be better off with no expansion.

But each player has an incentive to choose “small” if it believes the other pipeline would choose “no expansion.” b) The game tree below depicts the scenario in which Starline moves first. Pipetran’s best responses give any action by Starline are indicated by **. Given Pipetran’s best responses, Starline’s best strategies are the ones associated with the payoffs in bold numbers. Over all, Starline’s best response is to build large first, inducing Pipetran not to expand. It would earn 38 million euros by moving first, instead of the 32 million euros it would have earned at the

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simultaneous play game in part (a). To be credible, it needs to announce that it has signed an irrevocable agreement committing to build the large plant before Pipetran can move.

c) The game tree below depicts the scenario in which Pipetran moves first. Starline’s best responses give any action by Pipetran are indicated by *. Given Starline’s best responses, Pipetran’s best strategies are the ones associated with the payoffs in bold numbers. Over all, Pipetran’s best response is to build small first, inducing Pipetran to build small. It would earn 16 million euros by moving first, the same amount it would have earned at the simultaneous play game in part (a). But, by moving first, it can avoid the scenario in (b) in which Starline moves first, in which case Pipetran would earn only 10 million euros. So Pipetran should commit immediately and irrevocably to a contract to build small.

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14.23 ABC and XYZ are the two cereal manufacturers contemplating entry into a South American market. Each will be able to build one plant, and that plant can be used to make either a cereal that is high in fiber and low in calories (High Fiber) or a less healthy cereal with a sweet taste (Sweet). Once a plant is chosen to produce one kind of cereal, it will be prohibitively expensive to switch production to the other type. The following table shows the annual profit (in millions of pesos) that each firm would earn given the production choices of the two firms. ABC’s profit is the left number in each cell; XYZ’s profit is the right number. For example, if ABC makes the sweet cereal and XYZ produces the highfiber cereal, annual profits will be 50 million pesos for ABC and 60 million pesos for XYZ.

a) If the two firms choose the type of plant simultaneously, is there a unique Nash equilibrium? If so, what is it? If not, why not? b) Would ABC have a first-mover advantage if capacities were chosen sequentially? If so, briefly explain how it might credibly implement this strategy. c) Would XYZ have a first-mover advantage if capacities were chosen sequentially? If so, briefly explain how it might credibly implement this strategy. a) Best responses for ABC are indicated with *; best responses for XYZ are indicated with **.

XYZ ABC

Sweet High Fiber

High Fiber 50*, 60** 20, 30

Sweet 30, 40 40*, 60**

So there are two Nash equilibria, one in which ABC chooses High Fiber, and XYZ produces Sweet, and the second in which ABC chooses Sweet, and XYZ produces High Fiber. b) ABC would like to end up in the equilibrium in the upper left had corner, where it earns 50 million pesos, instead of at the equilibrium in the lower right hand corner, where it would earn only 40 million pesos. So it should move first, entering into an irrevocable agreement to build the plant that produces sweet cereal. XYZ would then choose to make high fiber cereal. c) XYZ would earn 60 million pesos at either of the Nash equilibria. So it has no first mover advantage. (In this discussion we assume that each firm cares only about its own profits, and not the profits of the rival. One could also consider a different game in each firm places value on its own profits as well as those of its rival.)

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14.24. Cities A, B, and C are located in different countries. The only airline serving the market between A and B is Ajax Air. Its total cost is CAjax = 20QAB. The airfare between A and B is PAB. Also, the only carrier serving the market between B and C is Sky Air. Its total cost is CSky = 20QBC. The airfare between B and C is PBC. The two airlines do not serve any other markets.

All traffic on the network flows between A and C, using B only as a point to interconnect with the other airline. (In other words, no traffic originates or terminates at B.) The demand for passenger service between A and C is QAC = 220 - PAC, where Q is the number of units of passenger traffic demanded when PAC, the total airfare between A and C, is PAB + PBC.

a) The preceding table shows the profits for each carrier for various combinations of airfares. The upper left number in a cell shows Ajax’s profit; the lower right number shows Sky’s profit. Suppose Ajax charges PAB = 100 and Sky charges PBC = 90. Determine the profit for each of the two carriers, and enter your calculation in the table. b) Currently, Ajax and Sky are not allowed to coordinate prices. They must act noncooperatively when setting their fares. Using the preceding table, find the Nash equilibrium fares. Explain how you arrived at your answer. c) The two airlines have been lobbying antitrust authorities to allow them to merge, an act that would enable them to price jointly as a monopolist. The merged airline would still stop at B for refueling. The cost and demand curves would not change if the carriers merged.

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Use the table to determine what price the merged entity would charge for a trip between A and C, and explain your reasoning clearly. a) Ajax’s profit = (PAB – MC)(220 – PAB – PBC) = (100 – 20)(220 – 100 – 90) = 2400 Sky’s profit = (PBC – MC)(220 – PAB – PBC) = (90 – 20)(220 – 100 – 90) = 2100

b) To find the possible noncooperative equilibria, we first show the reaction functions. The lightly shaded areas in the table show Ajax’s best responses to any price that Sky charges.

Similarly, the darkly shaded areas in the table show Sky’s best responses to any price that Ajax charges. Since the two reaction functions intersect in only one cell there is a unique Nash equilibrium, with each carrier charging 80. The total fare paid by consumers for a trip between A and B is therefore 160. c) If the carriers merge, they will set prices that maximize the sum of profits. The sum of profits is maximized at 3 cells in the table; they all show a total profit of 8100. The cells are not unique because the passenger cares only about the sum of the prices, which will be 130 in each cell. Thus the cooperative fare is 130.

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Chapter 15 Risk and Information Solutions to Review Questions 1. Why must the probabilities of the possible outcomes of a lottery add up to 1?

As a general rule, the sum of the probabilities of all possible outcomes is equal to one. This rule ensures that all possible outcomes are accounted for. If the probabilities for the possible outcomes summed to a number less than one, it would imply there were other possible outcomes that had not been included.

2. What is the expected value of a lottery? What is the variance?

The expected value of a lottery is a measure of the average payoff the lottery will generate. The variance of a lottery characterizes the average squared deviation between the possible outcomes of the lottery and the expected value of the lottery. The variance measures the risk associated with the lottery; a smaller variance implies greater certainty associated with the expected payoff.

3. What is the difference between the expected value of a lottery and the expected utility of a lottery?

The expected value of a lottery measures the average payoff of the lottery in monetary terms.

The expected utility of a lottery takes into account how much the decision maker values the expected payoff, particularly in terms of the risk associated with the expected payoff. For example, while the expected payoff from one lottery may exceed the expected payoff from a second lottery, if the second lottery has less risk associated with it, a decision maker might prefer the second lottery to the first.

4. Explain why diminishing marginal utility implies that a decision maker will be risk averse.

A utility function that exhibits diminishing marginal utility will imply the decision maker is risk averse. This is because with a utility function with diminishing marginal utility a decision maker will prefer a sure thing to a lottery with the same expected value. By preferring the sure thing, the decision maker prefers less risk, implying the decision maker is risk averse.

5. Suppose that a risk-averse decision maker faces a choice of two lotteries, 1 and 2. The lotteries have the same expected value, but Lottery 1 has a higher variance than Lottery 2.

What lottery would a risk-averse decision maker prefer?

A risk-averse decision maker, when comparing lotteries with the same expected value, will prefer the lottery with lower risk. In this instance, since Lottery 1 has a higher variance, Lottery 1 will have more risk associated with it. Thus, the decision maker will prefer Lottery 2.

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6. What is a risk premium? What determines the magnitude of the risk premium?

A risk premium is the difference between the expected value of a lottery and the payoff from a sure thing so that the decision maker is indifferent between the lottery and the sure thing. A key determinant of the risk premium is the variance associated with the lottery. If two lotteries have the same expected value but one has a higher variance, the risk premium associated with the lottery that has a higher variance will be larger. The decision maker is requiring a greater premium to take on greater risk.

7. What is fair insurance? Why will a risk-averse consumer always be willing to buy full insurance that is fair?

A fair insurance policy is one in which the insurance premium is equal to the expected value of the damage being covered. A risk-averse individual will always prefer to purchase a fair insurance policy that provides full insurance against a loss in order to eliminate risk.

8. What is the difference between a chance node and a decision node in a decision tree?

In a decision tree, a decision node indicates a particular decision that the decision maker faces.

Each branch from a decision node corresponds to a possible alternative that the decision maker might choose. A chance node indicates a particular lottery that the decision maker faces. Each branch from a chance node corresponds to a possible outcome of the lottery.

9. Why does perfect information have value, even for a risk-neutral decision maker?

Perfect information has value because it allows the decision maker to tailor its decisions to the underlying circumstances it faces. By knowing the outcome of the lottery, the decision maker can select the decision alternative best suited to the outcome of the lottery.

10. What is the difference between an auction in which bidders have private values and one in which they have common values?

In an auction in which bidders have private values, bidders have their own personal valuation of the object. The bidder knows how he values the object but is unsure how other potential bidders value the object. In an auction with common values, the value of the object is the same for all buyers, but no buyer knows exactly what that value is.

11. What is the winner’s curse? Why can the winner’s curse arise in a common-values auction but not in a private-values auction?

The winner’s curse arises in auctions in which bidders have common values. The bidder who wins the auction must have submitted the highest bid and therefore must have had the most optimistic estimate of the value of the object. The winning bidder will have almost surely overestimated the value of the object he or she is bidding on. Thus, the winner may suffer from the winner’s curse – bidding more for the object than the item’s intrinsic value.

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12. Why is it wise to bid conservatively in a common values auction?

A bidder should anticipate that if she wins a common values auction it is because she had the highest estimate of the object’s value. To avoid the winner’s curse, paying more for the object than the object’s intrinsic value, the bidder should act as if her value is something less than her initial estimate. This new estimate then becomes the starting point when trying to determine the bid; this is the starting point because the bidder may still want to scale down her bid even more, recognizing that other bidders may scale down their bids in a similar way.

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Solutions to Problems 15.1. Consider a lottery with three possible outcomes: a payoff of -10, a payoff of 0, and a payoff of +20. The probability of each outcome is 0.2, 0.5, and 0.3, respectively. a) Sketch the probability distribution of this lottery. b) Compute the expected value of the lottery. c) Compute the variance and the standard deviation of the lottery. a)

b) 𝐸𝑉 = 0.2(−10) + 0.5(0) + 0.3(20) 𝐸𝑉 = 4.0 c) 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 0.2(−10 − 4)2 + 0.5(0 − 4)2 + 0.3(20 − 4)2 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 124 Standard Deviation = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 Standard Deviation = √124 Standard Deviation = 11.14 15.2. Suppose that you flip a coin. If it comes up heads, you win $10; if it comes up tails, you lose $10. a) Compute the expected value and variance of this lottery. b) Now consider a modification of this lottery: You flip two fair coins. If both coins come up heads, you win $10. If one coin comes up heads and the other comes up tails, you neither win nor lose—your payoff is $0. If both coins come up tails, you lose $10. Verify that this lottery has the same expected value but a smaller variance than the lottery with a single coin flip. (Hint: The probability that two fair coins both come up heads is 0.25, and the probability that two fair coins both come up tails is 0.25.) Why does the second lottery have a smaller variance?

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a) 𝐸𝑉 = 0.5(10) + 0.5(−10) 𝐸𝑉 = 0.0 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 0.5(10 − 0)2 + 0.5(−10 − 0)2 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 100 EV = 0.25(10) + 0.5(0) + 0.25( −10) b) EV = 0 Variance = 0.25(10 − 0) 2 + 0.5(0 − 0) 2 + 0.25( −10 − 0) 2 Variance = 50

This second lottery has a smaller variance because the probabilities associated with winning or losing $10, which are 0.25, are smaller than the probabilities associated with winning or losing $10 in the first lottery, which are 0.50. In the second lottery there is a 50% chance of a $0 payoff and that reduces the overall variance of the lottery.

15.3. Consider two lotteries. The outcome of each lottery is the same: 1, 2, 3, 4, 5, or 6. In the first lottery each outcome is equally likely. In the second lottery, there is a 0.40 probability that the outcome is 3, and a 0.40 probability that the outcome is 4. Each of the other outcomes has a probability 0.05. Which lottery has the higher variance?

The first lottery has the higher variance. This can be verified by direct calculation. It can also be seen in the fact in the second lottery, it is much more certain that the outcome will be confined to one of two numbers, 3 and 4 (each of which lies in the middle of the distribution), whereas in the first lottery any number is equally likely.

15.4. Consider a lottery in which there are five possible payoffs: $9, $16, $25, $36, and $49, each occurring with equal probability. Suppose that a decision maker has a utility function given by the formula U = √I. What is the expected utility of this lottery?

The expected utility is: 0.20√9 + 0.20√16 + 0.20√25 + 0.20√36 + 0.20√49 = 5.

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15.5. Suppose that you have a utility function given by the equation U = √50I. Consider a lottery that provides a payoff of $0 with probability 0.75 and $200 with probability 0.25. a) Sketch a graph of this utility function, letting I vary over the range 0 to 200. b) Verify that the expected value of this lottery is $50. c) What is the expected utility of this lottery? d) What is your utility if you receive a sure payoff of $50? Is it bigger or smaller than your expected utility from the lottery? Based on your answers to these questions, are you risk averse? a)

b) 𝐸𝑉 = 0.75(0) + 0.25(200) 𝐸𝑉 = 50 c) Expected Utility = 0.75√50(0) + 0.25√50(200) Expected Utility = 25 d) Utility = √50(50) Utility = 50 The utility associated with the certain payoff of 50 is higher than the expected utility of the lottery with the same expected payoff. Thus, with this utility function the decision maker is risk averse since the decision maker prefers the sure thing to a lottery with the same expected payoff.

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15.6. You have a utility function given by U = 2I + 10√I. You are considering two job opportunities. The first pays a salary of $40,000 for sure. The other pays a base salary of $20,000, but offers the possibility of a $40,000 bonus on top of your base salary. You believe that there is a 0.50 probability that you will earn the bonus. a) What is the expected salary under each offer? b) Which offer gives you the higher expected utility? c) Based on your answer to (a) and (b), are you risk averse? a) The expected salary under the first job offer is $40,000. The expected salary under the second job offer is also $40,000: 0.5($20,000) + 0.5($60,000) = $40,000. b)

Your expected utility under the first offer is

𝑈 = 2(40000) + 10√40000 = 82,000 Your expected utility under the second offer is 𝑈 = .5(2(20000) + 10√20000) + .5(2(60000) + 10√60000) = 81,932.

The first offer gives you the higher expected utility. c) The two offers have the same expected value. Since you prefer the certain salary to the risky salary, it follows that you are risk averse.

15.7. Consider two lotteries, A and B. With lottery A, there is a 0.90 chance that you receive a payoff of $0 and a 0.10 chance that you receive a payoff of $400. With lottery B, there is a 0.50 chance that you receive a payoff of $30 and a 0.50 chance that you receive a payoff of $50. a) Verify that these two lotteries have the same expected value but that lottery A has a bigger variance than lottery B. b) Suppose that your utility function is U = √I + 500. Compute the expected utility of each lottery. Which lottery has the higher expected utility? Why? c) Suppose that your utility function is U = I √ 500. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral, or risk loving? d) Suppose that your utility function is U = (I √ 500)2. Compute the expected utility of each lottery. If you have this utility function, are you risk averse, risk neutral, or risk loving?

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a) 𝐸𝑉𝐴 = 0.90(0) + 0.10(400) 𝐸𝑉𝐴 = 40 𝐸𝑉𝐵 = 0.50(30) + 0.50(50) 𝐸𝑉𝐵 = 40 Thus, both lotteries have the same expected value. 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐴 = 0.90(0 − 40)2 + 0.10(400 − 40)2 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐴 = 14,400 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐵 = 0.50(30 − 40)2 + 0.50(50 − 40)2 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝐵 = 100 Thus, Lottery B has a smaller variance than Lottery A. b) Expected Utility𝐴 = 0.90√0 + 500 + 0.10√400 + 500 Expected Utility𝐴 = 23.13 Expected Utility𝐵 = 0.50√30 + 500 + 0.50√50 + 500 Expected Utility𝐵 = 23.24 Thus, Lottery B has the higher expected utility. In general, when two lotteries have the same expected value but different variance, a risk-averse decision maker will have a higher expected utility from the lottery with the lower variance. c) Expected Utility𝐴 = 0.90(0 + 500) + 0.10(400 + 500) Expected Utility𝐴 = 540 Expected Utility𝐵 = 0.50(30 + 500) + 0.50(50 + 500) Expected Utility𝐵 = 540 With this utility function both lotteries have the same expected value and same expected utility. In general, when two lotteries have the same expected value and different variances, a risk-neutral decision maker will be indifferent between the two lotteries, i.e., will have the same expected utility for both lotteries. Thus, this utility function corresponds with a risk-neutral decision maker. d) Expected Utility𝐴 = 0.90(0 + 500)2 + 0.10(400 + 500)2 Expected Utility𝐴 = 306,000 Expected Utility𝐵 = 0.50(30 + 500)2 + 0.50(50 + 500)2 Expected Utility𝐵 = 291,700 With this utility function the decision maker has a higher expected utility for Lottery A than for Lottery B. In general, when two lotteries have the same expected value but different variances, a risk-loving decision maker will prefer the lottery with the higher variance, Lottery A in this case.

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15.8. Consider two lotteries A and B. With Lottery A, there is a 0.8 probability that you receive a payoff of $10,000 and a 0.2 chance that you receive a payoff of $4,000. With Lottery B, you will receive a payoff of $8,800 for certain. You should verify for yourself that these two lotteries have the same expected value, but that Lottery A has a higher variance. For each of the utility functions below, please fill in the table below:

Utility function

𝑈 = 100√𝐼 𝑈=𝐼

𝑈=

𝐼2 10000

Expected utility lottery A

Expected utility lottery B

Which lottery gives the highest expected utility?

9,264.91 8,800

9,380.83 8,800

8,320

7,744

Lottery A Both give the same expected utility Lottery B

Does the utility function exhibit risk aversion, risk neutrality, or risk loving?

Risk aversion Risk neutrality

Risk loving

15.9. Sketch the graphs of the following utility functions as I varies over the range $0 to $100. Based on these graphs, indicate whether the decision maker is risk averse, risk neutral, or risk loving: a) U = 10I – (1/8)I2 b) U = (1/8)I2 c) U = ln (I + 1) d) U = 5I

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a)

Since this utility function increases at a decreasing rate, the decision maker will prefer a sure thing to a lottery with the same expected value. Thus, this utility function corresponds to a risk-averse decision maker. b)

Since this utility function increases at an increasing rate, the decision maker will prefer a lottery to a sure thing with the same payoff. Thus, this utility function corresponds to a risk-loving decision maker.

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c)

Since this utility function increases at a decreasing rate, the decision maker will prefer a sure thing to a lottery with the same expected value. Thus, this utility function corresponds to a risk-averse decision maker. d)

Since this utility function increases at a constant rate, the decision maker will be indifferent between a sure thing and a lottery with the same expected value. Thus, this utility function corresponds to a risk-neutral decision maker.

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15.10. a) Write down the equation of a utility function that corresponds to a risk-neutral decision maker. (Note: there are many possible answers to this part and the next two parts.) b) Write down the equation of a utility function that corresponds to a risk-averse decision maker. c) Write down the equation of a utility function that corresponds to a risk-loving decision maker. a) Examples of a risk-neutral utility function would be: U = 10I, U = 20 + 5I, or U = I/50.

The key point is that the marginal utility is constant in income b) Examples of a risk-averse utility function would be 𝑈 = √𝐼, U = log I, or U = 1 – I-2. In all of these cases, marginal utility decreases in I. As an example, consider U = 1 – I-2. Marginal utility for this utility function is MU = 2I-3. If you graph this function, you will see that marginal utility decreases in I. c) Examples of a risk-loving utility function would be U = I2, U = 5I3, U = 2I + 3I2. In all of these cases, marginal utility increases in I. As an example, consider U = 2I + 3I2. Marginal utility for this utility function is MU = 2 + 6I. This is clearly increasing in I.

15.11. Suppose that I represents income. Your utility function is given by the formula U = 10I as long as I is less than or equal to 300. If I is greater than 300, your utility is a constant equal to 3,000. Suppose you have a choice between having an income of 300 with certainty and a lottery that makes your income equal to 400 with probability 0.5 and equal to 200 with probability 0.5. a) Sketch this utility function. b) What is the expected value of each lottery? c) Which lottery do you prefer? d) Are you risk averse, risk neutral, or risk loving?

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a)

b) The expected value of each option is 300. c) Your (certain) utility under the first option is 3,000. Your expected utility under the second option is 0.5U(200) + 0.5U(400) = 0.5(2,000) + 0.5(3,000) = 2,500. Thus, you prefer the sure thing to the lottery. d) Since the sure thing and the lottery have the same expected value, but since you prefer the sure thing to the lottery, it follows that you are risk averse.

15.12. Suppose that your utility function is U = √ I. Compute the risk premium of the two lotteries described in Problem 15.7.

If your utility function were 𝑈 = √𝐼, then the risk premium associated with Lottery A would be 0.90 0 + 0.10 400 = 40 − RPA 40 − RPA = 2 40 − RPA = 4 RPA = 36

The risk premium associated with Lottery B would be 0.50 30 + 0.50 50 = 40 − RPB 40 − RPB = 6.27 40 − RPB = 39.36 RPB = 0.64

Lottery A has a risk premium of 36 and Lottery B has a risk premium of 0.64.

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15.13. Suppose you are a risk-averse decision maker with a utility function given by U(I) = 1 – 10I-2, where I denotes your monetary payoff from an investment in thousands. You are considering an investment that will give you a payoff of $10,000 (thus, I = 10) with probability 0.6 and a payoff of $5,000 (I = 5) with probability 0.4. It will cost you $8,000 to make the investment. Should you make the investment? Why or why not?

If you do not make the investment, your utility is: 1 – 10(8)-2 = 0.84375 If you make the investment, your utility is: (0.6)(1 – 10(10)-2) + (0.4)(1-10(5)-2) = (0.6)(0.9) + (0.4)(0.6) = 0.78 Since the expected utility from the investment is less than the utility from not making the investment, you should not make the investment.

15.14. You have a utility function given by U = 10 lnI where I represents the monetary payoff from an investment. You are considering making an investment which, if it pays off, will give you a payoff of $100,000, but if it fails, it will give you a payoff of $20,000. Each outcome is equally likely. What is the risk premium for this lottery?

The expected payoff of this lottery is given by 0.5(100,000) + 0.5(20,000) = 60,000.

The risk premium RP of for this lottery is the solution to the equation 0.5[10*ln(100,000)] + 0.5[10*ln(20,000)] = 10*ln(60,000 – RP) which is equivalent to 107.08 = 10*ln(60,000 – RP) Solving this equation tells us that RP = $15,278.64.

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15.15. In the upcoming year, the income from your current job will be $90,000. There is a 0.8 chance that you will keep your job and earn this income. However, there is 0.2 chance that you will be laid off, putting you out of work for a time and forcing you to accept a lower paying job. In this case, your income is $10,000. The expected value of your income is thus $74,000. a) If your utility function has the formula 100I - 0.0001I2, determine the risk premium associated with this lottery. b) Provide an interpretation of the risk premium in this particular example. a) The risk premium RP solves the following equation 0.8*U(90,000) + 0.2U(10,000) = U(74,000 – RP) Now: U(90,000) = 8,190,000 U(10,000) = 990,000 0.8*U(90,000) + 0.2U(10,000) = 6.750,000 U(74,000 – RP) = 100(74000 – RP) -0.0001(74,000 – RP)2 Thus, RP is the solution to: 6,750,000 = 100(74000 – RP) -0.0001(74,000 – RP)2 This is a quadratic equation that has two solutions. It can be verified that one solution is negative and one is positive. We can ignore the negative solution (which doesn’t make economic sense).

The positive solution is (approximately), RP = 1,200 b) In this problem, the RP of $1,200 can be interpreted as the maximum amount that you would be willing to pay to receive unemployment insurance that fully replaces your income loss if you are laid off.

15.16. Consider a household that possesses $100,000 worth of valuables (computers, stereo equipment, jewelry, and so forth). This household faces a 0.10 probability of a burglary. If a burglary were to occur, the household would have to spend $20,000 to replace the stolen items. Suppose it can buy an insurance policy for $500 that would fully reimburse it for the amount of the loss. a) Should the household buy this insurance policy? b) Should it buy the insurance policy if it cost $1,500? $3,000? c) What is the most the household would be willing to pay for this insurance policy? How does your answer relate to the concept of risk premium discussed in the text?

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a) If you remain uninsured, you face a lottery in which you have 10% chance of $80,000 in valuables and a 90% chance of $100,000 in valuables. The expected value of valuables is thus $98,000.

If you purchase the insurance policy for $500, then with no burglary you have 100,000 − 500 = $99,500 and with a burglary you have 100,000 − 500 − 20,000 + 20,000 = $99,500. The expected value if you purchase the policy is therefore $99,500. Since the expected value at year end with insurance exceeds the expected value at year end without insurance, you should purchase the insurance policy for $500. b) We can set up a table that shows the possible outcomes. The values in the table represent the value of valuables at year end depending on the corresponding row and column situations. Here is the $1500 case.

No Insurance Insurance Probability

Burglary $80,000 $98,500 0.10

No Burglary $100,000 $98,500 0.90

Expected Value $98,000 $98,500

If the policy costs $1,500, you are $500 better off with the policy.

Here is the $3,000 case.

No Insurance Insurance Probability

Burglary $80,000 $97,000 0.10

No Burglary $100,000 $97,000 0.90

Expected Value $98,000 $97,000

If the policy costs $3,000, you are $1,000 better off without the policy. c) Since without the policy you would have an expected value $2,000 less than the value of the valuables, the most you would be willing to pay for an insurance policy that fully reimburses for loss is $2,000.

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15.17. If you remain healthy, you expect to earn an income of $100,000. If, by contrast, you become disabled, you will only be able to work part time, and your average income will drop to $20,000. Suppose that you believe that there is a 5 percent chance that you could become disabled. Furthermore, your utility function is U = √ I. What is the most that you would be willing to pay for an insurance policy that fully insures you in the event that you are disabled?

If you do not purchase insurance, your expected utility is . 95√100,000 + 0.05√20,000 = 307.49 If you do purchase insurance at a price P, your expected utility is √100,000 − 𝑃. The highest price that you would be willing to pay is P such that: √100,000 − 𝑃 = 307.49, or 100,000 – P = 94,548, which implies that P = $5,451. Thus, the most you’d be willing to pay for this insurance policy is $5,451.

15.18. You are a risk-averse decision maker with a utility function U(I) = 1-3200I-2, where I denotes your income expressed in thousands. Your income is $100,000 (thus, I =100).

However, there is a 0.2 chance that you will have an accident that results in a loss of $20,000. Now, suppose you have the opportunity to purchase an insurance policy that fully insures you against this loss (i.e., that pays you $20,000 in the event that you incur the loss).

What is the highest premium that you would be willing to pay for this insurance policy?

If you do not buy insurance, your expected utility is: 0.2[1 – 3200(100 – 20)-2] + 0.8[1 – 3200(100)-2] = (0.2)(0.5) + (0.68)(0.5) = 0.644 If you do buy insurance at price P, you have no risk and your utility is 1 – 3200(100 – P)-2

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The most you would be willing to pay for insurance would be just a shade less than the P that makes your utility with insurance equal to your expected utility with no insurance. In terms of equations: 1 – 3200(100 – P)-2 = 0.644 3200(100 – P)-2 = 0.356 (100 – P)-2 = 0.356/3200 (100 – P)2 = 3200/0.356 100 – P = [3200/0.356](1/2) 100 – P = 94.81 P = 5.19.

Thus, the most that you would be willing to pay for the insurance, would be a shade less than $5.19 per thousand dollars of coverage.

15.19. You are a relatively safe driver. The probability that you will have an accident is only 1 percent. If you do have an accident, the cost of repairs and alternative transportation would reduce your disposable income from $120,000 to $60,000. Auto collision insurance that will fully insure you against your loss is being sold at a price of $0.10 for every $1 of coverage. Finally, suppose that your utility function is U = √I. You are considering two alternatives: buying a policy with a $1,000 deductible that essentially provides just $59,000 worth of coverage, or buying a policy that fully insures you against damage. The price of the first policy is $5,900. The price of the second policy is $6,000. Which policy do you prefer?

Your expected utility if you buy the first policy is 0.01√120,000 − 60,000 + 59,000 − 5900 + 0.99√120,000 − 5900 = 337.77.

If you buy the second policy, your expected utility is 0.01√120,000 − 60,000 + 60,000 − 6000 + 0.99√120,000 − 6000 = 337.64 Your expected utility is higher when you buy the first policy.

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15.20. Consider a market of risk-averse decision makers, each with a utility function U = √I. Each decision maker has an income of $90,000, but faces the possibility of a catastrophic loss of $50,000 in income. Each decision maker can purchase an insurance policy that fully compensates her for her loss. This insurance policy has a cost of $5,900.

Suppose each decision maker potentially has a different probability q of experiencing the loss. a) What is the smallest value of q so that a decision maker purchases insurance? b) What would happen to this smallest value of q if the insurance company were to raise the insurance premium from $5,900 to $27,500? a)

If an individual purchases insurance, her (certain) utility is

√90,000 − 5,900 = 290.

If an individual does not purchase insurance, her expected utility is 𝑞√90,000 − 50,000 + (1 − 𝑞)√90,000 = 200𝑞 + 300(1 − 𝑞) = 300 − 100𝑞 An individual will purchase insurance if 290 ≥ 300 − 100𝑞, or 𝑞 ≥ 0.10.

In other words, individuals that are 90 percent or more certain that they will not experience the loss will not purchase insurance. b) If the insurance premium is increased to $27,500, an individual who purchases insurance will achieve a certain utility of √90,000 − 27,500 = 250.

The individual still receives an expected utility of 300 – 100q if it does not purchase insurance.

Thus, an individual will purchase insurance if 250 ≥ 300 − 100𝑞, or 𝑞 ≥ 0.50.

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15.21. An insurance company is considering offering a policy to railroads that will insure a railroad against damage or deaths due to the spillage of hazardous chemicals from freight cars. Different railroads face difference risks from hazardous spills. For example, railroads operating on relatively new tracks face less risk than railroads with relatively older right of ways. (This is because a key cause of chemical spills is derailment of the train, and derailments are more likely on older, poorer tracks.) Discuss the difficulties that the insurance company might face in offering this type of policy; that is, why might it be difficult for the insurance company to make a profit from this type of policy?

There are two potential problems that might make it difficult for the insurance company to make a profit.

The first is the adverse selection problem. As noted in the problem, not all railroads are equally risky, but the insurance company may find it difficult to discern a railroad’s risk characteristics.

For example, the railroad will probably be much better informed about the condition of its track than the insurance company. This makes it difficult to tailor the terms of the insurance policy to the risk characteristics of the railroad. This is a problem because railroads whose track is in good conditions may choose to go without insurance. They may choose to “self insure” by taking precautions against derailment (which might not be very costly, since derailment risk for these railroads is low anyway), or they may simply do without insurance altogether. This means that the market for this insurance coverage might be primarily made up of railroads whose track is in poor condition and whose risk of derailment is correspondingly higher. The insurance company might not be able to make much profit from this pool of high-risk railroads. To make matters worse, if the insurance company tries to raise price to increase profit, the railroad that are most likely to “drop out” of the market in response to a now-more expensive insurance policy, are those which (among the pool of high-risk railroads) have the lowest risks. Hence, the insurance company makes things worse by trying to raise price!

The second problem is moral hazard. Once a railroad is insured against the risk of a chemical spill, it may become less careful in preventing such skills. Train operators may operate the train faster, maybe even violating speed limits, thus increasing the risk of a derailment. The company may skimp on investments in new track, new track ties, or ballast, also making a derailment more likely. With less care exerted by the railroad and the risk of derailment increased, the insurance company may find it difficult to make a profit on this particular insurance product.

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15.22. A firm is considering launching a new product. Launching the product will require an investment of $10 million (including marketing expenses and the costs of new facilities).

The launch is risky because demand could either turn out to be low or high. If the firm does not launch the product, its payoff is 0. Here are its possible payoffs if it launches the product.

a) Draw a decision tree showing the decisions that the company can make and the payoffs from following those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the firm acts as a risk-neutral decision maker, what action should it choose? What is the expected payoff associated with this action? a)

b) If the firm launches the product, its expected payoff is 0.5(20) + 0.5(–10) = $5 million.

Because this number is bigger than the payoff from not launching, a risk-neutral firm should launch the new product.

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15.23. A large defense contractor is considering making a specialized investment in a facility to make helicopters. The firm currently has a contract with the government, which, over the lifetime of the contract, is worth $100 million to the firm. It is considering building a new production plant for these helicopters; doing so will reduce the production costs to the company, increasing the value of the contract from $100 million to $200 million. The cost of the plant will be $60 million. However, there is the possibility that the government will cancel the contract. If that happens, the value of the contract will fall to zero. The problem (from the company’s point of view) is that it will only find out about the cancellation after it completes the new plant. At this point, it appears that the probability that the government will cancel the contract is 0.45. a) Draw a decision tree reflecting the decisions the firm can make and the payoffs from those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the firm is a risk-neutral decision maker, should the firm build a new plant? What is the expected value associated with the optimal decision? c) Suppose instead of finding out about contract cancellation after it builds the plant, the firm finds out about cancellation before it builds the plant. Draw a new decision tree corresponding to this new sequence of decisions and events. Again assuming that the firm is a risk-neutral decision maker, should the firm build the new plant? a & b) The decision tree for this situation is shown below. The chance nodes are circles, and the decision node is square.

Build

Contract is cancelled (probability = 0.45)

$0 - $60 million = -$60 million

Contract is not cancelled (probability = 0.55)

$200 - $60 million = $140 million

Contract is cancelled (probability = 0.45) Do not build Contract is not cancelled (probability = 0.55)

$0

$100 million

Since the decision maker is risk neutral, we can evaluate payoffs using expected values. The expected value if you build the plant is: (0.45)(-$60 million) + (0.55)($140 million) = $50 million

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The expected value if you do not build the plant: (0.45)($0) + (0.55)($100 million) = $55 million.

Not building the plant is the best course of action. c) The answer to question of whether the firm should build the plant is: it depends! The decision tree for the revised sequence of decisions and event is shown below.

Build $0 - $60 million = -$60 million

Contract is cancelled (probability = 0.45)

Do not build Build Contract is not cancelled (probability = 0.55)

$0

$200 - $60 million = $140 million

Do not build

$100 million

We see from the tree that if the contract is cancelled, the best action is not to build the plant, which results in a payoff of $0. However, if the contract is not cancelled, it’s better to build the plant than to not build the plant. Hence, the decision to build the plant depends on the circumstances the firm faces, in particular whether the contract is cancelled.

To finish the decision tree analysis, the picture below shows the folded back tree. There are no further decisions to be evaluated so all that needs to be done is to compute the expected vaue associated with this situation. That expected value is: (0.45)($0) + (0.55)($140) = $77 million.

Note that the firm’s expected value when the plant-building decision is made after the status of the contract is know is bigger ($77 million versus $55 million) than its value when the plant decision must be made when the contract status is still uncertain. This difference reflects the value of having perfect information about the status of the contract in the second situation.

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$0

Contract is not cancelled (probability = 0.55) $200 - $60 million = $140 million

Folded back decision tree

15.24. A small biotechnology company has developed a burn treatment that has commercial potential. The company has to decide whether to produce the new compound itself or sell the rights to the compound to a large drug company. The payoffs from each of these courses of action depend on whether the treatment is approved by the Food and Drug Administration (FDA), the regulatory body in the United States that approves all new drug treatments. (The FDA bases its decision on the outcome of tests of the drug’s effectiveness on human subjects.) The company must make its decision before the FDA decides. Here are the payoffs the drug company can expect to get under the two options it faces:

a) Draw a decision tree showing the decisions that the company can make and the payoffs from following those decisions. Carefully distinguish between chance nodes and decision nodes in the tree. b) Assuming that the biotechnology company acts as a risk-neutral decision maker, what action should it choose? What is the expected payoff associated with this action?

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Besanko & Braeutigam – Microeconomics, 6th edition

a)

Solutions Manual

A is a decision node; B and C are chance nodes.

Approves Sell rights

B Does not approve Approves

A Produce Self

C

Does not approve

10 2 50 –10

b) The expected payoff for “Sell rights” is 0.20(10) + 0.80(2) = 3.60. The expected payoff for “Produce yourself” is 0.20(50) + 0.80(–10) = 2.0. Therefore, the risk-neutral company should sell the rights, for an expected payoff of 3.60.

15.25. Consider the same problem as in Problem 15.24, but suppose that the biotech company can conduct its own test—at no cost—that will reveal whether the new drug will be approved by the FDA. What is the biotech company’s VPI? If the firm does not conduct a test, it should launch the new product, resulting in an expected value of $5 million.

If the outcome of the test marketing is that demand is high, the firm should launch the new product, and in so doing it receives a payoff of $20 million.

If the outcome of the test marketing is that demand is low, the firm should not launch the new product, and in so doing, it receives a payoff of $0.

The expected value from conducting the test is thus: 0.5($20 million) + 0.5(0) = $10 million.

VPI = expected value from conducting costless test – expected value from not conducting the test = $10 million – $5 million = $5 million.

If the test indicates the FDA will approve, the company will choose “Produce yourself” and earn a payoff of 50. If the test indicates the FDA will not approve, then the company will choose “Sell the rights” and earn 2. The expected payoff from conducting the costless test is therefore 0.20(50) + 0.80(2) = 11.60.

The VPI is the difference between the expected payoff with the test and the expected payoff without the test. Thus, VPI = 11.60 – 3.60 = 8.00.

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Chapter 15-25

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

15.26. You are bidding against one other bidder in a first-price sealed-bid auction with private values. You believe that the other bidder’s valuation is equally likely to lie anywhere in the interval between $0 and $500. Your own valuation is $200. Suppose you expect your rival to submit a bid that is exactly one half of its valuation. Thus, you believe that your rival’s bids are equally likely to fall anywhere between 0 and $250. Given this, if you submit a bid of Q, the probability that you win the auction is the probability that your bid Q will exceed your rival’s bid. It turns out that this probability is equal to Q/250. (Don’t worry about where this formula comes from, but you probably should plug in several different values of Q to convince yourself that this makes sense.) Your profit from winning the auction is profit = (200 - bid) x probability of winning. Show that your profit maximizing strategy is bidding half of your valuation.

From the given information, the profit from winning the auction is (assuming you bid 𝑄) 𝑄 ) 250 𝜋 = (0.80 − 0.004𝑄)𝑄 𝜋 = (200 − 𝑄) (

At the optimal bid marginal profit equals zero. Thus, at the optimum, the bid must satisfy 0.80 − 0.008Q = 0 0.008Q = 0.80 Q = 100

The optimal bid is therefore 100, which is equal to one-half of your true valuation, 200. Thus, a strategy of bidding one-half of your valuation is a Nash equilibrium; it is the best you can do given the other player’s strategy.

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Chapter 15-26

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 16 General Equilibrium Theory Solutions to Review Questions 1. What is the difference between a partial equilibrium analysis and a general equilibrium analysis? When analyzing the determination of prices in a market, under what circumstances would a general equilibrium analysis be more appropriate than a partial equilibrium analysis?

A partial equilibrium analysis studies the determination of price and output in a single market, taking as given the prices in all other markets. In general equilibrium analysis, we study the determination of price and output in more than one market at the same time.

One would employ a partial equilibrium analysis in situations where the concerns focused on a single market; for example, how does an increase in rainfall affect the price of corn? One would use general equilibrium analysis when one was concerned with how changes in price and output in one market affect the price and output in another market; for example, how does an increase in the price of natural gas affect the price and output for electric furnaces?

2. In a general equilibrium analysis with two substitute goods, X and Y, explain what would happen to the price in market X if the supply of good Y increased (i.e., if the supply curve for good Y shifted to the right). How would your answer differ if X and Y were complements?

If the supply of good 𝑌 increased, the equilibrium price of good 𝑌 would fall. Since 𝑋 and 𝑌 are assumed to be substitutes, when the price of good 𝑌 falls relative to good 𝑋, the demand for good 𝑋 will fall, lowering the equilibrium price and quantity for good 𝑋.

If 𝑋 and 𝑌 are complements, when the price of good 𝑌 falls, the demand for good 𝑋 will increase, increasing the equilibrium price and quantity for good 𝑋.

3. What role does consumer utility maximization play in a general equilibrium analysis?

What is the role played by firm cost minimization in a general equilibrium analysis?

In a general equilibrium, demand for finished products comes from utility maximization by households, while demand for inputs comes from cost minimization by firms. The supply of finished products comes from profit maximization by firms, while the supply of inputs comes from profit maximization by households.

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Chapter 16-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4. What is Walras’ Law? What is its significance?

Walras’ Law implies that a general equilibrium analysis will only be able to determine prices in 𝑁 − 1 of the markets being studied. This implies that a general equilibrium determines the prices of all goods and inputs relative to the price of another good or input, rather than determining the absolute levels of all prices.

5. What is an economically efficient allocation? How does an economically efficient allocation differ from an inefficient allocation?

An allocation of goods and inputs is economically efficient if there is no other feasible allocation of goods and inputs that would make some consumers better off without hurting other consumers. In contrast, an allocation of goods and inputs is economically inefficient if there is an alternative feasible allocation of goods and inputs that would make all consumers better off as compared with the initial allocation.

6. What is exchange efficiency? In an Edgeworth box diagram, how do efficient allocations and inefficient allocations differ?

Exchange efficiency occurs when a fixed amount of consumption goods cannot be reallocated among consumers in an economy without making at least some consumers worse off. Efficient allocations in an Edgeworth Box occur at points where the indifference curves for different consumers are tangent. Inefficient allocations occur at points where the indifference curves for different consumers intersect.

7. How does exchange efficiency differ from input efficiency? Could an economy satisfy the conditions for exchange efficiency but not the conditions for input efficiency?

Input efficiency occurs when a fixed stock of inputs cannot be reallocated among firms in an economy without reducing the output of at least one of the goods that is produced in the economy. It is quite possible that an economy could enjoy exchange efficiency, where the amount of goods available in the economy is allocated so that no consumer can be made better off without making some other consumer worse off, and not enjoy input efficiency, where the amount of inputs could be reallocated to produce more of all goods.

8. Suppose an economy has just two goods, X and Y. True or False: If the condition of input efficiency prevails, we can increase the production of X without decreasing the production of Y. Explain your answer.

False. If an economy has input efficiency, then the inputs cannot be reallocated among firms in an economy without reducing the output of at least one of the goods that is produced in the economy. That is, input efficiency implies that an expansion of output in one industry necessitates a reduction in output in another industry.

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Chapter 16-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

9. What is the production possibilities frontier? What is the marginal rate of transformation? How does the marginal rate of transformation relate to the production possibilities frontier?

The production possibilities frontier describes the combinations of consumption goods that can be produced in an economy given the economy’s available supply of inputs. The points on the frontier satisfy input efficiency, while the points inside the frontier have input inefficiency.

The marginal rate of transformation is the absolute value of the slope of the production possibilities frontier at some point. This measures the amount of one good the economy must give up in order to gain one additional unit of output for some other good.

10. Explain how consumers in an economy can be made better off if the marginal rate of transformation does not equal consumers’ marginal rates of substitution.

If the marginal rate of transformation is not equal to consumer’s marginal rate of substitution, then consumers in the economy can be made better off. As an example, suppose the marginal rate of substitution was 3 and the marginal rate of transformation was 1 for goods 𝑥 and 𝑦. By producing one more unit of 𝑥, the economy would need to sacrifice one unit of 𝑦. Consumers are willing to sacrifice 3 units of 𝑦, however, to get one additional unit of 𝑥. Therefore, if the economy produces one more unit of 𝑥 and one fewer unit of 𝑦, consumers would be better off.

11. Explain how the conditions of utility maximization, cost minimization, and profit maximization in competitive markets imply that the allocation arising in a general competitive equilibrium is economically efficient.

In a general competitive equilibrium, the economy will satisfy exchange, input, and substitution efficiency. This implies that all consumers are maximizing utility given the prices of goods in the economy and producers are maximizing profit at the point where prices equal marginal costs. 𝑃 That is, 𝑀𝑅𝑆𝑥,𝑦 = 𝑃𝑥 , 𝑝𝑥 = 𝑀𝐶𝑥 , and 𝑃𝑦 = 𝑀𝐶𝑦 . Together these imply that 𝑦

𝑃𝑥 𝑀𝐶𝑥 𝑀𝑅𝑆𝑥,𝑦 = = = 𝑀𝑅𝑇𝑥,𝑦 𝑃𝑦 𝑀𝐶𝑦 That is, utility maximization by consumers and profit maximization by producers implies the marginal rate of substitution will equal the marginal rate of transformation. This guarantees substitution efficiency is satisfied at the general competitive equilibrium. In other words, the allocation that arises in a general competitive equilibrium is economically efficient.

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Chapter 16-3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

12. What is comparative advantage? What is absolute advantage? Which of these two concepts is more important in determining the benefits from free trade?

Comparative advantage implies that one country has a lower opportunity cost in the production of some good, expressed in units of some other good forgone, than another country.

Absolute advantage implies that one country can produce a product at a lower cost in terms of units of some input, labor for example, than another country.

In determining the benefits from free trade, one must compare the opportunity costs to identify the benefits. For example, in a two-good world, while one country may have an absolute advantage in the production of both goods, it is still likely that each country has a comparative advantage in the production of different goods, and through specialization and trade, each country can be made better off.

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Chapter 16-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 16.1. Consider the markets for butter (B) and margarine (M), where the demand curves are Q = 20 – 2PM + PB and Q = 60 – 6PB + 4 PM and the supply curves are QM = 2PM and QB = 3PB. a) Find the equilibrium prices and quantities for butter and margarine. b) Suppose that an increase in the price of vegetable oil shifts the supply curve of margarine to QM = PM. How does this change affect the equilibrium prices and quantities for butter and margarine? Using words and graphs, explain why a shift in the supply curve for margarine would change the price of butter. a) In equilibrium we must have quantity supplied equal to quantity demanded in both the butter and margarine markets. This implies in equilibrium we will have

QMd = QMs QBd = QBs Substituting in the given curves implies 20 − 2 PM + PB = 2 PM 60 − 6 PB + 4 PM = 3PB

Solving for 𝑃𝐵 in the first equation and substituting into the second equation implies 60 + 4 PM = 9(4 PM − 20) 60 + 4 PM = 36 PM − 180 PM = 7.5

When 𝑃𝑀 = 7.5, 𝑃𝐵 = 10. At these prices, 𝑄𝑀 = 15 and 𝑄𝐵 = 30. b)

𝑠 When the supply curve for margarine shifts to 𝑄𝑀 = 𝑃𝑀 , we have

20 − 2 PM + PB = PM 60 − 6 PB + 4 PM = 3PB

Solving the first equation for 𝑃𝐵 and substituting into the second equation implies 60 + 4 PM = 9(3PM − 20) 60 + 4 PM = 27 PM − 180 PM = 10.43

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Chapter 16-5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

When 𝑃𝑀 = 10.43, 𝑃𝐵 = 11.30. At these prices, 𝑄𝑀 = 10.43 and 𝑄𝐵 = 33.91. The increase in the price of vegetable oil increases the price of margarine and decreases the quantity of margarine consumed. As consumers switch to butter, the price of butter rises and the quantity of butter consumed goes up.

The price of butter rises when the price of vegetable oil rises because butter and margarine are substitutes. The effects can be seen in the following graphs.

Because the goods are substitutes, when the supply of margarine shifts inward from S to S’, raising the price of margarine, consumers substitute butter for margarine, shifting demand for butter outward from D to D’. This raises both the equilibrium price and quantity of butter.

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Chapter 16-6

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.2. Suppose that the demand curve for new automobiles is given by QA = 20 – 0.7PA – PG where QA and PA are the quantity (millions of vehicles) and average price (thousands of dollars per vehicle), respectively, of automobiles in the United States, and PG is the price of gasoline (dollars per gallon). The supply of automobiles is given by Q5A = 0.3PA. Suppose that the demand and supply curves for gasoline are QdG = 3 – PG and QSG = PG. a) Find the equilibrium prices of gasoline and automobiles. b) Sketch a graph that shows how an exogenous increase in the supply of gasoline affects the prices of new cars in the United States. a) In equilibrium, the quantity supplied and the quantity demanded for both goods will be equal. This implies 𝑄𝐴𝑑 = 𝑄𝐴𝑠 𝑄𝐺𝑑 = 𝑄𝐺𝑠 Substituting in the given curves implies 20 − 0.7 PA − PG = 0.3PA 3 − PG = PG

Here we have two equations and two unknowns. Solving the second equation for 𝑃𝐺 yields 𝑃𝐺 = 1.5. Substituting into the first equation results in 20 − 0.7 PA − 1.5 = 0.3PA PA = 18.5

At these prices, 𝑄𝐴 = 5.55 and 𝑄𝐺 = 1.5.

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Chapter 16-7

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) If the supply of gasoline increases, the supply curve for gasoline will shift to the right lowering the equilibrium price of gasoline as seen in the graph below.

Because gasoline is a complement good for autos, the reduction in the price of gasoline will increase the demand for autos. This will shift the demand curve to the right, increasing the equilibrium price and quantity for autos as seen in the following graph.

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Chapter 16-8

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.3. Studies indicate that the supply and demand schedules for ties (t) and jackets (j) in a market are as follows:

The estimates of the schedules are valid only for prices at which quantities are positive. a) Find the equilibrium prices and quantities for ties and jackets. b) Do the demand schedules indicate that jackets and ties are substitute goods, complementary goods, or independent goods in consumption? How do you know? a)

In equilibrium (1) the supply and demand for ties will be equal

410 – 5Pt – 2Pj = –60 + 3PK , and (2) the supply and demand for jackets will be equal 295 – Pt – 3Pj = –120 + 2Pj Solving these two simultaneous equations, we find that Pj = 75 and Pt = 40. Also, using either the demands or supply schedules, we calculate that the equilibrium quantity of jackets is 30, and the equilibrium quantity of ties is 60. b) The demand function for ties shows that a higher price of jackets decreases the demand for ties. Similarly the demand function for jackets shows that a higher price of ties decreases the demand for jackets. Ties and jackets are therefore complements in consumption.

16.4. Suppose that the demand for steel in Japan is given by the equation QdS = 1200 – 4PS + PA + PT, where QS is the quantity of steel purchased (millions of tons per year), PS is the price of steel (yen per ton), PA is the price of aluminum (yen per ton), and PT is the price of titanium (yen per ton). The supply curve for steel is given by QSS = 4PS. Similarly, the demand and supply curves for aluminum and for titanium are given by QdA = 1200 – 4PA + PS + PT (demand curve for aluminum), QSA = 4PA (supply curve for aluminum), QdT = 1200 – 4PT + PS + PA (demand curve for titanium), and QST = 4PT (supply curve for aluminum). a) Find the equilibrium prices of steel, aluminum, and titanium in Japan. b) Suppose that a strike in the Japanese steel industry shifts the supply curve for steel to QSS = PS. What does this do to the prices of steel, aluminum, and titanium? c) Suppose that growth in the Japanese beer industry, a big buyer of aluminum cans, fuels an increase in the demand for aluminum so that the demand curve for aluminum becomes QdA = 1500 – 4PA + PS + PT. How does this affect the prices of steel, aluminum, and titanium?

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Chapter 16-9

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) In equilibrium the quantity supplied will equal the quantity demanded in all three markets. Algebraically this implies 𝑄𝑆𝑑 = 𝑄𝑆𝑠 𝑄𝐴𝑑 = 𝑄𝐴𝑠 𝑄𝑇𝑑 = 𝑄𝑇𝑠 Substituting in the given curves implies 1200 − 4𝑃𝑆 + 𝑃𝐴 + 𝑃𝑇 = 4𝑃𝑆 1200 − 4𝑃𝐴 + 𝑃𝑆 + 𝑃𝑇 = 4𝑃𝐴 1200 − 4𝑃𝑇 + 𝑃𝑆 + 𝑃𝐴 = 4𝑃𝑇 Solving the first equation for 𝑃𝑇 and substituting into the second equation implies 1200 − 4 PA + PS + (8 PS − PA − 1200) = 4 PA 9 PS = 9 PA PS = PA

Substituting these results into the third equation implies 1200 − 4(8 PA − PA − 1200) + PA + PA = 4(8 PA − PA − 1200) 10,800 = 54 PA PA = 200

At 𝑃𝐴 = 200, 𝑃𝑆 = 200 and 𝑃𝑇 = 200. The equilibrium quantities are 𝑄𝐴 = 800, 𝑄𝑆 = 800, and 𝑄𝑇 = 800. b)

Substituting the new supply curve for steel into the equilibrium condition implies

200 − 4𝑃𝑆 + 𝑃𝐴 + 𝑃𝑇 = 𝑃𝑆 1200 − 4𝑃𝐴 + 𝑃𝑆 + 𝑃𝑇 = 4𝑃𝐴 1200 − 4𝑃𝑇 + 𝑃𝑆 + 𝑃𝐴 = 4𝑃𝑇 Again solving for 𝑃𝑇 in the first equation and substituting into the second equation implies 1200 − 4 PA + PS + (5 PS − PA − 1200) = 4 PA 6 PS = 9 PA PS = 1.5 PA

Substituting these results into the third equation implies

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Chapter 16-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

1200 − 4(5(1.5 PA ) − PA − 1200) + 1.5 PA + PA = 4(5(1.5 PA ) − PA − 1200) 10,800 = 49.5 PA PA = 218.18

At 𝑃𝐴 = 218.18, 𝑃𝑆 = 327.27 and 𝑃𝑇 = 218.18. At these prices, the equilibrium quantities are 𝑄𝐴 = 872.72, 𝑄𝑆 = 327.27, and 𝑄𝑇 = 872.72.

The shift in the supply of steel raises the equilibrium price for all three goods, lowering the equilibrium quantity of steel and raising the equilibrium quantities of aluminum and titanium.

This last effect comes as a result of the demand curves for aluminum and titanium increasing in response to the shift in the steel supply curve. c)

Returning to the original equilibrium, this shift in the demand for aluminum implies

200 − 4𝑃𝑆 + 𝑃𝐴 + 𝑃𝑇 = 4𝑃𝑆 1500 − 4𝑃𝐴 + 𝑃𝑆 + 𝑃𝑇 = 4𝑃𝐴 1200 − 4𝑃𝑇 + 𝑃𝑆 + 𝑃𝐴 = 4𝑃𝑇 Solving the first equation for 𝑃𝑇 and substituting into the second equation implies 1500 − 4 PA + PS + (8 PS − PA − 1200) = 4 PA 9 PS + 300 = 9 PA PS = PA − 33.33

Substituting these results into the third equation implies 1200 − 4(8( PA − 33.33) − PA − 1200) + ( PA − 33.33) + PA = 4(8( PA − 33.33) − PA − 1200) 12,900 = 54 PA PA = 238.89

At 𝑃𝐴 = 238.89, 𝑃𝑆 = 205.56 and 𝑃𝑇 = 205.56. At these prices, the equilibrium quantities are 𝑄𝐴 = 955.56, 𝑄𝑆 = 822.24, and 𝑄𝑇 = 822.24.

An increase in the demand for aluminum will raise the equilibrium prices and quantities in all three markets. The price and quantity in the steel and aluminum industries increase because as the price of aluminum rises, the demand for steel and titanium increases.

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Chapter 16-11

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.5. Consider a simple economy that produces two goods, beer (denoted by x) and quiche (denoted by y), using labor and capital (denoted by L and K, respectively) that are supplied by two types of households, those consisting of wimps (denoted by W) and those consisting of hunks (denoted by H). Each household of hunks supplies 100 units of labor and no units of capital. Each household of wimps supplies 10 units of capital and no units of labor.

There are 100 households of each type. Both beer and quiche are produced with technologies exhibiting constant returns to scale. The market supply curves for beer and quiche are

where w denotes the price of labor and r denotes the price of capital. The market demand curves for beer and quiche are given by

where X and Y denote the aggregate quantities of beer and quiche demanded in this economy and IW and IH are the household incomes of wimps and hunks, respectively.

Finally, the market demand curves for labor and capital are given by

There are four unknowns in our simple economy: the prices of beer and quiche, Px and Py, and the prices of labor and capital, w and r. Write the four equations that determine the equilibrium values of these unknowns.

First, in equilibrium, the quantity supplied of beer and quiche must equal the quantity demanded of beer and quiche. This implies 20𝐼𝑊 + 90𝐼𝐻 𝑋 80𝐼𝑊 + 10𝐼𝐻 3/4 1/4 𝑤 𝑟 = 𝑌 𝑤 1/6 𝑟 5/6 =

Now, since each hunk supplies 100 units of labor and no units of capital and each wimp supplies 10 units of capital and no units of labor, 𝐼𝑊 (𝑤, 𝑟) = 10𝑟 𝐼𝐻 (𝑤, 𝑟) = 100𝑤

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Chapter 16-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Substituting these into the conditions above yields our first two equations: 200𝑟 + 9000𝑤 𝑋 800𝑟 + 1000𝑤 𝑤 3/4 𝑟 1/4 = 𝑌 𝑤 1/6 𝑟 5/6 =

Second, in equilibrium, the quantity supplied of labor and capital must equal the quantity demanded of labor and capital. Since there are 100 households of each type, we will have 𝐿 = 100(100) = 10,000 and 𝐾 = 100(10) = 1,000. Setting these equal to demand yields the third and fourth equations:

10, 000 =

Xr   6  w

5/ 6

1/ 4

+ 1/ 6

5X  w  1, 000 =   6 r

3Y  r    4  w

Y  w +   4 r 

3/ 4

16.6. In an economy, there are 40 “white-collar” households, each producing 10 units of capital (and no labor); the income from each unit of capital is r. There are also 50 “bluecollar” households, each producing 20 units of labor (and no capital); the income from each unit of labor is w.

Each white-collar household’s demand for energy is XW = 0.8MW/PX, where MW is income in the household. Each white-collar household’s demand for food is YW = 0.2MW/PY. Each blue-collar household’s demand for energy is XB = 0.5MB/PX, where MB is income in the household. Each blue-collar household’s demand for food is YB = 0.5MB/PY. Energy is produced using only capital. Each unit of capital produces one unit of energy, so r is the marginal cost of energy. The supply curve for energy is described by PX = r, where PX is the price of a unit of energy. Food is produced using only labor. Each unit of labor produces one unit of food, so w is the marginal cost of food. The supply curve for labor is described by PY = w, where PY is the price of a unit of food. a) In this economy, show that the amount of labor demanded and supplied will be 1,000 units. Show also that the amount of capital demanded and supplied will be 400 units. b) Write down the supply-equals-demand conditions for the energy and food markets. c) In equilibrium how will the price of a unit of energy compare with the price of a unit of food? d) In equilibrium how will the income of each white-collar family compare with the income of each blue-collar family?

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Chapter 16-13

Besanko & Braeutigam – Microeconomics, 6th edition

a)

Solutions Manual

The total amount of capital produced (all by white collar households together) is

(40 households)(10 units/household) = 400 units.

The total amount of labor produced (all by blue collar households together) is (50 households)(20 units/household) = 1000 units. b) The income in each white collar household is MW = 10r.

The income in each blue collar household is MB = 20w.

The aggregate demand for energy will be X = [50(0.5MB) + 40(0.8MW)]/PX = [500w + 320r]/PX The aggregate demand for food will be Y =[50(0.5MB) + 40(0.2MW)]/PY = [500w + 80r]/PY. The supply-equals demand condition in the energy market is r = [500w + 320r]/X = [500w + 320r]/400, or r = 6.25w The supply-equals demand condition in the food market is w = [500w + 80r]/Y= [500w + 80r]/1000, or, as before r = 6.25w (same as above by Walras’ Law) c) PX = r and PY = w. Since r = 6.25w, the price of a unit of energy is 6.25 times as large as the price of a unit of food. d) The income of a blue collar family is MB = 20w. The income of a white collar family is MW = 10r = 10(6.25w) = 62.5w. So a white collar family has an income 3.125 (= 62.5/20) times larger than that of a blue collar family.

16.7. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that price labor will be 25/52 ≈ 0.48 of the price of capital, as illustrated in Figure 16.10.

When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 =

5000𝑤+2500𝑟 𝑋

.

Solving for the quantity of energy 𝑋 , we find that 𝑋=

5000𝑤+2500𝑟 w1/3 r2/3

. Call this equation A.

Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2 =

2000𝑤+2500𝑟 𝑌

.

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Chapter 16-14

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Solutions Manual

Solving for the quantity of food 𝑌 , we find that 𝑌=

2000𝑤+2500𝑟 w1/2 r1/2

. Call this equation B.

When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 𝑋

𝑟 2/3

7000 = 3 (𝑤) 2000𝑤+2500𝑟 2w

=

𝑌

𝑟 1/2

+ 2 (𝑤)

=

16000𝑤+12500𝑟

5000𝑤+2500𝑟 3w1/3 r2/3

𝑟 2/3

(𝑤)

+

2000𝑤+2500𝑟 2w1/2 r1/2

𝑟 1/2

(𝑤)

=

5000𝑤+2500𝑟 3w

+

6w 𝑤

25

Thus: 42000w = 16000𝑤 + 12500𝑟, which can be reduced to 𝑟 = 52  0.48.

16.8. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that the ratio of the price of energy to the price of capital is about 0.78, as illustrated in Figure 16.10.

When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 =

5000𝑤+2500𝑟 𝑋

.

Solving for the quantity of energy 𝑋 , we find that 𝑋=

5000𝑤+2500𝑟 w1/3 r2/3

. Call this equation A.

Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2 =

2000𝑤+2500𝑟 𝑌

.

Solving for the quantity of food 𝑌 , we find that 𝑌=

2000𝑤+2500𝑟 w1/2 r1/2

. Call this equation B.

When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 𝑋

𝑟 2/3

7000 = 3 (𝑤) 2000𝑤+2500𝑟 2w

=

𝑌

𝑟 1/2

+ 2 (𝑤)

=

16000𝑤+12500𝑟

5000𝑤+2500𝑟 3w1/3 r2/3

𝑟 2/3

(𝑤)

+

2000𝑤+2500𝑟 2w1/2 r1/2

𝑟 1/2

(𝑤)

=

5000𝑤+2500𝑟 3w

+

6w

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Chapter 16-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

𝑤

25

Thus: 42000w = 16000𝑤 + 12500𝑟, which can be reduced to 𝑟 = 52  0.48.

Finally, from the relationship describing the supply of energy we know that 𝑤 1/3

Px = w1/3r2/3 = ( 𝑟 )

25 1/3

r = (52)

r  0.78 r

16.9. One of the implications of Walras’ Law is that the ratios of prices (rather than the absolute levels of prices) are determined in general equilibrium. In Learning-By-Doing Exercise 16.2, show that the ratio of the price of food to the price of capital is about 0.69, as illustrated in Figure 16.9.

When we equated the supply and demand for energy, we eliminated the price of energy to derive Equation 16.4: w1/3r2/3 =

5000𝑤+2500𝑟 𝑋

.

Solving for the quantity of energy 𝑋 , we find that 𝑋=

5000𝑤+2500𝑟 w1/3 r2/3

. Call this equation A.

Similarly, when we equated the supply and demand for food, we eliminated the price of food to derive Equation 16.5: w1/2r1/2 =

2000𝑤+2500𝑟 𝑌

.

Solving for the quantity of food 𝑌 , we find that 𝑌=

2000𝑤+2500𝑟 w1/2 r1/2

. Call this equation B.

When we substitute Equations A and B into the supply-equals-demand in the labor market (Equation 16.6), we find that 𝑋

𝑟 2/3

7000 = 3 (𝑤) 2000𝑤+2500𝑟 2w

=

𝑌

𝑟 1/2

+ 2 (𝑤)

=

16000𝑤+12500𝑟

5000𝑤+2500𝑟 3w1/3 r2/3

𝑟 2/3

(𝑤)

+

2000𝑤+2500𝑟 2w1/2 r1/2

𝑟 1/2

(𝑤)

=

5000𝑤+2500𝑟 3w

+

6w 𝑤

25

Thus: 42000w = 16000𝑤 + 12500𝑟, which can be reduced to 𝑟 = 52  0.48.

Finally, from the relationship describing the supply of food we know that 𝑤 1/2

PY = w1/2r1/2 = ( 𝑟 )

25 1/2

r = (52)

r  0.69 r

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Chapter 16-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.10. Two consumers, Josh and Mary, together have 10 apples and 4 oranges. a) Draw the Edgeworth box that shows the set of feasible allocations that are available in this simple economy. b) Suppose Josh has 5 apples and 1 orange, while Mary has 5 apples and 3 oranges.

Identify this allocation in the Edgeworth box. c) Suppose Josh and Mary have identical utility functions, and assume that this utility function exhibits positive marginal utilities for both apples and oranges and a diminishing marginal rate of substitution of apples for oranges. Could the allocation in part (b)—5 apples and 1 orange for Josh; 5 apples and 3 oranges for Mary—be economically efficient? a) & b)

c) To be economically efficient, the two consumers must have identical marginal rates of substitution at the allocation. While we are not given the MRS for each consumer, we are told that each has an identical utility function. This implies that at an efficient allocation where the MRS for each consumer is the same, the ratio of apples to oranges must be the same. Since at the current allocation Josh has a ratio of apples to oranges equal to 5 and Mary has a ratio of 1.67, this allocation cannot be efficient. The contract curve in this case will be a straight line between the origins for each consumer.

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Chapter 16-17

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.11. Ted and Joe each consume peaches, x, and plums, y. The consumers have identical utility functions, with Together, they have 10 peaches and 10 plums. Verify whether each of the following allocations is on the contract curve: a) Ted: 8 plums and 9 peaches; Joe: 2 plums and 1 peach. b) Ted: 1 plum and 1 peach; Joe: 9 plums and 9 peaches. c) Ted: 4 plums and 3 peaches; Joe: 6 plums and 7 peaches. d) Ted: 8 plums and 2 peaches; Joe: 2 plums and 8 peaches.

To be on the contract curve, an allocation must yield identical marginal rates of substitution for each consumer. a) b) c) d)

MRSTed = 80/9 < MRSJoe = 20/1. Not on the contract curve.

MRSTed = 10/1 = MRSJoe = 90/9. On the contract curve.

MRSTed = 40/3 > MRSJoe = 60/7. Not on the contract curve.

MRSTed = 80/2 > MRSJoe = 20/8. Not on the contract curve.

16.12. Two consumers, Ron and David, together own 1,000 baseball cards and 5,000 Pokémon cards. Let xR denote the quantity of baseball cards owned by Ron and yR denote the quantity of Pokémon cards owned by Ron. Similarly, let xD denote the quantity of baseball cards owned by David and yD denote the quantity of Pokémon cards owned by David. Suppose, further, that for Ron, MRSRx,y = yR/xR, while for David, MRSDx,y = yD/2xD. Finally, suppose xR = 800, yR = 800, xD = 200, and yD = 4,200. a) Draw an Edgeworth box that shows the set of feasible allocations in this simple economy. b) Show that the current allocation of cards is not economically efficient. c) Identify a trade of cards between David and Ron that makes both better off. (Note: There are many possible answers to this problem.) a)

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Chapter 16-18

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Solutions Manual

b) To be economically efficient, the MRS for the two consumers must be equal. At this allocation we have 𝑦𝑅 800 = =1 𝑥𝑅 800 𝑦𝐷 4200 𝐷 𝑀𝑅𝑆𝑥,𝑦 = = = 10.5 2𝑥𝐷 2(200) 𝑅 𝑀𝑅𝑆𝑥,𝑦 =

Since MRSD > MRSR, the current allocation is not economically efficient. c) At the current allocation Ron is willing to trade one baseball card for one Pokemon card, and David is willing to trade one baseball card for 10.5 Pokemon cards. If David gives Ron 9 Pokemon cards in exchange for one baseball card, both consumers will be better off.

Or, in other words, Ron thinks a baseball card is worth just one Pokemon card while David thinks it is worth 10.5 Pokemon cards. So both will be better off if Ron sells David a baseball card for anything more than one Pokemon card and less than 10.5 Pokemon cards.

16.13. There are two individuals in an economy, Joe and Mary. Each of them is currently consuming positive amounts of two goods, food and clothing. Their preferences are characterized by diminishing marginal rate of substitution of food for clothing. At the current consumption baskets, Joe’s marginal rate of substitution of food for clothing is 2, while Mary’s marginal rate of substitution of food for clothing is 0.5. Do the currently consumed baskets satisfy the condition of exchange efficiency? If not, describe an exchange that would make both of them better off.

Since the marginal rates of substitution are not equal for both people, the current consumption baskets do not satisfy exchange efficiency. Joe would be willing to give up 2 units of food to get 1 additional unit of clothing. Mary would be willing to give up 0.5 units of food to get 1 additional unit of clothing; put another way, Mary would be willing to give up 2 units of clothing to get 1 additional unit of food.

One exchange that would make both better off would be for Joe to give 1 unit of food to Mary in exchange for 1 unit of clothing. Joe is better off (he would have been willing to give up 2 units of food to get 1 additional unit of clothing). But what about Mary? To get the 1 additional unit of clothing, she would have been willing to give up 2 units of clothing; but with the proposed exchange, she only had to give up 1 unit of clothing. So the proposed exchange also makes her better off.

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Chapter 16-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.14. Consider an economy that consists of three individuals: Maureen (M), David (D), and Suvarna (S). Two goods are available in the economy, x and y. The marginal rates of substitution for the three consumers are given by MRSMaureenx,y = 2yM/xM, MRSDavidx,y = 2yD/xD and MRSSuvarnax,y = yS/xS. Maureen and David are both consuming twice as much of good x as good y, while Suvarna is consuming equal amounts of goods x and y. Are these consumption patterns economically efficient?

To be economically efficient, the marginal rates of substitution for all consumers must be equal.

From the given information we know 𝑥𝑀 = 2𝑦𝑀 , 𝑥𝐷 = 2𝑦𝐷 , and 𝑥𝑆 = 𝑦𝑆 . Substituting into the marginal rates of substitution we have 2𝑦𝑀 =1 2𝑦𝑀 2𝑦𝐷 𝐷𝑎𝑣𝑖𝑑 𝑀𝑅𝑆𝑥,𝑦 = =1 2𝑦𝐷 𝑦𝑆 𝑆𝑢𝑣𝑎𝑟𝑛𝑎 𝑀𝑅𝑆𝑥,𝑦 = =1 𝑦𝑆 𝑀𝑎𝑢𝑟𝑒𝑒𝑛 𝑀𝑅𝑆𝑥,𝑦 =

Thus, each consumer has an identical marginal rate of substitution. This consumption pattern is therefore economically efficient.

16.15. Two firms together employ 100 units of labor and 100 units of capital. Firm 1 employs 20 units of labor and 80 units of capital. Firm 2 employs 80 units of labor and 20 units of capital. The marginal products of the firms are as follows: Firm 1: MP1l = 50, MP1k = 50; Firm 2: MP2l = 10, MP2k= 20. Is this allocation of inputs economically efficient?

To satisfy input efficiency, the marginal rates of technical substitution must be equal across firms. Here we have 𝑀𝑃𝑙1 50 = =1 𝑀𝑃𝑘1 50 𝑀𝑃𝑙2 10 2 𝑀𝑅𝑇𝑆𝑙,𝑘 = = = 0.5 𝑀𝑃𝑘2 20 1 𝑀𝑅𝑇𝑆𝑙,𝑘 =

Thus, the allocation of inputs is not economically efficient.

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Chapter 16-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.16. There are two firms in an economy. Each of them currently employs positive amounts of two inputs, capital and labor. Their technologies are characterized by diminishing marginal rate of technical substitution of labor for capital. At the current operating basket, Firm A’s marginal rate of technical substitution of labor for capital is 3, while Firm B’s marginal rate of technical substitution of labor for capital is 1. Do the current production baskets satisfy the condition of input efficiency? If not, describe an exchange of inputs that would improve efficiency.

Since the marginal rates of technical substitution are not equal for both firms, the current production baskets do not satisfy input efficiency. Firm A would be willing to give up 3 units of capital to get one additional unit of labor. Firm B would be willing to give up 1 unit of capital to get 1 additional unit of labor; put another way, Firm B would be willing to give up 1 unit of labor to get 1 additional unit of capital.

One exchange that would make both firms better off would be for Firm A to give 2 units of capital to Firm B in exchange for 1 unit of labor. Firm A is better off (it would have been willing to give up 3 units of capital to get 1 additional unit of labor). But what about Firm B? To get the 2 additional units of capital, it would have been willing to give up 2 units of labor; but with the proposed exchange, Firm B only had to give up 1 unit of labor. So the proposed exchange also makes Firm B better off.

16.17. Two firms together employ 10 units of labor (l) and 10 units of capital (k). The marginal rate of technical substitution of each firm is given by: MRTS1lk = k1/l1 and MRTS2lk = 4k2/l2. Which of the following input allocations satisfy the condition of input efficiency? a) Firm 1 uses 5 units of labor, 5 units of capital; Firm 2 uses 5 units of labor, 5 units of capital. b) Firm 1 uses 5 unit of labor, 8 units of capital; Firm 2 uses 5 units of labor; 2 units of capital. c) Firm 1 uses 9 units of labor, 9 units of capital; Firm 2 uses 1 unit of labor; 1 unit of capital. d) Firm 1 uses 2 units of labor; 5 units of capital; Firm 2 uses 8 units of labor; 5 units of capital.

To satisfy input efficiency, the marginal rates of technical substitution must be equal across firms. a)

MRTS1 = 5/5 = 1 < MRTS2 = 4(5)/5 = 4. The allocation does not satisfy input efficiency.

b)

MRTS1 = 8/5 = 1.6 = MRTS2 = 4(2)/5 = 1.6. The allocation satisfies input efficiency.

c)

MRTS1 = 9/9 = 1 < MRTS2 = 4(1)/1 = 4. The allocation does not satisfy input efficiency.

d)

MRTS1 = 5/2 = 2.5 = MRTS2 = 4(5)/8 = 2.5. The allocation satisfies input efficiency.

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Chapter 16-21

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.18. Two firms together employ 20 units of labor and 12 units of capital. For Firm 1, which uses 5 units of labor and 8 units of capital, the marginal products of labor and capital are MP1l = 20 and MP1k = 40. For Firm 2, which uses 15 units of labor and 4 units of capital, the marginal products are MP2l = 60 and MP2k = 30. a) Draw an Edgeworth box for inputs that shows the allocation of inputs across these two firms. b) Is this allocation of inputs economically efficient? Why or why not? If it is not, identify a reallocation of inputs that would allow both firms to increase their outputs. a)

b)

To satisfy input efficiency we must have

MRTS L1 , K = MRTS L2, K MPL1 MPL2 = MPK1 MPK2 Substituting in the given information implies 20 60  40 30 0.5  2

Since the MRTS are not equal, the current allocation of inputs is not economically efficient.

At the current allocation, Firm 1 can trade 2 units of labor for 1 unit of capital without changing output. By giving up one unit of labor to receive one unit of capital the firm can increase its output. At the current allocation Firm 2 can trade 2 units of capital for one unit of labor without affecting output. By giving up only one unit of capital in exchange for one unit of labor Firm 2 can increase its output. Therefore, by reallocating one unit of capital from Firm 2 to Firm 1 and one unit of labor from Firm 1 to Firm 2, both firms can produce more output.

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Chapter 16-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.19. Consider an economy that produces two goods: food, x, and clothing, y. Production of both goods is characterized by constant returns to scale. Given current input prices, the marginal cost of producing clothing is $10 per unit, while the marginal cost of producing food is $20 per unit. What is the marginal rate of transformation of x for y? How much clothing must the economy give up in order to get one additional unit of food?

In general equilibrium, MRTx,y = MCx / MCy = 20/10 = 2. To get one additional unit of food (x), the economy must sacrifice 2 units of clothing (y).

16.20. An economy consists of two consumers (Julie and Carina), each consuming positive amounts of two goods, food and clothing. Food and clothing are both produced with two inputs, capital and labor, using technologies exhibiting constant returns to scale. The following information is known about the current consumption and production baskets: The marginal cost of producing food is $2, and the price of clothing is $4. The wage rate is 2/3 the rental price of capital, and the marginal product of capital in producing clothing is 3. In a general competitive equilibrium, what must be a) The price of food? b) The marginal rate of transformation of food for clothing? c) The shape of the production possibilities frontier for the economy? d) The marginal product of labor in producing clothing? a) In a competitive equilibrium, price must equal marginal cost for each good. Thus, the price of food must be $2. b)

By the same reasoning as in (a) the marginal cost of clothing must be $4.

MRTfood,clothing = MCfood / MCclothing = $2/$4 = 0.5 c) Because all production occurs with constant returns to scale, the marginal costs will be constant, and thus the MRTfood,clothing is always 0.5. On a graph with food on the horizontal axis and clothing on the vertical axis, the Production Possibilities Frontier will be a straight line with a slope of -0.5. d)

With input efficiency, w/r = MPL/MPK ; thus, 2/3 = MPL/3. So MPL = 2.

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Chapter 16-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.21. Consider an economy that uses labor and capital to produce two goods, beer (x) and peanuts (y), subject to technologies that exhibit constant returns to scale. The marginal cost of a 12-ounce can of beer is $0.50. The marginal cost of a 12-ounce tin of peanuts is $1.00.

Currently, the economy is producing 1 million 12-ounce cans of beer and 2 million 12ounce tins of peanuts. The marginal rates of technical substitution of labor for capital in the beer and peanut industries are the same. Moreover, there are 1 million identical consumers in the economy, each with a marginal rate of substitution of beer for peanuts given by MRSx,y = 3y/x. a) Sketch a graph of the economy’s production possibilities frontier. Identify the economy’s current output on this graph. b) Does the existing allocation satisfy substitution efficiency? Why or why not? a) Because the technologies exhibit constant returns to scale, the production possibilities frontier will be a straight line with slope 𝑀𝑅𝑇𝑥,𝑦 =

𝑀𝐶𝑥 0.50 = = 0.50 𝑀𝐶𝑦 1.00

Here is a graph of the production possibilities frontier for this economy.

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Chapter 16-24

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) To achieve substitution efficiency we must have 𝑀𝑅𝑇𝑥,𝑦 = 𝑀𝑅𝑆𝑥,𝑦 . At the current allocation this implies 𝑀𝐶𝑥 𝑥 = 𝑀𝐶𝑦 3𝑦 At the current allocation we have

0.50 1  1.00 3(2) 0.50  0.17 Since MRT < MRS, consumer utility would go up if more resources were devoted to beer production (x) and less resources were devoted to peanut production (y).

16.22. The United States and Switzerland both produce automobiles and watches. The labor required to produce a unit of each product is shown in the following table:

a) Which country has an absolute advantage in the production of watches? In the production of automobiles? b) Which country has a comparative advantage in the production of watches? In the production of automobiles? a) From the information given in the table, the U.S. has an absolute advantage in the production of watches because the production of one watch takes only 50 hours per watch in the U.S. compared with 60 hours per watch in Switzerland.

The U.S. also has an absolute advantage in the production of automobiles since the U.S. spends only 5 hours per auto produced compared with 20 hours per auto in Switzerland. b) In the U.S. the opportunity cost of producing one watch is 10 autos. In Switzerland the opportunity cost of producing one watch is 3 autos. Because the opportunity cost is lower for Switzerland than the U.S., Switzerland has a comparative advantage in the production of watches.

In the U.S. the opportunity cost of producing one auto is 1/10 of a watch. In Switzerland, the opportunity cost of producing one auto is 1/3 of a watch. Because the opportunity cost is lower for the U.S., the U.S. has a comparative advantage in the production of autos.

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Chapter 16-25

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

16.23. Brazil and China can produce cotton and soybeans. The labor required to produce a unit of each product is shown in the following table:

a) Which country has an absolute advantage in the production of cotton? In the production of soybeans? b) Which country has a comparative advantage in the production of cotton? In the production of soybeans? a) From the information given in the table, Brazil has an absolute advantage in the production of cotton because it takes only 10 hours of labor per unit of cotton in Brazil compared with 20 hours per unit cotton in China.

Brazil also has an absolute advantage in the production of soybeans since it spends only 80 hours of labor per unit of soybeans compared with 100 hours per unit of soybeans in China. b) In Brazil the opportunity cost of producing one unit of cotton is 8 soybeans. In China the opportunity cost of producing one unit of cotton is 5 soybeans. Because the opportunity cost is lower for China than Brazil, China has a comparative advantage in the production of cotton.

In Brazil the opportunity cost of producing one unit soybeans is 1/3 of a unit of cotton. In China, the opportunity cost of producing one unit of soybeans is 1/5 of a unit of cotton. Because the opportunity cost is lower for Brazil than China, Brazil has a comparative advantage in the production of soybeans.

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Chapter 16-26

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Chapter 17 Externalities and Public Goods Solutions to Review Questions 1. What is the difference between a positive externality and a negative externality?

Describe an example of each.

With a negative externality, the marginal private cost of a good is less than the marginal social cost of a good. For example, the private costs associated with driving to work on the highway include personal time, gasoline, wear and tear on the vehicle, etc. In addition, by entering the highway, a vehicle creates congestion that increases the time it takes all other drivers to get where they are going. Thus, the social cost exceeds the private cost and entering the highway creates a negative externality.

With a positive externality, the marginal private benefit is less than the marginal social benefit.

For example, when parents immunize a child they reduce the risks of the child contracting a disease. In addition, by immunizing the child, the child is less likely to pass on certain diseases to other people. Thus, the social benefits from immunizing exceed the private benefits and immunization creates a positive externality.

2. Why does an otherwise competitive market with a negative externality produce more output than would be economically efficient?

A competitive market with a negative externality produces more output than is socially optimal.

This occurs because the firms in the industry do not take into account the external costs associated with production; they only take into account their private costs. Because they view the cost as lower than it actually is, they produce more than would be produced if they were forced to take into account the external costs.

3. Why does an otherwise competitive market with a positive externality produce less output than would be economically efficient?

A competitive market with a positive externality produces less output than is socially optimal.

This occurs because consumers do not take into account the external benefits associated with consumption; they only take into account their private benefits. Because they view the benefit as lower than it actually is, they consume less than they would if they were forced to take into account the external benefits.

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Chapter 17-1

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

4. When do externalities require government intervention, and when is such intervention unlikely to be necessary?

Negative externalities may require government intervention when there is significant disparity between the socially optimal production level of a good and the unregulated equilibrium production level. To limit production, the government might impose taxes on production or a production quota limiting production. If property rights are clearly defined and bargaining is costless, the market may reach the socially efficient level of production without government intervention.

Positive externalities may also require government intervention when there is significant disparity between the socially optimal production level of a good and the unregulated equilibrium production level. To encourage production, the government might provide production subsidies. If property rights are clearly defined and bargaining is costless, the market may reach the socially optimal level of production without government intervention.

5. How might an emissions fee lead to an efficient level of output in a market with a negative externality?

With a negative externality, an emissions fee might lead to an efficient level of output. By imposing a fee on production, producers are forced to take into account not only their private costs but also the external costs (as measured by the emissions fee) of production. This has the effect of raising the firm’s costs and reducing the firm’s production. If the level of the emissions fee is set so that, for the last unit produced, the fee equals the external cost, this fee could lead to an efficient level of output.

6. How might an emissions standard lead to an efficient level of output in a market with a negative externality?

An emissions standard could lead to an efficient level of output. By setting a standard and only selling the rights to a limited amount of emissions, the government can reduce the level of emissions. In addition, by implementing a system whereby the rights can be traded, the government could reduce emissions and distribute the rights so that abatement costs are as low as possible.

7. What is the Coase Theorem, and when is it likely to be helpful in leading a market with externalities to provide the socially efficient level of output?

The Coase Theorem states that, regardless of how property rights are assigned with an externality, the allocation of resources will be efficient when the parties can costlessly bargain with each other. This Theorem will be helpful in leading a market with externalities to the socially efficient level when the cost of bargaining is low and when all parties involved can agree on the costs and benefits associated with the externality.

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Chapter 17-2

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

8. How does a nonrival good differ from a nonexclusive good?

A nonexclusive good is one that no consumer can be prevented from consuming. A nonrival good is one that a consumer’s consumption does not eliminate or prevent another consumer’s consumption.

9. What is a public good? How can one determine the optimal level of provision of a public good?

A public good is any good that is nonexclusive and nonrival. To determine the optimal level of provision of a public good, one should determine the marginal social benefits from the public good, which is equal to the sum of the marginal private benefits for the individual consumers, and equate that to the marginal cost of providing the public good. Units of the good should be provided as long as the marginal social benefit exceeds the marginal cost. This will occur up to the point where the marginal social benefit equals the marginal cost.

10. Why does the free-rider problem make it difficult or impossible for markets to provide public goods efficiently?

It is difficult to provide public goods efficiently when free riders exist. Free riders will consume the good, but will pay nothing for the good, anticipating that others will pay. It may therefore prove difficult to raise funds to finance a project with a public good, leading to an underproduction of the good, or possibly even no provision of a good with positive net benefits.

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Chapter 17-3

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Solutions to Problems 17.1. Why is it not generally socially efficient to set an emissions standard allowing zero pollution?

If the government were to set an emissions standard requiring zero pollution, this standard would probably not be socially efficient. By setting the standard at zero, the government could reduce pollution by preventing polluting industries from producing goods that society values. By setting the standard at zero, however, the government will also eliminate the benefits to society from production of these goods. In general, the social benefits from producing will likely exceed the social costs up to some non-zero level of production (pollution) implying the socially efficient level of production is non-zero.

17.2. Education is often described as a good with positive externalities. Explain how education might generate positive external benefits. Also suggest a possible action the government might take to induce the market for education to perform more efficiently.

Education is a good that might generate positive external benefits. For example, when an individual furthers her education she benefits directly in terms of higher income. In addition, this individual, because of her increased education, might be able to develop a new technology that benefits all of society. Thus, while the education helped the individual, by allowing the development of the new technology (because she’s smarter!) many people benefited from her education.

To induce the market to perform more efficiently, the government would like to entice more individuals to further their education since education generates positive externalities. The government could do this by providing grants or low interest student loans, for example.

17.3. a) Explain why cigarette smoking is often described as a good with negative externalities. b) Why might a tax on cigarettes induce the market for cigarettes to perform more efficiently? c) How would you evaluate a proposal to ban cigarette smoking? Would a ban on smoking necessarily be economically efficient? a) For one, by smoking in public, smokers force other individuals to breathe air with smoke, known as second-hand smoke. In addition, the health problems associated with smoking force society to pay higher health care costs to pay for smoking related illnesses, both for smokers and for those who breathe second-hand smoke, than if no one in society smoked.

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Chapter 17-4

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) By imposing a tax on cigarettes the government increases the marginal private cost of smoking and forces the individual to take into account (at least some of) the negative externality associated with smoking. This would likely reduce the level of smoking in society, pushing the equilibrium toward the socially efficient level of smoking. c) A ban on smoking entirely is probably not socially efficient. To evaluate such a ban, one would need to compare the marginal benefits with the marginal social costs. The ban would only be socially efficient if the marginal social costs exceed the marginal benefits at a level of zero.

This would not necessarily be socially efficient because it is possible that the marginal benefit of smoking exceeds the marginal social cost for low levels of smoking.

17.4. Consider Learning-By-Doing Exercise 17.2, with a socially efficient emissions fee.

Suppose a technological improvement shifts the marginal private cost curve down by $1. If the government calculates the optimal fee given the new marginal private cost curve, what will happen to the following? a) The size of the optimal tax b) The price consumers pay c) The price producers receive a) If the marginal private cost shifts down by $1, we have 𝑀𝑃𝐶 = 1 + 𝑄. With demand 𝑃𝑑 = 24 − 𝑄 and marginal external cost 𝑀𝐸𝐶 = −2 + 𝑄, the social optimum occurs where P d = MPC + MEC 24 − Q = (1 + Q ) + ( −2 + Q ) Q = 8.33

At 𝑄 = 8.33, the price is 𝑃 = 15.67.

The size of the optimal tax is the difference between the equilibrium price, 15.67, and the 𝑀𝑃𝐶 at the socially efficient quantity of 8.33. At this quantity the marginal private cost is 𝑀𝑃𝐶 = 9.33. Thus, the optimal tax is 𝑇 = 15.67 − 9.33 = 6.34. b) With this tax, consumers will pay the socially efficient price of 15.67. c) Producers will receive the difference between the price consumers pay, 15.67, and the tax, 6.34. Thus, producers will receive 9.33.

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Chapter 17-5

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.5. Consider the congestion pricing problem illustrated in Figure 17.5. a) What is the size of the deadweight loss from the negative externalities if there is no toll imposed during the peak period? b) Why is the optimal toll during the peak period not $3, the difference between the marginal social cost and the marginal private cost when the traffic volume is Q5? c) How much revenue will the toll authority collect per hour if it charges the economically efficient toll during the peak period? a) In Figure 17.5, the deadweight loss is area ABG. This is deadweight loss because for every vehicle beyond the optimum, Q4, the marginal social cost exceeds the marginal benefit. Area ABG is approximately (assuming the demand and 𝑀𝑃𝐶 curves are nearly straight lines over this part of the graph ) 0.5(𝑄5 − 𝑄4 )(8 − 5) = 1.5(𝑄5 − 𝑄4 ). b) The socially efficient traffic volume occurs where the marginal social cost curve intersects the marginal benefit curve. In Figure 17.5 this occurs at Q4. At Q4, the marginal benefit is $5.75 and the marginal private cost is $4.00. To achieve the social optimum the toll should be set so that the marginal benefit equals the marginal private cost plus the toll, effectively forcing the driver to take into account the external cost of entering the highway. At Q4, this is $5.75 − $4.00 = $1.75.

The toll is not $3.00 because the toll should be set to force the driver at Q4 to observe the external costs imposed by entering the highway. By setting the toll at $3.00, the difference between the MPC and MB at Q5, the toll would be set to force the driver at Q4 to observe the external costs imposed from the driver at Q5 entering the highway. But this cost is unimportant because at the optimum the driver at Q5 will not be on the highway. The $3.00 toll would create a level of traffic below the social optimum. c) If the toll authority sets a toll at the economically efficient level of $1.75, it will earn revenue equal to the toll multiplied by the number of drivers. In this case, revenue will be $1.75Q4.

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Chapter 17-6

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.6. The accompanying graph (on next page) shows the demand curve for gasoline and the supply curve for gasoline. The use of gasoline creates negative externalities, including CO2, which is an important source of global warming. Using the graph and the table below, identify: • The equilibrium price and quantity of gasoline • The producer and consumer surplus at the market equilibrium • The cost of the externality at the free-market equilibrium • The net social benefits arising at the free-market equilibrium • The socially optimal price of gasoline • The consumer and producer surplus at the social optimum • The cost of the externality at the social optimum • The net social benefits arising at the social optimum • The deadweight loss due to the externality

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Chapter 17-7

Besanko & Braeutigam – Microeconomics, 6th edition

Consumer surplus Private producer surplus - Cost of externality Net social benefits Deadweight loss

Solutions Manual

Equilibrium price and quantity = P2 and Q2

Social optimum price and quantity = P1 and Q1

A+B+G+K E+F+R+H+N -R-H-N-G-K-M A+B+E+F-M M

A B+E+F+R+H+G -R-H-G A+B+E+F Zero

Difference between social optimum and equilibrium -B-G-K B+G-N M+N+K M M

17.7. The graph below shows conditions in a perfectly competitive market in which there is some sort of externality. In this market, a consumer purchases at most one unit of the good.

There are many such consumers, and they have different maximum willingnesses to pay.

Assume that the graph is drawn to scale. a) What type of externality is present in this market: positive or negative? b) What is the maximum level of social surplus that is potentially attainable in this market? c) What is the deadweight loss that arises in a competitive equilibrium in this market? d) Suppose a subsidy is given to producers: What is the magnitude of the subsidy per unit that would enable this market to attain the socially efficient outcome?

For the remaining questions, please indicate whether the following government interventions would increase social efficiency relative to the competitive equilibrium outcome with no government intervention, decrease social efficiency, or keep it unchanged: e) A subsidy per unit equal to 0F given to consumers who purchase the good. f ) The government replaces private sellers and offers the good at a price of zero. (Assume that government has no inherent cost advantage or disadvantage relative to private producers. Assume, too, the government’s cost of production is financed by levying taxes.) g) The government imposes a price ceiling that sets a maximum price for the good equal to 0D. h) The government imposes a tax equal to NR on consumers who do not purchase the good.

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Chapter 17-8

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) The externality is positive. We can see this because the marginal social cost at any quantity is less than the marginal private cost. b) The maximum level of social surplus that is potentially attainable in this market is AR0. The socially efficient outcome is at the intersection of marginal social cost and marginal social benefit (point R), and the maximum level of social surplus is the area between the SMC and SMB curves to the left of this point. c) The deadweight loss that arises in a competitive equilibrium in this market is KLR. The equilibrium point is K. The deadweight loss is the area between MSB and MSC, from the equilibrium quantity 0M to the efficient quantity 0S. d) The subsidy to producers that would enable this market to attain the socially efficient outcome is 0F (or equivalently, IH, LK, RN, VU, EC, DB). e) A subsidy equal to 0F given to consumers who purchase the good increases social efficiency.

In fact, the efficient outcome is achieved. It does not matter that the subsidy is given to consumers, not producers. A subsidy to consumers shifts the demand curve upward by 0F so that it intersects the supply curve at N, which is the new equilibrium point. The equilibrium quantity is the efficient quantity 0S. f) If the government offers the good at a price of zero then social efficiency decreases. If the good is provided at a price of zero, consumers will purchase quantity 0W. The deadweight loss is RVW, which is the area between SMC and SMB over the range between the socially efficient quantity 0S and 0W. This deadweight loss is larger than the deadweight loss KLR with no government intervention, so efficiency is reduced.

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Chapter 17-9

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

g) A price ceiling that sets a maximum price for the good equal to 0D decreases social efficiency.

At this price, the consumers may wish to purchase the efficient quantity 0S, but the producers are only willing to supply quantity 0J, and the consumers are unable to buy more than that. The deadweight loss increases to (at least) IRG. h) If the government imposes a tax equal to NR on consumers who do not purchase the good, then social efficiency increases. This is a monetary incentive to purchase the good. By purchasing a unit of the good a consumer gets two things --- the value from the good and avoidance of the tax. Thus, a tax on consumers who do not purchase the good makes it “as if” the demand curve is NR above the actual demand curve. The result is an equilibrium in which the quantity is the socially efficient quantity 0S. Thus, this incentive is equally strong as the subsidy from part e), and it has the same effect.

17.8. A competitive refining industry produces one unit of waste for each unit of refined product. The industry disposes of the waste by releasing it into the atmosphere. The inverse demand curve for the refined product (which is also the marginal benefit curve) is Pd = 24 Q, where Q is the quantity consumed when the price consumers pay is Pd. The inverse supply curve (also the marginal private cost curve) for refining is MPC = 2 + Q, where MPC is the marginal private cost when the industry produces Q units. The marginal external cost curve is MEC = 0.5Q, where MEC is the marginal external cost when the industry releases Q units of waste. a) What are the equilibrium price and quantity for the refined product when there is no correction for the externality? b) How much of the chemical should the market supply at the social optimum? c) How large is the deadweight loss from the externality? d) Suppose the government imposes an emissions fee of $T per unit of emissions. How large should the emissions fee be if the market is to produce the economically efficient amount of the refined product? a) If there is no correction for the externality, the equilibrium will occur at the point where the marginal benefit curve, 𝑃𝑑 = 24 − 𝑄, intersects the marginal private cost curve, 𝑀𝑃𝐶 = 2 + 𝑄. This occurs at 24 − Q = 2 + Q Q = 11

At 𝑄 = 11, price is 𝑃 = 13.

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Chapter 17-10

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

b) At the social optimum marginal benefit, 𝑃𝑑 = 24 − 𝑄, will equal marginal social cost, 𝑀𝑆𝐶 = 𝑀𝑃𝐶 + 𝑀𝐸𝐶. This occurs where 24 − Q = (2 + Q) + 0.5Q Q = 8.80

Thus, the social optimum is to produce 𝑄 = 8.80. c) At the uncorrected equilibrium, the marginal social cost is 𝑀𝑆𝐶 = 2 + 1.5(11) = 18.5.

Thus, the deadweight loss will be 0.5(11 − 8.80)(18.5 − 13) = 6.05. d) The emissions fee of $𝑇 should be set to shift the 𝑀𝑃𝐶 curve so that it intersects the marginal benefit curve at 𝑄 = 8.80, the socially optimal quantity. At 𝑄 = 8.80 the marginal benefit is 𝑃 = 15.2 and the marginal private cost is 𝑀𝑃𝐶 = 2 + 8.80 = 10.80. Therefore, the optimal tax is 𝑇 = 15.2 − 10.8 = 4.4.

17.9. Consider a manufactured good whose production process generates pollution. The annual demand for the good is given by Qd = 100 - 3P. The annual market supply is given

by Qs = P. In both equations, P is the price in dollars per unit. For every unit of output produced, the industry emits one unit of pollution. The marginal damage from each unit of pollution is given by 2Q. a) Find the equilibrium price and quantity in a market with no government intervention. b) At the equilibrium you computed, calculate: (i) consumer surplus; (ii) producer surplus; (iii) total dollars of pollution damage. What are the overall social benefits in the market? c) Find the socially optimal quantity of the good. What is the socially optimal market price? d) At the social optimum you computed, calculate: (i) consumer surplus; (ii) producer surplus; and (iii) total dollars of pollution damage. What are the overall social benefits in the market? e) Suppose an emissions fee is imposed on producers. What emissions fee would induce the socially optimal quantity of the good? a)

100 – 3P = P  P = 25 and Q = 25.

b)

See below

c) MEC = 2Q, while MPC = Q. Thus, MSC = 3Q. To find the optimal quantity, we equate MSC to inverse demand, or 3Q = 100/3 – Q/3, or Q = 10. The socially optimal price would equal the marginal social cost at the optimal quantity, or P = 3(10) = $30. d)

See below

e) The optimal emissions fee is equal to the difference between MSC and MPC at the socially optimal quantity. Since MSC = 3Q and MPC = Q, and Q = 10, the optimal emissions fee equals: 3(10) – 1(10) = $20 per unit.

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Chapter 17-11

Besanko & Braeutigam – Microeconomics, 6th edition

Price 75

Solutions Manual

MSC = 3Q

MPC = Q M

100/3 30

A B

25

G

K N

E

H

F

P = 100/3 – Q/3

R

Quantity 10

Consumer surplus Private producer surplus - Cost of externality Net social benefits Deadweight loss

25

Equilibrium price Social optimum Difference and quantity = P2 and price and quantity = between social Q2 P1 and Q1 optimum and equilibrium A+B+G+K = A= -B-G-K = $104.167 $16.67 -$87.497 E+F+R+H+N = B+E+F+R+H+G = B+G-N= $312.50 $250 -62.5 -R-H-N-G-K-M = -R-H-G = -$100 M+N+K= $625 $525 A+B+E+F-M = A+B+E+F= M= $208.33 $166.67 $375 M Zero M

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Chapter 17-12

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.10. The demand for widgets is given by P = 60 - Q. Widgets are competitively supplied according to the inverse supply curve (and marginal private cost) MPC = c. However, the production of widgets releases a toxic gas into the atmosphere, creating a marginal external cost of MEC = Q. a) Suppose the government is considering imposing a tax of $T per unit. Find the level of the tax, T, that ensures the socially optimal amount of widgets will be produced in a competitive equilibrium. b) Suppose a breakthrough in widget technology lowers the marginal private cost, c, by $1.

How will this affect the optimal tax you found in part (a)? a) The socially optimal level of output occurs when P = MPC + MEC, or 60 – Q = c + Q, which implies Q = 30 – 0.5c. At this output level, P = 30 + 0.5c. The optimal tax is the difference between price and MPC at this output level: T = 30 + 0.5c – c = 30 – 0.5c. b) If marginal private cost falls to MPC = c – 1, then the optimal tax becomes T = 30 – 0.5(c – 1) = 30.5 – 0.5c. That is, the optimal tax rises by $0.50. You can also see this since ΔT/Δc = –0.5 in part (a).

17.11. The market demand for gadgets is given by Pd = 120 - Q, where Q is the quantity consumers demand when the price that consumers pay is Pd. Gadgets are competitively supplied according to the inverse supply curve (and marginal private cost) MPC = 2Q, where Q is the amount suppliers will produce when they receive a price equal to MPC. The production of gadgets releases a toxic effluent into the water supply, creating a marginal external cost of MEC = Q. The government wants to impose a sales tax on gadgets to correct for the externality. When producers receive a price equal to MPC, the amount consumers must pay is (1 + t)MPC, where t is the sales tax rate. Find the level of the tax rate that ensures the socially optimal amount of gadgets will be produced in a competitive equilibrium.

The socially optimal output level occurs where P = MPC + MEC, or 120 – Q* = 2Q* + Q*, implying Q = 30. With the sales tax, the equilibrium output level occurs where Pd = (1 + t)MPC, or 120 – Q = (1 + t)(2Q). As a function of t, the equilibrium output level is then Q = 120/(3 + 2t).

The optimal sales tax level ensures that Q = Q*, or 120/(3 + 2t) = 30. Solving, we have t = 0.50.

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Chapter 17-13

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.12. Amityville has a competitive chocolate industry with the (inverse) supply curve Ps = 440 + Q. While the market demand for chocolate is Pd = 1200 - Q, there are external benefits that the citizens of Amityville derive from having a chocolate odor wafting through town. The marginal external benefit schedule is MEB = 6 - 0.05Q. a) Without government intervention, what would be the equilibrium amount of chocolate produced? What is the socially optimal amount of chocolate production? b) If the government of Amityville used a subsidy of $S per unit to encourage the optimal amount of chocolate production, what level should that subsidy be? a) The equilibrium level of output occurs where Pd = Ps, or 1200 – Q = 440 + Q. Equilibrium output is then Q = 380. Taking into account the positive externality, the social optimal amount of production sets Pd + MEB = Ps, or 1200 – Q* + 60 – 0.05Q* = Q* + 440, yielding Q* = 400. b) With a subsidy of $S, equilibrium occurs where Pd + S = Ps or 1200 – Q + S = 440 + Q. To get Q = Q* = 400 the subsidy must satisfy 1200 – 400 + S = 440 + 400 or S = 40.

17.13. The only road connecting two populated islands is currently a freeway. During rush hour, there is congestion because of the heavy traffic. The marginal external cost from congestion rises as the amount of traffic on the road increases. At the current equilibrium, the marginal external cost from congestion is $5 per vehicle. Would a toll charge of $5 per vehicle lead to an economically efficient amount of traffic? If not, would you expect the economically efficient toll to be larger than, or less than $5?

The graph below demonstrates that the optimal toll would be less than $5. Equilibrium with no toll occurs at Q3, although the socially optimal amount of driving would occur at Q2 < Q3.

Although the marginal external cost is $5 at Q3, the socially optimal toll occurs at MEC(Q2), which is less than $5 since MEC is upward sloping.

As an alternative interpretation, you can notice that imposing a $5 toll would cause equilibrium to occur at Q1, where demand intersects the curve MPC + $5. By definition, at Q3 it must be true that MSC = MPC + MEC = MPC + $5. Thus the curve MPC + $5 intersects the MSC curve at Q3. Since MEC is upward sloping, MSC < MPC + $5 at output levels less than Q3 so the MPC + $5 curve must intersect demand to the left of Q2.

P

Demand MSC MPC + $5 MPC

$5

MEC Q1 Q2 Q3

Q

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Chapter 17-14

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.14. A firm can produce steel with or without a filter on its smokestack. If it produces without a filter, the external costs on the community are $500,000 per year. If it produces with a filter, there are no external costs on the community, and the firm will incur an annual fixed cost of $300,000 for the filter. a) Use the Coase Theorem to explain how costless bargaining will lead to a socially efficient outcome, regardless of whether the property rights are owned by the community or the producer. b) How would your answer to part (a) change if the extra yearly fixed cost of the filter were $600,000? a) If the firm installs the filter, the community benefits by $500,000 while the firm incurs a cost of $300,000. The socially efficient outcome is for the firm to install the filter.

If the firm possesses the right to pollute, the community will have an incentive to pay the firm some price above $300,000 (perhaps $499,999) to induce the firm to install the filter.

Afterwards, the community would be $1 better off and the firm would be $199,999 better off.

On the other hand, suppose the community possesses the right to prevent the firm from polluting.

The firm then has two choices: it could install the filter at a cost of $300,000, or it could compensate the community for the costs of its pollution by paying them $500,001. Since it’s clearly cheaper to install the filter, the firm will have a strong incentive to choose this socially efficient outcome. b) If the filter costs $600,000, then the socially efficient outcome is for the firm to not install the filter, since the costs exceed the benefits.

If the firm possesses the right to pollute, it will do so. Although the community is willing to pay the firm up to $500,000 to stop polluting, that does not exceed the firm’s costs of installing the filter.

If the community possesses the right to prevent the firm from polluting, then the firm will have a strong incentive to compensate the community for the costs of polluting rather than installing the filter. The community could demand that the firm either install the filter or pay it some price less than $600,000 (say $599,999). Then the firm would be better off (by $1) not installing the filter, while the community also benefits (by $99,999) since the payment exceeds the costs of living with the pollution.

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Chapter 17-15

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.15. Two farms are located next to each other. During storms, sewage from Farm 1 flows into a stream located on Farm 2. Farm 2 relies on this stream as a source of drinking water for its livestock, and when the stream is polluted with sewage, the livestock become sick and die. The annual damage to Farm 2 from this form of pollution is $100,000 per year. It is possible that Farm 1 can prevent the runoff of sewage by installing storm drains. The cost of the storm drains is $200,000. a) Provide an argument that the Coase Theorem holds in this situation. b) Suppose that the damage to Farm 2 is $500,000 per year, not $100,000 per year (with the cost of storm drains remaining fixed at $200,000). Provide an argument that the Coase Theorem holds in this case. a) Suppose the property rights are assigned to Farmer 1. Farmer 2 can either pay for storm drains at Farm 1 for $200,000 or live with the damage of $100,000. Farmer 2 will not find it worthwhile to pay for the storm drains, and the run-off from Farm 1 will continue.

Suppose the property rights are assigned to Farmer 2. Farmer 1 can either spend $200,000 to prevent the run-off, or can pay $100,000 in compensation for the pollution damage. Farmer 1 will find it worthwhile to pay for the damage, and the run-off will continue.

With either property rights assignment, the outcome is the same: the run-off will continue. It is not economically efficient to build a storm drain because the storm drain costs more than the damage due to the run-off. b) Suppose the property rights are assigned to Farmer 1. Farmer 2 can either pay for storm drains at Farm 1 for $200,000 or live with the damage of $500,000. Farmer 2 will find it worthwhile to pay for the storm drains, and the run-off from Farm 1 will be prevented.

Suppose the property rights are assigned to Farmer 2. Farmer 1 can either spend $200,000 to prevent the run-off, or can pay $500,000 in compensation for the pollution damage. Farmer 1 will find it worthwhile to pay for the storm drain, and the run-off will be prevented.

With either property rights assignment, the outcome is the same: the storm drain will be built, and the run-off will be abated. It is economically efficient to build a storm drain because the storm drain costs less than the damage due to the run-off.

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Chapter 17-16

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.16. Suppose a factory located next to a river discharges pollution that causes $2 million worth of environmental damage to the residents downstream. The factory could completely eliminate the pollution by treating the water on location at a cost of $1.6 million.

Alternatively, the residents could construct a water purification plant just upstream of their town, at a cost of $0.8 million, which would not completely eliminate the environmental damage to them but reduce it to $0.5 million. Under current law, the factory must compensate the town for any environmental damage the factory causes. Bargaining between the factory owner and the town is costless. What would the Coase Theorem imply about the outcome of bargaining between the town and the factory owner?

There are three scenarios that could emerge: (a) status quo in which pollution damage is $2 million; (b) full elimination of pollution at a cost of $1.6 million; (c) construction of a water purification plant near the town for $0.8, which reduces pollution damage to $0.5 million. The Coase theorem predicts that bargaining would result in the option with the lowest total cost -damage cost plus abatement cost -- which is (c), and that furthermore, this option would emerge irrespective of which party has the property rights.

In this case, because the town has the property rights, it could indeed force the factory to either pay $2 million in damages or to completely eliminate the pollution at a cost of $1.6 million. This suggests that the firm would be forced to eliminate the pollution for $1.6 million. But if bargaining is costless, we would imagine that the factory would approach the town and say: “Look, we have a better idea than this expensive solution. We will build you a purification plant near your town for $0.8 million. We know that this will not fully clean up the damage, and so we will compensate you $0.5 million for the remaining damage. We have reduced the damage by $1.5 million and compensated you for the remaining damage of $0.5 million. You are better off, and by the way, we are better off, too, because this solution only costs us $1.3 million, rather than $1.6 million. This deal is win-win for both us.”

17.17. The demand for energy-efficient appliances is given by P = 100/Q, while the inverse supply (and marginal private cost) curve is MPC = Q. By reducing demand on the electricity network, energy-efficient appliances generate an external marginal benefit according to MEB = eQ. a) What is the equilibrium amount of energy-efficient appliances traded in the private market? b) If the socially efficient number of energy-efficient appliances is Q = 20, what is the value of e? c) If the government subsidized production of energy efficient appliances by $S per unit, what level of the subsidy would induce the socially efficient level of production?

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Chapter 17-17

Besanko & Braeutigam – Microeconomics, 6th edition

a)

Solutions Manual

Equilibrium occurs where P = MPC, or 100/Q = Q. Thus Q = 10.

b) If the socially efficient number of appliances is Q = 20, then P + MEB = MPC at Q = 20, or 100/20 + e*20 = 20. Solving, we get e = 0.75. c) With the subsidy, equilibrium occurs where P + S = MPC, or 100/Q + S = Q. Since the efficient amount of appliances is Q = 20, the proper subsidy would solve 100/20 + S = 20 or S = 15.

17.18. The demand for air-polluting backhoes in Peoria is PD = 48 - Q. The air pollution creates a marginal external cost according to MEC = 2 + Q. Supply of backhoes is given by PS = 10 + cQ. If the socially efficient level of backhoes is Q* = 12, find the tax that induces the socially efficient level of backhoes in equilibrium and the value of c.

The socially efficient quantity Q* solves PD = PS + MEC while the optimal tax T solves PD = PS + T. Thus, we know that the optimal tax is equal to the marginal external cost at Q* = 12, or T = 2 + Q*. Thus T = 14.

The value of c can be determined since we know that at Q* = 12, PD = PS + MEC or 48 – 12 = 10 + c*12 + 2 + 12. Solving, we get c = 1.

17.19. The town of Steeleville has three steel factories, each of which produces air pollution.

There are 10 citizens of Steeleville, each of whose marginal benefits from reducing air pollution is represented by the curve p(Q) = 5 - Q/10, where Q is the number of units of pollutants removed from the air. The reduction of pollution is a public good. For each of the three sources of air pollution, the following table lists the current amount of pollution being produced along with the constant marginal cost of reducing it.

a) On a graph, illustrate marginal benefits (“demand”) and the marginal costs (“supply”) of reducing pollution. What is the efficient amount of pollution reduction? Which factories should be the ones to reduce pollution, and what would the total costs of pollution reduction be? In a private market, would any units of this public good be provided? b) The Steeleville City Council is currently considering the following policies for reducing pollution: i. Requiring each factory to reduce pollution by 10 units ii. Requiring each factory to produce only 30 units of pollution iii. Requiring each factory to reduce pollution by one fourth

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Chapter 17-18

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

Calculate the total costs of pollution reduction associated with each policy. Compare the total costs and amount of pollution reduction to the efficient amount you found in part (a).

Do any of these policies create a deadweight loss? c) Another policy option would create pollution permits, to be allocated and, if desired, traded among the firms. If each factory is allocated tradeable permits allowing it to produce 30 units of pollution, which factories, if any, would trade them? (Assume zero transactions costs.) If they do trade, at what prices would the permits be traded? d) How does your answer in part (c) relate to that in part (a)? Explain how the Coase Theorem factors into this relationship. a) The market demand curve is created by adding the individual demand curves vertically: P(Q) = 10*p(Q) = 50 – Q. Demand intersects supply at a price of $20, associated with Q = 30 units of pollution reduction. This entails factory A reducing pollution by 20 and factory B reducing pollution by 10.

The total cost of pollution reduction would be 20*$10 + 10*$20 = $400. In a private market, each consumer would pay no more than $5 for the first unit of pollution reduction while no factory would reduce pollution for less than $10. Thus, no amount of this public good would be provided in a private market. b)

Under (i), total costs are 10*$10 + 10*$20 + 10*30 = $600.

Under (ii), A does not have to reduce pollution at all. B must reduce pollution by 10 units, and C must reduce pollution by 30 units. Thus total costs under this policy are 10*$20 + 30*$30 = $1100.

Under (iii), A reduces pollution by 20/4 = 5 units, B by 40/4 = 10 units, and C by 60/4 = 15 units. Total costs of pollution reduction under this policy are thus 5*$10 + 10*$20 + 15*$30 = $700.

While each policy achieves the efficient amount of pollution reduction (Q = 30), it does so in a way that is more costly than the efficient allocation in part (a). In particular, the policies lead to deadweight losses of $200, $700, and $300 respectively. c) With the permits, C must still reduce pollution by 30 units and B by 10 units, while A is only creating 20 units of pollution in the first place and so does not need to reduce pollution at all. However, Factory A could eliminate all its pollution (at a marginal cost of $10 per unit) and sell its permits to Factory C at a price of up to $30 per unit. Thus, Factory C would buy all of Factory A’s permits, at a price between $20 and $30 per unit (note that Factory B would purchase some of them if the price were less than $20). d) By defining property rights over pollution and allowing them to be traded, the socially efficient outcome (where A reduces pollution by 20 units and B reduces its pollution by 10 units) is achieved, just as the Coase Theorem predicts.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-19

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.20. A chemical producer dumps toxic waste into a river. The waste reduces the population of fish, reducing profits for the local fishing industry by $100,000 per year. The firm could eliminate the waste at a cost of $60,000 per year. The local fishing industry consists of many small firms. a) Using the Coase Theorem, explain how costless bargaining will lead to a socially efficient outcome, regardless of whether the property rights are owned by the chemical firm or the fishing industry. b) Why might bargaining not be costless? c) How would your answer to part (a) change if the waste reduces the profits for the fishing industry by $40,000? (Assume, as before, that the firm could eliminate the waste at a cost of $60,000 per year.) a) If property rights are assigned to the chemical producer, the fisherman will pay $60,000 to the firm to eliminate the toxic waste. If property rights are assigned to the fisherman, the chemical producer will clean up the waste since this is cheaper than compensating the fisherman.

Thus, regardless of whom property rights are assigned to, the toxic waste gets cleaned up because this is less costly than the damage. b) Because the waste harms many fishermen, it may not be easy to organize them to bargain about compensation. Organizing the fishermen may be costly. In addition, if the fishermen and the firm have different perceptions regarding the costs of the externality, they might not reach an efficient solution. c) If the fishermen’s profits were reduced by $40,000 rather than $100,000, then if property rights were assigned to the chemical producer, the fishermen would not find it worthwhile to pay for cleanup. The fishermen will receive no compensation. If property rights are assigned to the fishermen, the chemical producer would compensate the fishermen $40,000 rather than paying the cleanup costs. Thus, regardless of whom property rights are assigned to, the waste will not get cleaned up. This is economically efficient because cleanup costs more than the damage.

17.21. Consider an economy with two individuals. Individual 1 has (inverse) demand curve for a public good given by P1 = 60 - 2Q1, while Individual 2 has (inverse) demand curve for the public good given by P2 = 90 - 5Q2. The prices are measured in $ per unit. Suppose the marginal cost of producing the public good is $10 per unit. What is the efficient level of the public good?

The marginal social benefit curve is the vertical sum of the individual consumer’s inverse demand curves. When we sum vertically we add prices (i.e., willingness to pay.) Thus, letting Q denote the quantity of the public good, we have: MSB = (60 – 2Q) + (90 – 5Q) = 150 – 7Q. Equating MSB to MC we have: 150 – 7Q = 10, or Q = 20. This is the socially efficient quantity of the public good.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-20

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.22. There are three consumers of a public good. The demands for the consumers are as follows: Consumer 1: P1 = 60 - Q Consumer 2: P2 = 100 - Q Consumer 3: P3 = 140 - Q where Q measures the number of units of the good and P is the price in dollars.

The marginal cost of the public good is $180. What is the economically efficient level of production of the good? Illustrate your answer on a clearly labeled graph.

The economically efficient level of output occurs where 𝑀𝑆𝐵 = 𝑀𝐶. Since this occurs where all three consumers are in the market we have (60 − Q) + (100 − Q) + (140 − Q) = 180 3Q = 120 Q = 40

17.23. Suppose that the good described in Problem 17.22 is not provided at all because of the free rider problem. What is the size of the deadweight loss arising from this market failure?

If the good is not provided at all, the deadweight loss would be the area under the demand curve (the 𝑀𝑆𝐵 curve) and above the marginal cost curve, or 0.5(300 – 180)(40) = 2400. This is a deadweight loss because it measures the potential net economic benefits that would disappear if the good were not offered.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-21

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

17.24. In Problem 17.22, how would your answer change if the marginal cost of the public good is $60? What if the marginal cost is $350?

When marginal cost is 𝑀𝐶 = 60, the 𝑀𝑆𝐵 and 𝑀𝐶 curves intersect at a level of output at which only Consumers 2 and 3 have a positive willingness to pay. This implies the efficient level of production will occur where (100 − Q) + (140 − Q) = 60 2Q = 180 Q = 90

If the marginal cost was $350, the marginal cost would exceed the marginal social benefits at all levels of output. Therefore, at a marginal cost of $350, the economically efficient level of output would be zero.

17.25. A small town in Florida is considering hiring an orchestra to play in the park during the year. The music from the orchestra is nonrival and nonexclusive. A careful study of the town’s music tastes reveals two types of individuals: music lovers and intense music lovers.

If forced to pay for an outdoor concert, the demand curve for music lovers would be Q1 = 100 - (1/20)P1, where Q1 is the number of concerts that would be attended and P1 is the price per (hypothetical) ticket (in dollars) to the concert. The demand curve for intense music lovers would be Q2 = 200 - (1/10)P2. Assuming the marginal cost of a concert is $2800, what is the efficient number of concerts to offer each year?

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-22

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

To find the equation of the MSB curve, we need to add the inverse demand curves. Thus, as the first step, we determine the inverse demand curves of each type of individual: Music lovers: Q1 = 100 – (1/20)P1  P1 = 2000 – 20Q1.

Intense music lovers: Q2 = 200 – (1/10)P2  P2 = 2000 – 10Q2.

Letting Q denote the (common) quantity of orchestra performances, the MSB is determined as follows: MSB = 100(2000 – 20Q) + 50(2000 – 10Q) = 300,000 – 2500Q. Since the marginal cost of a concert is $250,000, the socially efficient number of performances is: 300,000 – 2500Q = 250,000  Q = 20.

17.26. Some observers have argued that the Internet is overused in times of network congestion. a) Do you think the Internet serves as common property? Are people ever denied access to the Internet? b) Draw a graph illustrating why the amount of traffic is higher than the efficient level during a period of peak demand when there is congestion. Let your graph reflect the following characteristics of the Internet: i. At low traffic levels, there is no congestion, with marginal private cost equal to marginal external cost. ii. However, at higher usage levels, marginal external costs are positive, and the marginal external cost increases as traffic grows. c) On your graph explain how a tax might be used to improve economic efficiency in the use of the Internet during a period of congestion. d) As an alternative to a tax, one could simply deny access to additional users once the economically efficient volume of traffic is on the Internet. Why might an optimal tax be more efficient than denying access?

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-23

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) The Internet can be viewed as common property because virtually anyone has access to it. In practice, people are sometimes denied access, particularly when the congestion is great and consumers cannot connect to it. b)

The graph might be very similar to Figure 17.5.

Price MSC = MPC + MEC D2

MPC

D1 Pb

MEC

Pa

Q2

Volume of interconnections to the internet

When the demand for connections to the internet is D1, there is no congestion. However, when the demand is high at D2, congestion creates a positive marginal external cost. c) When the demand is large, a tax equal to (Pb – Pa) would lead users to demand the efficient number of connections Q2. d) A tax would ensure that users who value connections the most would be able to connect.

If access is denied to some users, some users with a higher value for an interconnection might be unable to connect, while other users with a lower value for a connection might be able to go online. This would not be economically efficient because the scarce resource (connections) would not necessarily be allocated to consumers who value connection the most.

17.27. There are two types of citizens in Pulmonia. The first type has an inelastic demand for public broadcasting at Q = 8 hours per day; however, they are willing to pay only up to $30 per hour for each hour up to Q = 8. The second type demands public broadcasting according to P = 60 - 3Q. a) Suppose the marginal cost of public broadcasting is MC = 15. What is the economically efficient level of public broadcasting? Hint: it will help if you draw a careful sketch of the demand curve of each type of citizen. b) Repeat part (a) for MC = 45.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-24

Besanko & Braeutigam – Microeconomics, 6th edition

Solutions Manual

a) Marginal social benefits are given by P = 90 – 3Q for Q < 8 and P = 60 – 3Q for Q > 8.

In the graph below, marginal social benefits are the curve ABCD. At MC = 15, the efficient amount of production is Q = 15.

P $90 $60

A B MC = 45

$30

C

8 10

b)

15

MC = 15 D 20 Q

At MC = 45, the efficient amount of production is Q = 8.

Copyright © 2020 John Wiley & Sons, Inc.

Chapter 17-25