Universidad Carlos III de Madrid June Microeconomics Grade

Transcription

1 Universidad Carlos III de Madrid June 2017 Microeconomics Name: Group: Grade You have 2 hours and 45 minutes to answer all the questions. 1. Multiple Choice Questions. (Mark your choice with an x. You get 2 points if your answer is correct, points if it is incorrect, and zero points if you do not answer.) 1.1. The preferences % over consumption bundles in R 2 + de ned as (x; y) % (x 0 ; y 0 ) if and only if fx > x 0 ; or x = x 0 and y y 0 g do not satisfy axiom A:1 (completeness) do not satisfy axiom A:2 (transitivity) do not satisfy axiom A:3 (monotonicity) satisfy axioms A:1; A:2 and A: At the prices (p x ; p y ) = (1; 1); the optimal bundle of a consumer with income I = 4 and preferences as those described in question 1.1 is (2; 1) (4; 0) (0; 2) (3; 1=2) A consumer s preferences are represented by the utility function u(x; y) = minf2x; yg. The prices are (p x ; p y ) = (1; 1): Hence the signs of the substitution e ect (SE) and income e ect (IE) over the demand of good x of an increase in the price of x to p 0 x = 2 are SE < 0; IE = 0 SE = 0; IE > 0 SE = 0; IE < 0 SE = 0; IE = 0: 1.4. If the prices of goods x; y were (p x ; p y ) = (2; 1) in 2016, and are (p x ; p y ) = (2; 2) in 2017, then Laspeyres consumer price index for an individual whose consumption in 2016 was (x; y) = (2; 2) is:

2 1.5. If the preferences of the consumer of the previous question are represented by the utility function u(x; y) = 2x + y, then her true consumer price index is: If the preferences over lotteries of a consumer are represented by the Bernoulli utility function u(x) = p x, then her expected utility and risk premium for the lottery l = (x; p) that pays x = (1; 4; 16) with probabilities p = ( 1 2 ; 1 4 ; 1 4 ) are Eu(l) = 1; RP (l) = 9 2 Eu(l) = 2; RP (l) = 9 2 Eu(l) = 2; RP (l) = 3 2 Eu(l) = 1; RP (l) = 3 2 : 1.7. Lolita, the competitive cow of Holstein that produces milk using oats (O) and hay (H) according to the production function F (O; H) = 2O + p H; has constant returns to scale increasing returns to scale decreasing returns to scale undetermined returns to scale A rm that produces a good with average cost CM e (q) = 2 p q has economies of scale diseconomies of scale constant returns to scale increasing returns to scale If a rm s output in the short run competitive equilibrium price is positive, then its marginal cost is greater than or equal to its average cost the market price is greater than or equal to its average variable cost the market price is greater than or equal to its average cost its marginal cost is decreasing The Lerner index of a monopoly that produces the good with cost C(q) = 4 + 3q and faces the demand D(p) = maxf12 2p; 0g is

3 2. A consumer s preferences for food (x) and clothing (y) are represented by the utility function u(x; y) = x + 2 ln y. (a) (10 points) Calculate her demand for food and clothing, x(p x ; p y ; I) and y(p x ; p y ; I). (Verify the possible existence of interior and corner solutions to the consumer s problem.) Calculate and graph the consumer s budget set, her optimal bundle and her utility level at prices and income (p x ; p y ; I) = (1; 2; 4). Solution: Since RMS(x; y) = 1 2 y = y 2 ; an interior solution to the consumer s problem solves the system y = p x 2 p y p x x + p y y = I: Solving the system we get x(p x ; p y ; I) = I p x 2; y(p x ; p y ; I) = 2 p x p y : For (p x ; p y ; I) such that I > 2p x these functions have positive values, and therefore they provide the solution to the consumer s problem. For (p x ; p y ; I) such that I 2p x the problem has the corner solution x(p x ; p y ; I) = 0; y(p x ; p y ; I) = I=p y. For (p x ; p y ; I) = (1; 2; 4) the budget constraint is Since 4 = I > 2p x = 2; the optimal bundle is and the consumer s utility is The graph below illustrates these calculations. x + 2y = 4: (x ; y ) = (2; 1); u(2; 2) = ln 1 = 2 y u(x,y)= x 3

4 (b) (10 points) At prices and income (p x ; p y ; I) = (1; 2; 4); calculate the income and substitution e ects over the demand of good x of an increase of its price to p 0 x = 2 Solution: In order to calculate the substitution e ect we solve the system y = p0 x 2 p y x + 2 ln y = 2: The solution is y = 2; x = 2 2 ln 2. Hence the substitution e ect is SE = (2 2 ln 2) x(1; 2; 4) = 2 ln 2 < 0: The total e ect is T E = x(2; 2; 4) x(1; 2; 4) = 0 2 = 2 < 0: Therefore the income e ect is IE = T E SE = 2 ( 2 ln 2) = (2 2 ln 2) < 0: 4

5 3. (15 points) A individual s preferences over leisure (h; measured in hours) and consumption (c; measured in euros) are represented by the utility function u(h; c) = hc 2 : The individual has H = 16 hours available for leisure and labor activities, and a monetary income of M = 48 euros. The wage is 8 euros/hour. (Note that p c = 1:) Calculate the individual s consumption and leisure assuming that there is a tax of 25% on labor income. Calculate also his consumption and leisure if the labor tax is replaced by an equivalent xed tax T: (Thus, T is the total taxes paid by the consumer with the 25% labor tax.) Determine whether the individual would be better or worse o with the xed tax T than with the 25% labor tax. Solution: Withe the 25% labor tax the e ective wage is w = 8(1 problem is max h;c hc2 s.t. c 6(16 h) h 16; c 0: A solution to this problem solves the system c 2h = 6 c + 6h = 144: 1=4) = 6 and the individual s Solving the system we obtained h = 8 and c = 96: The individual s tax bill is 1 8(16 h ) = 16 euros. 4 If the labor tax is replaced with a xed tax T = 16, the individual s budget constraint becomes c + 8h 8(16) + 48 T: A solution to the new individual s problem solves the system c 2h = 8 c + 8h = 160 Solving the system we get h = 20=3 and c = 320=3: The individual s utility with this tax is u = = > u = 8 (96) 2 = 3 27 Hence his welfare is greater than with the 25% labor tax = :

6 4. A competitive rm produces a good using labor, L, and capital, K, according to the production function F (L; K) = p L 3p K. The prices of labor and capital are w = 3 and r = 2, respectively. (a) (10 points) Calculate and graph the total, average and marginal cost functions, and determine if the rm has economies or diseconomies of scale. Calculate also the rm s supply. Solution: In order to calculate the costs functions, we calculate the conditional demands of inputs. To do this, we solve the system MRT S(L; K) = w r q = p L 3p K: Since MRT S(L; K) = 3K=2L; the solution to the system is The total cost function is The average and marginal cost functions are L(q) = K(q) = q 6 5 : C(q) = wl(q) + rk(q) = 5q 6 5 : AC(q) = C(q) q = 5q 1 5 ; MC(q) = 6q 1 5 : Since AC(q) is increasing, the rm has diseconomies of scale. In order to calculate the rm s supply, we solve the rst order condition for pro t maximization p = CMa(q) ) q = p 5 : Since the cost function is convex ( C 00 (q) = 6q 4=5 =5 > 0); the second order condition for pro t maximization holds. Moreover, since p = MC(q) = 6q 1 5 > 5q 1 5 = AC(q); the closing down condition also holds. Therefore, the rm s supply is p 5 S(p) = : 6 cost 20 C(q) costs MC(q) AC(q) q q 6

7 (b) (10 points) Calculate and graph the short run total, average and marginal cost functions and the rm s supply assuming that K = 8. Solution: We calculate the conditional demand of labor in the short run: The short run total cost function is q = F (L; 9) = 2 p L ) L(q) = q2 4 : C(q) = 8r + w L(q) = q2 : The short run average, average variable, and marginal cost functions are AC(q) = 16 q q; AV C(q) = 3 4 q; MC(q) = 3 2 q: Solving the rst order condition for pro t maximization we get p = MC(q) ) q = 2 3 p: Since M C is increasing, the second order condition for pro t maximization holds. Moreover, since AV C(q) = 3 4 q < 3 2 q = MC(q) the closing condition also holds. Therefore the short run supply of the rm is S(p) = 2 3 p: 35 C(q) 15 MC(q),S(p) AVC(q) AC(q) q q, p 7

8 5. A rm produces a good with cost C(q) = q 2 =2 + 2 and monopolizes a markets in which the demand of young consumers is D 1 (p) = f18 2p; 0g and the demand of old consumers is D 2 (p) = maxf12 p; 0g. (a) (5 points) Calculate the equilibrium assuming that the monopoly can price discriminate young and old consumers. Solution: The inverse demands young and old consumers are P 1 (q 1 ) = maxf9 q 1 =2; 0g and P 2 (q 2 ) = maxf12 q 2 ; 0g., respectively. Therefore the monopoly chooses q 1 18 and q 2 6 by solving the problem max (q 1 ;q 2 )2R 2 + P 1 (q 1 )q 1 + P 2 (q 2 )q 2 C(q 1 + q 2 ): An interior solution to this problem is identi ed by the system of equations 9 q 1 = q 1 + q q 2 = q 1 + q 2 whose solution is q1 = q 2 = 3: The equilibrium prices are p 1 = 9 3=2 = 7:5 and p 2 = 12 3 = 9. 8

9 (b) (10 points) Calculate the monopoly equilibrium in the absence of price discrimination, and determine who is better/worse o (monopoly, young/old consumers) relative to the equilibrium of part (a). Solution: The aggregate demand is 8 < 30 3p if p 9 D(p) = D 1 (p) + D 2 (p) = 12 p if 9 < p 12 : 0 if p > 12: Hence the inverse demand is 8 < 12 q if q < 3 P (q) = 10 q=3 if 3 q 30 : 0 if q > 30: The monopoly chooses q to solve max q2r + P (q)q C(q): A solution to this problem satis es For q < 3; P 0 (q)q + P (q) = MC(q): P 0 (q)q + P (q) = 12 2q > 6 > q = MC(q): Hence the solution to the monopoly problem involves 3 q 30, and is identi ed by the equation P 0 (q)q + P (q) = MC(q), 1 3 q + 10 q 3 = q: The solution to is q = 6: Therefore the monopoly equilibrium without price discrimination is q = 6 and p = P (q ) = 8: Without price discrimination young consumers pay a larger price and are worse o, and old consumers pay a smaller price and are better o, than with price discrimination. Also, without price discrimination the monopoly sells the same output than under discrimination, but at a smaller average price, and therefore its pro t is smaller. 9

10 (c) (10 points) Assume that the monopoly is the result of a patent that is about to expire. Once it does, any rm may produce the good using the technology of the established rm, and sell it in the market. Determine price and total output, as well as the number of rms, in the long run competitive equilibrium. Solution: The average cost is Solving the equation AC(q) = q q dac(q) dq = q 2 = 0; we nd that the level of output that minimizes the average cost is q = 2; and the minimum average cost is AC(q) = 2: Thus, the long run equilibrium price in both markets is p = 2: At this price the aggregate demand D(2) = 24, and the total number of rms producing the good is n = D(p) q = 24 2 = 12: 10