ECON 301 Intermediate Microeconomic Theory I - Practice and Tutorial Questions and answers– updated

ECON 301 Intermediate Microeconomic Theory I Practice and Tutorial Questions and answers– updated for Feb 2026 exam prep Concordia University

Practice and Tutorial Questions

Econ 301 Fall 2024 Sections A, B and C Content updated is yellow highlighted

• This file is updated frequently throughout the Fall 2024 semester. Please check the yellow highlighted date to ensure that you have the newest version.

• Please feel free to delete all the answers in this file, save a copy, and work on the questions to test your understanding.

• Practice and tutorial questions are organized by chapter.

• Practice questions – answers and brief explanations are posted below. You can also ask the TA if you have any questions about deriving the answers to these questions.

• Tutorial questions – the TA will derive the answers step-by-step and will answer your questions. The answers will not be posted here.

• Tutorials: - Location: H 920 - Schedule: Friday, 2:45-3:45 p.m. (from September 20th, 2024 to November 29th, 2024) - TA: Omer Altun - Format: The TA will derive answers to the tutorial questions and will also answer your questions related to the course. The number of questions that the TA will discuss in each tutorial session will depend on the pace of the tutorial as well as the lecture pace across the three lecture sections. In the unlikely situation that the tutorial coverage significantly lags behind the lecture coverage, the TA may skip certain questions and the solutions will be added into this file instead.

• This file is created by Carol Lau (Section A).

• Please directly email me at carol.lau@concordia.ca if you spot any typos in this file.

• I wish to thank the generous contributions from Concordia University Professors Nicolas Lepage Saucier, Anca Alecsandru, David Desjardins, Szilvia Papai, Rouzbeh Ghouchani and Asefeh Salarinezhad. I also consulted various online resources in the creation of this file, including Boğaziçi University, University of California (Davis), British Columbia Institute of Technology, University of California (Los Angeles), York University, MIT Khan Academy, University of Victoria, Stanford University, University of California (Santa Barbara), Princeton University, Simon Fraser University, Syracuse University, University of British Columbia, University of Massachusetts (Amherst), University of Nebraska (Lincoln), Harvard University, Oakland University, LIUC Cattaneo University and Universitat de Valencia.

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Math Review and Exercises 1. Exponents

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2. Derivative rules

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Ans: a)

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3. Partial derivative rules

Ans: a)

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Chapter 2 Budget Practice Question #1 If you could afford either 4 units of x and 24 units of y, or 9 units of x and 4 units of y, then if you spent all of your income on y, how many units of y could you buy?

Ans: 40 units of y, see tutorial question #1 for explanation.

Practice Question #2 Lopez consumes coffee (C) and pizza (P). His budget line was described by the equation 20 = (1)P + 2C, where m = $20, price of P = $1, price of C = $2. a) Draw the budget line with coffee on horizontal axis and pizza on vertical axis.

Pizza

Coffee b) The maximum amount of coffee Lopez can afford is . c) The maximum amount of pizza Lopez can afford is . d) The slope of the budget line is equal to . The slope can be interpreted as the opportunity cost of an extra unit of coffee, which in this case is pizza(s). e) If the price of pizza increases, the budget line becomes (flatter, steeper). f) If Lopez’s income decreases, the budget line shifts (outward, inward). g) Suppose the government imposes a percentage tax of 25% on the price of pizza. The slope of the budget line is now and the maximum number of pizza Lopez can afford is . h) Suppose the government offers an income subsidy of $10 to Lopez. This will shift his budget line (outward, inward) with the slope (increasing in absolute value, decreasing in absolute value, unchanged). If the subsidy were $0.4 per unit of coffee consumed, the new budget line slope is and the maximum number of coffee Lopez can afford is .

Ans: a) Budget line is a straight line, with slope equals – 2. b) $20/$2 = 10, the horizontal intercept. c) P = 20, the vertical intercept. d) slope -2; 2 e) flatter f) inward g) slope -2/1.25 = -1.6; $20/1.25 = 16 h) outward; unchanged; -(2-0.4)/1 = -1.6; $20/1.6 = 12.5 6

Practice Question #3: While traveling in Tokyo, Cindy spent all the money in her wallet to buy 5 pieces of unagi (eel) and 6 sushi rolls. Unagi costs 80 yen per piece and Cindy had 1,000 yen. If u denotes the number of pieces of unagi and s denotes the number of sushi rolls purchased, which of the following is her budget constraint? a) 20u + 16s = 1,000 b) 16u + 8s = 1,000 c) 80u + 100s = 1,000 d) 50u + 60s = 1,000 Ans: C; 400 + 6s = 1,000 yen, so price of s = 100 yen.

Practice Question #4: If you have an income of $30, x1 costs $5 per unit and x2 costs $10 per unit, then the budget line is a) 6x1 + 11x2 = 30. b) (x1 + x2)/15 = 30. c) x1 + 2x2 = 6. d) x1/5 + x2/10 = 30.

Ans: C; 5x1 + 10x2 = 30, divide everything by 5.

Practice Question #5: Maggie has an income of $500 and she only consumes bread (B) and butter (T). The price of B is $5 per unit and the price of T is $8 per unit. Suppose her income rises to $2,000. If she tells you that she does not care much for this income increase because she cannot afford to buy more of either goods.

This means the price of B has become and the price of T has become . a) $10; $16 b) $15; $24 c) $18; $28 d) $20; $32 Ans: D; Quadrupled income but also quadrupled prices leave Maggie the same affordability set of B and T. Practice Question #6: Lance consumes only potatoes and sardines. When the price of potatoes was $9 per sack and the price of sardines was $5 per pound, he spent his entire income to buy 5 sacks of potatoes and 10 pounds of sardines per month. Now the government subsidizes potatoes. Market prices have not changed, but consumers get a subsidy of $5 for every sack of potatoes consumed. To pay for this subsidy, the government introduced an income tax. Lance pays an income tax of $20 per month. If p is the number of sacks of potatoes and s is the number of pounds of sardine, what is Lance’s new budget equation? a) 9p + 5s = 100. b) 14p + 5s = 75. c) 4p + 5s = 95. d) 4p + 5s = 75.

Ans: D; initially 5*$9 + 10*$5 = $95, then it becomes 5*($9-sub=$5) + 5*s = $95 – tax=$20, so 4p + 5s = 75.

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Practice Question #7: Suppose that the price of a pen is always $3. The price of a pencil is $1 for each of the first 10 pencils a person buys. After that, each pencil costs $2. On a graph with pencils on the vertical axis and pens on the horizontal axis, the budget line will a) Have a kink at 10 pens and will be steeper to the right of the kink than to the left of the kink. b) Have a kink at 10 pens and will be flatter to the right of the kink than to the left of the kink. c) Have a kink at 10 pencils and will be steeper to the right of the kink than to the left of the kink. d) Have a kink at 10 pencils and will be flatter to the right of the kink than to the left of the kink.

Ans: C; If pencils are on the vertical axis, the slope of the budget line is equal to the price of pens divided by the price of pencils. From 0 to 10 pencils, the slope is -3, which is to the right of the kink at pencils = 10. For 10+ pencils, the slope = -1.5, which is to the left of the kink at pencils = 10. The upper portion of the budget line will have a flatter slope than the lower portion.

Practice Question #8: The Canadian Goods and Services Tax (GST) rate has been reduced from 6% to 5%, effective January 1, 2008. The two goods in your consumption bundle are basic food (no GST) and a composite good (GST applies). If we put basic food on the vertical axis and the composite good on the horizontal axis, the tax rate change a) shifts your budget constraint out. b) shifts your budget constraint in. c) makes your budget constraint flatter without changing its vertical intercept. d) makes your budget constraint flatter without changing its horizontal intercept.

Ans: C; The vertical intercept does not change since the tax change does not affect the maximum amount of basic food you can buy. The horizontal intercept pivots to the right since the price of the composite good has decreased. The budget constraint flattens.

Practice Question #9: Continue with the previous question on GST: This tax rate change has a) increased the opportunity cost of basic food. b) increased the opportunity cost of the composite good. c) decreased the opportunity cost of the composite good. d) not changed any opportunity cost because the tax cut does not affect income.

Ans: A; The price of basic food has effectively increased relative to the price of the composite good because the composite good is now cheaper with a lower tax rate. This means that the opportunity cost of basic food has increased. For example, the price of basic food is $10 and the price of the composite is $5. Suppose the price of the composite is now $4+ because of the tax cut.

Each unit of food means giving up 10/4+ = 2+ units rather than 10/5 = 2 units of the composite.

Practice Question #10: Suppose a store offers volume discount for the sale of x1. Anyone who purchases up to “d” units of x1 the price each unit is p1. If the consumer purchases more than “d” units, each of the subsequent units will be p1/2, or 50% off. Draw the consumer budget line.

Ans:

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Tutorial Question #1 If you spent your entire income, you could afford either 3 units of x and 9 units of y or 9 units of x and 3 units of y. If you spent your entire income on x, how many units of x could you buy?

Tutorial Question #2 Suppose you have been hired to examine whether the government should offer coupons or vouchers to low-income households. As a first step, you want to incorporate such policies into the consumer’s budget constraints. For simplicity, the consumer purchases only x1 and x2 and has an income m.

Coupon: - Case #1: The consumer gets 20% off x1 and x2, regardless of the quantity purchased.

- Case #2: The consumer gets 20% off x1 only, regardless of the quantity purchased.

- Case #3: The consumer gets 20% off x1 for the first A units only.

Voucher: - Case #4: The consumer is given a voucher that offers three units of x1 for free.

Depict the four cases graphically. Label the horizontal and vertical intercepts.

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Chapter 3 Preferences Practice Question #1: If Tim cannot decide whether he likes a North Face tent better than a Columbia tent or vice versa, or whether he is indifferent between these two tents, his preferences violate the assumption of a) non-satiation. b) transitivity. c) completeness. d) reflexivity.

Ans: C; If preferences are complete, an individual must be able to identify the bundle or product he likes better or if he is indifferent between the two bundles or products. We do not have enough information to analyze the validity of transitivity, reflexivity and non-satiation.

Practice Question #2: Suppose you put five M&M’s chocolates close to each other on a 5-inch plate and give them to your two-year old niece. She thinks there are too few chocolates and begins to cry. You turn around, hide the plate from her view and then spread out the five M&M’s on the same plate and present them to your niece again. She is now happy with getting five chocolates and smiling. Your niece’s preferences violate the assumption of a) non-satiation. b) transitivity. c) completeness. d) reflexivity.

Ans: D; The first set of five chocolates is not as good as the second set of five chocolates!

Practice Question #3: Offered a choice between apple pie and chocolate cake, I choose chocolate cake. Offered a choice between chocolate cake and vanilla ice cream, I choose vanilla ice cream. Offered a choice between vanilla ice cream and apple pie, I choose apple pie. Which assumption do my choices violate? a) completeness b) reflexivity c) transitivity d) substitutivity Ans: C; I should prefer vanilla ice cream to apple pie.

Practice Question #4: Ken is a photographer who always frames three 4”x4” pictures in one frame. He wants to frame some of the pictures from this year’s trip to Ireland. He has $120, a frame costs $25 and printing one 4”x4” photograph at the Sakura Photo Centre in Montreal costs $5. Pictures and frames are . Ken will purchase frames and he will print pictures. a) perfect substitutes; 2;6. b) perfect substitutes; 3;9. c) perfect complements; 2;6. d) perfect complements; 3;9.

Ans: D; pictures and frames are perfect complements. He will spend $120 on three frames and nine pictures.

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Practice Question #5: The Quebec government is worried about the increased consumption of fentanyl. The consumers have access to two goods, namely, money and fentanyl. The more fentanyl they consume, the more money they are willing to give up for it. These preferences deviate from the classical assumption that . Hint: Imagine fentanyl is on the horizontal axis and money on the vertical axis. a) people do not consume drugs. b) money and drugs are perfect substitutes. c) consumers maximize their utility. d) the marginal rate of substitution between two goods is diminishing.

Ans: D; Instead of decreasing, the MRS of money for drugs increases with the amount of drugs people purchase. This violates the diminishing MRS assumption.

Practice Question #6: Ben consumes goods x1 and x2. His indifference curves are described by x2 = k/(x1 + 6), where k is a constant. A higher k means a higher indifference curve. Which of the following is CORRECT? a) Ben prefers bundle (4, 5) to bundle (5, 4). b) Ben likes good x2 and hates good x1 . c) Ben prefers bundle (10, 9) to bundle (9, 10). d) Ben likes good x1 and hates good x2 .

Ans: A; k = 50 > k = 44.

Practice Question #7: A diminishing MRS implies that a) people prefer variety in their consumption choices. b) indifference curves are convex. c) any bundle that is a weighted average between two equally preferred extreme bundles will be preferred to those extreme bundles. d) All of the above are correct.

Ans: D; these are well-behaved preferences.

Practice Question #8: 6Y Kate’s marginal rate of substitution of good X for good Y is MRS = . She is currently consuming 4X 6 units of Y and 3 units of X. Should she accept an offer to trade 2 units of Y for 1 unit of X? a) Yes, because she values Y more than X. b) No, because this would put her on a lower indifference curve. c) No, because she cannot afford this trade. d) Yes, because what she has to give up for the additional unit of X is less than her MRS. Ans: D; MRS = 3. This means that she is willing to give up three units of Y for one unit of X. Practice Question #9: Sara has convex preferences. As Sara drinks additional cups of milk at night, her a) marginal utility from milk decreases. b) total utility from milk increases. c) total utility from milk decreases. d) both answers A and B are correct.

Ans: D; MU positive but decreasing as she drinks more milk.

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Practice Question #10: Suppose that Ana is indifferent between getting a bundle with 6 magazines and 2 books and a bundle with 2 magazines and 6 books. If Ana prefers a bundle with 6 magazines and 2 books to one with 4 magazines and 4 books, we could conclude that a) Ana's preferences are not monotonic. b) Ana's preferences are not transitive. c) Ana's preferences are not complete. d) Ana's preferences are not convex.

Ans: D; If Ana's preferences were convex and she is indifferent between two bundles, she should prefer an average of those two bundles.

Practice Question #11: Bicycle frames and wheels are perfect complements: you need two wheels for every bicycle frame you own. If frames are on the X-axis and wheels are on the Y-axis, your indifference curves for bicycle frames and wheels have which of the following shape? a) L-shape, with kinks at (1F, 2W), (2F, 4W), (3F, 6W), and so on. b) L-shape, with kinks at (2F, 1W), (4F, 2W), (6F, 3W), and so on. c) a 45-degree line from the origin. d) downward-sloping straight line.

Ans: A; once you own a bike frame and two wheels, your utility remains unchanged if you get more wheels and no additional frames, or vice versa. The bundle (1F, 2W) is on the same indifference curve as (2F, 2W), (3F, 2W), (4F, 2W) and so on, and (1F, 3W), (1F, 4W), (1F, 5W) and so on, which gives us an L-shaped indifference curve with a kink at (1F, 2W).

Practice Question #12: Bundle D represents which of the following features of a well-behaved utility function?

a) convexity b) transitivity c) completeness d) none of the answers above is correct.

Ans: A; bundle D is a weighted average of bundles A and B and it is preferred to these two bundles because it lies on a higher indifference curve.

Practice Question #13: Mary’s preferences can be described by x2 = k – 4x1 1/2, where k is a constant. If x1 is drawn on the horizontal axis and x2 on the vertical axis, Mary’s MRS is when her consumption bundle is (9, 5). a) –0.67 b) –8 c) –9/5 d) –5/9 12

Ans: A; MRS = slope = -2/(x1)0.5 Practice Question #14: Draw indifferences curves to represent each of the following type of preferences and write down the MRS and relate to opportunity costs. a) Quarters-dollars: A consumer is always happy to change 4 quarters for 1 dollar. b) Bicycle frames-tires: A consumer always wants 2 tires to go with 1 frame.

Ans: a) Perfect substitutes – consumer indifferent across the two; MRS = - 4/1 = - 8/2 = - 4; consumer is willing to give up 1 dollar in exchange for 4 quarters.

b) Perfect complements – consumer has to consume the two in fixed proportions; MRS = infinity on the vertical segment; MRS = 0 on the horizontal segment; MRS undefined at kink; consumer cannot substitute across tires and frames.

Tutorial Question #1: Coach Sheth likes his players to be big, fast, and obedient. If player A is better than player B in two of these three characteristics, Sheth will prefer A to B. Three players try out for goalie. Will (W) weighs 320 pounds, runs very slowly, and is quite obedient. Han (H) weighs 240 pounds, runs extremely fast, and is extremely disobedient. Jaxon (J) weighs 150 pounds, runs at average speed, and is extremely obedient. Does Coach Sheth have transitive preferences? Explain your answer.

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Chapter 4 Utility Practice Question #1: Sara thinks a 16-ounce can of Pepsi is just as good as two 8-ounce cans. Joni only drinks 8 ounces at a time and hates stale Pepsi. She thinks a 16-ounce can is no better or worse than an 8-ounce can. a) Write a utility function that represents Sara’s preferences. Let x1 stand for the number of 8-ounce cans and x2 for the number of 16-ounce cans. b) Write a utility function for Joni’s preferences. c) Monotonic transformation: Does U = 100x1 + 200x2 represent Sara’s preferences? Find the MRS. d) Monotonic transformation: Does U = (5x1 + 10x2)2 represent Sara’s preferences? Find the MRS. e) Monotonic transformation: Does U = x1 + 4x2 represent Sara’s preferences? Find the MRS. Ans: a) U = x1 + 2x2 b) U = x1 + x2 c) Yes…proportion of x1 and x2 preserved; MRS = 0.5. d) Yes…proportion of x1 and x2 preserved; MRS = 0.5. e) No… proportion of x1 and x2 not preserved; MRS = ¼.

Practice Question #2: Tom always begins his day with a strawberry milkshake. He makes it by mixing milk (x) with five strawberries (y). The secret of a really good milkshake lies in the optimal proportion of milk and fruit: one glass always comes with five strawberries. a) Plot Tom's indifference curves. Depict three indifference curves that pass through the following bundles (5, 1), (10, 10) and (15, 4). What is the MRS at each of these points? b) Monotonic transformation: A utility function that represents these preferences can be U(x, y) = min{5x, y}. Multiply this utility function by ten and add two to it. Call this new utility function V(x, y). How did the indifference curves change? How was the level of utility associated with each indifference curve affected?

Ans: a) MRS = 0 in each case, see graph below. b) We have, V = 10 min{5x, y} + 2. Since v is a monotonic transformation of u, shape of indifference curve and preference rankings will not be affected. But, the numbers utility levels assigned on each indifference curve will change. We have V(5, 1) = 12, V(10, 10) = 102, and V(15, 4) = 42.

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Practice Question #3: For each of the following utility functions, find the marginal utilities (MUx and MUy) and the MRS, and explain whether MRS is diminishing. Also indicate whether the preferences are strictly convex.

NOTE: A straight line is convex but not strictly convex. a) U(x, y) = 3x + y b) U(x, y) = x1/2y c) U(x, y) = (x + 2y)2 d) U(x, y) = x1/3y2/3 e) U(x, y) = x2 + 3y – 2 f) U(x, y) = x1/2 + y g) U(x, y) = x2+ y2 √ Ans: a) MUx = 3, MUy = 1, MRS = 3, constant, so convex but not strictly convex. y b) MUx = 0.5x−0 .5y, MUy = x0 .5, MRS = 0.5 , diminishing, so strictly convex. x c) MUx = 2(x + 2y), MUy = 4(x + 2y), MRS = 1/2, constant, so convex but not strictly convex. y 1 −2/ 3 2 /3 2 1 /3 −1 /3 d) MUx = ( ) x , MUy =( , MRS = 0.5 , diminishing, so strictly convex. y )x y x 3 3 2x e) MUx = 2 x, MUy =3, MRS = , increasing, so not convex.

3 1 −1 /2 1 −1 /2 , diminishing, so strictly convex. f) MUx = ( ) x , MUy =1, MRS = ( ) x 2 2 −1 −1 g) MUx = ( 1 )( x2+ y2) 2 (2 x), MUy == ( 1 )( x2+ y2) 2 (2 y ), MRS = x , increasing, so not convex. y 2 2 Practice Question #4: Show that all three utility functions have the same marginal rate of substitution. a) U(x, y) = xy b) U(x, y) = x2y2 c) U(x, y) = ln(x) + ln(y) Ans: All three have MRS = y/x. They are monotonic transformations of each other.

Practice Question #5: xy Ben buys two goods, x and y. His utility function is U(x, y) = . The prices of the goods are px and x+ y py and income is m. Does Ben’s utility function satisfy the requirements of monotonicity and convexity?

2 2 2 Ans: MUx = y 2, MUy = x 2, MRS = y2 , both MU’s are positive so the more Ben (x + y ) (x + y ) x consumes, he get additional utility – satisfies more is better, monotonicity. MRS diminishing as x rises, so strictly convex.

Practice Question #6: Dave’s utility function for Labatt (L) and Guinness (G) beer is U(L, G) = 3L + 6G. This means that Dave would give up bottle(s) of Guinness for bottle(s) of Labatt. a) one; two b) two; one 15

c) one; four d) two; three Ans: A; To see this, let U = 6 or any other number. To get U = 6, we need either L = 2 or G = 1.

This means Dave is indifferent between getting either 1G or 2L. Practice Question #7: Which of the following represents a Cobb-Douglas utility function? a) U(x1, x2) = ax1/x2 b) U(x1, x2) = a ln(x1/x2) c) U(x1, x2) = a ln(x1 + x2) d) U(x1, x2) = a (ln x1) + b (ln x2) – see math review on how to differentiate natural log functions.

Ans: D; a Cobb-Douglas utility function with monotonic transformation is U = x 1ax 2b.

Practice Question #8: Don consumes x1 and x2 together in fixed proportions. He always consumes 2 units of x1 for every unit x2. One utility function that describes his preferences is a) U = 2x1x2. b) U = x1 + 2x2. c) U = min{2x1, x2}. d) U = min{x1, 2x2}.

Ans: D; Not C! Let U = 2, we need x1 = 2 and x2 = 1. See tutorial for explanation.

Practice Question #9: Pam’s utility function is U(A, B) = AB, where A and B are the numbers of apple and banana, respectively, that she consumes. When Pam is consuming 15 apples and 60 bananas, if we put apple on the horizontal axis and banana on the vertical axis, Pam’s MRS at her current consumption levels is a) - 6. b) - 8. c) - 4. d) - 0.25.

Ans: C; slope = B/A = - 60/15 = - 4.

Practice Question #10: Luther consumes goods 1 and 2. No matter how much of each good he is consuming, he always finds that 2 units of good 1 is a perfect substitute for 5 units of good 2, and vice versa. Which of the following utility functions could represent Luther’s preferences? a) U(x1, x2) = min{2x1, 5x2} b) U(x1, x2) = 20x1 + 8x2 c) U(x1, x2) = x1 + 5x2 + 1,000 d) U(x1, x2) = 2x1 + 5x2 Ans: B; see question above for explanation.

Practice Question #11: Helen consumes goods 1 and 2. She finds 4 units of good 1 to be a perfect substitute for 3 units of good 2, and vice versa. Which of the following utility functions could represent Helen’s preferences? a) U(x1, x2) = 4x1 + 3x2 b) U(x1, x2) = 6x12 + 8x1x2 + 8x22 c) U(x1, x2) = 3x1 + 4x2 + 20 d) U(x1, x2) = x1 + 2x2 16

Ans: C; Not A! We want the utility from good 1 = utility from good 2. We want ax1 = bx2 or 4a = 3b, so a possibility is a = 3 and b = 4, or a = 6 and b = 8. The 20 is a monotonic transformation that does not change the a/b ratio. Check: Helen will enjoy U = 12 + 20 regardless of whether she consumes 4 units of good 1 or 3 units of good 2. Also see tutorial for discussion.

Practice Question #12: Eugene’s utility function is U(x, y) = xy.

Megan’s utility function is U(x, y) = 1,000xy.

Elizabeth’s utility function is U(x, y) = –1/(xy + 1).

Fergie’s utility function is xy – 10,000.

Margaret’s utility function is x/y.

Which of these people has/have the same preferences as Eugene? a) Margaret only b) Megan and Fergie c) Megan, Fergie, and Elizabeth d) All of them Ans: C; Find the MRS for each function and check if it is the same as Eugene’s. See tutorial for explanation. See practice Question #15.

Practice Question #13: A person's utility from goods x and y is given by U(x; y) = 3x3 + 4y2. What is the marginal utility from x when x is equal to 2 and y is equal to 2? a) 40. b) 52. c) 36. d) 60.

Ans: MUx = 9x2 and x = 2.

Practice Question #14: Suppose a consumer spends all of her income on only two goods, x and y. Her preferences over these two goods are represented by the utility function U(x, y) = 2x0.5 + 3y. a) Find the marginal rate of substitution, MRS. Is this preference convex? Explain. b) Find the relationship between x and y that gives us the indifference curve for the utility level u = 12.

Then calculate MRS when y = 2 without using MU’s.

Ans: 1 x−1/ 2 a) MRS = = −1 /2 . MRS is diminishing as x increases, so it is convex.

3 3x b) 12 = 2x0.5 + 3y, and if y = 12, x = 9. Substitute x = 9 into MRS to get MRS = 1/9 in absolute value.

Practice Question #15: Which of the utility functions below reflect preferences that violate the assumption of monotonicity? a) Eugene’s utility function is U(x, y) = xy. b) Megan’s utility function is U(x, y) = 1,000xy. c) Elizabeth’s utility function is U(x, y) = –1/(xy + 1). d) Fergie’s utility function is xy – 10,000. e) Margaret’s utility function is x/y.

Ans: a) MUx = y, MUy = x, does not violate because MU’s positive – more is better, higher total utility.

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b) MUx = 1000y, MUy = 100x, same as above. c) MUx = (-1)(-1)(y)(xy + 1)-2, MUy = (-1)(-1)(x)(xy + 1)-2, same as above. d) Same as a). e) MUx = 1/y, MUy = (-1)(x)(y)-2 < 0, violates, more y decreases total utility.

Tutorial Question #1: Lori’s utility function is U(x, y) = xy.

Kirk’s utility function is U(x, y) = 1,000xy + 2,000.

Tammy’s utility function is U(x, y) = xy(1 – xy).

Dean’s utility function is U(x, y) = –1/(10 + 2xy).

Marj’s utility function is U(x, y) = x(y + 1,000).

Pat’s utility function is U(x, y) = 0.5xy – 10,000.

Lane’s utility function is U(x, y) = x/y.

Francis’ utility function is U(x, y) = –xy.

Who has/have the same preferences as Lori?

Tutorial Question #2: Suppose you like to consume bread (B, in pieces) and jam (J, in spoonful), in the following fixed proportions for a given weekday. a) Monday – you want to consume 1 B with 1 J. b) Tuesday – you want to consume 1 B with 0.5 J. c) Wednesday – you want to consume 5 B with 1 J. d) Thursday – you want to consume 0.5B with 2.5 J. e) Friday – you want to consume α amount of B and β amount of J. - Write down a utility function that describes your preference for each weekday. Hint: Let utility = a constant, such as 1, and figure out the values of B and J. - Draw an indifference curve for each weekday and label the slope. Put B on the horizontal axis and J on the vertical axis.

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Chapter 5 Choice Practice Question #1: Barbara’s utility function is U(x, y) = x + 46y – 2y2. Her income is $135. If the price of x is $1 and the price of y is $18, how many units of good x will Barbara demand? a) 5 b) 9 c) 12 d) 16 Ans: B; MRS gives 46 – 4y = 18, y = 7, substitute into income to get x = 9.

Practice Question #2: Ann likes shrimp and rice. The price of shrimp is 10 cents per unit and the price of rice is 20 cents per unit. Her marginal utility of shrimp is 6 and her marginal utility of rice is 4. Which of the following is true of the relationship between Ann’s utility and her expenditures? a) Without changing her total expenditures, she could increase her utility by consuming more shrimp and less rice. b) Without changing her total expenditures, she could increase her utility by consuming more rice and less shrimp. c) Without changing her total expenditures on shrimp and rice, she could not increase her utility. d) She should spend more money on both shrimp and rice.

Ans: A; MRS = 1.5 and price ratio is 0.5, should consume more shrimp so that its MU drops and consume less rice so that its MU rises until the MRS becomes 0.5.

Practice Question #3: Samuel has a utility function U(C, T) = CT, the price of cheese (C) is $1, and the price of tea (T) is $2.

If Samuel’s income were $120, how many units of tea and cheese would he consume if he chose the bundle that maximized his utility subject to his budget constraint? a) 6 T and 30 C b) 15 T and 10 C c) 30 T and 60 C d) 40 T and 20 C Ans: C; MRS = T/C, price ratio = 0.5, so T = 0.5C, substitute into m = $120, T = 30 and C = 60.

Practice Question #4: Manuel likes donut (D) and Gatorade (G). His utility function is U(D, G) = 4D + 2G . His weekly budget is $50, one donut costs $5 and one Gatorade costs $2.5. His demand bundle is . a) (8D, 4G) b) (4D, 8G) c) (10D, 5G) d) (12D, 1G) 19

Ans: A; The MU of donut is 4, MU of Gatorade is 2, MRS = 2, slope of budget constraint = 2. He consumes D/G at the ratio 2 and bundle A satisfies this and is affordable. The other choices either do not have a ratio of 2 or cost more than $50.

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Practice Question #5: Suppose that Dani only consumes goods x and y and her marginal utility from x and marginal utility from y are given by MUx = 3y and MUy = 6x. If the price of y is $25 and the price of x is $50, Dani's optimal bundle will contain a) x = 2y. b) x = 0.5y. c) x = 4y. d) x = 0.25y.

Ans: D; Dani consumes at MRS = y/2x, and the price ratio is 2, setting these equal and simplify gives us 4x = y, or x = 0.25y, so the amount of y will be four times larger than the amount of x.

Practice Question #6: Jim always consumes hot dogs (H) and buns (B) in a one-to-one ratio. He gets no additional utility from an extra hot dog unless he also gets an extra bun. Also, he gets no additional utility from an extra bun unless he also gets an extra hot dog. a) Write down a utility function that represents Jim's preferences. b) Suppose that the price of a hot dog is $4 and the price of a bun is $2. Jim has $42 to spend. Draw a graph with buns on the horizontal axis and hot dogs on the vertical axis that shows his budget line, indifference curve and optimal consumption bundle. Also label the intercepts of the budget line and the optimal bundle with their numerical values.

Ans: a) U(H, B) = min {H, B} b) He should consume H = B, plug into income and simplify to get H = B = 7.

Practice Question #7: Suppose Kai’s utility function is U = (x + 2)(y + 1) and the price of x and y is $1. Her income is $201. a) What is her MRS? b) How much x and y does she demand?

Ans: a) y+ 1 MRS = x +2 b) x = 100 and y = 101

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Practice Question #8: Cobb-Douglas: Suppose Harry likes apples (xA) and bananas (xB). His utility function is U = xAxB. Let the price of apples be $1 and the price of bananas be $2 per unit. Harry’s income is $80. Find the optimal number of apples and bananas that Harry will demand.

Ans: MRS = xB/xA, pA/pB = 0.5, xA = 40 and xB = 20.

Practice Question #9: Perfect complements: Suppose Eno’s utility function is given by U = min {x, y2}. a) Suppose the price of x is $1 and the price of y is $2 and his income is $24. What are his demands for x and y? Both x and y are positive values. b) Suppose the price of x is $4 and the price of y is $2 and he buys 100 units of x. What is Eno’s income? Hint: First figure out how many units of y he will buy.

Ans: a) x = y2, 24 = x + 2y, 24 = y2 + 2y, which is a quadratic equation, solving gives x = 16, y = 4. b) y = 10, income = $420.

Practice Question #10: You are consuming hot dogs and Cokes and believe you are maximizing utility for your entire budget of $6, and hot dogs and Cokes each cost $1. Suppose the price of hot dogs rises to $2 and the price of Cokes remains at $1. You adjust your consumption of hot dogs and Cokes and believe you are once again receiving maximum total utility from the meal. Which one of the following can you conclude? a) The marginal utility of hot dogs must rise compared to the marginal utility of Cokes. b) You are purchasing 1 hot dog and 4 Cokes. c) You are purchasing 2 hot dogs and 2 Cokes. d) The marginal utilities of hot dogs and Cokes must be equal.

Ans: A; MRS = price ratio = 1 initially for Coke versus hot dogs. You consume identical quantities of each. Now hot dogs have become twice as expensive, MU from hot dogs has to rise to justify buying hot dogs.

Practice Question #11: Quasi-linear: Suppose Sam’s utility function is U(x1, x2) = x 11/2 + x2. Find the two demand equations.

2 m 1 p2 1 p 2 Ans: x1 = ( ) and x2 = − ( ) p 2 4 p1 4 p1 Practice Question #12: Max has a utility function U(x, y) = 2xy + 1. The prices of x and y are both $1 and his income is $20. a) How much of each good will he demand? b) A tax is placed on x so that x now costs $2 while his income and the price of y stay the same. How much of good x does he now demand? How much tax does he pay in total? c) Would Max be as well off as he was before the tax if when the tax was imposed, his income rose by an amount equal to $1 times the answer to part b), such as if x in part b)? (Such as if x = 8… not the correct x in part b)… and Max’s income rises by $8 but price of x stays at $2). Compare the utility levels from parts a), b) and c).

Ans: a) MRS = 1, x = y = 10. b) MRS = 2, x = 5, y = 10, tax = $5.

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c) Ua = 201 and Ub = 101, MRS = 2 but income = $20 + $1*5 = $25, x = 6.25, y = 12.5, Uc = 157.25, still worse off than in part a) with no tax in spite of the fact that he gets a tax refund.

Practice Question #13: Squid costs $2/kilo and octopus costs $1/kilo. Jon buys only octopus and gets 10 units of utility from the 10th kilo. If he has maximized his utility, his marginal utility from the 1st kilo of squid is a) more than 20. b) less than 20. c) zero. d) less than 10.

Ans: B; MU from the 1st kilo of squid has to be 20 if he were indifferent between octopus and squid and buys both.

Practice Question #14: Perfect substitutes: Suppose Tina’s preferences are described by U(x, y) = 3x + 5y. Her income is $15 and the price of both x and y is $1. Find MRS and optimal bundle.

Ans: MRS = 3/5, px/py = 1, x = 0, y = 15.

Quantity of lipstick

Practice Question #15: Putting calculus aside… we can still get the results we want. Janelle has $16 that she can spend on either cappuccinos or lipsticks. Below is a graph that shows Janelle’s budget lines and Janelle’s consumer maximization points for each of the three budget lines. Using this information, derive Janelle’s demand curve for cappuccinos. Plot this demand curve and the equation. Assume Janelle’s demand curve for cappuccinos is linear.

Ans: 4. 5

4

3. 5

3

2. 5

2

1. 5

1 0. 5 0 0

1

2

3

4

5

6

7

8

9

10

Quantity of Cappuccinos

Equation: P = 10-2Qd, since with BC1, PL=$4, Pc=$2, so the combination is (Pc=$2, C=4); with BC2, PL=$4, Pc=$4, so the combination is (Pc=$4, C=3); with BC3, PL=$8, C=1, so the combination is (Pc=$8, C=1). The slope of the demand is therefore –2. Since when Pc=$8 and C=2, we can interpolate that when Pc=$10, C=0.

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Price of Cappucicnos

Demand for Cappuccinos

9

8

8 7 6 5

4

4 3

2, 4

2 1 0 0.5

1

1.5

2

2.5

3

3.5

4

4.5

Quantity of Cappuccinos

Tutorial Question #1: Jacob spends all his income on movie tickets (good 1) and magazines (good 2). His utility function can be represented by U(x1, x2) = x1x2. His income is m = $100. The prices are p1 = $10 and p2 = $5. a) Derive Jacob's optimal consumption bundle. Show the calculations and illustrate your answer graphically (draw the indifference curves and the budget constraint). b) How would your answer in a) change if Jacob's utility function were given by U(x1, x2) = 7x 12x 22? c) What is Jacob's marginal utility of movie tickets when he buys 5 movie tickets and 8 magazines?

Compute this marginal utility for both utility functions given above. d) Now suppose the government levies a 25% tax on each movie ticket purchased. Derive Jacob's consumption bundle and illustrate it in the same graph that you drew in a). e) Would Jacob prefer a $10 income tax (i.e., lump-sum tax) to the 25% value tax on movie tickets?

Which scenario would the government prefer? Explain your answers.

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Tutorial Question #2: Suppose that your total income is $200. Books cost $5 each. Your utility from books (B) and from all other goods (O) is given by: U(B, O) = 4B + 2O a) Graph your budget constraint on a graph with books on the horizontal axis and other goods on the vertical axis. You can assume that the price of other goods, po, is $1. b) Derive expressions for the marginal utility of books (MUB), the marginal utility of other goods (MUO) and the marginal rate of substitution (MRS). c) Find the combination of books and other goods that maximizes your utility given your budget constraint. (You can consume fractions of books and fractions of other goods.) d) Now suppose you are given a $50 gift card for the bookstore. The gift card can only be spent on books. Graph your new budget constraint on a graph with books on the horizontal axis and other goods on the vertical axis. e) Find your new optimal combination of books and other goods. f) By how much did the gift card increase your utility?

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Tutorial Question #3: Art is a history major. He has to take two courses, one in philosophy and the other in mathematics. He is relatively good at philosophy, but not so much at math. He needs to spend 1 hour studying for philosophy in order to improve his grade by one point, while he needs to spend 3 hours studying for math in order to do the same. Art has a total of 60 hours to study for the two subjects. Suppose that he would get 50 points in each course if he didn't study at all. a) We want to analyze Art's problem with consumption theory. What are the goods, prices, and income? Write down the budget constraint. Illustrate it with a graph. (Hint: consider the improvements on Art's grades. That is, the goods are improvements in philosophy and math, and the prices are the cost of improvement measured in hours.) b) Art's satisfaction from his academic record is solely determined by the lower of his grades in philosophy and mathematics. For example, he is indifferent between 60 in philosophy and 70 in math, and 60 in philosophy and 60 in math. What kind of preferences does Art have over improvements in his philosophy and mathematics grades? Find a utility representation for these preferences. Derive his optimal choice of grades. Illustrate it in a graph.

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Chapter 6 Demand Practice Question #1: Ed consumes two commodities, x and y, and his utility function is min{x + 2y, y + 2x}. He chooses to buy 8 units of good x and 16 units of good y. The price of good y is $0.50. What is his income? a) $16 b) $40 c) $24 d) $32 Ans: A; Set x + 2y = y + 2x, so x = y, U = min {40, 32}, so his utility is 32. This utility is achieved by consuming y = 16 + (2)8 = 32 units of y. Each y costs $0.5 so income is $16.

Practice Question #2: Grace buys only two goods and her preferences can be described by a Cobb-Douglas utility function.

Her demand functions have which of the following features? a) Her demand for one of the two goods does not depend on income. b) Her demand for both goods does not depends on income. c) Her demand for each of the goods depends on income and on the prices of both goods. d) Her demand for each of the two goods depends only on her income and on the price of that good itself.

Ans: D; demand has the form x1 = (am)/p1, where a is the portion of income that Grace will spend on x1. Same idea for x2 but the share of income is (1-a).

Practice Question #3: Regardless of his income and regardless of prices, Alan always spends 40% of his income on housing, 25% on food, 20% on transportation and 10% on savings and 5% on recreation. This behavior is consistent with which of the following? a) Alan considers housing, food, savings and recreation as normal but he views food an inferior good. b) Alan has a Cobb-Douglas utility function for all of the goods and services described. c) Alan consumes all of the goods and services described in fixed proportions. d) All of the answers above are correct.

Ans: B; Demand for something is always a constant share of income (such as “a” or 0.4 for housing), and the units of this good this share of income can purchase is a*income/price.

Practice Question #4: Jada has homothetic preferences. Which of the following is CORRECT? a) Her preferences can be represented by a utility function that is homogeneous to degree 0. b) The utility function that describes her preferences will yield a demand function that is homogeneous of degree 1. c) She will spend a constant portion of her income on each good. d) All of the answers above are correct.

Ans: C; Homothetic preferences, or equivalently the corresponding utility function that describes such preferences are HD1 and the associated demand is HD0. She spends a fixed portion of her income on each good.

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Practice Question #5: Jada has homothetic preferences. When her income was $500, she bought 20 candles and 50 matchbooks. When her income increased to $1,000 and prices did not change, she buys a) 30 candles and 60 matchbooks. b) 40 candles and 100 matchbooks. c) 75 candles and 100 matchbooks. d) 100 candles and 200 matchbooks.

Ans: B; She spends a constant portion of her income on each good. Now her income doubles, so she consumes double the amount of each good.

Practice Question #6: Jada has homothetic preferences. When her income was $500, she bought 20 candles and 50 matchbooks. When the price of candles doubles and her income and price of matchbooks do not change, she buys a) 10 candles and 50 matchbooks. b) 12 candles and 50 matchbooks. c) 15 candles and 80 matchbooks. d) 18 candles and 60 matchbooks.

Ans: A; She spends a constant portion of her income on each good. Now the price of candles doubles, she buys 10… imagine candles = 0.2*m/price of candles, the denominator doubles.

Practice Question #7: Uma’s utility function is U(x1, x2) = 2(ln x1) + x2. Given her current income and the current relative prices, she consumes 10 units of x1 and 15 units of x2. If her income doubles, while prices stay constant, how many units of x1 will she consume after the change in income? a) 5 b) 8 c) 10 d) 20 Ans: C; She has a quasilinear utility function and her demand for x1 is independent of income.

Practice Question #8: “Demand functions are homogeneous to degree zero in all prices and income.” This means a) a proportional increase in all prices and income will leave quantities demanded unchanged. b) a doubling of all prices will not alter consumption decisions. c) prices directly enter individuals’ utility functions. d) an increase in income will cause all quantities demanded to increase proportionately.

Ans: A; If prices of everything double and income also doubles, consumers do not change their buying plans.

Practice Question #9: Suppose Jon’s preferences can be described by a Cobb-Douglas utility function of HD1. If the prices of all goods and income increase by the same proportion, Jon’s quantity demanded of all goods will a) decrease. b) increase. c) remain unchanged. d) increase or decrease depending on functional form of the demand equations.

Ans: C; Demands are HD0 if the utility function is HD1.

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Practice Question #10: If Akiko’s utility function is U = xA0.25xB0.75, the price of sushi rice is pA, the price of raw tuna is pB, and her income is m, then Akiko’s demand for sushi rice will be a) 0.75m/pA. b) m/(pA + pB). c) 0.25m/pA. d) 0.5m/pA. Ans: C; xA = [(0.25)/(0.25+0.75)]m/pA, xA = 0.25m/pA, so sushi rice takes up 25% of income.

Practice Question #11: The demand curve for good x and the Engel curve for good x have the same sign if x is a good. a) Inferior b) Normal c) Luxury d) Both B and C are correct.

Ans: A; demand slopes downward and if it is inferior, income rises and quantity demanded falls, so Engel curve also slopes downward.

Practice Question #12: Coffee and sugar are complements. As the price of sugar rises, the quantity demanded for coffee a) Rises b) Falls c) Does not change d) None of the answers above is correct.

Ans: B; Because they are consumed together, we can imagine they as being identical goods. If the price of coffee rises, consumers typically consumes less coffee.

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Practice Question #13: Which of the following utility functions represent preferences of a consumer who does not have homothetic preferences? a) U(x, y) = x0.5y0.5. b) U(x, y) = x + 2y. c) U(x, y) = x + y0.5. d) U(x, y) = min{x, y}.

Ans: C; Homothetic preferences give rise to utility functions that are homogeneous to degree 1 (HD1). Option a is HD1 because U = (tx)0.5(ty)0.5 = tx0.5y0.5. Option b is HD1 because U = (tx) + (2y)t = t (x + 2y). Option c is not HD1 because U = (tx) + (ty)0.5. For Option d, U = min{tx, ty} = t min{x, y} (plug some numbers in, if you double x and y (t=2), min U doubles).

Practice Question #14: The relationship between changes in income and the quantity demanded of a good indicates a) whether the good is a luxury or necessity. b) whether the good is normal or inferior. c) whether the good is a complement or substitute to other goods. d) Both a) and b) are correct.

Ans: D; Inferior good, higher income leads to lower quantity demanded; the other three types have higher quantity demanded.

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Practice Question #15: If income doubles and the quantity demanded of good more than doubles, then this good is a(n) a) Giffen good. b) inferior good. c) necessity good. d) luxury good.

Ans: D; A 50% increase in income leads to more than 50% increase in quantity demanded.

Practice Question #16: Two goods, x and y, are substitutes if a) an increase in px causes more y to be bought. b) an increase in px causes less y to be bought. c) an increase in py causes more y to be bought. d) an increase in income causes more of both x and y to be bought.

Ans: A; x and y compete with each other for consumers. When x is more expensive, consumers switch to y because both goods serve similar purposes.

Practice Question #17: Two goods, X and Y, are complements if a) an increase in px causes more y to be bought. b) an increase in px causes less y to be bought. c) an increase in py causes more y to be bought. d) an increase in income causes more of both x and y to be bought.

Ans: B; think of x and y as the same good.

Practice Question #18: The demand for jello (J) in terms of the price of jello (pj ), the price of condensed milk (pc) and income (m) is given by the following: J = 4 pm+ p j

c

Which of the following is true? a) Jello is a normal good, and jello and condensed milk are substitutes. b) Jello is a normal good, and jello and condensed milk are complements. c) Jello is an inferior good, and jello and condensed milk are substitutes. d) Jello is an inferior good, and jello and condensed milk are complements.

Ans: B; From the demand function, we can see that if income increases, demand for jello goes up so jello is a normal good. We can also see that if the price of condensed milk increases, the demand for jello goes down. Therefore, jello and condensed milk are complements.

Practice Question #19: If good x and good y are the only two goods a person consumes, and the demands are homogenous to degree zero in prices and income. This means doubling the prices of both goods and income will a) Double the quantity demanded for both goods. b) Increase the quantity demanded for both goods but less than double. c) Leave the quantity demanded for both goods unchanged. d) More than double the quantity demanded for both goods.

Ans: C; The vertical and horizontal intercepts and slope of the budget constraint do not change and neither will the indifference curve and budget line tangency point.

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Practice Question #20: For x1, derive and draw the Engel curve, x1(m), the demand curve, x1(p1) and briefly discuss if x1 is a normal or inferior good. a) U(x1, x2) = x 11/3 x22/3 b) U(x1, x2) = x1 2+ x 22 c) U(x1, x2) = 2 x1 +3 x2 Ans: m a) The “direct” demand is x1 = 3 p , and to plot price on the vertical axis, rearrange to get the 1 m m “inverse” demand p1 = 3 x , and the slope is , downward sloping. As for the Engel curve, 3 x 12 1 we rearrange to get m = 3x1p1, and the slope is 3p1, so normal.

b) If p1 > p2, spend all income on x2. If p1 = p2, any amount of x1 between 0 and m/p1. If p1 < p2, spend all income on x1, so x1 = m/p1. Normal if buys x1.

c) If p1/p2 < 2/3, then buy only x1. If p1/p2 = 2/3, then buy any amount of x1 between 0 and m/p1. If p1/p2 > 2/3, spend all income on x2, so x1 = 0. Normal if buys x1.

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Tutorial Question #1: Determine if the following functions are homogeneous and if so, to what degree. a) x12x2 + 3x1x22 + x23 b) x17x2x32 + 5x16x24 – x25x35 c) 4x12x23 – 5x1x22 d) 5x1 + 10x2 + 3x3 + x4

Tutorial Question #2: Suppose the utility function is U(x1, x2) = x 10.5+2 x 2. For x1, derive and draw the Engel curve, x1(m), the demand curve, x1(p1) and briefly discuss if x1 is a normal or inferior good.

Tutorial Question #3: Isaac buys music albums both in the form of CDs and as digital downloads. His utility from albums on CD (C) and digital albums (D) is given by: U(C, D) = 8C + 10D The price of a CD is $5 and the price of a digital album is $6 and Isaac's income is $30. Draw two graphs, one showing the Engel curve for CDs and one showing the Engel curve for digital albums. Be certain to clearly label slopes and intercepts with their values and to label the axes.

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Chapter 8 Slutsky Equation Practice Question #1: Kai’s utility function is U = x 12x2, income is $600, and p 2 = $1. The price of x 1 falls from $10 to $5. a) How much does the consumption of x1 change due to this price change? b) Find the substitution and income effects on x1, using Slutsky compensation? c) Find the substitution and income effects on x1, using Hicksian compensation?

Ans: a) MRS = 2x2/x1, old price ratio is 10, so x1 = 40, x2 = 200, U = 320,000. With the new price ratio = 5, x1 rises to 80, so the total rise in x1 = 40. b) Mark was buying 40 units, now each unit is $5 cheaper, so we need to adjust income by $200. The “new” income is $400. With this new income and new price $5, x1 = 53.3333, so SE = 13.3333. The YE = 40 – 13.3333 = 26.6666. c) Old U = 320,000, new price ratio is 5, so x2 = 2.5x1, substitute back into U = 320,000 = 2.5 x 13, x1 = 50.3968, so the Hicksian SE = 10.3968, the YE = 29.6032.

Practice Question #2: Jim’s preferences are represented by the utility function u(x₁, x₂) = min{5x₁, x₂}. His income is $200 and the price of good x₂ is $1. Suppose that the price of x₁ drops from $15 to $5. a) What is the total change in consumption of x₁? b) Find the total change into substitution effect and income effect, using Hicksian compensation.

Ans: a) Jim should consume at 5x₁ = x₂. Plug this into the budget constraint 200 = 15x₁ + x₂, which becomes 200 = 15x₁ + 5x₁ = 20x₁. This gives us x1 = 10, and x2 = 50. After the price change, 200 = 5x₁ + x₂, we get 5x₁ + 5x₁ = 10x₁ = 200. This gives us x1 = 20, and x2 = 100. b) Because Jim cannot substitute across the two goods, he consumes them at the ratio of 5x₁ = x₂.

Jim would not buy more of x₁ or x₂ if the prices change. Jim will only buy more if the “new” income allows him to buy more of both in the ratio of 1-to-5. There is only YE with this type of utility function and in this case it is doubling the consumption quantities of both goods.

Practice Question #3: Kelly buys two goods, food (F) and clothing (C), with the utility function U(F, C) = FC + F. She has an income of $20, and the price of clothing is $4 per unit. Food is on the horizontal axis. a) Compute the optimal consumption bundle when the price of food is $1 per unit. b) Suppose the price of food rises to $4 per unit. What is the new optimal bundle? c) Find the Slutsky SE and YE. d) Find the Hicksian SE and YE. 33

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Ans: a) MRS = (C+1)/F and the p ratio is 0.25, so F = 4C + 4, sub into income to get F = 12 and C = 2. b) MRS = (C+1)/F and the price ratio is 1, F = 3 and C = 2. Total drop in F = 9. c) Change in income is 12*3 = $36. This means Kelly needs an additional $36 to offset the price increase. With m = $56, F = 4C + 4, then F = 7.5. The SE is a drop in the consumption of F by 4.5 units. The YE is the remaining drop of 4.5 because the final F = 3 and the old F = 12. d) Initial F and C were (12, 2), so initial U = 24 + 12 = 36. To keep U = 36 but price ratio is now 1, so F = C+1. Substitute this into U = C(C+1) + (C+1), or U = (C + 1)2 = 36, so C = 5 and F = 6.

The SE is 12 dropping to 6, so SE = -6 and YE = -3.

Practice Question #4: Continue with Question #3: Now suppose Kelly’s utility function is U(F, C) = 2F0.5 + C. She has an income of $20, and the price of clothing is $4 per unit. Food is on the horizontal axis. a) Compute the optimal consumption bundle when the price of food is $1 per unit. b) Suppose the price of food rises to $4 per unit. What is the new optimal bundle? c) Find the Slutsky SE and YE. d) Find the Hicksian SE and YE. Ans: a) MRS = 1/F0.5, price ratio = 0.25, so F = 16, C = 1, U = 9. This is a quasi-linear function. b) MRS = 1/F0.5, price ratio = 1, so F = 1, C = 4, so F dropped by 15. c) She was buying F = 16, price has risen by $3, so she needs an additional income of $48. Her new income is $68. Use this income and MRS = 1/F0.5, price ratio = 1, so SE = -15 and YE = 0. d) We expect SE = -15 and YE = 0. To check, U = 9 = 2F0.5 + C, MRS = 1/F0.5, price ratio = 1. This gives us F = 1, so indeed the consumption of F has dropped by 15.

Practice Question #5: When price increases, the income effect a) is negative for normal goods. b) is negative for inferior goods. c) is negative for a Giffen good. d) is positive for a luxury good.

Ans: A; price increase reduces real income and the quantity demanded for normal good falls.

Practice Question #6: If Sam buys only two goods that are perfect complements, such as 1 flashlight with 2 batteries, then _ a) there is no substitution effect from a change in the price of flashlight. b) there is no income effect from a change in the price of flashlight. c) an increase in income will not affect battery purchases. d) None of the answers above is correct.

Ans: A; Because Sam would not want to buy more flashlights if she cannot also buy two more batteries. Of course, this is assuming that she cannot buy cheaper flashlights and store them.

Practice Question #7: If a good is a normal good and its price rises, the quantity demanded a) may or may not fall. b) will always fall. c) will always rise. d) will remain unchanged.

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Ans: B; Higher price means lower real income, and the demand for normal goods falls when income falls. SE and YE move in the same direction.

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Practice Question #8: Sam’s utility function is U = AB. The price of A was $1, the price of B was $2, and his income was $40. If the price of A rises to $5, the substitution effect will reduce Sam’s consumption of A by . a) 4 b)13 c) 8 d) 16 Ans: C; A was 20, falls to 4, so total drop is 16. Of this 16, SE = drop of 4 with “new” income = $120 and MRS = price ratio gives us B = 2.5A. Practice Question #9: Sam’s utility function is U = AB. The price of A was $1, the price of B was $2, and his income was $40. If the price of A rises to $2, the substitution effect will reduce Sam’s consumption of A by . a) 3 b) 5 c) 6 d) 4 Ans: B; A was 20, falls to 10, so total drop is 10. Of this 10, SE = drop of 5 with “new” income = $60 and MRS = price ratio gives us B = A. Practice Question #10: Suppose that Pam has $465 to spend on tickets for her trip. She intends to spend the entire amount on tickets and prefers traveling first class to traveling second class. She needs to travel a total of 1,500 miles. Suppose that the price of first-class tickets (F) is $0.40 per mile and the price of second-class tickets (S) is $0.20 per mile. How many miles will Pam travel by second class? a) 225 b) 675 c) 550 d) 1,000 Ans: B; $0.4(1,500 – S) + $0.2S = $465 where 1,500 – S = F, so S = 675.

Practice Question #11: Suppose that Pam has $825 to spend on tickets for her trip. She intends to spend the entire amount on tickets and prefers traveling first class to traveling second class. She needs to travel a total of 2,000 miles. Suppose that the price of first-class tickets (F) is $0.60 per mile and the price of second-class tickets (S) is $0.30 per mile. How many miles will Pam travel by first class? a) 0 b) 400 c) 350 d) 250 Ans: A; see previous question. Pam will travel exclusively first-class.

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Practice Question #12: Ben purchases goods x and y. The graph below shows how his preferred bundle changes as his income rises from m1 to m2 to m3. His initial consumption bundle is (x1,y1). Which of the following statements is CORRECT?

a) Both goods x and y are normal. b) Both goods x and y are inferior. c) Good x is normal and good y is inferior. d) Good x is inferior and good y is normal.

Ans: D; quantity demanded for x falls and quantity demanded for y rises.

Practice Question #13: Amy buys goods x and y. Her initial bundle is A. The graph below shows how she adjusts her bundles after an increase in px. The substitution effect is and the income effect . The total effect is . These are changes.

a) XA – XC; XA – XB; XB – XA; Hicksian b) XA – XC; XA – XB; XB – XA; Slutsky c) XC – XA; XB – XC; XB – XA; Hicksian d) XC – XA; XB – XC; XB – XA; Slutsky Ans: C; Bundles A and C are on the original indifference curve after this price increase.

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Practice Question #14: Continue with the previous question: The substitution effect for good y is represented by income effect for good y is represented by . The total effect is represented by .

. The

a) YC – YA; YB – YC; YB – YA. b) YA – YB; YB – YC; YA – YC. c) YA – YC; YA – YB; YB – YA. d) YC – YB; YA – YB; YA – YC. Ans: A; As x becomes more expensive, x falls from A to C and correspondingly, y rises from A to C. As the higher px leads to a fall in real income, quantity demanded for x falls from C to B, which corresponds to y falling from C to B. The total change is therefore only A to B. Practice Question #15: Consider the graph below: A consumer’s initial choice of goods x and y is A. After a decrease in the price of good x the consumer chooses a different utility maximizing bundle. The change in consumption for good x can be split into a substitution effect equal to and an income effect equal to . The total effect for good x is .

a) 3; 2; 5. b) -3; -2; -5. c) 2; 3; 5. d) -2; -3; -5.

Ans: A; SE is points D versus A, which corresponds to 6 – 3 = 3 units. YE is points B versus D, which corresponds to 8 – 6 = 2 units. The total change is 2 + 3 = 5 units.

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Practice Question #16: Jon only eats apples and oranges. Bundle A (3A, 11O), in the graph below shows his initial consumption. His income on budget constraint I1 is $10. After a decrease in the price of apples his budget constraint rotates to I2 and Jon chooses B (8A, 9O). On budget line I3 Jon has $8, and on budget line I4 he has $12. Budget line I3 was drawn so that the utility level is the same as at point A and the prices are the same as at point B. Budget line I4 was drawn so that the utility level is the same as at point B and the prices are the same at point A. Suppose the government does not want apples to become cheaper because this could lower the income of farmers. The government sets the price of apples to their original price. To compensate Jon, the government can give him .

a) $2 b) $3. c) $4. d) $5 Ans: A; The subsidy has to move Jon’s preferred bundle from point A to point C. At point C prices are the same as at A, and Jason is as happy as at B. The monetary difference between the two points is $12 (I4) - $10 (I1) = $2.

Practice Question #17: Continue with the previous question: Suppose the government does not want to give out cash. How many apples and oranges should the government give out instead? a) 1 apple and 1 orange b) 2 apples and 1 orange c) 1 apple and 2 oranges d) 2 apples and 2 oranges Ans: B; Compares A and C, Jon has to get two apples and one orange.

Practice Question #18: Continue with the previous questions: Now assume that some producers charge the high price for apples and others charge the low price. The maximum fee the low-price producers can charge Jon for allowing him to buy from them, such as an entrance fee for their farmers’ market, is . a) $6 b) $5 c) $4 d) $2 Ans: D; The highest fee farmers can charge is the one that would put Jon at point D, which has the same utility level as point A. They cannot make him pay more because this would put him on a lower utility level than at point A and he would prefer to pay the high price for apples and no entrance fee. The monetary difference between I1 and I3 is $10 - $8 = $2.

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Practice Question #19: Refer to the graph below. Price of X has decreased. Which of the following statements is CORRECT?

a) Both goods are normal goods. b) Only Y is a normal good. c) The income effect moves the best affordable consumption bundle from A to B. d) The substitution effect moves the best affordable consumption bundle from A to B. Ans: D; SE shows the consumption bundle with old price, “new” income.

Practice Question #20: Given below are two of Alex’s indifference curves. When Alex’s income is $160, and the price of plays is $20 and the price of basketball games is $40, he has budget line represented by BC1 and his optimal consumption point is A. If Alex’s income stays at $160, but the price of plays goes up to $80 and the price of basketball games stays at $40, he has budget line given by BC2 and his optimal consumption point is B. Budget line BC3 represents what Alex’s budget line would be if he had income $320 and the price of plays was $80 and the price of basketball games was $40. In this case his optimal consumption point would be C.

a) What happens to the optimal quantity of plays when the price goes from $20 to $80? b) What portion of this change is due to substitution effect? c) What portion of this change is due to income effect? d) Based on your previous answers, are plays a normal or inferior good for Alex?

Ans: a) Alex’s optimal choice of plays changes from 4 plays to 1 movie, for a decrease of 3 plays. b) A decrease of 2 plays. c) A decrease of 1 play. d) A normal good because when income goes down, the consumption of plays goes down as well.

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Tutorial Question #1: Ena splits her time on the weekend between baking bread and playing games. Her utility function is given by the following: U (B, G) = B2G where B is the loaves of bread she consumes and G is the hours of games she plays. It takes Ena two hours to bake a loaf of bread. a) Suppose that Ena has 18 hours of free time on the weekend to devote to bread baking and playing games. Write down a budget equation that captures Ena's affordable combinations of B and G. b) Given your budget constraint from part a) and Ena's utility function, determine how many loaves of bread Ena will bake and how many hours she will spend playing games. You can assume that she has no other use for her time that will increase her utility. c) Suppose that Ena gets better at baking bread. Each loaf now only takes one hour to bake. What is her new optimal consumption bundle of bread loaves and hours of game playing? d) How much of the change in bread loaves was due to the income effect? How much was due to the substitution effect?

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Tutorial Question #2: David’s utility from shirts (S) and hats (H) is given by the following utility function: U(S, H) = 5S3/4 H 3/4 a) Assuming that David has $400 to spend on hats and shirts, derive the demand for hats with pH as the price of hats and m as income. Also derive the demand for shirts with pS as the price of shirts. b) Let the price of a shirt be $10 and the price of a hat be $10. What are the quantities demanded for hats and shirts, respectively? Also find the level of utility. c) Slutsky SE and YE: The price of a shirt decreases to $5. Derive the SE and YE for shirts. d) Hicksian SE and YE: The price of a shirt decreases to $5. Derive the Hicksian SE and YE for shirts.

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Chapter 14 Consumer Surplus Practice Question #1: The difference between what consumers are willing to pay for each unit of product and what they actually pay for that unit is called a) producer surplus. b) consumer surplus. c) reservation price. d) equilibrium price.

Ans: B; Consumer surplus is “good bargain”.

Practice Question #2: Graphically, consumer surplus is measured as a) the area above the demand curve. b) the area under the demand curve and above the equilibrium price. c) the area above the supply curve and below the equilibrium price. d) the area under the supply curve.

Ans: B; A typical triangle under the demand curve, with the base of triangle determined by the price and the corresponding quantity.

Practice Question #3: represents payment received but not required in order for producers to be willing to supply a specific amount of a product. a) Producer surplus b) Consumer surplus c) Price ceiling d) Price floor Ans: A; We can think of producer surplus loosely as “profit”. The precise definition of producer surplus depends on fixed costs. We will discuss this more in subsequent chapters.

Practice Question #4: Suppose demand is Q = 12-2P and supply is Q = P. Producer surplus in equilibrium is equal to a) $8 b) $12 c) $16 d) $18 Ans: A; The triangle under the price and above the supply curve.

Practice Question #5: Suppose demand is Q = 12-2P and supply is Q = P. The value of consumer surplus is a) $4 b) $8 c) $12 d) $16 Ans: A; The triangle under the demand curve and above the price.

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Practice Question #6: At the market equilibrium that is Pareto efficient, a) producer surplus exceeds consumer surplus by the greatest possible amount. b) total consumer surplus is zero. c) the sum of consumer surplus and producer surplus is maximized. d) total producer surplus is zero.

Ans: C; PE means that no one else can be made better off without making someone else worse off. If CS and PS are maximized, there is no DWL and all efficiencies have been achieved.

Practice Question #7: Jen’s annual demand function for movie tickets is Q = 24 - 2P. A movie ticket costs $8. The local movie theatre decides to reduce the price of a ticket to $6 and require an access card that would have to be renewed annually. Without the card Jen would not be allowed to buy any movie tickets. The maximum amount Jen would pay for the access card is a) $10. b) $16. c) $20. d) $36.

Ans: C; The maximum amount she would pay for the access fee is the difference between her consumer surplus when the price is $6 per movie and the consumer surplus when the price is $8 per movie. This is represented by areas B and C in the graph below. B + C = 16 + 4 = 20.

Practice Question #8: Jen’s annual demand function for movie tickets is Q = 24 - 2P. A movie ticket costs $8. Jen’s consumer surplus is equal to a) 4. b) 8. c) 16. d) 32.

Ans: C; Jen’s consumer surplus is given by area A in this figure, and it is equal to (12– 8)(8)/2=32/2=16. See graph in the previous question.

Practice Question #9: Dino loves guavas. His demand for guavas is p = 25 – 2x, where x is the number of guavas he consumes. If the price of guavas rises from $1 to $3 per fruit, Dino’s consumer surplus drops by a) $15. b) $25. c) $24. d) $23.

Ans: D; at $1 each, quantity demanded is 12; at $3 each, quantity demanded is 11; the drop in CS is the area of the trapezoid, i.e., (11 + 12)/2*2 = $23.

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Practice Question #10: The question below is about the market for eggs and government intervention. a) Suppose the demand for eggs, measured in cartons of 120 eggs, is given by P = 24 - Q and the supply curve by P = $12. What are the market equilibrium price and quantity? b) What is the consumer surplus at the equilibrium? c) If the government decides to implement a price floor at a value of P = $14, what are the new quantity traded and the loss in consumer surplus?

Ans: a) Q = 12 cartons and P = $12. b) CS = (24 – 12)*12/2 = $72. c) Q =10, CS = (24 – 14)*10/2 = $50, so loss = $22.

Practice Question #11: The question below is about the market for USB storage keys and government intervention.

The supply curve for USB storage keys is given by P = 1 + 0.2Q. The demand is given by P = $17. a) What are the equilibrium price and quantity, respectively? b) What is the supplier surplus at the equilibrium? c) If the government imposes a price ceiling in this market at a price P = $15, what are the new quantity supplied and the new supplier surplus? d) What is the loss in surplus to the suppliers?

Ans: a) Q = 80 and P = $17. b) PS = (17 – 1)*80/2 = $640. c) Q = 70, PS = (15 – 1)*70/2 = $490. d) $150.

Practice Question #12: Suppose that the market for milk can be represented by the following equations: Demand: P = 12 – 0.5Q Supply: P = 0.1Q where P is the price per gallon, and Q represents the gallons of milk consumed per day. a) Calculate the equilibrium price and quantity of milk. Also calculate the values for consumer surplus (CS) and producer surplus (PS). Let the sum of these two values be termed social surplus (SS). b) To help dairy farmers, the government sets a minimum price of $2.50 per gallon of milk. What is the new quantity of milk sold in the market? c) Calculate the new CS, PS and SS after the price supports are implemented. Compare your new SS and old SS from part a). The difference is what introductory micro courses call deadweight loss. What is the value of this loss?

Ans: a) Q = 20, P = $2, CS is (12 – 2)*20/2 = $100, PS = 2*20/2 = $20, SS = $120. b) Substitute P = 2.5 into the demand or supply equation, Q = 19. c) CS = (12 – 2.5)/2*19 = $90.25. PS is a bit tricky to find. First, find the price on the supply curve when Q = 19. This price is $1.9. You can treat the area as a trapezoid and find (0.6+2.5)/2*19 = $29.45 or treat it as the sum of a rectangle and a triangle, which is 0.6*19 + 1.9*19/2 = 11.4 + 18.05 = $29.45. The new SS is $119.7, which is $0.3 lower than in part a). This is the deadweight loss.

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Practice Question #13: Consider the market for whale watching boat rides. In this market, the whale watching boat drivers are producers and the whale watching riders are consumers. The demand and supply equations are Demand: P = 10 – 0.5Q

Supply: P = 0.5Q

a) Find the equilibrium price and quantity, CS, PS and SS. Assume now that the government in Tofino imposes a quota of 8 rides per year. To implement this quota, the government requires the drivers to purchase an operating license that allows them to build and operate a whale watching boat. b) Calculate the price that consumers will be charged per ride if they purchase only 8 rides per year.

Calculate the price producers must receive in order to be willing to sell 8 rides. Use this information to calculate how much the government can charge drivers (per ride) for the right to operate a whale watching boat (this is called a quota rent). c) Calculate the consumer surplus, producer surplus, and the total quota rent. Compare the sum with the social surplus of the market you obtained in part a). Is this policy desirable from an efficiency standpoint? Explain your answer.

Ans: a) P = $5 per whale watching ride, Q = 10 whale watching rides. PS = $25, CS = $25, SS = $50. b) Pconsumer = $6 per whale watching ride, Pproducer = $4 per whale watching ride, quota rent = $2 per whale watching ride. Drivers are willing to pay up to $2 per whale watching ride to purchase the right to operate a boat. c) PS = $16, CS = $16, total quota rent: $16 (quota rent per ride times the quantity sold and bought), SS = $48. This policy is not desirable from an efficiency standpoint. The government ends up creating a distortion in the market. Note that the equilibrium quantity is reduced by 2 units. The deadweight loss (a measure of the inefficiency of the program) equals $2.

Practice Question #14: m Suppose the demand is x = and m = $100. Calculate CS for when p =4 falls to p =2.

1 1 1 3 p1 Gentle reminder: To find the area under the demand curve, we need to integrate along the demand m x = curve because the demand equation is not linear. If the demand equation is , then we need to 1 3 p1 find its antiderivative or integrate this equation. Antiderivatives are covered in the final chapters in Math 209 and equivalent.

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Ans: Integrating the demand gives us 4 m ∫ 3 p d p1, which can be simplified to 2 1 4 100 ∫ 3 p d p 1, 2

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100 1 ∫ p d p1, 3 100 2 1 [ ln( p )], evaluated at p =4 and p =2, 1 1 1 3 100 [ ln( p =4 )−ln( p =2)], 1 1 3 which is equal to CS = $23.105.

Practice Question #15: 20 Suppose the demand p = . Calculate CS for when x =0 and x =3.

1 1 1 x1+1 Ans: The vertical intercept is p1 = 20, x1 = 3, p1 = 5.

We are integrating along the horizontal axis.

3 ∫ x 20+1 d x1, 0

1

3

20∫

1

dx, x 1 +1 1 20[ ln ( x1 +1)], evaluated at x1=0 and x1=3, 20[ ln( x1 +1=4)−ln ( x1+ 1=1)], which is equal to the yellow area = $27.7259.

0

CS is the difference between the yellow area and the green area, the triangle above $5. The green area is expenditure, $5*3 = $15. CS = $12.7259.

Practice Question #16: The questions below ask you to decide whether the CV or EV measurement should be applied. Please note that there is a degree of subjectivity in the answers. Supplement your choice with an explanation.

CV: “Old utility, new prices” - After change – CV is the payment/compensation in order to for you to accept a change… CV brings you back to the old utility, given the (price) change.

- Price increase – CV is how much we pay you to offset the bad effects from the price increase.

- Price decrease – CV is how much you are willing to pay to continue enjoying this price decrease.

EV: “New utility, old prices” - Before change – EV is the payment/compensation if the change has not happened… EV brings you to the new utility, without the (price) change.

- Price increase – EV is how much you are willing to pay to prevent this price increase.

- Price decrease – EV is how much we have to pay you if you cannot enjoy this price decrease.

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a) CV or EV: Proposal to construct or extend the Keystone Pipeline which could lower gasoline prices but kill birds. For simplicity, you do not care about birds and you drive gasoline-fueled cars. How much do environmental groups have to pay you to support not having this project? b) CV or EV: Windmills installed in fields kill off the bee population and this leads to a rise in the price of honey. You love honey and do not care for green energy. How much do the operators of the windmills need to pay you to co-exist with the windmills? c) CV or EV: Loblaw and other grocery chains recently admitted to price-fixing bread over the past 16 years. Price-fixing means the stores kept prices high and refrained from offering sales. Loblaw’s parent company settled a $500 million class-action lawsuit. Is this likely a CV or EV measurement? d) CV or EV: The government ponders on whether to subsidize first-time home buyers or give a rebate to home-owners on the first three years of home ownership. CV or EV measurement for each choice?

Ans: a) EV – project not in place yet, you need to be paid to not enjoy the price drop. b) CV – installed, you need payment to offset the higher price of honey and accept the windmills. c) CV – have been paying 10 cents more per loaf, which amounts to around $2,000 for a family of four with an average consumption of five loaves of bread each week for over 16 years. d) EV for subsidy (prior to purchase) and CV for home-owners (already bought).

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Practice Question #17: Suppose your utility function is U(x, y) = x0.3y0.7. If income is $1,000, px = $2 and py = $5. a) Find the initial equilibrium quantities for x and y and utility level. b) Find the value for compensating valuation (CV) if px rises to $4.

Ans: a) With the initial prices, x = 0.3*1,000/2 = 150, y = 0.7*1,000/5 = 140, and U = 142.93. b) If px = $4, x = 0.3m’/4 and y = 140 and U = 142.93. Solving for m’ gives $1,231, so CV = $231.

Practice Question #18: You enjoy phoning your friends in Asia and drinking coffee. Each hour of long-distance phoning costs $2 and each cup of coffee costs $1. Your utility function is U(x, y) = xy, where x is the hours of phoning and y the number of cups of coffee you drink. Your income is m = $20. a) Calculate your quantity demanded for x and y and the utility level. b) Now the price of telephone calls drops to $1 (a price decrease). Calculate your new utility level. c) Calculate the compensating variation (CV) for the price decrease. d) Calculate the equivalent variation (EV) for the price decrease.

Ans: a) With m = 20, px = 2, and py = 1, x = 5, y = 10. The resulting utility level is 50. b) With m = 20, px = 1, and py = 1, x = 10, y = 10. The resulting utility level is 100. c) CV is based on “old utility, new prices”: U = 50, px = 1, py = 1, x = 0.5m/px, y = 0.5m/py, so U = 50 = 0.25(m’)(m’), so the new income m’ = 14.1421. CV is 20 – 14.14 = $5.8579. This is the amount that we can take away from you and because of the price decrease. d) EV is based on “new utility, old prices”: U = 100, px = 2, py = 1, x = 0.5m’/2, y = 0.5m’, so U = 100 = 0.125(m’)(m’), so m’ = $28.2843. EV is the income that we have to pay you if you cannot enjoy this price decrease. You should receive in order to enjoy U = 100 when px = 2 is an additional $8.2843.

Practice Question #19: Suppose your utility function is U(x, y) = xy. Let the price of x = $1 and leave the price of y be py. a) Derive the Marshallian demand for x and y. b) Suppose m = $100, initial py = $1. Find x, y and initial utility. c) Now suppose m = $100, new py = $4 (a price increase). Find x, y and this new utility. d) Calculate the compensating variation (CV). e) Calculate the equivalent variation (EV).

Ans: a) x = 0.5m, y = 0.5m/py. b) x = 50, y = 50, U = 2,500. c) x = 50, y = 12.5, U = 625. d) CV is “old utility, new prices”: The new x = 0.5m’, y = 0.5m’/4, and U = 2,500, so 2,500 = 0.0625(m’)(m’), so m’ = $200. This means we need to pay you $100 if you were to enjoy your initial utility because of the py increase. e) EV is “new utility, old prices”: The new x = 0.5m’, y = 0.5m’/1, and U = 625, so 625 = 0.25(m’) (m’), so m’ =$50. This means we can take away from you $50 if we prevent py from rising.

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Practice Question #20: Suppose your utility function is U(x, y) = x0.5y0.5. Let the price of y = $1 and the price of x = $1. a) Derive the Marshallian demand for x and y. b) Suppose m = $100, initial px = $1. Find x, y and initial utility. c) Now suppose m = $100, new px = $2 (a price increase). Find x, y and this new utility. d) Calculate the compensating variation (CV). e) Calculate the equivalent variation (EV). f) Depict your results graphically.

Ans: a) x = 0.5m/px, y = 0.5m/py. b) x = 50, y = 50, U = 50. c) x = 25, y = 50, U = 35.3553. d) CV is “old utility, new prices”: The new x = 0.5m’/2, y = 0.5m’, and U = 50, so m’ = $141.42.

This means we need to pay you $41.42 if you were to enjoy your old utility with the increase in py. e) EV is “new utility, old prices”: The new x = 0.5m’, y = 0.5m’, and U = 35.3553, so m’ =$70.71.

This means we can take away from you $29.29 if we prevent py from rising. f)

Practice Question #21: Consumers for DVDs have a downward sloping demand curve and are currently buying 100 DVDs. If the price of a DVD drops by $1, consumer surplus will a) Go up by less than 100. b) Go up by 100. c) Go up by more than 100. d) Go down.

Ans: C; The consumer surplus goes up by 100 on the DVDs they are already buying and the lower price will increase quantity demanded. Consumer surplus increase by more than 100.

Practice Question #22: If the demand is horizontal, consumer surplus is . If the supply is vertical, producer surplus is a) Zero; Zero b) Zero; undefined c) Undefined; zero d) Undefined; undefined Ans: A; Try drawing both curves and finding the CS and PS.

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Tutorial Question #1: There have been continuing efforts in various countries to open up Alaska for energy exploration.

There are two goods, natural gas (x1) and other goods (x2). Suppose your income is m = $25,000. The current price of natural gas p1 = $5 and the price of other goods is p2 = $1. If exploration in Alaska is allowed, the price of gas will fall to $1. However, a group of environmentalists is outraged about this idea. Suppose you are not an environmentalist and your utility function is U = 30x10.5x20.5. a) Find the quantity demanded for each good and your initial utility level. b) Suppose the exploration takes place and the price of gas does fall. Find the new quantity demanded for each good and your new utility level. c) How much of the change in the quantity demanded for gas is due to the substitution effect? What about the income effect? d) Scenario #1: Suppose the exploration is approved and is scheduled to begin but it has yet to proceed. How much do the environmentalists have to pay you if they want you to not protest against their efforts to stop this project? Hint: CV or EV? See Practice Question #16. e) Scenario #2: Suppose the environmentalists are unsuccessful in stopping the project. The drilling completes and the price of gas is $1. The “green” media is gaining momentum and there are constant protests against the operation of the mine. How much would you be willing to pay for keeping the mine open and gas price staying at $1? Hint: CV or EV? See Practice Question #16.

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Tutorial Question #2: Consider a consumer with the following Cobb-Douglas utility function for two goods, 1 and 2, U = x11/3x22/3 Assume that the consumer faces a price of $1 for good 2 and this price stays constant. The consumer’s income is represented by m = $100. The price of good 1 decreases from $4 to $2. a) Find consumer surplus (CS). See Practice Questions 14 and 15 for review. b) Find the compensating variation (CV). c) Find the equivalent variation (EV).

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Chapter 15 Market Demand Practice Question #1: There are two consumers, 1 and 2, in the market for coffee (C). Their individual inverse demand curves are given by C1 = 10 – P1 and C2 = 20 – P2. Graph the market demand curve. Make certain to label all slopes, intercepts and kinks with the appropriate values.

Ans: Step 1: Rewrite both equations with price on the left-hand side. We get P1 = 10 – C1 and P2 = 20 – Q2.

Step 2: Because the demands have different intercepts, there is a kink in the market demand.

The vertical intercept of the market demand is the higher value (P = $20). The kink is at P = $10.

Step 3: The upper portion of the market demand, which is from P = $20 down to P = $10, the slope is -1. Only Consumer 1’s demand is relevant. For the portion that is below P = $10, we need to add across the two demands but make sure that we add across the Q’s, not the P’s. This gives us Q = 30 – 2P, or P = 15 – 0.5Q. The slope of the market demand below $10 is -0.5. The horizontal intercept is when P = 0 for both consumers, giving us 10 and 20.

Practice Question #2: The demand for apple pie for Alan is given by DA = 20 − 2P and for Ben is given by DB = 10 − 3P where P is the price of apple pie. The market supply of apple pie is given by S = −1 + 2P. a) Derive the market demand for apple pie with these two consumers. b) What is the price elasticity demand for apple pie when the price is $5?

Ans: a) Step 1: Alan’s demand is PA = 10 – 0.5Q and Ben’s demand is PB = 3.33 – 0.33Q. Step 2: Alans’ demand is relevant only if P is below $10. For Ben, it is $3.33. There is a kink at the lower price, which is at P = $3.33.

Step 3: Because demands in the question are already expressed in the form that quantity is on the left, we can add across the two demands directly. This gives us D = 30 – 5P. Rewriting gives us P = 6 – 1/5Q, and the useful information we get from this equation is that the market demand has a slope of -1/5, or -0.2 for the portion of the market demand that Ben is also demanding.

Therefore, the market demand is P = 10 – 0.5Q from P = $10 and P = $3.33, and the market demand is P = 3.33 – 0.2Q for all prices below $3.33. b) P = $5 puts us onto the top segment of the market demand curve with only Alan’s demand.

P P P dQ (−2) = -1 (−2) Ɛ= Ɛ= Ɛ= Q dP Q=20−2 P 10 53

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Practice Question #3: The demand for rice in Canada is P = 5 - 0.5Q and in Japan it is P = 5 - 0.25Q. The total demand is . a) P = 10 - 0.75Q b) P = 10 - 1.5Q c) P = 5 - 0.167Q d) P = 4 - 0.25Q Ans: C; Because both demands have the same intercept, the total demand does not have a kink.

This simplifies our derivations. We add demands by adding quantities. Rearrange the equations to add the two Q’s. This gives us Q = 10 – 2P and Q = 20 – 4P, total Q = 30 – 6P or P = 5 – 1/6Q. Practice Question #4: There are four consumers for Mars candy bar. For each consumer, if the price of Mars falls by $1, quantity demanded rises by 2. The slope of each demand is and the slope of the market demand is . a) -1/2; -1/8 b) -1/2; -1/2 c) -2; -4 d) -2; -8 Ans: A; For a single customer, for every change in price of -1, demand rises by 2. The slope of the demand curve is -1/2. For the market demand curve, for every change in price of -1, demand rises by 8 (four individuals). The slope of the market demand curve is -1/8.

Practice Question #5: A market has two consumers and each consumer has a demand curve with a slope of -2. a) The market demand may be upward-sloping. b) The market demand will have a slope of -2. c) The market demand will have a slope between 0 and -2. d) The market demand will be steeper than the individual demand curves.

Ans: C; The market demand is flatter with a slope of smaller than -2 (absolute value).

Practice Question #6: The cross-price elasticity of demand for apples and okra is −1.6. Which of the following is CORRECT? a) Apples and okra are substitutes. b) Demand for apples in elastic. c) Apples and okra are complements. d) Demand for okra is inelastic.

Ans: C; A negative cross price elasticity means that when the price of one good goes up, demand for the other good goes down. This tells us that the two goods are complements.

Practice Question #7: The demand for wheat bread is likely to be , in general. a) More elastic than the demand for bread b) More inelastic than the demand for bread c) Less elastic than the demand for bread d) Just as elastic as the demand for bread Ans: A; There are more close substitutes for wheat bread than for bread in general. This will mean that demand for wheat bread will tend to be more elastic than demand for bread in general (if the price of wheat bread goes up, people will likely substitute a different type of bread).

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Practice Question #8: A kink in the market demand curve typically occurs when a) The price drops below some consumer's maximum willingness to pay. b) An additional firm enters the market. c) A change in demand occurs on along the intensive margin. d) A firm leaves the market.

Ans: A; See Practice Questions #1 and #2.

Practice Question #9: If a sugar shack raises the price of a can of maple syrup by 2%, revenue falls by 1%. This means a) The demand for maple syrup is elastic. b) The demand for maple syrup is inelastic. c) The demand for maple syrup is perfectly elastic. d) The demand for maple syrup is perfectly inelastic.

Ans: A; Elasticity is %∆Q/%∆P, TR = P*Q or in linear approximation form, %∆TR = %∆P + %∆Q. This means %∆Q, in absolute value, is larger than %∆P. Practice Question #10: If sweaters are a normal good, we can say for certain that a) The income elasticity for sweaters is greater than 1. b) The income elasticity for sweaters is less than 1. c) The income elasticity for sweaters is less than -1. d) None of the above is correct.

Ans: D; Normal only means income elasticity is positive but we do not know its magnitude.

Practice Question #11: If you raise the price of lemonade in your stand from $1 to $2.5, total revenue doubles. This means a) The demand for lemonade is elastic. b) The demand for lemonade is inelastic. c) The demand for lemonade is unit elastic. d) You should lower prices.

Ans: B; %ΔTR = %ΔP + %ΔQ, so %ΔQ = -50%. Elasticity is %ΔQ/%ΔP = -0.33.

Practice Question #12: If the price elasticity of demand is -0.5, then a 30% price hike will lead to a a) 5% drop in quantity demanded. b) 15% drop in quantity demanded. c) 20% drop in quantity demanded. d) 40% drop in quantity demanded.

Ans: B; -0.5 = %ΔQ/%ΔP = %ΔQ/30, so %ΔQ = -15%.

Practice Question #13: If beer and pizzas are complementary goods, then we can conclude that a) The cross-price elasticity of demand is negative. b) The cross-price elasticity of demand is positive. c) The cross-price elasticity of demand is zero. d) Beer is an inferior good.

Ans: A; Think of beer and pizzas as “one” good. Own-price elasticity is typically negative.

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Practice Question #14: Some economists have argued that one of the reasons for the very significant rise in public expenditure on health and education is due to rising average income levels. These services are a) Normal, income elastic services. b) Normal, income inelastic services. c) Inferior services. d) Income-neutral services.

Ans: A; Not only normal but also a luxury good; Income could rise by 50% and quantity demanded for health and education rises by more than 50%.

Practice Question #15: As income increases during economic upturns, the demand for potatoes declines, yet as income falls during economic downturns, consumption of potatoes rises. The most likely explanation is a) Changing tastes for potatoes. b) Negative income elasticity of demand. c) Very low price elasticity of demand. d) Very high price elasticity of demand.

Ans: B; inferior good.

Practice Question #16: On a downward-sloping linear demand curve, total revenue would be at a maximum at the a) midpoint of the demand curve. b) lower end of the demand curve. c) upper end of the demand curve. d) vertical intercept of the demand curve.

Ans: A; This is when own-price elasticity is equal to -1.

Practice Question #17: Tina and Brian work for the same recording company. Tina claims they earn more revenue by increasing the price of their CDs while Brian claims they would be better off by decreasing the price.

We can conclude that a) Tina thinks the demand for CDs has price elasticity of 0 and Brian thinks price elasticity equals 1. b) Tina thinks the demand for CDs has price elasticity equal to 1 and Brian thinks price elasticity of 0. c) Tina thinks the demand for CDs is price elastic and Brian thinks it is price inelastic. d) Tina thinks the demand for CDs is price inelastic and Brian thinks it is price elastic.

Ans: D; Inelastic means price rises, quantity demanded falls a bit only, so total revenue rises.

Practice Question #18: The thrift clothing store estimates that the price elasticity of demand for hats is -0.40. If the store decreases the price of hats by a small amount, a) The store will sell fewer hats and earn less revenue. b) The store will sell fewer hats and earn more revenue. c) The store will sell more hats and earn less revenue. d) The store will sell more hats and earn more revenue.

Ans: C; Demand is currently inelastic so lowering price would decrease revenue even though it would lead to more units being sold.

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Practice Question #19: Stan and Wendy consume candies. Stan’s demand is pS = 5 – q/2. Wendy’s demand is pW = 4 - q. a) Find their aggregate demand. b) Draw the aggregate demand on a graph, showing all relevant values on the axes. Show the loss in consumer surplus when the price rises from p = $3 to p = $4. c) If the price rises from p = $3 to p = $4, what is the impact on the aggregated consumer surplus of Stan and Wendy?

Ans: a) Because the demands have different intercepts, there is a kink in the total demand when P = $4. The vertical intercept of the total demand is the higher of the two individual demand intercepts, which is $5. To find the slope of the total demand, we need to add the two individual demands across quantities, not prices. First, rearrange the equations to get qS = 10 – 2p and qW = 4 – p. The total demand is Q = 14 – 3p. Graphically, with price on the vertical axis, we get P = 14/3 – (1/3)Q. The intercept of the total demand is 5, with a slope of -1, until price drops to 4, and the slope becomes -1/3. b)

c) The total loss in CS is (4 – 3)*2 + (4 – 3)*(5 – 2)/2 = 2 +3/2 = $3.5.

Practice Question #20: The demand function for a good is given by x = m/p2. a) Determine whether demand is elastic or inelastic for this good. b) Would a price drop increase or decrease revenue from selling this good? Explain. c) Is this a normal good? Is this a luxury or necessary good? Justify your answers.

Ans: −2 m ∗p P dx p3 a) The price elasticity is Ɛ = = -2 = x dP m p2 Since |Ɛ| = 2 > 1, demand is elastic. Note that the price elasticity is constant for this demand function. b) Given that demand is elastic (|Ɛ| > 1), the revenue increases when the price drops.

1 m dx p2 ∗m = 1.

The income elasticity is Ɛm = = c) x dm m p2 Thus, Ɛm > 0, which shows that this is a normal good. However, this good is neither a luxury good (for which Ɛm > 1) nor a necessary good (for which Ɛm < 1). An income elasticity of 1 indicates that preferences are homothetic, which is the special case between a luxury and a necessary good.

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Tutorial Question #1: The two graphs below give the demand curves for two different consumers, Adam and Bob. Both demand curves are straight lines. The horizontal and vertical intercepts are labeled on the graphs.

a) Draw a graph of the market demand curve. Label all intercepts, kinks and slopes. b) Write down the equation for market demand with quantity on the left-hand side. c) Suppose that the supply function for this market is given by Q = 10p. Solve for the equilibrium market price, the number of units Adam buys at the equilibrium price and the number of units Bob buys at the equilibrium price.

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Tutorial Question #2: There are two types of consumers of coffee.

Each consumer of type A has the following demand for coffee: CA = 20 - 2p where CA is the number of cups of coffee demanded when the price of a cup of coffee is p.

Each consumer of the type B has the following demand function: CB = 10 – 0.5p where CB is the number of cups of coffee demanded when the price of a cup of coffee is p.

There are 10 consumers of type A and 10 consumers of type B. a) Calculate the price elasticity of demand for each consumer type at a p = $4. b) Would you expect the price elasticity of demand for all beverages (including coffee) to be larger or smaller in magnitude (absolute value) than the elasticities you found in part a)? Explain. c) Derive the market demand and graph the demand curve, labeling all values. d) Using the market demand you found in part c), is the price elasticity of demand for the total market at a p = $4 larger or smaller (absolute value) than the elasticities you found for the individual consumers? Explain why your result makes sense.

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Chapter 16 Equilibrium Practice Question #1: Suppose that the market for Chippy (a brand of barbecue corn chips that taste like no other!) has a negatively sloped linear demand curve and a positively sloped linear supply curve. A quantity tax placed on consumers will generate a deadweight loss a) at least as large as the revenue generated from the tax. b) smaller than the revenue generated from the tax. c) equal to the revenue generated from the tax. d) that can be larger or smaller than the revenue generated from the tax Ans: D; At small levels of the tax, the deadweight loss will likely be smaller than the tax revenue.

However, the deadweight loss will continue to increase as the tax goes up while the tax revenue will eventually start to fall. At some point, deadweight loss will be greater than the tax revenue.

Practice Question #2: When a $6 quantity tax is placed on consumers, one third of the burden of the tax falls on consumers while two thirds fall on producers. If the tax is instead placed on producers, the portion of the tax burden falling on consumers will be: a) $1. b) $2. c) $3. d) $4.

Ans: B; The tax has the same effect regardless of whether it is placed on consumers or producers.

Practice Question #3: Suppose that the government proposes two types of taxes. One is a quantity tax on cigarettes and the other is a value tax on cigarettes. Economists estimate that both taxes will raise the same amount of tax revenue. Which of the following is CORRECT? a) The change in consumer surplus will be greater under the quantity tax than under the value tax. b) The change in consumer surplus will be greater under the value tax than under the quantity tax. c) The change in consumer surplus will be the same under either tax. d) There is no change in consumer surplus because the producers will pay all taxes.

Ans: C; If the two taxes generate the same revenue, then they will lead to the same after-tax quantity and prices for the consumer and producer.

Practice Question #4: Demand for insulin is perfectly inelastic. The supply of insulin is an upward sloping line. If a $5 quantity tax is placed on the producers of insulin, then the price consumers pay for insulin will a) stay the same. b) increase but by less than $5. c) increase by $5. d) increase by $2.5, which is half of the tax per unit.

Ans: C; If demand is perfectly inelastic, or vertical, the entire tax will be paid by the consumers.

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Practice Question #5: If supply is completely inelastic, then a quantity tax placed on consumers will a) Increase the equilibrium price paid by consumers but by less than the amount of the tax. b) Decrease the equilibrium price received by producers but by less than the amount of the tax. c) Increase the equilibrium price paid by consumers by the full amount of the tax. d) Decrease the equilibrium price received by producers by the full amount of the tax.

Ans: D; If supply is inelastic, or vertical, the entire tax will be paid by the producers.

Practice Question #6: The demand curve for Chippy is a downward sloping line and the supply curve is an upward sloping line. If a quantity tax is imposed, the dead weight loss generated by the tax will a) Increase as the tax increases. b) Depend on whether the tax was placed on consumers or producers. c) Be smaller than the revenue from the tax. d) Be larger than the revenue from the tax.

Ans: A; Increasing the tax will move us further from the initial equilibrium price and quantity and increase the deadweight loss. The deadweight loss will be the same regardless of whether the tax is placed on consumers or producers. At low levels of the tax, the deadweight loss will be smaller than the tax revenue but this can change as the size of the tax rises.

Practice Question #7: The demand for paintings is given by Q = 100 – p. The supply for paintings is perfectly inelastic, which is 50 paintings. This quantity is supplied regardless of the price of a painting. a) Find the equilibrium price of and quantity of paintings. b) Suppose that a $10 quantity tax is placed on consumers. Find the new equilibrium quantity, the new equilibrium price paid by consumers and the new equilibrium price received by painters. c) What was the change in consumer surplus resulting from the tax? How much is the tax revenue?

Ans: a) 100 – p = 50, so p = $50 and Q = 50. b) Demand will shift downward by 10 and intersect with the vertical supply at a lower price but same Q = 50. This means consumers pay $40 to the suppliers and $10 in taxes to the government for each painting. c) Consumer surplus does not change because they still pay $50 for each painting even though $40 goes to the suppliers and $10 to the government. Tax revenue is $500.

Practice Question #8: The demand for beer is Q = 2, 000 − 100P and the supply of beer is Q = −100 + 200P. Suppose the government imposes a $2 per unit sales tax on the suppliers of beer. a) What is the equilibrium price without the tax? b) What is the equilibrium price with the tax? c) What are the tax burdens on the consumer and producer, per unit?

Ans: a) 2,000 − 100P = −100 + 200P. Price = $7.00. b) 2,000 − 100P = 200(P − 2) – 100, 2, 000 − 100P = 200P – 500, 2, 500 = 300P, and P = 8.33 c) The consumers’ tax burden = $8.33 − $7.00 = $1.33. Out of the $8.33 the consumers pay, producers have to pay $2 to the government. They only keep $6.33. Compared to the previous price of $7, their tax burden is $0.67. Therefore, out of the $2 tax, consumers pay $1.33 and producers pay $0.67.

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Practice Question #9: The demand for pet turtles is Q = 2,600 − 20P, and the government places a tax of $4 per turtle. a) Calculate the deadweight loss of this tax when supply of pet turtles is Q = 400. b) Calculate the deadweight loss of this tax when supply of pet turtles is Q = 12P. c) Explain why the deadweight loss calculations differ between part a) and part b).

Ans: a) No change in the equilibrium quantity. No deadweight loss but producers pay all taxes. b) Before tax: 2,600 − 20P = 12P, P = $81.25, Q = 975. After tax: 2,600 − 20P = 12(P − 4), 2,600 − 20P = 12P – 48, 2, 648 = 32P, P = 82.75; Q = 945. The quantity change is 975 − 945 = 30, so the area of the deadweight loss triangle is (30*4)/2 = $60. c) Deadweight loss is caused by the drop in the equilibrium quantity because the tax discourages consumption and production. In part a), because supply was vertical, there was no change in quantity. When quantity does not change, the tax has caused no distortion.

Practice Question #10: The burden of a tax that is imposed on a good is said to fall completely on the consumers if the a) price paid by consumers for the good declines by the amount of the tax. b) price paid by consumers for the good increases by the amount of the tax. c) price paid by consumers does not change. d) wages received by workers who produce the good increase by the amount of the tax.

Ans: B; Demand must be vertical or completely inelastic.

Practice Question #11: Given any upward-sloping supply curve, the more elastic the demand curve, the equilibrium output will fall and the will be the deadweight loss when the government imposes an excise tax. a) more; smaller b) more; larger c) less; smaller d) less; larger Ans: B; Consumers are very sensitive to the price increase. See Practice Question #14.

Practice Question #12: Suppose the price elasticity of demand for yachts equals 4.04, while the price elasticity of supply for yachts equals 0.22. If the government introduces a luxury tax on yachts, who will pay more of the tax? a) Yacht builders will pay more. b) Yacht buyers will pay more. c) Yacht builders and buyers will pay equally. d) Yacht buyers will pay all of the tax.

Ans: A; Low supply elasticity means the producers cannot quickly and massively cut quantity supplied when the tax is imposed. They have to pay a large portion of the tax if they want to sell.

Practice Question #13: A tax usually leads to in consumer surplus and in producer surplus. a) an increase; an increase b) an increase; a decrease c) a decrease; an increase d) a decrease; a decrease Ans: D; Tax decreases both if quantity falls. This means either demand or supply is non-vertical.

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Practice Question #14: The demand for green tea is Q = 260 − 5P and the supply (prices are per kg) is Q = 8P. a) Find the equilibrium price and quantity. b) Now the government imposes a tax of $13 per kg of green tea on the consumers.

Calculate the equilibrium consumer price, producer price and tax revenue. d) Graph your results and label the values of the change in consum er surpl us ( CS), change in producer surplus (PS), tax revenue (TR) and deadweight loss (DWL).

Ans: a) P = $20, Q = 160 kg. b) QD = 260 − 5 × (P + 13), QD = 195 − 5PS,QD = QS, 195 − 5P = 8P, P = 15, PT = 15 + 13 = $28, Q = 8*15 = 120, tax revenue = 120 *13 = $1,560. c) ∆CS = −a − b = − (160 + 120)(28 − 20)/2 = −$1,120, ∆PS = −c − d = (160 + 120)(20 − 15) = $700, DWL = 13 × (160 − 120)/2 = $260.

Practice Question #15: “The size of the deadweight loss is determined by elasticities.” Correct or incorrect? Explain.

Ans: Correct – for a given tax, such as 10% of the price or $1 per unit, will discourage consumption and production. Deadweight loss is the loss in consumer surplus and producer surplus because of such lost quantity. If the demand (or supply) is very steep, or inelastic, consumers will cut consumption only by a bit. As a result, the loss is small. If quantity demanded falls a lot because the demand is elastic, or very flat, or very responsive to the price increase due to the tax, then the loss is large. See the two graphs below, with the good as gasoline.

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Tutorial Question #1: Demand for cab rides in Montreal is given by the following demand function: P = 10 -

1 10

R

where R is the number of cab rides at a price of p. The supply function for cab companies is: P=

1 10

R

a) Find the equilibrium price and quantity of cab rides in Montreal? b) Suppose that the City of Montreal decides to raise government revenue by putting a quantity tax of $2 per ride. Find the new equilibrium consumer price, producer price and quantity of rides. c) How much will the City of Montreal collect in taxes? d) How much deadweight loss does the tax generate? e) Suppose that Montrealers protest that it is unfair to tax cab rides that are essential, such as trips to the hospital. The city agrees to only impose the tax on non-essential cab rides but adjusts the size of the tax so that the same amount of tax revenue is generated. Will this change in policy lead to more or less deadweight loss than the original tax? Explain.

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Chapter 19 Technology Practice Question #1: Suppose that computers are made using machines and workers and the production function is convex and monotonic. If 2 workers and 8 machines can produce 20 computers and 10 workers and 4 machines can also produce 20 computers, then 6 workers and 7 machines can produce a) Exactly 20 computers. b) At least 20 computers. c) Less than 20 computers. d) Not enough information to determine.

Ans: B; Convexity tells us that 6 workers and 6 machines will produce at least 20 computers.

Because the technology is also monotonic, increasing the number of machines from 6 to 7 will either leave output unchanged or increase it.

Practice Question #2: If a firm using capital and labor as inputs has isoquants that are straight lines, then a) The firm's technology exhibits an increasing technical rate of substitution. b) The marginal product of labor equals the marginal product of capital. c) The ratio of the marginal product of capital to the marginal product of labor is constant. d) Both b) and c) are correct.

Ans: C; If the isoquants are straight lines, then the technical rate of substitution is constant, implying that the ratio of MPL to MPK is constant.

Practice Question #3: On a graph with labor on the horizontal axis and capital on the vertical axis, a steep isoquant means a) Marginal product of capital is low. b) Marginal product of labour is high. c) Marginal product of labour is zero. d) Both a) and b) are correct.

Ans: D; The magnitude of the slope is MPL/MPK. Practice Question #4: The isoquants for the production function f(L, K) = 2K + 3L will a) Be straight lines. b) Be L-shaped. c) Have a diminishing technical rate of substitution. d) Have a technical rate of substitution that is undefined at the kink.

Ans: A; MPL and MPK are constants and isoquants are straight line.

Practice Question #5: We use wood and glue to make chairs and we have to use them in fixed ratios. The isoquants are a) The isoquants will be negatively sloped straight lines. b) The isoquants will be upward sloping. c) The isoquants will be L-shaped. d) The isoquants will be horizontal.

Ans: C; For fixed input ratios, the isoquants are L-shaped.

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Practice Question #6: Which of the following is INCORRECT if a production technology using capital and labor as inputs is monotonic? a) The marginal product of capital is diminishing. b) The technical rate of substitution is negative. c) The marginal product of labor is negative. d) The marginal product of capital is increasing.

Ans: C; If the production technology is monotonic, increasing one input while holding everything else constant should either increase output or leave output constant.

Practice Question #7: A firm uses capital and labor in production. Suppose the firm improves its technology such that capital and labor produce twice as much output as before. Which of the following is INCORRECT? a) The marginal product of labor will be doubled. b) The marginal product of capital will be doubled. c) The technical rate of substitution will be doubled. d) All of the answers above are incorrect.

Ans: C; Notice that the new production function is just two times the old production. MPL and MPK are doubled and TRS is the same as before.

Practice Question #8: Suppose that capital and labor are the only two inputs a firm uses. The marginal product of capital and the marginal product of labor are both constant and the firm's technology is monotonic. Which of the following is CORRECT? a) Labour and capital are complements. b) The production function is not convex. c) The isoquants will be straight lines. d) All of the answers are correct.

Ans: C; MPL and MPK are constants and the inputs are perfect substitutes.

Practice Question #9: A firm uses an increasing returns to scale technology and decides to double output by scaling up all of its inputs by the same amount. This will require a) Doubling the number of inputs used. b) Increasing the number of inputs used but by less than double. c) Increasing the number of inputs used by more than double. d) No need to increase inputs.

Ans: B; Doubling inputs more than double output, and so we require less than doubling inputs.

Practice Question #10: If a production technology exhibits diminishing marginal product of labor and the 10th worker hired increases output by 20 units: a) The 20th worker hired will increase output by more than 20 units. b) The 20th worker hired will decrease output. c) The 10th worker hired will increase output by more than 20 units. d) The 15th worker hired will decrease output.

Ans: C; Diminishing product says the 10th worker is more productive than the subsequent workers.

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Practice Question #11: Suppose we produce output with two inputs, L and K. The production function is f(L, K) = L3 + K. This technology exhibits a) Constant returns to scale. b) Increasing returns to scale. c) Decreasing returns to scale. d) Insufficient information to determine the returns to scale.

Ans: B; f(tL; tK) = t3L3 + tK. This is greater than tL3 + tK or tf(L, K).

Practice Question #12: The principle of diminishing returns states that as more of an input is used, its marginal product must eventually a) Rise, if all other inputs are held constant. b) Rise, if more of other inputs are used. c) Fall, if more of other inputs are used. d) Fall, if all other inputs are held constant.

Ans: D; More inputs increase total output but by a decreasing rate.

Practice Question #13: A farm produces tomatoes using water and fertilizer. The pounds of tomatoes produced, T, is given by f(W, F) = W2F2, where W is the number of gallons of water used and F is the number of pounds of fertilizer used. The marginal product of water and the marginal product of fertilizer are MPW = 2WF2 and MPF = 2W2F. a) Discuss the following features of this production function: Diminishing technical rate of substitution, monotonicity and convexity. If plotted on a graph, F is on the horizontal axis and W is on the vertical axis. b) Draw the isoquant passing through the point (5 gallons of water; 15 pounds of fertilizer) and label the slope of the isoquant at this point with its numerical value.

Ans: a) - Increasing MPW: MPW is larger when W is larger (2WF2 increases as W increases).

- Increasing MPF: MPF is larger when F is larger (2W2F increases when F increases).

- Diminishing TRS: TRS = - MPF/MPW, TRS = - 2W2F/2WF2, TRS = - W/F. As we move from left to right along an isoquant, increasing F and decreasing W, the magnitude of the TRS falls (absolute).

- Monotonicity: Both marginal products are always greater than zero, so increasing either input will always lead to a level of output greater than the original level of output.

- Convexity: TRS is diminishing giving us convex-shaped isoquants. b) Isoquant is convex passing through the point (15, 5). The slope of the isoquant is TRS = -1/3.

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Practice Question #14: Vanier College uses instructors (I) and rooms (R) to produce lectures (L). The number of lectures the College can offer based on how many instructors and rooms it has is given by L = f (I, R) = min {10I, 25R} a) Explain why this production function is more sensible than that of Cobb-Douglas. b) If plotted on a graph, rooms are on the horizontal axis and instructors are on the vertical axis. Discuss the shape of the isoquants for 100 and 300 lectures.

Ans: a) Instructors and rooms are perfect complements. It would not work if there were more instructors but no room or vice versa. A Cobb-Douglas function allows substitutions across the two inputs, which in this case will not work. b) The isoquants should be L-shaped, with the corners corresponding to where the number of rooms and instructors is in the exact right proportion. For 100 lectures, this will occur at 10 instructors and 4 rooms. For 300 lectures, it will be at 30 instructors and 12 rooms.

Practice Question #15: For each scenario below, write down a production function that reflects the given description. a) Ali’s exam grade, G, is a product of how many classes she goes to, C, and how many hours she studies, H. Missing a class always leads to a five-point drop in her grade. She can always make up for this five-point drop with another two hours of studying. If Ali goes to 10 classes and does no studying, she will get an 80. b) Kicking Horse Company produces carriage rides R using drivers, D, and horses, H. Every carriage ride requires one driver and two horses. c) A-Cotton produces shirts, S, using weavers, W, and looms, L. Each weaver is more productive if given more looms to work with. Each loom is more productive with more weavers using it. Each additional weaver is less productive than the previous one. Each additional loom is just as productive as the previous loom. d) Second Cup coffee shop brews coffee using both Brazilian coffee beans and Arabica coffee beans. Customers cannot tell the difference between the beans so the coffee shop can always switch between the beans without the customer noticing and each bean can always brew just as much coffee as the previous bean. Arabica beans are a bit stronger than Brazilian beans, so it takes more Brazilian beans to brew a mug of coffee than Arabica beans. The coffee shop can always use one Arabica bean instead of three Brazilian beans. If the coffee shop only uses Brazilian beans, it would take 90 beans to brew each one-liter jug of coffee. C. Ans: a) Perfect substitutes, and G = 30 + 5C + 2.5H because every C, she earns 5 points and for every H, she earns 2.5 points. However, if she attends 10C, she earns 80. This is why we need to add the 30. b) Perfect complements, R = min {D, 0.5H}. Plug in R = 1 to convince yourself. c) Cobb-Douglas, because MP is diminishing for both inputs and can be substituted with each other. A possible function is S = W0.5L. The power of 0.5 captures each W is less productive than the previous one (MPW = 0.5L/W0.5, which is diminishing in L) and the power of 1 for L shows that its MP is constant (MPL = W0.5). d) Perfect substitutes, and to brew one jug of coffee, it needs C = min {1/30A, 1/90B}.

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Practice Question #16: Firm A wants to produce 100 units of output using L = 5 and K = 10. For each of the production functions given below, state whether it is feasible to produce the required output with the given input combination, and if it is feasible, state whether the input combination is efficient. a) f (L, K) = KL + 2L + 40 b) f (L, K) = 3KL c) f (L, K) = 8 min{3L, 2K} d) f (L, K) = L2K e) f (L, K) = L + 6K Ans: a) Q = 50 + 10 + 40 = 100 = 100. Feasible and efficient. b) Q = 150 > 100. Feasible and inefficient. c) Q = 8 min{15, 20} = 8*15 = 120 > 100. Feasible and inefficient. d) Q = 52*10 = 250 > 100. Feasible and inefficient. e) Q = 5 + 60 = 65 < 100. Not feasible.

Practice Question #17: For each of the production functions below, find MPL and MPK, discuss whether TRS = MPL/MPK is diminishing and whether it has increasing, decreasing or constant returns to scale. a) Q = f(L, K) = KL² + L b) Q = f(L, K) = 5L + 10K − 1 c) Q = f(L, K) = 3L1/3K d) Q = f(L, K) = min{2L, K} Ans: 2 KL +1 2 K 1 = + . As L goes up, a) MPL = 2KL + 1 and MPK = L2. Then |TRS| = 2 L L2 L both terms go down. TRS is diminishing. For a t > 1, f (tL, tK) = tK(tL)2 + tL = t3KL2 + tL > tKL2 + tL = t[KL2 + L] = tf (L, K). It exhibits IRS. b) MPL = 5 and MPK = 10. Then |TRS| = 5/10 = 0.5 , which is constant rather than diminishing. For a t > 1, f (tL, tK) = 5(tL) + 10(tK) − 1 > 5(tL) + 10(tK) − t = tf (L, K). It exhibits IRS. L' 2 /3 K K = c) MPL = L−2/3K and MPK = 3L1/3. Then |TRS|= . As L goes up, it decreases.

3 L1/ 3 3 L TRS is diminishing. For a t > 1, we have f (tL, tK) = 3(tL)1/3(tK) = t(1/3+1)3L1/3K = t4/3f (L, K) > tf (L, K). It exhibits IRS. d) If 2L > K, then f (L, K) = K, MPL = 0 and MPK = 1, where |TRS| = 0. If 2L < K, then f(L, K) = 2L, MPL = 2 and MPK = 0, where |TRS| = ∞. If 2L = K, then TRS is undefined. For any t > 1, f(tL, tK) = min{2tL, tK} = t min{2L, K} = tf(L, K). It exhibits CRS. Practice Question #18: K L Suppose production function is given as f (L, K) = [min( , ¿]c, where a, b, c > 0 are all positive a b constants. The input prices are r and w, for K and L, respectively. Does this production function exhibit increasing, decreasing or constant returns to scale, when a = 1/2 and b = 1/3?

Ans: We can pick an arbitrary pair (K, L) and assume for now that min (2K, 3L) = 2K (so 2K < 3L). For any t > 1, min{2tK, 3tL} = 2tK. We get f (tK, tL) = [min(2tK, 3tL)]c = (2tK)c = tc (2K)c = tc [min(2K, 3L)]c = tc f (K, L). Also try min(2K, 3L) = 3L (so 3L < 2K) and convince yourself 70

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that the results would not change. This production function exhibits DRS if c < 1, CRS if c = 1 and IRS if c > 1.

Tutorial Question #1: For each of the scenarios below, right down a production function that accurately captures the technology. Your production function should contain only the two inputs and numerical constants.

Also determine whether the production technology exhibits a diminishing technical rate of substitution (convexity) and whether it exhibits decreasing, constant or increasing returns to scale. a) A bakery uses apples (A) and bananas (B) to produce fruit salads (F ). Every fruit salad requires two apples and one banana. b) A restaurant uses cooks (C) and dishwashers (D) to produce meals (M ). Each additional cook increases the number of meals but by a smaller amount than the previous cook did. Each additional dishwasher also increases the number of meals but by a smaller amount than the previous dishwasher. The marginal product of a cook does not depend on the number of dishwashers and the marginal product of a dishwasher does not depend on the number of cooks. c) A firm uses metal (M) and wood (W) to produce chairs (C). They can make chairs entirely out of metal, entirely out of wood, or out of a combination of metal or wood. They can always replace two units of wood with one unit of metal or one unit of metal with two units of wood without affecting the level of output. If they use only metal, it takes four units of metal to produce a single chair.

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Chapter 20 Profit Maximization Practice Question #1: A firm's production function is Q = f(K, L) = 4K0.5L0.5. The wage rate is $1 and the rental rate is $2. In the short run, capital is fixed. If the firm is using 16 workers in the short run and is producing 400 units of output, what are the firm's fixed costs? Assume that the firm is maximizing profits. a) $625. b) $1,000. c) $1,250. d) $1,300.

Ans: C; In the short run, the fixed costs are coming from capital. We can plug L = 16 and f(K, L) = 400 into the above equation to find that K = 625. The cost of 625 units of capital is $1,250.

Practice Question #2: A firm uses capital and labor as inputs to produce output. In the short run, capital is fixed. A decrease in the rental rate will labor used in the short run and labor used in the long run. a) Increase, decrease. b) Decrease, increase. c) Not change, decrease. d) Not change, increase.

Ans: C; In the short run, the level of labor depends only on the wage and the price of output.

The firm cannot use less capital because it has committed to the amount which is why capital is fixed. In the long run, the higher rental rate will lead the firm to use more capital and less labor.

Practice Question #3: A firm uses capital and labor to produce output and has standard, convex isoquants. At the firm’s current output level, the slope of the isoquant is greater than the wage divided by the rental rate. The firm can maintain the same level of output and decrease costs by a) Increasing capital and increasing labor. b) Decreasing capital and decreasing labor. c) Increasing capital and decreasing labor. d) Decreasing capital and increasing labor.

Ans: D; If the MPL/MPK > w/r, then it should hire more L (which will decrease MPL) and less capital (which will increase MPK).

Practice Question #4: Suppose that capital and labor are the only inputs used by a firm and that capital is fixed in the short run. If each additional worker increases output by 3 units and the price of output is $10 a unit, what is the highest wage the firm is willing to pay in the short run? a) $3.33. b) $20. c) $30. d) $10.

Ans: C; The value of the output produced by an additional worker (P*MPL), or value of the marginal product of labour, will be $30, which is what the firm would be willing to pay up to $30 to hire an additional worker.

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Practice Question #5: If a firm uses inputs x and y to produce output and y is fixed in the short run, then the firm can maximize profits in the short run by setting the marginal product of x equal to a) The price of x divided by the price of y. b) The price of y divided by the price of x. c) The price of y divided by the price of output. d) The price of x divided by the price of output.

Ans: D; P*MPX = price of x; Let MPX = 10 and each unit of output this x produces sells for $6, the maximum that the firm will be willing to pay for this x input is $60. The firm “breaks even”.

Practice Question #6: Labour is our only variable input in the short run. The profit-maximizing level of labour is where a) The price of a unit output is equal to the price of a unit of labour. b) The marginal product of labour is equal to the price of a unit of labour. c) The dollar value of the marginal product of labour is equal to the price of a unit of output. d) The dollar value of the marginal product of labour is equal to the price of a unit of labour.

Ans: D; See explanation from previous question.

Practice Question #7: A firm’s production function is Q = f(L, K) = 8L0.5 + 8K0.5. The price of labour is $1 and the price of capital is $3. The price of the output is $6. What is the profit-maximizing output? a) 256. b) 512. c) 252. d) 516.

Ans: A; Max 6(8L0.5 + 8K0.5) – L – 3K; It hires L = 576 and K = 64, Q = 8*24 + 8*8 = 256.

Practice Question #8: Suppose a firm has the production function Q = L1/3K1/3 − 3. The prices of the output, labour and capital are p = 9, w = 3, r =1, respectively. a) Short run: The firm has K = 8. Find the labour amount hired. b) Long run: Find the labour amount hired and profit.

Ans: a) max π = pQ − wL − rK = 9[L1/3K 1/3− 3] − 3L − K = 9[L1/381/3 − 3] − 3L − 8 = 9[2L1/3 − 3] − 3L − 8.

The first order condition is 9[2/3(L−2/3)] = 3, L−2/3 = 1/2 or L2/3 = 2, so L SR=81/2 . b) Max π = pQ − wL – rK = 9[L1/3K1/3 − 3] − 3L − K. The first order conditions are as follows: For L, 9[(1/3)L−2/3K1/3] = 3, that is, L−2/3K1/3 = 1.

For K, 9[(1/3)L1/3K−2/3] = 1, that is, L1/3K−2/3 = 1/3.

Combining these two, K = 3L. The profit is 9[L1/3(3L)1/3 − 3] − 3L − 3L = 9[L2/3(3)1/3 − 3] − 6L. The first order condition is 9[(3)1/3(2/3)L−1/3] = 6, that is, (3)1/3L−1/3 = 1 that is L = 3, and K = 9. Profit is 9[31/391/3 − 3] − 9 − 9 = 9[31/33 − 3] − 18 < 0. The best that the firm can do is to get LLR = 0, KLR = 0 and produce zero in the long run (exit the market).

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Practice Question #9: Suppose a firm has the production function Q = K1/4L1/2. The output price is p. The prices of capital and labor are r and w, respectively. a) Find the unconditional factor demands for L and K. b) Find the profit maximizing output Q and profit equation.

1 c) In the short run, capital is fixed at K = . Find the short run labour demand and output equations.

64 Ans: a) We can first write profits as a function of the choice variables K and L: π = pQ − wL − rK = pK1/4L1/2 − wL − rK. Taking the first order conditions with respect to K and L, respectively, yields 1 p K−3 /4 L1/ 2=r 4 1 p K 1/ 4 L−1/ 2=w 2 Solving the two equations together for the two unknowns K and L yields K=

p4 64 r 2 w 2

p4 32 r w3 b) The equation for output is L=

Q=(

p4 )1 / 4 ( p4 )1/ 2 =( p3 ) 64 r2 w2 32 r w3 16 r w2

π = p( π=

p3 p4 p4 ) ) )… simplify and simplify again…

– w( – r( 16 r w2 32r w3 64 r2 w2 p4

64 r w2

… voila!

c) Capital is fixed at K =

1 64

. The profit function becomes

π = p K 1/4L1/2 – wL – rK π = p(

1 1 / 4 1 /2 −wL−r ( 1 ) ) L 64 64

The first-order condition with respect to L is 1 p¿ 2 Solving for L gives us

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L=

p2 32 w2

The short run output is then Q=¿ Practice Question #10: Clearview factory makes windows (W) with frames (F) and panes of glass (G). Each window requires one frame and five panes of glass. The cost of a frame is $10. The cost of each pane of glass is $2. The price of windows in the market is $50 each. a) Write down a production function f(F, G) that shows the number of windows it can produce. b) Short run: The firm has 30 window frames but can order more glass. Find the profit-maximizing amount of glass it will use, the number of windows it will produce, Clearview's total costs and profit. c) Long run: All inputs are variable. Suppose it wants to produce the same number of windows in the long run as the number it produced in part b). Will the costs of producing this many windows in the long run be higher, lower or the same as the costs you found in part b)?

Ans: a) It takes one frame and five panes of glass to make one window. The production function is Q = min {F, 0.2G}. To check, plug in Q = 1, we need F = 1 and G = 5. b) F = 30 in the short run, we need G = 150, total costs = $10*30 + $2*150 = $600, total revenue = $50*30 = $1,500, so profit is $900. c) Because we have to use glass and frame in a fixed ratio, there is no change in costs. We cannot substitute across the inputs and use more of the cheaper input to cut costs.

Practice Question #11: A firm uses capital and labor to produce towels. Its production technology is given by Q = K1/3L1/3. a) In the short run, capital is fixed at 8 units. The price of output is $2, the rental rate of capital is $2 and the wage $3. Find the profit-maximizing level of labor and profits. b) Does the production function exhibit increasing, decreasing or constant returns to scale?

Ans: a) With K = 8, we only need to find L. The optimal L is found by solving MPL = w/p = 3/2 = 1.5.

1 /3 MPL = K 2 /3 . Solving for L gives us L = 8/27. Profit π = p*f (K, L) − rK – wL, substitute K = 8 3L and L = 8/27 into the profit function we get π = - 128/9 = - $14.22. Notice that if the firm does not produce, it has to pay for fixed costs = $16 and π = - $16. It is better to produce and lose less. b) To check returns to scale, Q = (tK)1/3(tL)1/3, so Q = t2/3K1/3L1/3. Because t is raised to the power less than 1, this production function shows decreasing returns to scale.

Practice Question #12: A firm has a production function given by Q = L1/3K1/2 a) Does this production function exhibit increasing, decreasing, or constant returns to scale? b) Suppose in the short run, capital is fixed at K = 100. Find the unconditional demand for labour. c) In the long run both inputs are flexible. Find the unconditional demands for labour and capital.

Ans: a) Q = (tL)1/3(tK)1/2 = t5/6L1/3K1/2, since t is raised to the power less than 1, this is decreasing returns to scale.

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10 p 3/ 2 = w. This means L = ( ) .

3w c) π = pK1/2L1/3 − wL − rK, first-order conditions are p(1/3)L-2/3K1/2 = w and p(1/2)L1/3K-1/2 = r.

K 3w Simplifying we get = . Substitute this expression into the first-order conditions to get L 2r p6 p6 L= and K = .

3 24 36 r4 w 2 (6 wr) b) π = p10L

1/3

– wL − 100r, first-order condition is p(10/3)L

-2/3

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Tutorial Question #1: The Concordia Economics Department allows faculty to get either an Apple computer or a Dell computer for their research. These computers are used for doing programming, statistical analysis and writing, all in order to produce research articles. Faculty occasionally work with each other on research projects. When this happens, research is more productive if both faculty use the same type of computer. If faculty work by themselves, an Apple and a Dell are equally productive. Consider the following possible production functions describing the number of research articles (R) the department can produce as a function of the number of Apple (A) and Dell (D).

R=A 1/ 2+ D1/ 2 Equation (1) Equation (2) R=A 2 + D2 R=A + D Equation (3) a) In terms of the marginal product of an Apple computer, MPA, which of these production functions do you think would best capture the relationship between computers and research output? b) In terms of the technical rate of substitution, MPA/MPD, which of these production functions do you think would best capture the relationship between computers and research output? c) Regardless of your answers to parts a) and b), suppose that the production function is given by Equation (1) above. Does our Economics Department exhibit decreasing, constant or increasing returns to scale? Explain. d) Continue with Equation (1): Suppose that Apple computers cost $10 each and Dell computers cost $5 dollars each (unrealistic numbers but less messy to work with). Find an expression for the department's optimal number of Apple computers to use if they want to generate R research articles and both types of computers are variable inputs.

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Tutorial Question #2: Consider the technology Q = Kl/3Ll/3. The price of labor is w, the price of capital is r and the price of the output is p. a) Find its short run production function where capital is fixed at K . b) Based on your answer in part a), what is the impact of the capital cost, r, on the optimal short run production? Explain. c) Find its short run maximum profit function where capital is fixed at K . d) Find the firm's long run production function where L and K can be chosen freely. e) Find its long run maximum profit function where L and K are freely chosen.

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Tutorial Question #3: A firm's production function is f (K, L) = min {√ K , √ L } , the price of labor is w, the price of capital is r and the output price is p. a) Find its short run production function where capital is fixed at K . b) Find its long run production function where both L and K can be chosen freely.

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Chapter 21 Cost Minimization Practice Question #1: Suppose that a firm has a production function given by Q = f(L, K) = 16L1/2 + K, where L denotes labor and K denotes capital. The rental rate is r = 1 and the wage rate is w = 4. a) Suppose the capital is fixed at K . Find the short run total cost function. b) In the long run, both inputs are variable. Find the long run total cost function. c) Check: Verify that the long run cost of producing Q = 36 units is equal to the short run cost of producing it when the short run fixed capital level is the capital level you would pick in the long run (the one that you found in part b)).

Ans: a) Since rK is fixed, we minimize wL subject to Q = 16L1/2 + K. To produce Q units, the labor demand must be L = [(Q − K)/16]2 = [Q − K]2/256. The total cost is wL + rK = 4[Q − K]2/256 + K. The short run cost function is CSR = [Q − K]2/64 + K. w MPL = 8 . The tangency implies that 8 = 4, so L = 4. The production b) The |TRS| = 1/ 2 = 1 /2 L MPK L r function is 16L1/2 + K = Q, plugging L = 4, K∗ = Q − 32. The long run cost function is CLR = wL∗ + rK∗ = 4(4) + 1(Q − 32) = 16 + Q − 32 = Q – 16. c) The long run cost is simply CLR = Q − 16 = 36 − 16 = 20. The capital level in part b) is K = Q − 32 = 36 − 32 = 4. Plugging K = 4 and Q = 36, we get CSR = [Q − K]2/64 + K = [36 − 4]2/64 + 4 = 16 + 4 = 20. They are indeed the same.

Practice Question #2: Suppose that a firm has a production function given by f(L, K) = 2√ KL, where L denotes labor and K denotes capital. The rental rate is r = 1 and the wage rate is w = 4. a) Suppose the capital is fixed at K . Find the firm’s short run total cost function. b) Find the firm’s long run total cost function. c) Verify that the long run cost of producing Q = 20 units is equal to the short run cost of producing it when the short run fixed capital level is the capital level you would pick in the long run (the one that you found in part b)).

Ans: a) Since rK is fixed, we minimize wL subject to Q = 2√ KL. To produce Q units the labor demand must be L = Q2/4K. The total cost is wL + rK = 4[Q2/4K] + K. The short run cost function is CSR = [Q2/K] + K. K w K b) Now K is also variable. The |TRS| = MPL = . The tangency implies that = = 4, so K = L r MPK L 2 2 2 4L. The production function is 4LK = Q , and K = 4L, we get 16L = Q . Therefore, 4L = Q and L = Q/4, K = 4(Q/4) = Q. The long run cost function is CLR = w(Q/4) + rQ = 4(Q/4) + Q = 2Q. c) The long run cost is simply CLR = 2Q = 40. The capital level in part b) is K = Q = 20. We get CSR = [400/20]+20 = 20 + 20 = 40. They are indeed the same.

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Practice Question #3: Suppose a firm has the production function Q = f (L, K) = L + 2K. The output price is p. The prices of capital and labor are r and w, respectively. a) Find the cost minimizing L and K as functions of r and w. Suppose r ≠ 2w. b) Assume the prices are p = 12, w = 15 and r = 10. Suppose the firm can produce at most 100 units.

Find the profit maximizing quantity, input combination and the profit level. c) Suppose the firm produces Q = 100. Prices are the same as in part b). Show that the cost minimizing input level is the same level you found in part b).

Ans: a) Because the inputs are perfect substitutes, we focus on MPL/w and MPK/r, where MPL = 1 and MPK = 2. If r/w > 2, L = Q and K = 0. The cost function is c(Q) = wL = wQ. If r/w < 2, L = 0 and K = Q/2. The cost function is c(Q) = rK = rQ/2. b) Note that pMPL = 12 < 15 = w and pMPK = 24 > 10 = r. We produce all Q with K, so K = Q/2. The profit function is 12Q − 10K = 12Q − 10(Q/2) = 7Q. The firm can only produce 100 units so the profit is π = $700 and K = Q/2 = 50. c) We only use K as input so the cost minimizing levels are L = 0 and K = 50.

Practice Question #4: A firm has the production function Q = f(K, L) =√min {L , K }. The output price is p and the prices of capital and labor are r and w, respectively. Find the firm’s supply function Q (p, r, w).

Ans: We can rewrite the production function as min {K, L} = Q 2. We should use K = L to minimize cost. Therefore, the conditional factor demands are K = Q2 and L = Q2 and the cost function is C = (r + w)Q2. The profit function is pQ – (r + w)Q2, and the first-order p condition is gives us Q = .

2(r +w ) Practice Question #5: A firm has a production function Q = f(L, K) = 3LK + K and the input prices are w = 6 and r = 10. a) Suppose the firm wants to produce Q = 125. Find the long run cost minimizing labor and capital combination and the total cost function. b) Suppose the firm wants to produce Q = 20. Find the long run cost minimizing labor and capital combination and the total cost function. c) Suppose the wage rate increases and is now w = 10 and the rent is r = 10. Find the long run cost minimizing input combination for Q = 125. Compare them to your results in part a).

Ans: 3K w 6 a) MP = 3K and MP = 3L + 1. Then, |TRS| = = = . This gives us 5K = 3L + 1.

L K r 10 3 L+1 Plug this into the production function 125 = 3LK + K and solve for K. The results are K = 5 and L = 8, total cost = 98. b) Similar to part a), we find K = 2, L = 3 and the total cost = 38. c) Since w changes, the tangency condition also changes. Setting |TRS| = 1, we get 125 = 3LK + K, so K = 6.5, L = 6.1. Compared to part a), as w goes up, the firm substitutes some L with K.

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Tutorial Question #1: Producing ice cream requires both capital (K) and labor (L). Suppose that the production function for the ice cream is given by f (L, K) = KL2. a) If the wage rate is w = 40 and the rental rate of capital is r = 1, what is the least costly way (how much capital and how much labor) to produce 20 units of ice cream? b) What is the total cost using the least costly method? c) Suppose ice cream makers suddenly become more productive, changing the production function for the ice cream to f (L, K) = KL3. At the same time, the wage rate for the ice cream making workers also changed from w = 40 to a new wage rate of w. The firm producing ice cream finds that that the cheapest combination of capital and labor to produce 20 units of ice cream has not changed from before. That is, the least costly way of making 20 units is the same as in part a). What must the new wage level w be on order for this to be the case? d) Give some intuition for your answer to part c), comparing the new wage w to the old wage of 40. If the wages of the workers depended only on their productivity, would the scenario described in part c) be likely to occur? e) Suppose for the rest of the question that production has gone back to f (L, K) = KL2, the wage rate is back to w = 40, and the rental rate of capital is r = 1. Furthermore, the firm is stuck with having exactly 20 units of capital. What is the short run total cost curve for the firm when K = 20? What are its total costs when producing 20 units with capital fixed at K = 20? f) What is the long run total cost curve for makers ice cream? What is the total cost of producing 20 units when both capital and labor can be altered? g) Is the total cost in part e) (with K = 20 fixed) less, more, or the same as in parts b) and f) (in which K can be altered)? Explain why.

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Chapter 22 Cost Curves Practice Question #1: A firm faces a production function Q = 100L0.5K2. The wage rate is $10 and the rental rate is $5. a) L = 100 and it is fixed. Calculate the following functions: Total product curve for K (TP), marginal product curve for K (MP), variable cost curve (VC), average variable cost curve (AVC) and marginal cost curve (MC). b) K = 10 and it is fixed. Calculate the following functions: Total product curve for L (TP), marginal product curve for L (MP), variable cost curve (VC), average variable cost curve (AVC) and marginal cost curve (MC).

Ans: a) TP = 1,000K2, MP = 2,000K, VC = 5(Q/1,000)0.5, AVC = 5(1/1,000Q)0.5, MC = 2.5(Q/1,000)0.5 b) TP = 10,000L0.5, MP = 5,000L-0.5, VC = Q2/107, AVC = Q/107, MC = 2Q/107 Practice Question #2: Suppose a firm's short run total cost function is TC = 3Q2 + Q + 500. a) Find its variable cost (VC) and fixed cost (FC). b) Calculate its short run average total cost (ATC) and short run marginal cost (MC). c) Calculate the point where ATC = MC. What is the significance of this intersection?

Ans: a) VC = 3Q2 + Q, FC = 500. b) ATC = 3Q + 1 + 500/Q, MC = 6Q + 1. c) Q = 12.9, ATC = MC = 78.4 This is the point at which ATC is at its minimum.

Practice Question #3: For each of the total cost functions below, find the ATC, AVC, AFC and MC cost equations. a) TC = 2Q2 – Q + 4 b) TC = 1,000 + 50Q2 c) TC = Q3 + 1 Ans: a) ATC = 2Q – 1 + 4/Q, AVC = 2Q – 1, AFC = 4/Q, MC = 4Q – 1. b) ATC = 50Q + 1,000Q, AVC = 50Q, AFC = 1,000/Q, MC = 100Q. c) ATC = Q2 + 1/Q, AVC = Q2, AFC = 1/Q, MC = 3Q2.

Practice Question #4: A firm has a short run cost function C = Q3 – 2Q2 + 5Q + 6. Find its AVC and MC cost functions. Also find the level of output that AVC is minimized.

Ans: AVC = Q2 – 2Q + 5, MC = 3Q2 – 4Q + 5, set MC = AVC, solve for Q = 1 and MC = AVC = 4.

Practice Question #5: A firm has a production function Q = 2L1/4 K 1/4. Find the long run TC, ATC and MC functions.

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Ans: We minimize wL + rK subject to Q. We get L =

Q2 r1 /2

. Substitute these into 4 r1/ 2 Q2 w1/ 2 r1 /2 Q w1 /2 r1/ 2 wL + rK gives us TC = . Divide TC by Q gives us ATC = . Differentiating TC 2 2

by Q gives us MC = Q w1/2 r1/2.

Tutorial Question #1: Continue with Practice Question #18, Chapter 19.

K L Suppose production function is given as Q = f(L, K) = [min( , ¿]c, where a, b, c > 0 are all positive a b constants. The input prices are r and w, for K and L, respectively.

4 w1/ 2

and K =

Q2 w1/ 2

a) See Chapter 19, Practice Question #18 for a quick quiz. b) Without specifying a and b, find the conditional input demand functions K, Land the cost function. c) How are the conditional factor demands K, L and the cost function affected by r and w? d) Let us now fix r and w and rewrite the cost function c as a function of Q only. Assume there is no fixed cost. More specifically, define the constant A such that A = ra + wb and write the cost function as c(Q). Also find the marginal cost function and average cost function. Also indicate the slopes of the MC and ATC equations. e) Describe the relationship between returns to scale and the shape of the cost curves MC and ATC equations.

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