3. Other things being equal, a lump sum tax is at least as good for a consumer as a sales tax that collects the same revenue from him.

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1 Section I: or This section is worth a total of 10 marks. There are 10 questions worth 1 mark each; answer all of them. Simply indicate if you think the statement is true or false. 1. A consumer with convex preferences who is indifferent between the bundles (5, 1) and (11, 3) will like the bundle (8, 2) at least as well as either of the first two bundles. 2. Wanda Lott has the utility function U(x, y) = max{x, y}. Wanda s preferences are convex. 3. Other things being equal, a lump sum tax is at least as good for a consumer as a sales tax that collects the same revenue from him. 4. Isaiah is a net borrower when the interest rate is 5% and a net saver when the interest rate is 25%. An increase in the interest rate from 5 to 25% may make Isaiah worse off. 5. If a price changes, then changes in consumption at the intensive margin are changes that happen because consumers alter the amounts that they consume but do not either stop consuming or start consuming the good. 6. The demand curve, which is a downward-sloping straight line, crosses the supply curve, which is an upward-sloping straight line. If a tax is introduced where sellers must pay a tax of $2 per unit sold, then the equilibrium price paid by demanders will rise by more than $1 if the absolute value of the slope of the demand curve is greater than the absolute value of the slope of the supply curve. 7. If the production function is f(x, y) = x + min{x, y}, then there are constant returns to scale. 8. Just as in the theory of utility-maximizing consumers, the theory of profit-maximizing firms allows the possibility of Giffen factors. These are factors for which a fall in price leads to a fall in demand. 9. If the production function is f(x 1, x 2 ) = min{x 1, x 2 }, then the cost function is c(w 1, w 2, y) = min{w 1, w 2 }y 10. A competitive firm has a continuous marginal cost curve. It finds that as output increases, its marginal cost curve first rises, then falls, then rises again. If it wants to maximize profits, the firm should never produce at a positive output where price equals marginal cost and marginal cost decreases as output increases. 2

2 Section II: Multiple Choice This section is worth a total of 60 marks. There are 30 questions worth 2 marks each; answer all of them. Select the best answer from those available. 1. Angela consumes goods x and y. Her indifference curves are described by the formula y = k/(x + 3). Higher values of k correspond to better indifference curves. Which of the following statements is true? (a) Angela prefers bundle (8, 9) to bundle (9, 8) (b) Angela likes good y and hates good x (c) Angela prefers bundle (11, 9) to bundle (9, 11) (d) Angela likes good x and hates good y (e) More than one of the above statements are true 2. Toby Talkalot subscribes to a local phone service that charges a fixed fee of $10 per month and allows him to place as many local phone calls as he likes without further charge. Let good 1 be an aggregate of commodities other than local phone use and let good 2 be local phone use. (Measure good 1 on the horizontal axis and good 2 on the vertical axis.) On Monday, Toby didn t use the telephone at all. The slope m of his indifference curve at the consumption bundle he chose on Monday was (a) positive (b) less than or equal to 0 (c) 0 (d) greater than or equal to 0 (e) negative 3. Doreen has preferences represented by the utility function U(x, y) = 10x + 5y. She consumes 10 units of good x and 9 units of good y. If her consumption of good x is lowered to 1, how many units of y must she have in order to be exactly as well off as before? (a) 30 (b) 30 (c) 27 (d) 18 3

3 4. Seppo consumes brandy and saunas. Neither is an inferior good. Seppo has a total of $30 a day and 6 hours a day to spend on brandy and saunas. Each brandy costs $2 and takes half an hour to consume. Each sauna costs $1 and takes 1 hour to consume. (It is, unfortunately, impossible to consume a brandy in the sauna.) Seppo suddenly inherits a lot of money and now has $50 a day to spend on brandy and saunas. Since Seppo is a rational consumer, he will (a) increase brandy consumption only (b) increase sauna consumption only (c) increase consumption of both (d) consume the same amounts of both goods as before (e) We can t tell since we are told nothing about his preferences 5. Walt considers x and y to be perfect substitutes. They originally cost $10 and $9 respectively. His income is $720. One day the price of x drops to $8. (a) The income effect increases the quantity of y to 90 (b) The substitution effect increases the quantity of y to 80 (c) The substitution effect increases the quantity of x to 90 (d) The income effect increases the quantity of x to Neville s demand function for claret is q = 0.02m 2p, where m is income and p is price. Neville has a friend named Cedric who has the same demand function for claret as Neville. Cedric s income is $6,000 and he initially had to pay a price of $40 per bottle of claret. The price of claret rose to $60. The substitution effect of the price change (a) reduced his demand by 40 (b) reduced his demand by 56 (c) reduced his demand by 24 (d) increased his demand by 40 (e) reduced his demand by Peter consumes no commodities other than Miller Lite and Bud Light. His annual budget for these two commodities is described by the equation 5x + 30y = 300, where x is sixpacks of Miller Lite and y is cases of Bud Light. Peter considers 2 cases of Bud Light to be perfect substitutes for 6 sixpacks of Miller Lite. (a) He will consume 60 sixpacks of Miller Lite per year (b) He will consume 10 cases of Bud Light per year (c) He will consume 14 cases of Bud Light per year (d) He will consume 12 sixpacks of Miller Lite per year (e) He is indifferent between any two bundles that use up his entire income 4

4 8. Mary Granola consumes tomatoes and nectarines. Mary s indifference curves are kinky. When she is consuming more tomatoes than nectarines, she is just willing to trade 3 tomatoes for 1 nectarine. When she is consuming more nectarines than tomatoes, she is just willing to trade 4 nectarines for 1 tomato. Let P 1 be the price of nectarines, and P 2 the price of tomatoes. Mary maximizes her utility subject to her budget constraint. (Hint: Sketch one of her indifference curves.) (a) When P 1 > P 2, she must consume only tomatoes (b) When P 1 > P 2, she must consume 3 times as many tomatoes as nectarines (c) When P 1 > 3P 2, she must consume only tomatoes (d) When 4P 1 > P 2, she must consume only nectarines (e) She must consume equal numbers of both 9. Janet consumes x 1 and x 2 together in fixed proportions. She always consumes 2 units of x 1 for every unit x 2. One utility function that describes her preferences is (a) U(x 1, x 2 ) = 2x 1 x 2 (b) U(x 1, x 2 ) = 2x 1 + x 2 (c) U(x 1, x 2 ) = x 1 + 2x 2 (d) U(x 1, x 2 ) = min{2x 1, x 2 } (e) U(x 1, x 2 ) = min{x 1, 2x 2 } 10. Georgina consumes only grapefruits and pineapples. Her utility function is U(x, y) = x 2 y 8, where x is the number of grapefruits consumed and y is the number of pineapples consumed. Georgina s income is $105, and the prices of grapefruits and pineapples are $1 and $3, respectively. How many grapefruits will she consume? (a) 10.5 (b) 7 (c) 63 (d) 21 5

5 11. Twenty years ago, Dmitri consumed bread which cost him 10 kopeks a loaf and potatoes which cost him 20 kopeks a sack. With his income of 330 kopeks, he bought 9 loaves of bread and 12 sacks of potatoes. Today he has an income of 452 kopeks. Bread now costs him 22 kopeks a loaf and potatoes cost him 17 kopeks a sack. Assuming his preferences haven t changed (and the sizes of loaves and sacks haven t changed), when was he better off? (a) He was better off 20 years ago (b) He is better off today (c) He was equally well off in the two periods (d) From the information given here, we are unable to tell 12. Sam s utility function is U(x, y) = 2x + y, where x is the number of x s he consumes per week and y is the number of y s he consumes per week. Sam has $200 a week to spend. The price of x is $4. Sam currently doesn t consume any y. Sam has received an invitation to join a club devoted to the consumption of y. If he joins the club, Sam can get a discount on the purchase of y. If he belonged to the club, he could buy y for $1 a unit. How much is the most Sam would be willing to pay to join this club? (a) Nothing (b) $100 a week (c) $50 a week (d) $40 a week 13. Sir Plus has a demand function for mead that is given by the equation D(p) = 100 p. If the price of mead is $60, how much is Sir Plus s net consumer s surplus? (a) $40 (b) $1,600 (c) $400 (d) $3,900 (e) $ If the demand function for tickets to a play is q = 3, p, at what price will total revenue be maximized? (a) $80 (b) $40 (c) $20 (d) $10 (e) none of the above 6

6 15. Rollo would love to have a Mercedes. His preferences for consumption in the next year are represented by a utility function U(x, y), where x = 0 if he has no Mercedes and x = 1 if he has a Mercedes for the year and where y is the amount of income he has left to spend on other stuff. If U(0, y) = y 0.5 and U(1, y) = (10/9)y 0.5 and if Rollo s income is $50,000 a year, how much would he be willing to pay per year to have a Mercedes? (a) $5, (b) $5,000 (c) $12,200 (d) $9,500 (e) $10, The inverse demand function for mangos is defined by the equation p = 91 5q, where q is the number of crates that are sold. The inverse supply function is defined by p = 3 + 6q. In the past there was no tax on mangos but now a tax of $44 per crate has been imposed. What are the quantities produced before and after the tax was imposed? (a) 5 crates before and 5 crates after (b) 16 crates before and 9 crates after (c) 14 crates before and 7 crates after (d) 8 crates before and 4 crates after 17. Xaquane and Yullare are obscure but talented eighteenth-century painters. The world s stock of Xaquanes is 100 and the world s stock of Yullares is 70. The demand for each painter s work depends on its own price and the price of the other painter s work. If P x is the price of Xaquanes and P y is the price of Yullares, the demand function for Xaquanes is 101 3P x +2P y and the demand function for Yullares is 72+P x P y. What is the equilibrium price for Yullare s paintings? (a) $5 (b) $11 (c) $12 (d) $7 7

7 18. A firm has the production function f(x, y) = x y, where x is the amount of factor x it uses and y is the amount of factor y. On a diagram we put x on the horizontal axis and y on the vertical axis. We draw some isoquants. Now we draw a straight line on the graph and we notice that the slopes of all the isoquants that it meets have the same slope at the point where they meet this line. The straight line we drew was (a) vertical (b) horizontal (c) diagonal through the origin with slope 0.5 (d) diagonal with slope 2 (e) diagonal with slope greater than Suppose that the production function is f(x 1, x 2 ) = (x a 1 + x a 2) b, where a and b are positive constants. For what values of a and b is there a diminishing technical rate of substitution? (a) For any value of a if b < 1 (b) For any values of a and b if ab < 1 (c) For any values of a and b if a > b (d) For any value of b if a < A firm has the production function f(x, y) = x 0.9 y 0.8. This firm has (a) constant returns to scale (b) decreasing returns to scale and dimininishing marginal products for factor x (c) decreasing returns to scale and increasing marginal product for factor x (d) increasing returns to scale and decreasing marginal product of factor x 21. The production function is given by f(x) = 4x 1/2. If the price of the commodity produced is $60 per unit and the cost of the input is $10 per unit, how much profits will the firm make if it maximizes profits? (a) $1,440 (b) $718 (c) $2,884 (d) $1,425 (e) $723 8

8 22. When Farmer Hoglund applies N pounds of fertilizer per acre, the marginal product of fertilizer is 1 N/200 bushels of corn. If the price of corn is $4 per bushel and the price of fertilizer is $1.20 per pound, then how many pounds of fertilizer per acre should Farmer Hoglund use in order to maximize his profits? (a) 280 (b) 74 (c) 140 (d) 288 (e) A competitive firm uses two inputs, x and y. Total output is the square root of x times the square root of y. The price of x is $17 and the price of y is $11. The company minimizes its costs per unit of output and spends $517 on x. How much does it spend on y? (a) $766 (b) $480 (c) $655 (d) $ Using existing plant and equipment, Priceless Moments Figurines can be manufactured using plastic, clay, or any combination of these materials. A figurine can be manufactured by F = 4P + 2C, where P is pounds of plastic and C is pounds of clay. Plastic costs $2 per pound and clay costs $5 per pound. What would be the lowest cost of producing 40,000 figurines? (a) $20,000 (b) $100,000 (c) $60,000 (d) $10,000 (e) $40,000 9

9 25. The VCR manufacturing business is perfectly competitive. Suppose that currently, firms that manufacture VCRs utilize either technology 1 with total cost function C 1 (Q) = Q+Q 2 or technology 2 with total cost function C 2 (Q) = Q + Q 2. In the long run, assuming no new manufacturing technologies, what will happen in this industry? (a) Firms utilizing technology 1 and firms utilizing technology 2 will stay in business (b) Firms utilizing technology 1 will shut down, but firms utilizing technology 2 will stay in business (c) Firms utilizing technology 1 will stay in business, but firms utilizing technology 2 will shut down (d) Firms utilizing technology 1 and firms utilizing technology 2 will shut down 26. A profit-maximizing firm continues to operate even though it is losing money. It sells its product at a price of $100. (a) Average total cost is less than $100 (b) Average fixed cost is less than $100 (c) Marginal cost is increasing (d) Average variable cost is less than $100 (e) Marginal cost is decreasing 27. Suppose that Dent Carr s long-run total cost of repairing s cars per week is c(s) = 3s If the price he receives for repairing a car is $18, then in the long run, how many cars will he fix per week if he maximizes profits? (a) 4.50 (b) 0 (c) 6 (d) 3 (e) The bicycle industry is made up of 100 firms with the long-run cost curve c(y) = 2 + (y 2 /2) and 60 firms with the long-run cost curve c(y) = y 2 /10. No new firms can enter the industry. What is the long-run industry supply curve at prices greater than $2? (a) y = 400p (b) y = 420p (c) y = 200p (d) y = 300p (e) y = 435p 10

10 29. The demand for a monopolist s output is 6, 000/(p + 2) 2, where p is the price it charges. At a price of $3, the elasticity of demand for the monopolist s output is (a) 1 (b) 2.20 (c) 1.20 (d) 1.70 (e) A monopoly has the demand curve q = 10, p. Its total cost function is c(q) = 1, q. The government plans to tax the monopoly s profits at a rate of 50%. If it does so, the monopoly will (a) increase its price by 50% (b) increase its price by more than 50% (c) recover some but not all of the tax it pays by increasing its price (d) not change its price or the quantity it sells 11

11 Section III: Short Answer This section is worth 30 marks and contains 6 questions; answer all of them. Explain your answers and use appropriately labelled diagrams if necessary. 1. (5 marks) Jim takes two accounting exams. His overall grade for the course will be the maximum of his scores on the two exams. Jim decides to spend a total of 200 minutes studying for these two examinations. If he spends m 1 minutes studying for the first examination, his score on this exam will be x 1 = m 1 /5. If he spends m 2 minutes studying for the second examination, his score on this exam will be x 2 = m 2 /10. (a) On the graph below, draw a budget line showing the various combinations of scores on the two exams that Jim can achieve with a total of 400 minutes of studying. On the same graph, draw two or three indifference curves for Jim. On your graph, find the point on Jims budget line that gives her the best overall score in the course. score on exam score on exam 1 (b) Given that he spends a total of 200 minutes studying, Jim will maximize his overall score by achieving a score of g 1 on the first exam and g 1 on the second exam. What are g 1 and g 2? 12

12 2. (5 marks) The cost function c(w 1, w 2, y) of a firm gives the cost of producing y units of output when the wage of factor 1 is w 1 and the wage of factor 2 is w 2. Find the cost functions for the following firms: (a) A firm with production function f(x 1, x 2 ) = min{2x 1, 3x 2 } (b) A firm with production function f(x 1, x 2 ) = 2x 1 + 3x 2 (c) A firm with production function f(x 1, x 2 ) = max{2x 1, 3x 2 } 13

13 3. (5 marks) Ann s firm makes sweaters. Her production function is f(x 1, x 2 ) = x 1 + 4x 2, where x 1 is the amount of unskilled labor and x 2 is the amount of capital that she employs. (a) In the graph below, draw a production isoquant representing input combinations that will produce 40 units of output. Draw another isoquant representing input combinations that will produce 60 units of output. x 2 x 1 (b) Does this production function exhibit increasing, decreasing, or constant returns to scale? (c) If Ann uses only capital to produce sweaters, how much capital would she need in order to produce y sweaters? (d) If Ann faces factor prices (w 1, w 2 ), what will be the minimal cost of producing y units of output? 14

14 4. (5 marks) The demand for Craftmatic Adjustable Beds is described by Q c = P 1.40 c I 0.60 P 0.20 m A 0.25, where Q c is the number of Craftmatic Adjustable Beds demanded, P c is the price of a Craftmatic Adjustable Bed, I is per capita income, P m is the price of a battery-powered massage pillow, and A is the advertising budget. (a) If the marginal cost of producing a Craftmatic Adjustable Bed is $200, what is the profitmaximizing price? (b) Per capita income in the United States is forecast to rise by 3% next year. How will this impact Craftmatic s sales? (c) The price of battery-powered massage pillows suddenly fell by 10%. How will this impace Craftmatic s sales? 15

15 5. (5 marks) In the city of Nottoobright, the mayor has decided that the rents are too high. The long-run supply function of one-room rental apartments is given by q = p and the long-run demand function is given by q = p, where p is the rental rate in dollars per week. The authorities made it illegal to rent an apartment for more than 50 dollars per week. To avoid a housing shortage, the authorities agreed to pay landlords enough of a subsidy to make supply equal to demand. How much would the weekly subsidy per apartment have to be to eliminate excess demand at the ceiling price? Explain. 16

16 6. (5 marks) Belinda loves chocolate and always thinks that more is better than less. Belinda thinks that a few piano lessons would be worse than none at all, but if she had enough piano lessons to get good at playing the piano, she would prefer more lessons to less. Draw a graph with piano lessons on the horizontal axis and chocolate on the vertical axis. On your graph sketch two indifference curves for Belinda that would be consistent with this story. Label the better of the two indifference curves AA and the worse one BB. chocolate piano lessons 17

17 Extra Graph Paper 18

18 Reference Sheet f(x 1, x 2 ) df(x 1, x 2 ) dx 1 df(x 1, x 2 ) dx 1 2x 1 + 3x x 1 + 6x ax 1 + bx 2 a b x x 2 0.5x ln x 1 + x 2 1 x 1 1 v(x 1 ) + x 2 v (x 1 ) 1 x 1 x 2 x 2 x 1 x a 1x b 2 ax a 1 1 x b 2 bx a 1x2 b 1 (x 1 + 2)(x 2 + 1) x x (x 1 + a)(x 2 + b) x 2 + b x 1 + a x a 1 + x a 2 x a 1 + x a 2 (x a 1 + x b 2) c ax a 1 1 ax a 1 1 ax a 1 2 ax a 1 1 ax a 1 2 [ c(x a 1 + x b 2) c 1] bx b 1 2 [ c(x a 1 + x b 2) c 1] MRS = MU 1 MU 2 TRS = MP 1 MP 2 19