prelims_Micro Prelim June 2007

prelims_Micro Prelim June 2007 - University of California,...

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Unformatted text preview: University of California, Davis Date: June 25, 2007 Department of Agricultural and Resource Economics Department of Economics Time: 5 hours Microeconomics Reading Time: 20 minutes PRELIMINARY EXAMINATION FOR THE PILD. DEGREE Please answer four of the following five parts Part 1. Two firms by the riverside. We consider two firms, named Upstream and Downstream. Upstream produces an output, with quantities denoted q 3 0, that it can sell at a price of p per unit, Ct and its cost function is C(q) = q?, where A > 0 and 0: > 1. The production of output generates emissions 8, where we assume 0 5 e 3 q. Reducing emissions below its maximum level q imposes a cost [Mg — e), where 0 g [5 Ep. Think ofq — e 3 0 as the amount of emissions abated, or filtered out, by Upstream, at an abatement cost of B(q — e). Upstream’s emissions 9 negatively affect Downstream’s technology. Downstream can mitigate these negative effects by incurring adaptation costs, which reduce the effective amount of emissions entering Downstream’s technology below the amount 8 of emitted by Upstream. Denoting by k . 0 E k 3 e, the effective amount of emissions entering Downstream’s technology, Downstream’s profits are given by y — k2 - (e - k)2, where y is a large positive constant that makes Downstream’s profits positive in all relevant situations. (We can interpret it: as the damage that Downstream suffers from the effective emissions entering its technology, and (e - fit)2 as the adaptation costs incurred by Downstream when Upstream has omitted the amount (3 and Downstream reduces its effective amount to it.) (a) Downstream takes 9 as given, and chooses It in order to maximize y - k2 — (e — I02 subject to the constraints 0 E I: g (2. Call this Problem D. (a.1) Write the Kuhn-Tucker conditions for Problem D. Clearly argue from the Kuhn-Tucker conditions that, ife > 0, then the solution, denoted Jae) , satisfies 0 < like) < e. Interpret in words. (a.2) Compute the value function 1(a) of Problem I). (b) A planner’s objective is to maximize aggregate profits, net of abatement and adaptation costs. More precisely, she wishes to solve the following problem. Prohz’em C. Choose g, e and k in order to maximize (a '1 m— th—e)+r-k2-(e-t)1 subject to O :4 fr 3 e 5 q. (b.l) Write the Kuhn-Tucker conditions for Problem C. (b.2) Argue that the Kuhn—Tucker conditions for Problem C are sufficient. The planner also considers the following problem. Problem P. Choose q and e in order to maximize { U‘ m - BU; 6) + rte), subject to t) 5: e 3 q, where 12(9) is the function obtained in (a2) above. (b.3) Write the Kuhn-Tucker conditions for Problem P. (b.4) Argue, using ("b.3), that e > 0 at any solution to Problem P. (h.5) By appealing to (b2), argue that if(q*, 8*) solves Problem P, and k'* = Re) (as defined in (a.l)). then (92*, 6*, k‘") solves Problem C. Interpret in words. (0) The parameters p, 0., B and 1-: will be maintained fixed it what follows, but we will allow ,4 to vary along the positive halfline. Denote by (q*(A), e*(A)) the solution to Problem P for the parameter value A. It turns out that there is a set S C Ell. I with the property that, ifA E S, then 6*(A') = q*(A), (i. e, there is no abatement at the solution ofthe planner’s problem), whereas ifA e T: {A E‘Rq—i A E S}. then 9*(A) <1 (3*(A). (_e.l) How can we interpret a high A vs. a low A? (e.2) Characterize Sand Thy using (b3) and (b4). (0.3) Define k*(/1) as the solution to Problem D of (a) above when Downstream takes the emission level e*(_/1) as given. (In other words, k*(A) 2 file * (AJ) .) Graph, on the same graph, (3*(A }. e*('A) and WM) as functions ol‘A, with A on the horizontal axis. (You can set {1 = 2 in the graph.) Comment on the dependence of output, abatement and adaptation as a functions of A, in view ofyour interpretation ofA given in (01) above. Part 2. Insurance. A consumer has to choose the amount ofinsurance coverage. The bad state, called state 1, is characterized by a loss, and occurs with probability 7E. The complementary, or good, state, called state 2, occurs with probability 1 — TE. In the absence ofinsurance the consumer ends up with the level of consumption * to], where 033 - (01 P 0 is the amount ofthe loss, in the bad state, * 0); in the good state. The insurance market offers contracts with variable coverage or e [0, m2 — on], and linear premix. i.e.. the premium per unit of coverage, or premium rate, is o, and hence the total premium paid by the consumer is get. The consumer takes (,2 as given, and chooses the amount of coverage or. We denote byxl her consumption contingent to the bad state, and by X2 her consumption contingent to the good state. Assume that the preferences ofthe consumer satisfy the expected utility hypothesis (with state independence), and that her von Neumann-Morgenstern-Bemoulli (vNMB) utility function it satisfies: u’(.r') > 0 and u”(_r) 4 t). We assume throughout this part that the insurance premium is actuarially unfavorable to the consumer. i. e., that g P “I. but that nevertheless she always decides to contract positive amounts of coverage, so that any nonnegativity constraint on or can be disregarded. (:1) Write the consumer maximization problem, and its Kuhn-Tucker conditions. Argue that, under the above assumptions, her decision will be Characterized by a first-order equality condition. Write it. Assume that the second order condition holds with strict inequality. (b) lgnder the above assumptions. does the consumer choose to fully insure? Argue your answer. both analytically from the Kuhn-Tucker conditions, and graphically. (c) We are interested in the following exercise in local comparative statics. Suppose that the probability 7t ofthe loss increases, but the insurance premium rate q does not. (Perhaps the insurance companies do not have access to the same information as the consumer does.) Will the consumer decide to increase, decrease, or keep constant her coverage 0:? Precisely argue your answer, and offer a graphical illustration. (d) We now consider the same question as in (c), but in a more complex setup. Assume now that that the insurance companies know the probability it of the loss, and always set the premium rate if proportional to n, i. e., q(n) = in, where I is a positive parameter greater than one. Therefore. an increase in it now has two effects: it increases the probability ofthe loss, and it proportionally increases the premium rate paid by the consumer. ((1.1) Adapt to this case the first order condition of (a) above. ((1.2) Show that, under these assumptions, and if the utility function of the consumer displays constant absolute risk aversion, then the consumer will actually choose a smolt'er amount of coverage or as TI: increases. (M. The implicit differentiation yields some terms involving it” and some involving n“. To sign the sum ofthe terms that involve u", use the constancy of the absolute iisk aversion and the FCC. To sign that ofthe terms involving 11’, use the stnct concavity ofn”.) Can you give an example ofa vNMB utility function displaying constant absolute risk aversion? (d.3) Show that the conclusion of(d.2) remains valid ifthe coefficient of absolute risk aversion is increasing. Can you give an example of a vNMB utility function displaying increasing absolute risk aversion? (d.4) Argue that ifthe coefficient Ofrez’orive risk aversion is constant, then the previous argument is inconclusive. Can you give an example ofa VNMB utility function displaying constant relative risk aversion? (e) Compare (c) and (d). and comment. 3. “'0 consider eeononiies with two goods. goods 1 and 2. and two types of agents, [he Extremists and the Com-{*xes. All agents have the same endowment mi 2 [11 1). All Extremists have. the utility ii me t. i on 'rr..f';[3"1. .132] = {1113312 + [:£.'3)2 while all CUIIX'CXQH have the utility function r'.‘ \ ; J_ n'[;r1 .‘I'gi = (rm )“(mj a [J < a < 1 An tL‘coriorny with r Extremist-s and r CUIlYL‘thi-i is errlled a. r—repliea. of the basic economy. [2.1 j Derive the \V;-1lr‘r.—:sia.n demand for an agent of type E and for an agent of type. Oil—lint: look at the gemnetry of the optimal choice problem. Be sure to answer this question correctly: nil the questions that follow depend on it]. {b} Show geonretrietlily in the Edgeworth box that if -r- = 1, for any value of o the economy has no (1()1'l'J]')(‘EiTi\"C equilibrium. Assume o = 1,52. Sl'iow That ,0 — (1.1} is a competitive equilibrium price vector for the ‘2—repliea. eetmomy. [Hintr the two Extremists (lo nor. need to have the. same conslnnptionl {cl} Assume o lid. 0' positive integer. Show that. p — (l. l) is a. competitive equililn‘ium price vector for The I'll—replica economy. [ejl Suppose o — mid with n, (1 positive integers r} > :1. What is the only I'Joii'rntiul equilibrium hrice yccror for n replica. economy'-j Wl'lat is the minimum number i" such tljiat there is an equilibrium in the r replica economy? {it} Show inlornmlly that if r}: is H rational number and if r Lends lo infinity, the economy has on "approxiumte" competitive equilibrium. Is the ussurnption of convexity of preferences as essentia‘rl as you were [old in your general equilihrhnn course(s}‘.’ Part 4 In the kingdom of Evilunda there is no separation of the legislative,judicial and executive branches of govcrrunent. The king makes all the decisions. One thing he particularly likes to do, on Monday mornings. is to have three prisoners brought to him who have been found guilty (by him, ofcourse} ofthe worst crimes. He then decides who should be executed and who should be set free. He asks them to turn their backs to him and informs them that he is going to write either an "E" [for “to be executed") or an "F" (for “to be set free”) on each prisoner’s back. The prisoners can then look at each others“ backs (of course, no prisoner can see his own back). The king then has them wearjackets that cover their backs and invites the prisoners” wives to the room. He explains to the wives that he either put an “E” or an “F” on each prisoner’s back and allowed them to look at each others” backs. Now he publicly announces that today he was feeling generous and decider] not to have all three prisoners executed. He then asks prisoner i “do you know whether you are going to be executed or set free?” and warns him that if he claims to know. when in fact he couldn’t know, then he is going to be executed and so is his wife. Prisoner 1 answers “No”. The king then asks prisoner 2 "do you know whether you are going to be executed or set free?" (and gives him the same warning). Prisoner 2 answers “Yes, I do". Hearing this, one of the wives screams "my husband is going to be executed!” and faints. Whose wife is she"? You will answer this question as you go through (a)~(g) below. Another will: moans “I can’t Stand the uncertainty! l wish I knew what was going to I11 happen to my husband . Whose wife is she? You will answer this question as you go through (at—(g) below. Another wife jumps Lip and down and yells “i don’t have to wash the dishes tonight! He’ll be home to do it for me!". Whose wife is she? You will answer this question as you go through (a)-t'g) below. it might help you to know that the crime committed by the prisoners was to get a PhD in game theory and to marry women who had a PhD in logic. In order to answer this question to the king’s satisfaction, you need to (a) represent. by means ofin Formation partitions, the prisoners’ state of knowledge after they have seen each others” backs but before the king announces that he had decided not to ex ecutc all three, (b) represent, by means of information partitions, the prisoners’ state of knowledge after the king announces that he had decided not to execute all three, (0) represent, by means of information partitions, the prisoners” state of knowledge after prisoner l answers the first question, (d) represent. by means of information partitions, the prisoners” state of knowledge after prisoner 2 answers the second question, (e) write in words what is common know-tedge at this stage (ie after prisoner 2 answers the second question) among the three prisoners, (0 represent what the wives know after prisoner 2 answers the second question, [g) in the case under (e) find a state a and an event 6 such that, at a, 2 and 3 know G and know that 1 knows G but it is not true that l knows that 2 knows G. Part5 There are two types of workers: low-productivity workers (type L) and high productivity workers (type H). For any worker utility depends on three things: the wage w, the speed of work 5 and the parameter 1, as follows: U(w,s,r) : w—f. For type H workers t = {H and for type L P _ . w i _ . workers 1 — it, w 1th (H > It > 0. (One could interpret I as the maiginal cost of speed, so that high-productivity workers find higher speed of work less costly). The utility of not working is 0. Assume that, if indifferent between working and not“ working, each worker wih’ choose to work and. ifinriifl‘i’rcn! benreen rwo contracts with dgj’erent mines ofs. a type L worker wih’ choose the one i-i-‘irh for-yer 3 and a (t-‘pe H i-t-‘orker it'iii choose the one with the higher 3. There are many firms competing for workers. Each firm runs exactly one assembly line at a chosen speed s 2 0 (s = 0 means that the worker can work at her own pace). The output of a worker of type r at a firm with speed s is given by 00,3) = r. Thus a worker’s productivity increases with her type, but speed does not affect productivity. The speed of a fimi’s assembly line is chosen by the firm; it is not subject to worker discretion (there is no moral hazard). The labor costs are a 1i rm's only costs and there are no fixed costs. The price of the firm‘s product is 1. An employment contract is a pair C = (“t-19,3) specifying a total wage w and a speed s. No firm can distinguish workers' types and there is team production, so a firm cannot condition a worker’s wage on her output. Let P. e (0,1) be the fraction of low-productivity workers in the population. (a) For the case where I}, = l, iH = 4 and 2'. = 0.?5 draw two diagrams in (aw) space (with .5“ on the horizontal axis), one illustrating the indifference curves ofeach type ofworker and the other illustrating the zero-profit line for each firm in the following three cases: (i) all the workers are oftype IL, (2) all are oftype I” and (3) "35% are oftype ti. (b) For the general case (that is, removing the hypothesis ti = 1, 3H = 4 and i. = 0.?5), define a zero-profit pooling eq uilibiiuin as a situation where all firms offer the same contract and every firm makes zero expected profits (do not worry about “profitable deviations" for the moment; thus we should call it a candidate for an equilibrium). (b.1) What contract are the firms offering? [Hint: you need to consider three cases] (b2) What is this contract when (L = 1. {H = 4 and it = 0.35? (c) For the general case, define a zero-profit separating equilibrium as a situation where some firms offer contract CI and attract only type-L workers, while other firms offer contract C I, I, and attract only type—H workers and all firms make zero profits (again, do not worry about "profitable deviations" for the moment; thus, as before, we should call it a candidate for an equilibrium). (c.1) What contracts could the firms be offering? [Write down the incentive compatibility conditions and the zero—profit conditions] (c.2) Give an example ofa separating equilibrium when IL = l, I” = 4, 2. = 0.?5 and Si takes on the lowest possible value (subject to the incentive compatibility constraints). (c.3) For the case where ti 2 1, r = 4, 2 = 0.?5 and both 5L and 5” take on the lowest If possible values (subject to the incentive compatibility constraints), show the separating- equilibriuni contracts in [l\-',S) space (with s on the horizontal axis). ((1) Now let us add the “no profitable deviation" condition for a separating equilibrium, that is, (f) we add the condition that there is no new contract that, if introduced by a firm, would yield positive profits to that firm (given that all the other firms are offering either contract C}, or contract Cg). For the case where IL = l, = 4 and the speed 3 always takes on the lowest H possible value (subj ect to the incentive compatibility constraints), (d.l) calculate the values offlt for which a separating equilibrium exists, and (d.2) give a graphical illustration in (112,5) space (with s on the horizontal axis). If rib, is sufficiently close to I}, then a separating equilibrium (satisfying the “no profitable deviation” condition) will not exist, regardless ofhow large 21 is. Continue to assume that the speed 5 always takes on the lowest possible value (subject to the incentive compatibility constraints) and consider the general case where I), and I” can be any numbers (subject to {H :5 ti 3- l) ). Show non—existence ofa separating equilibrium by showing that iftH is sufficiently close to (I, the H—type worker always prefers the peeling contract, no matter how large it is (subject to 0 < xi *1 l). [Hint differentiate the relevant expressions with respect to t” and evaluate at {H = r1 (£1) For the case where IL = l and xi. = 0.25, illustrate in a graph the utility that the H—type gets from the C” contract and the utility that he gets from a pooling contract, as functions of r! 1,. (£2) Briefly explain the intuition for the phenomenon proved in part (e). ...
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prelims_Micro Prelim June 2007 - University of California,...

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