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CHAPTER 1

Economic Models A. Summary This chapter provides an introduction to the book by showing why economists use simplified models. The chapter begins with a few definitions of economics and then turns to a discussion of such models. Development of Marshall's analysis of supply and demand is the principle example used here, and this provides a review for students of what they learned in introductory economics. The notion of how shifts in supply or demand curves affect equilibrium prices is highlighted and is repeated in the chapter’s appendix in a somewhat more formal way. The chapter also reminds students of the production possibility frontier concept and shows how it illustrates opportunity costs. The chapter concludes with a discussion of how economic models might be verified. A brief description of the distinction between positive and normative analysis is also presented.

B. Lecture and Discussion Suggestions We have found that a useful way to start the course is with one (or perhaps two) lectures on the historical development of microeconomics together with some current examples. For example, many students find economic applications to the natural world fascinating and some of the economics behind Application 1.1, might be examined. The simple model of the world oil market in Application 1A.3 is also a good way to introduce models with real world numbers in them. Application 1.6: Economic Confusion provides normative distinction and to tell a few economic jokes (several Internet sites offer such jokes if your supply is running low).

C. Glossary Entries in the Chapter • Diminishing Returns • Economics • Equilibrium Price • Microeconomics • Models • Opportunity Cost • Positive Normative Distinction • Production Possibility Frontier • Supply-Demand Model • Testing Assumptions • Testing Predictions

Chapter 1: Economic Models

APPENDIX TO CHAPTER 1

Mathematics Used in Microeconomics A. Summary This appendix provides a review of basic algebra with a specific focus on the graphical tools that students will encounter later in the text. The coverage of linear and quadratic equations here is quite standard and should be familiar to students. Two concepts that will be new to some students are graphing contour lines and simultaneous equations. The discussion of contour lines seeks to introduce students to the indifference curve concept through the contour map analogy. Although students may not have graphed such a family of curves for a many-variable function before, this introduction seems to provide good preparation for the economic applications that follow. The analysis of simultaneous equations presented in the appendix is intended to illustrate how the solution to two linear equations in two unknowns is reflected graphically by the intersection of the two lines. Although students may be familiar with solving simultaneous equations through substitution or subtraction, this graphical approach may not be so well known. Because such graphic solutions lead directly to the economic concept of supply-demand equilibrium, however, I believe it is useful to introduce this method of solution to students. Showing how a shift in one of the equations changes the solutions for both variables is particularly instructive in that regard. In that regard, some material at the end of the appendix makes the distinction between endogenous and exogenous variables – a distinction that many students stumble over. The appendix also contains a few illustrations of calculus-type results. Depending on student preparation, instructors might wish to pick up on this and use a few calculus ideas in later chapters. But this is not a calculusbased text, so there is no need to do this.

B. Lecture and Discussion Suggestions Since much of the material in this appendix is self-explanatory, most instructors may prefer to skip any lecture on this topic. For those who feel a lecture is useful, we would suggest developing a specific numerical example together with graphic and tabular handouts for students. The presentation should, focus primarily on linear equations since these are most widely used in the book and since students will be most familiar with them.

C. Glossary Entries in the Chapter • Average Effect • Contour Lines • Dependent Variable • Functional Notation

Chapter 1: Economic Models

• Independent Variable • Intercept • Linear Function • Marginal Effect • Simultaneous Equations • Slope • Statistical Inference • Variables

SOLUTIONS TO CHAPTER 1 PROBLEMS 1.1

Yes, the points seem to be on straight lines. For the demand curve: P = 1 Q = –100 Q 100 at P = 1,Q = 700, so a = 8 and Q P =8− or Q = 800 −100P 100 P=a−

Chapter 1: Economic Models

For the supply curve, the points also seem to be on a straight line: P 1 = Q 200 Q

If P = a + bQ = a +

200 at P = 2, Q = 300, 2 = – a + 1.5, or a = 0.5. Hence the equation is P = 0.5 + c,d

Q or Q = 200P −100 200

For supply Q = 200P – 100 If P = 0, Q = –100 = 0 (since negative supply is impossible). If P = 6, Q = 1100. For demand Q=800-100P When P = 0, Q = 800. When P = 6, Q = 200. Excess Demand at P = 0 is 800. Excess supply at P = 6 is 1100 – 200 = 900

1.2

Supply: Q = 200P – 100 Demand: Q = –100P + 800 Supply = Demand: 200P – 100 = –100P + 800 300P = 900 or P = 3 When P = 3, Q = 500.

At P = 2, Demand = 600 and Supply = 300. At P = 4, Demand = 400 and Supply = 700.

New demand is Q = –100P + 1100.

Chapter 1: Economic Models

Supply = Demand: 200P – 100 = –100P + 1100. 300 P = 1200 P = 4, Q = 700.

Supply is now Q = 200 P – 400.

Supply = Demand when QS = QD 200P – 400 = –100P + 800 300P = 1200 P = 4, Q = 400

At P = 3, QS = 200, QD = 500 ; this is not an equilibrium price. Participants would know this is not an equilibrium price because there would be a shortage of orange juice.

1.3 a. Excess Demand is the following at the various prices

P = 1 ED = 700 −100 = 600

P = 2 ED = 600 − 300 = 300 P = 3 ED = 500 − 500 = 0 P = 4 ED = 400 − 700 = −300 P = 5 ED = 300 − 900 = −600 The auctioneer found the equilibrium price where ED = 0. b. Here is the information the auctioneer gathers from calling quantities:

Q = 300 PS = 2 PD = 5 Q = 500 PS = 3 PD = 3 Q = 700 PS = 4 PD = 1

So, the auctioneer knows that Q = 500 is an equilibrium.

Chapter 1: Economic Models

c. Many callout auctions operate this way – though usually quantity supplied is a fixed amount. Many financial markets operate with “bid” and “asked” prices which approximate the procedure in part b.

1.4

The complaint is essentially correct – in many economic models price is the independent variable and quantity is the dependent variable. Marshall originally chose this approach because he found it easier to draw cost curves (an essential element of supply theory) with quantity on the horizontal axis. In that case, quantity can legitimately be treated as the independent variable. a.

The restrictions on P are necessary with linear functions to ensure that quantities do not turn negative.

The following graph has P on the vertical axis. Equilibrium P is found by −P +10 = P − 2  2P = 12  P = 6, Q = 4 .

The following figure graphs the demand and supply curves with P on the horizontal axis. Solution proceeds as in Part b.

Chapter 1: Economic Models

The equations can be graphed either way and will y

Both graphs yield the same solution

Reasons for preferring one over the other are not readily apparent in these drawings. As we shall see, however, developing demand and supply curves from their underlying theoretical foundations does provide some rationale for Marshall’s choice.

1.51.5

The algebraic solution proceeds as follows: a.

Chapter 1: Economic Models

QD = −2P + 20 QS = 2P − 4 QD = QS  4P = 24 P = 6, QD = QS = 8 b.

QD = −2P + 24 QD = QS  4P = 28 P = 7, QD = QS = 10.

1.6

P = 8, Q = 8 (see graph)

T = .01 I

2 2

I = 10, T = .01(10) = 1 2

I = 30, T = .01(30) = 9 2

I = 50, T = .01(50) = 25

Taxes = $1,000 Taxes = $9,000 Taxes = $25,000

I = 100, T = 100. b. I = 10,000 I = 30,000 I = 50,000

Average Rate 10% 30% 50%

Marginal Rate 20% 60% 100%

c. I

10,000

1,000

10,001

1,000.20

30,000

9,000

30,001

9,000.60

Marginal Tax Rate .20 .60

Chapter 1: Economic Models

1.7

1.8

50,000

25,000

50,001

25,001

1.00

Both these points lie below the frontier.

This point lies beyond the frontier.

Opportunity cost of 1Y is 2X independent of production levels.

If Y = 0, X = 10 If X= 0, Y = 5

X 2

Y 24

4 6

X 2 Y2 + = 1 is a quarter of an ellipse since both X and Y are positive. 100 25 c.

The opportunity cost depends on the levels of output because the slope of a frontier is not constant.

Chapter 1: Economic Models

The opportunity cost of X is the change in Y when one more unit of X is produced. Example: X0 = 3, X1 = 4 When X0 = 3,Y0 = 9½ When X1 = 4,Y1 = [Y1 – Y0] = .187

.187 units of Y are "given up" to produce one more unit of X at X = 3. 1.9

X + 4Y = 100 2

If X = Y, then 5X = 100 and X =

20 and Y =

20 .

X = 10, can consume where any X, Y combination such that X + Y = 10.

Since prefers X = Y, will choose X = Y = 5.

The cost of forgone trade is 5 –

20 = 5 – 2

5 = 1.52 units of both X and Y.

Chapter 1: Economic Models

1.10

This problem provides practice with contour lines. a.

If Y = X  Z the Y = 4 is the same line as “Y = 2” in Figure 1A.5.

If X = 8 − 4Z , Y = X  Z = 8Z − 4Z 2 = 4. This has a solution of Z = 1, X = 4.

None of the other points on the Y = 4 contour line obey the linear equation. This is so because the contour line is convex and hits the straight line at only a single tangency.

If X = 10 − 4Z , Y = 10Z − 4Z 2 = 4 or 4Z 2 −10Z + 4 = 0 . Using the quadratic formula yields Z = (10  100 − 64) / 8 or Z = 2, 0.5 . Hence the line intersects the contour in two places. These points of intersection are Z = 2, X = 2, and Z = 0.5, X = 8.

Yes, many points on the line X + 4Z = 10 provide a higher value for Y (any points between the two identified in part d do). The largest value for Y is at the point X = 5, Z = 5/4. In this case Y = 25/4 = 6.25.

As we shall see, this problem is formally equivalent to a utility maximization problem (see Chapter 2) in which utility is given by U ( X , Z ) = X  Z , the price of good X is 1, the price of good Z is 4, and income is either 8 or 10.

CHAPTER 2

Utility and Choice A. Summary Chapter 2 introduces many new concepts to the student and for that reason it is one of the more difficult chapters in the text. The central concept of the chapter is the indifference curve and its slope, the Marginal Rate of Substitution (MRS). The MRS formalizes the notion of trade-off and is (in principle) measurable. For those reasons it is superior to a “marginal utility” introduction to consumer theory. The definition provided for the MRS in Chapter 2 needs to be approached carefully. Here the concept is defined as the Marginal Rate of Substitution of “X for Y” by which is meant X is being substituted for Y. In graphic terms the individual is moving counter-clockwise along an indifference curve and the MRS measures how much Y will be willingly given up if one more X becomes available. The pedagogic convention of always using counter-clockwise movements along an indifference curve is helpful because the MRS does indeed diminish for movements in that direction. Students’ primary difficulty with the material in Chapter 2 is in confusing the MRS (a slope concept) with the ratio of the amounts of two goods. Unfortunately, that confusion is increased by some examples based on the Cobb-Douglas utility function, which make it appear that the two concepts are interchangeable. To avoid this confusion, some instructors may wish to give further emphasis to the marginal utility definition of MRS, which is presented in footnote 2 of the chapter. This might be followed by greater use of the utility maximization principle (the “equi-marginal principle”) from footnote 5. The soft drink-hamburger example that runs throughout Chapter 2 is intended to provide an easy, mildly amusing introduction to the subject for students. In general, the example seems to work well and is, we believe, definitely superior to introducing the concepts through general goods X and Y. Note also that this chapter includes analyses of 4 specific kinds of goods (useless goods, economic bads, perfect substitutes, and perfect complements). Examining the utility maximizing conditions in these cases (Figures 2-5 and 2-9) should help students to visualize what the conditions mean in cases where the results should be obvious.

B. Lecture and Discussion Suggestions The challenge in lecturing on Chapter 2 is to avoid mere repetition of the text. One way to do that is to offer a somewhat more mathematical treatment. The use of calculus involved in such a treatment may, however, prove too difficult for students to grasp, especially if it involves introducing the Lagrangian technique. An alternative approach would be to start from one point in the X-Y plane and ask how an indifference curve might look. Proceeding from one point to the next in this way reinforces the concept of the trade-off and (on a more sophisticated level) demonstrates Samuelson’s integrability

Chapter 2: Utility and Choice

problems. Once a single indifference curve has been traced out, a second can be constructed to the northeast of the first by using the “more is better” assumption and proceeding with an identical construction. Utility maximization can be approached in the same way by starting at the Y-intercept on the budget constraint and inquiring whether the individual would make various trades along the constraint. Discussions of Chapter 2 material might focus on real world illustrations of both economic and non-economic choices that people make. To approach these, students might be asked to theorize what budget constraint faces people in unusual situations (e.g., what is the cost of shopping for bargains or for wearing seat belts). The instructor can then ask whether there is evidence that individuals respond to changes in the relative costs associated with such activities (that is, do they search more for bargains in high priced items, or are certain types of people less likely to wear seatbelts). Application 2.6 Loyalty Programs also offers a number of discussion possibilities that would help to illustrate the actual shape of budget constraints.

C. Glossary Entries in the Chapter • Budget Constraint • Ceteris Paribus Assumption • Complete Preferences • Composite Good • Indifference Curve • Indifference Curve Map • Marginal Rate of Substitution (MRS) • Theory of Choice • Transitivity of Preferences • Utility

SOLUTIONS TO CHAPTER 2 PROBLEMS 2.1

$8.00

= 20 apples can be bought.

$.40/apple b.

$8.00

= 80 bananas can be bought.

$.10/banana c.

10 apples cost: 10 apples × $.40/apple = $4.00, so there is $8.00 – $4.00 = $4.00 left to spend on bananas which means $4.00 = 40 bananas can be bought. $.10/banana

One less apple frees $.40 to be spent on bananas, so $.40 = 4 more bananas can be bought. $.10/banana

$8.00 = $.40  number of apples + $.10  number of bananas = .40A + .10B.

Chapter 2: Utility and Choice

2.2

A • B =

5 • 80 =

U = 20 = 10 . B so 400 = 10 . B,

400 = 20.

40 = B. c.

U = 20 =

20 . B , so 400 = 20 . B, 20 = B.

From the budget part d, an individual can buy 10 apples and 40 oranges.

One less apple: U =

9  44 =

One more apple: U =

11  36 =

396 <

400 =20

396 <

400

At both endpoints of the budget constraint: U = 0 =

20  0 =

0  80

Graph shown in d. 2.3

To graph the indifference curves, use U 2 instead of U. U = 10 means U 2 = 100 = C  D . Hence, indifference curves are hyperbolas. a.

See Figure in Solutions to odd-numbered problems.

See graph.

Chapter 2: Utility and Choice

2.4

D=10, U = 10  0 = 0

If, say, spent half of income on D, half on C, would buy D=5, C=20. Utility would be U = 5  20 = 10 which is less than 20. Trial and error shows that any other budgetary allocation provides even less utility than this.

As in part d, Paul can buy 20 C and utility will be 10.

Any other allocation yields less utility (see graph).

Tangency is the same in either case.

Costs are: i. $520 ii. $290 iii. $205 iv. $200 v. $250 vi. $425

The bundle C = 20, D = 5 (option iv in part b) is the least costly of those that provide utility of 10. This is the same solution as in problem 2.3.

2.52.5

The indifference curves here are straight lines with slope -4/3. Hence, the MRS is a constant 4/3. The goods are perfect substitutes

Because one unit of tea provides more utility than a unit of coffee, she will spend all of her income on tea when the prices are equal: T = 4, C = 0.

The graph shows that the indifference curves (which have a constant slope) are always steeper than the budget constraint, so maximum occurs on the T axis.

With more income she would continue to buy only tea. If coffee prices fall to $2, coffee is now a cheaper way to obtain utility – one unit of coffee yields 3 units of utility at a cost of $2 so utility costs $2/3 per unit of utility. With tea, utility costs $3/4 per unit of utility.

Chapter 2: Utility and Choice

2.6

Each meal consists of PB=2, C=1. This costs 4(2)+2(1)=10. With an income of $100 she can buy 10 meals per month – or PB=20, C=10.

Now each meal costs 5(2)+2(1)=12. She can buy 100/12 = 8.33 meals.

To restore Vera’s ability to buy 10 meals she would need Food Stamps to buy 1.67 meals. These would cost 1.67x12 = 20.

These preferences allow no substitution of PB for C in response to changing prices. A graph of this utility function would resemble that shown for Right Shoes and Left Shoes in Figure 2.5d.

2.7

Income subsidy is cheaper since AB < A´B´. This result occurs because the housing subsidy encourages people to buy more housing though housing is not really cheaper. 2.8

This person will participate in the Food Stamp program if (as in graph) he or she can reach a utility level higher than U0 by doing so. With cash, the post-transfer constraint would extend the line to the nonfood axis, making it desirable for all to participate. 2.9

a, b.

Chapter 2: Utility and Choice

The figure shows that an unconstrained choice will yield utility level U1 with choices of C*, H*. If the government requires purchase of H**, utility would fall to U0. Low-income consumers are most likely to be constrained by H  H**. c.

To restore this person to U1 would require extra income to shift the budget constraint outward to I’.

A housing subsidy would permit this person to reach U1 with budget constraint I”.

In problems 2.2 and 2.3  =  = 0.5 .

Utility maximization requires PX PY = MRS = Y  X = Y (1−  ) X .

2.10

Some algebraic manipulation yields: (1−  )PX X =  PY Y . Substituting this into the budget constraint yields: PX X + (1−  )PX X  = I or PX X =  I . c.

Because this person spends  I on good X, this amount does not change unless I changes.

Because spending on X is given by  I , changes in the price of Y will not affect this spending.

If Income doubles, spending on both X and Y must double because income is split evenly between the two goods. But prices have not changed, so the quantities of X and Y must double.

Chapter 2: Utility and Choice

CHAPTER 3

Demand Curves A. Summary This chapter provides a complete development of the demand curve concept. It begins with the traditional analysis of the effects of changes in income and prices on the quantities of goods one person demands. Most of the analysis deals with reactions to price changes: income and substitution effects are stressed. Considerable emphasis is given to investigating reasons why these individual demand curves might shift. The purpose of such a detailed investigation is to plant firmly in students’ minds the distinction between movement along a demand curve and shifts in a demand curve. Only by understanding the way in which demand curves are constructed and the ceteris paribus assumptions that are implied is it possible to grasp this distinction completely. Consumer surplus is shown using the usual (Marshallian) demand curve rather than with a compensated demand curve. The compensated demand curve notion is mentioned only in the problems. Market demand curves are developed in the second half of Chapter 3 by summing the individual curves. This construction demonstrates the notion of price-taking behavior that lies behind such demand curves. The summing technique is also used to demonstrate how shifts in market curves are brought about by shifts in individuals’ curves. Finally, the chapter introduces the general concept of elasticity and shows its application to demand theory. Only point elasticity is mentioned. This raises some problems in providing a precise definition without using calculus, but the approach seems preferable to introducing all the algebra that arc elasticity requires. An extended section seeks to clarify the relationship between slope and elasticity for a linear demand curve. We believe students also should learn about log-linear (constant elasticity) demand curves too, but these are covered only in a footnote (footnote 8). The relationship between total expenditures and price elasticity is analyzed in the chapter, but the concept of marginal revenue is not explicitly introduced until Chapter 8. The reason for this is that marginal revenue is not relevant to demanders—it is a concept that should be encountered in connection with the discussion of firms’ goals.

B. Lecture and Discussion Suggestions Comparative statics methodology should be the principal focus of the lecture for this chapter. It is essential that students understand why one compares equilibrium (utility-maximizing) positions to analyze behavior. In addition to repeating the discussion of income and substitution effects (including perhaps the Slutsky equation) there are two other approaches to this chapter that might also make the point. First, one could introduce the revealed preference concept (through a 2-good graphical approach) and show that the axioms of rationality require that the substitution effect be negative. This proof would

Chapter 3: Individual Demand Curves

take about one class and would be a useful supplement to material in the text. A second approach to teaching comparative statics would be to offer an extended example in lecture. Going over the Lump Sum Principle (Figure 3-6) should reinforce the distinction between income and substitution effects. Policy applications of the elasticity concept provide the most interesting points of departure for discussions based on this chapter. The health insurance example (Application 3.7) raises a number of issues about whether elasticities can provide a guide for policy actions (whether services with high demand elasticities should be covered by insurance is an important issue for all health reform plans, for example). The housing and electricity estimates offer the opportunity to develop a similar set of questions. For students who have had a fairly broad exposure to economics, the housing and charity estimates might also be used as a way to introduce notions of “optimal” (nondistorting) taxes and subsidies—that is, one might ask how tax- and subsidyinduced price effects might be minimized and whether this would make sense from an overall perspective.

C. Glossary Entries in the Chapter • Complements • Consumer Surplus • Cross Price Elasticity of Demand • Demand Function • Elasticity • Giffen’s Paradox • Homogeneous Demand Function • Income Effect • Income Elasticity of Demand • Increase or Decrease in Demand • Increase or Decrease in Quantity Demanded • Inferior Good • Market Demand • Market Demand Curve • Normal Good • Price Elasticity of Demand • Substitutes • Substitution Effect (in Consumption Theory

SOLUTIONS TO CHAPTER 3 PROBLEMS 3.1

I = $200 S = J. Ps S + PJ J = 20S + 20S = 200

40S = 200

S=5J=5 b.

PS.S + PJ.J = I S=4J=4

20 . S + 30 . S = 200

50S = 200.

Chapter 3: Individual Demand Curves

Elizabeth’s indifference curves are L-shaped since she gains utility only when shoes and jeans are purchased in a one to one proportion. 10 shoes and 5 pairs of jeans yield the same utility as 5 sweaters and 5 pairs of jeans. d.

The change from U2 to U1 is entirely attributable to the income effect. There is no substitution effect due to Elizabeth’s insistence on a fixed proportion of jeans and shoes.

S = J throughout because of her preferences. 20S + PJ S = 200 S=J=

200 20 + PJ

The following choices will be made: S=J PJ 30 20 10 5 f.

4 5 62/3 8

Chapter 3: Individual Demand Curves

300 20 + PJ S=J

30 20 10 5

6 7.5 10 12

Now S = J =

More J is demanded at each price (see graph in part f). h.

Now: S = J =

200 30 + PJ

This will shift both demand curves inward.

3.2

These price changes still allow Paula to afford her initial choices. Hence the budget constraint rotates around this point (T = 5, L = 4) .

Because the new budget constraint is no longer tangent to the indifference curve, Paula can make choices that improve utility. The figure shows why this choice will involve more L and less T.

This effect allows utility to increase whereas we have defined the substitution effect as constituting a move along a single indifference curve.

If the substitution effect is defined as the result of a rotation around the initial consumption bundle, the “income effect” would be measured by the effects of parallel shifts in the budget constraint from this point. The end result would be the same under either disaggregation.

3.3

Chapter 3: Individual Demand Curves

PB = 2J and .05PB + .1J = 3 5PB + 10J = 300 PB + 2J = 60 4J = 60 b.

J = 15,

PB = 30

PJ = $0.15

PB = 2J

.05PB + .15J = 3

5PB + 15J = 300

25J = 300

J = 12, PB = 24

To continue buying J = 15, PB = 30, David would need to buy 3 more ounces of jelly and 6 more ounces of peanut butter. Should increase his allowance by: 3(.15) + 6(.05) = $.75.

Since David N uses only PB and J to make sandwiches (in fixed proportions), and because bread is free, it is just as though he buys sandwiches where Psandwich = 2PPB + PJ . In part a, PS = .20, QS = 15. In part b, PS = .25, QS = 12. In general, QS = 3 PS

Chapter 3: Individual Demand Curves

f. 3.4

3.5

3.6

There is no substitution effect due to the fixed proportion nature of David’s preferences. A change in price results only in an income effect.

Function is homogeneous because a doubling of I and P leaves Q unchanged 60 b. Graph of Q = P

See graph.

Since Q=0 for P>10, CS=0. Equation gives same result.

If P=3, Q = 20 and CS=180 – 60lnP = 180 – 60 = 120. This is the amount that Irene would pay for the right to buy pizza at a price of 3.

With P=4 Q=15 and CS=174 – 60ln(4) = 174 – 83 = 91. She would be willing to pay 39 less for the right to buy pizza at 4.

a. This is true by definition. The person starts from the same place under either concept. b.

The compensated demand curve incorporates only substitution effects. Because Marshallian demand also incorporates income effects, demand will generally be more price-responsive under the Marshall concept..

Because utility varies along the Marshallian demand curve, each point provides a new utility level from which to construct a different compensated demand curve.

There are no substitution effects in this case so the compensated demand curve will be vertical. The Marshallian demand curve will be sloped, however, because of income effects.

U2009 =

40  40 = 40

Chapter 3: Individual Demand Curves

U2010 = b.

20  80 = 40

I2009 ('09 prices) = 1(40) +1(40) = 80 I2010 ('09 prices) = 1(20) +1(80) = 100 “real income” has risen.

I2009 ('10 prices) = 4(40) +1(40) = 200 I2010 ('10 prices) = 4(20) +1(80) = 160 “real income” has fallen.

3.7

Results of calculations depend on which prices are used. It may be necessary to use some combination of the two indices to conclude (correctly) that utility (real income) has not changed. Notice that the product of the real income ratios does give the correct solution (though this is a special case). 100 160  = 1.0 . 80 200

Q = 20 b.

Q = 0 when P = 20

P=1

Q = 19 P  Q = 19

P=2

Q = 18 P  Q = 36

P=3

Q = 17 P  Q = 51

P=9

Q = 11 P  Q = 99

P = 10

Q = 10 P  Q = 100

P = 11 Q = 9

P  Q = 99

P = 19

P  Q = 19

Q=1

Highest total expenditures are 100 when P = 10.

Since 40 – 2P = 2(20 – P), Q will be twice as large at each price. Total expenditures are still as large as possible when P = 10.

Chapter 3: Individual Demand Curves

3.8

Tom

Dick

Harry

Total

P = 50

= 35

= 25

135

= 10

120

100

300

= 0

100

160

150

410

“Total” column in part a.

3.9

Above graph.

Because the market demand curve is the horizontal sum of each individual’s demand curve, the total area of consumer surplus triangles for each person will equal the area of the consumer surplus triangle in the market. This is easiest to show for a small price increase of P . Let initial quantities be denoted by asterisks, post-change quantities by primes. Then, for each person the loss of consumer surplus is Pxi' + 0.5P(x*i − xi i) . Summing over all individuals yields PX ' + 0.5P( X * − X ' ) which is what one would get from the market demand curve.

Chapter 3: Individual Demand Curves

The loss of consumer surplus is larger in the inelastic case because consumers do not reduce quantity purchased by very much in response to the price increase. With small substitution effects consumers can not “get out of the way” of the price increase whereas with larger ones they can.

3.10

This would be literally true only if demand were completely inelastic. With a more elastic demand total spending total spending may even fall in response to a price increase though there will still be a loss of consumer surplus.

b. c.

−a when Q = 0. b

−a −a Q * −a −Q * − P* = − = . b b b b Q P P*  = b because the demand curve is linear with slope b (note eQ,P = P Q Q* this is a negative number). Y=

d. Now use the result that X = P ,Y =

−Q* b

eQ,P =

X = −Q* Y b

For movements downward along a linear demand curve, distance X falls and distance Y increases. Hence the demand curve becomes less elastic.

One could draw a linear demand curve tangent to any demand curve. This tangent demand curve would have the same elasticity as the original one.

CHAPTER 4

Uncertainty A. Summary Chapter 4 provides a foundation for students to help understand the important role that uncertainty and information theory has come to play in microeconomics. This material appears early in the text for two reasons: first, so that it can be used occasionally in subsequent chapters (including for example the chapter on game theory, where uncertainty comes up in the discussion of mixed strategies), and second, because it can be viewed as an extension of the basic theory of consumer choice to environments involving uncertainty, it should naturally appear right after the chapters on utility and choice. The basic goal of the chapter is to show why individuals are generally risk-averse and are therefore willing to pay something to reduce the risks they face. That point is made initially using the Friedman-Savage utility of wealth analysis and followed up with a discussing of various methods for reducing risk and uncertainty including insurance, diversification, options, and information. Financial applications are highlighted in a separate section. The appendix introduces a new model—a two-state model based on Rothschild-Stiglitz—which we show can be applied to understand all the previous concepts. Making use of the “certainty line” in this model of consumer choice is an especially intuitive way of illustrating risk aversion. Most of the material on asymmetric information is a bit more advanced and so is provided in a later chapter (Chapter 15).

B. Lecture and Discussion Suggestions With the huge volume of material that needs to be covered in an intermediate microeconomics course, the instructor needs to pare down what is covered to fit into a sensible one-semester course. There is a great temptation to omit this chapter, but we would urge the instructor to reconsider this choice. Uncertainty is an extremely important topic and may be covered in no other course that the undergraduate takes. If time constraints are severe, the instructor could cover the very basics, say on risk aversion and insurance (Section 4.2 and the first entry in Section 4.3). With more time, the instructor could include coverage of diversification, options, and information. Of course a second semester course covering advanced topics in microeconomics could spend a lot more time on uncertainty and could have a whole unit on uncertainty, including a deeper treatment of Chapter 4 combined with the material on asymmetric information in Chapter 15. Courses in business schools may want to include coverage of Section 4.4 on financial applications. The appendix model essentially goes through all the same material a second time using a different model. Many instructors will choose to omit the appendix entirely. Another possibility is just to introduce the bare essentials (say just the text surrounding Figures 4A.1 to 4A.3) to supplement the earlier material. A third option, for instructors who particularly favor the

Chapter 4: Uncertainty

two-state model, is to focus their lectures entirely around the appendix and have the students read the earlier material as background. The concepts in this chapter are perhaps a bit more difficult than in some other chapters. More repetition of material in the book may be required in lectures to help guide students to an understanding (for example, Figure 4.3 is very complex, as are some of the graphs for the two-state model in the appendix), or certain issues can simply be omitted. Discussion topics on uncertainty and information are virtually unlimited. Issues about the stock market (efficient market theory, the role of investment advisors, and so forth), the reform of insurance for healthcare in the U.S., and crime and punishment (with imperfect and so uncertain enforcement) are all of great interest to students.

C. Glossary Entries in the Chapter • Diversification • Expected value • Fair gamble • Fair insurance • Market line • Option contract • Probability • Real option • Risk aversion • Risk neutral

D. Notes on Review Questions Review Question 3. Some background might help the instructor if students have questions with this one. We generated the example assuming that gamble 1 has a 50-50 chance of paying off 0 or 100. Gamble 2 has a ¾ chance of paying off 0 and ¼ chance of paying off 300. The assumed utility function is U (I ) = I . These assumptions are all internally consistent. So, yes, it is possible for the person’s ranking based on expected utility to reverse the ranking based on expected payoffs. This is the whole rationale for inserting the extra layer of the utility function into the analysis of the problem, as Bernoulli did originally. To complete the answer to the question, you should make your decision based on which gamble provides the higher expected utility. This is gamble 1. It involves a lower expected payoff but since it also involves much less risk, given your assumed risk preferences you prefer it to gamble 2.

SOLUTIONS TO CHAPTER 4 PROBLEMS 4.1

Given that these are actual gambles offered in Las Vegas, you shouldn’t be surprised to learn that they are unfair in the casino’s favor. To verify this claim, we can compute the expected payoff from gamble 1,

Chapter 4: Uncertainty

−2,000  18   20   −52.6,  (+1,000) +  (−1,000) = 38  38   38  and from gamble 2, −1,000 1   37  17,500) + −500) =  −26.3.  (  ( 38 38 38     They are both negative, not zero as required of fair gambles. b.

To figure out which gamble Wen would take, compute the expected utility from gamble 1,  18   20   99.61,   10,000 +1,000 +   10,000 −1,000  38   38  and from gamble 2,  1   37   99.27.   10,000 +17,500 +   10,000 − 500  38   38  The first is higher, so Wen would choose gamble 1.

4.2

The expected utility from not taking either gamble is the same as the utility from current income ($10,000) with certainty: 10, 000 = 100 . This is higher than the expected utility from either gamble, verifying that Wen, who is risk averse, wouldn’t want to take fair gambles, let alone the unfair gambles offered in Las Vegas.

E(1) = .50(100) + .50(–100) = 0 E(2) = .75(100) + .25(–300) = 0 E(3) = .90(100) + .10(–900) = 0

Assume current income is $1,000. Then utility of income graph is:

Bet 1 will be preferred since it has smaller variability.

Chapter 4: Uncertainty

4.3

Expected utility without insurance is .75 log(10,000) + .25 log(9,000) = 3.9886.

Expected utility with insurance is log (9750) = 3.9890. This is greater than the expected utility from part a.

The individual will pay up to point where expected utility with insurance equals the expected utility without. The expected utility with insurance having premium cost P is log (10,000 – P). Therefore, we set log (10,000 – P) = 3.9886 to find the highest premium Mr. Fogg is willing to pay. Raising both sides to the power of 10, 103.9886 = 10,000 – P = 9.741, implying the maximum premium is $259.

4.4

a. b.

The fair insurance premium equals the expected loss: E(L) = .30  1,000 = $300. Since $300 > $259, he will not buy this insurance even though it is fair. This is an example of moral hazard.

I = 0.5 45,000 + 0.5 55,000 = 446.65 I = 49,875 . ln(I ) = 0.5 ln(45,000) + 0.5 ln(55,000) = 10.815.

I = e10.815 = 49762. −1 c.

−.5

=−

45,000 55,000

 −5 = 2.02 10 I

105

2.02

49,504 .

The functions exhibit increasingly greater risk aversion. This is an illustration of the general principle that the degree of (relative) risk aversion for the utility function I R / R U (I ) =  ln(I )

R 1 R=0

is given by 1 – R.

4.5

U = ln($18,000) = 9.798.

U = ln($18,300) = 9.815.

If Molly invests $100 in the trip, she will have a wealth of $17,900 if Crazy Eddie does not have the set and $18,200 if he does. E(U) = .5 ln(17,900) + .5 ln(18,200) = 9.801. Since this exceeds the utility from part a, it is worth the trip.

4.6

Strategy one: