Universidad Carlos III de Madrid June Microeconomics Grade

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1 Universidad Carlos III de Madrid June 05 Microeconomics Name: Group: 5 Grade You have hours and 5 minutes to answer all the questions. The maximum grade for each question is in parentheses. You should manage your time accordingly.. Multiple Choice Questions. (Mark your choice with an x. You get points if your answer is correct, points if it is incorrect, and zero points if you do not answer.).. If an individual s preferences over goods x and y are monotonic (axiom A:), then its indi erence curves do not cross are decreasing are increasing are convex... If the prices of the goods increase by % and the consumer s income increases by %, then the budget line rotates over its intersection with the x-axis rotates over its intersection with the y-axis maintains its position shifts parallel towards the origin... A consumer s monetary income is I = and the prices are p x = ; p y = : The consumer is considering buying the bundle (x; y) = (; 0) : If her marginal rate of substitution at this bundle is MRS (; 0) =, then the consumer should buy more x and less y buy more x and more y buy more y and less x buy the bundle (; 0)... If a consumer s preferences are represented by the utility function u(x; y) = minfx; yg; then a decrease in the price of x causes a decrease in the demand of x an ambiguous e ect on the demand of x an income e ect equal to zero a substitution e ect equal to zero.

2 .5. In 0 the prices of goods were (p x ; p y ) = (5; ), and in 05 they are (p 0 x; p 0 y) = (; ). If the consumption bundle of the representative consumer is (x; y) = (; ), then the Consumer Price Index calculated as a Laspeyres Index is :.6. If a rm s production function is F (L; K) = L + p LK; then the rm s returns to scale are increasing decreasing constant indeterminate..7. Identify the certainty equivalent (CE) and the risk premium (RP ) of the lottery l = (; ; 6; ; ; 6 ) for an individual whose preferences are represented by the Bernoulli utility function u(x) = p x. EC(l) = ; P R(l) = 0 EC(l) = ; P R(l) = EC(l) = ; P R(l) = EC(l) = ; P R(l) = :.8. If a rm has constant returns to scale, then its total cost function is concave its total cost function is convex its average cost is constant its marginal cost is less than its average cost..9. If the total cost function of a competitive rm is C(Q) = Q Q + 8Q, and the long run equilibrium price is P = 0, then the rm obtains losses obtains pro ts produces Q = 0 units produces Q = units..0. The Lerner Index of a monopoly is inversely proportional to the absolute value of demand elasticity directly proportional to the absolute value of demand elasticity larger the larger is the monopolist total cost larger the smaller is the monopolist pro t.

3 . María has a daily endowment of hours (to work or use for leisure activities) and a monetary (non-labor) income of M euros. Her preferences are represented by the utility function u(h; c) = ln h + ln c; where h denotes the number of hours of leisure and c denotes her consumption. Set the price of consumption to p c = ; and denote by w the hourly wage. (a) (5 points) Describe María s problem, and calculate her demands of leisure and consumption, as well as her labor supply as functions of M and w. The consumer s problem is max h;c u(h; c) = ln h + ln c c + wh M + w 0 h ; c 0 Interior solution: in order to nd María s demands for consumption and leisure, we calculate the marginal rate of substitution MRS(h; c) = c h ; and solve the system, The solution to this system is h = M w + 8; c = w c h = w c + wh = M + w: M w + 8 = M + w: In order for 0 h to hold, we need M=w ; that is, w M=6: If w < M=6; then the solution to María s problem is h = ; c = M: Therefore, demand functions are h = h (w; M) = M w + 8 if w M=6 if w < M=6 ; c = c (w; M) = M + w if w M=6 M if w < M=6 : María s labor supply function is M s(m; w) = h(m; w) = w if w M=6 0 if w < M=6

4 (b) (5 points) Using your results in (a), represent María s budget set and calculate her optimal consumption-leisure bundle for M = 6 and w = : Budget line: c + h = 6 + () Optimal bundle: (h ; c ) = (h (6; ) ; c (6; )) = (9; 8) : (c) (0 points) With the data in part (b), calculate income and substitution e ects over the demand of leisure caused a 5% tax on labor income. Solution: The tax imposed on labor income is equivalent to a reduction of the wage from w = to w 0 = ( 0; 5) = euros per hour. The total e ect is T E = h (6; ) h(6; ) = 8 9 = : In order to nd the substitution e ect, we solve the system ln 9 + ln 8 = ln h + ln c c h = ; where ln 9 + ln 8 = u(9; 8) is María s utility with the optimal bundle identi ed in part (b). Substituting c = h= from the second equation we get which can be written as Hence ~ h = q 9 (8) ' 9; 9: The substitution is ln 9 + ln 8 = ln h + ln h ; ln[9 (8)] = ln[ h ]: ES = 9; 9 9 = 0; 9: The income e ect is IE = T E SE = 0:56

5 . (0 points) In a multiple choice exam there are possible answers and only one is correct. A student who does not know the correct answer believes that the answers are correct with the same probability. A correct answer is worth points, an incorrect is worth m points (m 0), and not answering entails 0 points. (a) Describe the decision problem of a student who does not know the correct answer. If the student is risk-neutral, what is the maximum value of m for which it would be optimal for her to answer randomly? Is this value smaller or larger for a risk-averse student? (Justify your answer.) Solution. The student faces the lotteries no answer, l NA = (0; ), and answer, l A (m) = (; m; =; =) : The expected value of these lotteries are E(l NA ) = 0 and E(l A (m)) = ( m)=: The optimal decision can be identi ed by comparing the certainty equivalence of these lotteries, which for a risk-neutral student coincide with the expectations of the lotteries. The value m is the maximum for which EC(l A (m)) = E(l A (m)) = m 0 = E(l NA ) = EC(l NA ): The value m = =; for which the condition holds with equality, is the maximum value for which answering randomly is optimal. If the student is risk-averse, then EC(l A (m)) < E(l A (m)), which EC(l NA ) = E(l NA ) (since l NA is a degenerate lottery). Hence EC(l C (m )) < E(l C (m )) = 0 = E(l NC ) = EC(l NC ): Therefore for m answering randomly is not optimal for a risk-averse student. The maximum value of m for which a risk-averse student would answer randomly is less than m. (b) Assume that m = = and that the student is risk-neutral. If the student knew with certainty that one of the answers is incorrect, would it be optimal to choose randomly one of the three remaining answers? What if he knew that two of the answers are incorrect? Solution. For m = = > = the risk-neutral student prefers the lottery l NA. However, if the student can dismiss one answer as incorrect, then the lottery answer would be la (=) = (; =; =; =) which certainty equivalence is EC(l A(=)) = E(l A(=)) = () = 6 > 0 = E(l NA) = EC(l NA ): Hence la (=) (choose one of the three possible answers randomly) is optimal. If the student can dismiss two answers, then the lottery answer is even more favorable, la (=) = (; =; =; =) and EC(lC(=)) = E(lC(=)) = () = 5 8 > 0 = E(l NC) = EC(l NC ) and therefore it would be the optimal decision. 5

6 . In a competitive market there are three rms producing the good with a total cost given by the functions C (q) = 5q + 0q + 0; C (q) = q + 6q + ; C (q) = q + 6q + ; respectively. We know that at the competitive equilibrium rm is producing at its minimum average cost. (a) (0 points) Calculate the rms supply, and compute the market price and the output and pro t of each rm at the competitive equilibrium. We have AC (q) = 5q =q; therefore, the company produces q = q such that dcme (q) dq = 5 0 q = 0; i.e., q = : Besides the rm is competitive, (q ; p) = (; p) is in its supply curve, that we obtain of the equation CM (q ) = p: Therefore, the supply curve of the rm is S (p) = 0 The market price p we have S (p) = ; i.e., (p 0) if p 0 0 if p < 0 ; (p 0) = : 0 Solving this equation we get p = 0: The pro t of the rm = 0 () 5 () + 0 () + 0 = 0 Since the rms and are also competitive we get their supply functions from the equations MC (q ) = p and MC (q ) = p; i.e., q + 6 = p and 6q + 6 = p: Solving for q we get S (p) = S (p) = 6 (p 6) if p 6 0 if p < 6 ; (p 6) if p 6 0 if p < 6 ; The amounts supplied are in equilibrium are S (0) = 6 and S (0) = : So, the pro ts of these rms are = 0 (6) (6) + 6 (6) + = 0 and = 0 () () + 6 () + = 6: 6

7 (b) (0 points) Compute the supply function for each rm and the market equilibrium in the long run, assuming that there is free entry and all three technologies can be freely adopted. The supply functions are calculated in (a). In the long run all rms get zero pro t and produce at the minimum average cost. The minimum average costs for rms and are obtained by solving the system ACe (q) = q q ACe (q) = q q The minimum average costs are derived by solving the system q q = 0 = 0: whose solution is q = ; q = : Therefore, the average minimum costs of these companies: ACe = ; and ACe = 8: As ACe = 0; in the long-run the market price p = 8 and only companies with the technology of the company that produces and two units remain on the market. The number of such companies in the market is the need to serve the demand for a price p = 8, which is given by the expression D(8)=: 7

8 5. A rm monopolizes a market for a textbook whose demand is D(p) = maxf50 p=; 0g. The rm s total cost function is C(q) = q =. (a) (0 points) Calculate the monopoly equilibrium and the Lerner index. The inverse demand function is P (q) = maxf00 Thus, for q 50; the monopoly s revenue is R(q) = P (q)q = 00 q; 0g: q q; and its marginal revenue is MR(q) = 00 The rst order condition for pro t maximization is q: that is, The solution to this equation is MR(q) = MC(q); 00 q = q: q M = 50 < 50: Therefore, q M = 50 is the monopoly equilibrium output. The equilibrium price is The Lerner Index is p M = 00 L = pm C 0 (q M ) p M = 00 (50) = 00 (50) 00 = 8

9 (b) (0 points) Determine the price and output that would result if the government regulates the monopoly with the objective of maximizing the total surplus subject to the condition that the monopoly does not incur losses. Solution: The marginal cost function is MC(q) = C 0 (q) = q: The maximum total surplus is obtained when the output level is such that price coincides with marginal cost, that is, P (q) = MC(q): Assuming q < 50; this equation can be written as 00 q = q: The solution to the equation is q = 75: For this output level, the market equilibrium price is and the monopoly pro t is p = P (q) = 00 (75) = 50 p q C(q ) = 50 (75) (75) = 875 > 0: Therefore (p ; q ) = (50; 75) maximizes the total surplus (and satis es the condition that the monopoly does not incur losses). 9