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prelims_Micro Prelim Sept 2005

prelims_Micro Prelim Sept 2005 - University of California...

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Unformatted text preview: University of California, Davis Date: September 1, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE PILD. DEGREE Please answer four of the following five questions QUESTION 1 Throughout this question we assume that consumers have preferences over the quantities of two market goods, x1 3 0 and x2 2 0, as well as the size g 3 0 of a park. A consumer has no control over 3, and therefore takes the value g as given, but he or she spends all his or her (after-tax) wealth w, also given to him or her, buying x1 and x; in the market at the given prices (131, 1);). In summary, in his or her Walrasian demand the consumer takes (p1, p2 , w, g) as given and chooses x1 and 2:; subject to the budget constraint plx, + pgx2 s w. Throughout this question we assume that wealth is always large enough so that at the chosen point the amount of x; is strictly positive. 1.1. Consumer Karl’s preference relation on ER: can be represented by the utility function 12:53: —>iR+ :r’i(x,,x2,g)=2,,l(g—i-I)x1 +x2. (1) 'l.1(a). Compute Karl’s Walrasian demand functions 55, (pl, ,02 , w, g) and 3', (p1, p2 , w, g) , and indirect utility function. Comment. 1.1(b)_. Let (p], p2) = (1, 1), and let Karl’s initial wealth be 10. The government is considering .a program that would improve g from 0 to 3, and would require Karl to pay a tax of 2. The prices (p1, p2) would not change as a result of the program. Would Karl benefit from the program? 1.2. Karl has a cousin, Priscilla, whose preference relation on ER: can be represented by the utility 2.r’(g+l)xI +.rl g+l function 1.2(3). ls Priscilla’s preference relation on ilii identical to Karl’s? 1.2(b). Compute Priscilla’s Walrasian demand functions for x1 and x2. 1.2(c). Would Priscilla benefit from the program of 1.1(b) ab0ve? (Her initial wealth is 15, and her tax contingent to the implementation of the program would also be 2). Comment. University of California Davis Date: September 1, 2005 Department of Economics Time: 4 hours Microecononfics Reading Time: 20 minutes 1.3. More generally, let a consumer’s preference-s be represented by a utility function of the form 5319?? —) SR : ti(x1,x2_,g) : <p(ti(xl,x2,g),g), where tp : Elli —> ‘Ji is a differentiable function with strictly positive first partial derivative, and Lil is the function given by (1) above. Without further assumptions: 1.3(a). What can you say abOut the Walrasian demand functions for an and x2 obtained in the UM AX problem for ti ‘? 1.3(b). What can you say about the consumer’s willingness to pay for g when his or her utility function is of the Li form? 1.4. Suppose now that the function ti , as defined in 1.3 above, satisfies the following assumption: 513(on g) _ 0, as For all (xyg) 6 ”Hi, i. e., the consumer does not care about g whenever he or she does not consume good 1. 1.4(a). Argue that now ii(x1 , xyg) : 31:20:”):2 , g)) for some increasing function r2. 1.4(b). What can you now say about the consumer’s Willingness to pay for g? Page 2 of 'i' University of California, Davis Date: September 1, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20' minutes QUESTION 2 We consider a technology where the first M goods are inputs, and the last L goods are outputs. The production set is denoted Y C Qif’ x ‘Jif _, and a production vector is denoted (2, q), where z 5 my and q e ‘J'i “" . We define the production possihiiity setfor the input vector: as P(z) E {q 6 iii" : (q,z) E Y} . We visualize .Y as the technology of a small country that has a fixed vector of inputs 2, faces world prices p E 9?: for its outputs, and tries to maximize the vaiue of production at world prices. Formally, we consider the REVMAX[ p, 2] problem: Given p 6 ER: and z e i}? f’ , choose q in order to maximize p o q subject to q e P(z) . We write r for the value function of this problem, to be called the revenue function, and Q (resp. if ) for its. solution correspondence (resp. function), to be called the Conditional suppiy correspondence (resp. function). 2.1. Is r homogenous of some degree in p? Argue your answer. 2.2. Show that r is convex in p. 2.3. Show that, ifr and if are differentiable with Ef(p,_z) >> O, and if P(z) is ofthe form 13(2) = {q 6 Eli " :y(q,z) S 0} for a differentiable function y with 81 > ON}, then q} 691 .2) ... . p = q_,,-(p,2).VJ- 6p ,— Comment. 2.4. Prove and discuss the generalized law ofcona'itionai suppiy “ Ap I At; 2 0 ,” where Ap E p‘ — p0 and Ag 3 (21 fit“, for q' e Q(p' Hand 9'0 e Q(p‘12). 2.5. As an illustration, let M = 1 and L = 2, and let the technology be described by the implicit production function : F: ‘J-i_ ><‘Ji2 —> ‘33: Hz, qt, :92) = (9'1)2 + (q; )2 + 2:, so that" any vector (2, on, Q; ) satisfyingzs 0 and (q1 )2 + (g; )2 + zg 0 is technologically feasible. (Moreover, the production set is assumed to satisfy the free disposal assumption.) 2.5(a). Graphically represent the revenue maximizing problem REVMAX in (9'1, (12) space. 256)). Compute r and if and check that the equality in 2.3 holds. Page 3 of 7 University of California, Davis Date: September 1, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes QUESTION 3 Public Good and Income Redistribution. Consider an economy with three goods — labor, 'a consumption good and a public good —and I consumers. All consumers have the same utility function 'u (x, E, n) : ln(x) + 11105") + ln(n_) , where x is the consumption of private good, i? the consumption of leisure and n the consumption of the public good. The only initial resource of a consumer is hislher labor time. We model the fact that consumers have different abilities by assuming that their endowments of labor time differ (in efficiency units), and let Us denote the total endowment of labor of consumer i. As usual, leisure is i the time not spent working. We assume that (If. > 2 and that Zen, = 31' . The consumption good is i=l produced from labor with constant returns, one unit of labor giving one unit of consumption good. In the same way the public good is produced from labor with constant returns, one unit of labor giving one unit of public good. We are interested in the implicit income redistribution that can accompany the use of taxes to finance the public good. 3.1. Compute the Lindahl equilibrium. 3.2. Suppose that the (if. "s are not observable, so that personalized prices cannot be used. Suppose that, instead, the government chooses the level of the public good and uses lump-sum taxes to finance it. If the of "s are not observable, then the lump-sum tax must be the same for all the agents. 3.2(a). Show that all Pareto optima of the economy in which all agents work have the same level ofpublic good. 3.2(b). Compute the competitive equilibrium allocation obtained when the government chooses this optimal level of the public good and finances it by a uniform lump-sum tax. 3.2(c). Using the case I = 2, argue that the allocation obtained in 3.2(b) is- farther from being- egalitarian than the Lindahl equilibrium. Explain why. Page 4 of 7" University of California, Davis Date: September 1, 2005 Department ol‘Economics Time: 4 hours Microeconomics Reading Time: 20 minutes QUESTION 4 There are two groups of individuals. All the individuals in Group 1 have the same utility function, which is as follows: [x + q if owns a car of quality 9! and $x 1 x if owns Sx but not car All the individuals in Group 2 have the same utility function, which is as follows: I + org if owns a car of quality (1 and $x .1: if owns $1" but not car withor> 1. All cars are owned by individuals in Group 1. Each car is of one of the qualities in the set Q = { 1,000, 2,000, 1,000a}, where n 2 2. There is an equal number of cars of each quality (i.e. as many cars of quality 1,0001' as cars of quality 1,000}, for every Lj 6 {1,2,. . .,n}). The quality of each car is known to the owner but cannot be determined by the buyer. Thus. there can be only one price for second-hand cars. Call this common price P. All agents are risk neutral. 4(a). For every integer s e {1, 2, ..., it}, give a necessary and sufficient condition on the value of o. fer there to be an equilibrium where all and only the cars of quality up to (and including) 1,0003 are traded at price P (assume that the number of individuals in Group 2 is sufficiently large for there to be a potential buyer for every car and that each member of this group has a sufficiently large endowment of meney). 403). For what values of or is there an equilibrium at which the final allocation of cars (and money) in the population is Pareto efficient? 4(c). Suppose that there are 9 quality levels (a = 9) and CL = 1.76. Furthermore, suppose that there are 2,000 cars of each quality (thus a total of 18,000 cars). What is the largest number of cars that can be traded at an equilibrium? Page 5 of?' University ot‘Catifornia, Davis Date: September I, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes QUESTION 5 Consider the following cases. Assume throughout that the firms are risk-neutral. CASE 1. Tomorrow, with equal probability, it will either rain or shine. Firms 1 and 2 have to make their production plans today not knowing what the weather will be like tomorrow. They simultaneously choose whether to produce umbrellas or sun lotion. Their profits are as follows: if they make the same choice (both choose umbrellas or both choose lotion) then they make a profit of 1 each, irrespective of the weather. If they make different choices, then the firm that makes the choice appropriate to the weather makes a profit of 4, while the other one makes a profit of 0 (cg, if firm 1 chooses lotion, firm 2 chooses umbrellas and the weather turns out to be rain, then firm 1 makes a profit of 0 and firm 2 makes a profit of 4). 5(3). Draw the extensive form of this game. 5(b). Write the corresponding normal-form game and find all the pure-strategy Nash cquilibria. CASE 2. Consider now the alternative situation where firm I knows with certainty what the weather will he like (e.g., because it has access to an accurate weather forecast), while firm 2 does not know (although it is common knowledge that firm 1 knows what the weather will be like). The production decisions are still made simultaneously (that is, in ignorance of what the other firm has chosen). [Hintz you can think of this situation as equivalent to the situation where production decisions are made after the realization of the weather and firm 1 gets to see what the weather is like, while firm 2 does not.) 5(c). Draw the extensive form of this game. 5(d). Write the corresponding normal-form game and find all the pure-strategy Nash equilibria. CASE 3. Now let‘s go back to Case 1, where both firms are uncertain about the weather, but modify it as follows: firm 1 chooses first (not knowing what the weather will be like) and then firm 2 learns what firm I chose and makes its choice second with the knowledge of what firm I did (but not knowing what the weather will be like)a Page 6 of? University of California, Davis Date: September 1, 2005 Department of Economics Time: 4 hours Microeconomics Reading Time: 20 minutes 5(8). Draw the extensive form of this game. 5(1). Write the corresponding normal-form game and find all the pure-strategy Nash equilibria. CASE 4. Now modify Case 2 (where firm 1 knows the weather before it. makes its decision) as follows: firm 2 knows firm 1’s decision but nOt what the weather is like when it makes its own decision. 5(g). Draw the extensive form of this game. 5(h). Write the corresponding normal-form game and find all the pure-strategy Nash equitibria. 5(i). Now compare Case 1 with Case 2: does firm 1 gain from the'additional information it has in Case 2 (the state of the weather) as compared to Case 1? Does firm 2 gain or suffer from the additional information given to firm 1‘? 5(j). Now compare Case 3 with Case 4: does firm 1 gain from the additional information. it has in Case 4 (the state of the weather) as compared to Case 3? Does firm 2 gain or suffer from the additional information given to firm 1? Try to explain why this is the case. Page 7' of? ...
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