Department of Economics Microeconomics PhD Qualifying ...

1 Department of EconomicsMicroeconomics PhD Qualifying ExaminationJune 5, 2006 SOLUTIONSI nstructions: Do not open the exam booklet until you are told to do so. Answer all four of the following problems. Each problem has equal weight. Think before you write. Do not spend too much time on any one problem. If you have to leave the room for any reason, please give the instructor your examination onthe way out. You will have 4 hours and 30 minutes to complete the exam. Answer each question in a SEPARATE answer booklet. Use a dark pencil or a pen (the exam will be photocopied). DO NOT WRITE OUTSIDE OF THE BOXES PROVIDED FOR THE ANSWERS. Theboxes probably provide much more space than you need for your answers, so do not use thesize of the box as a guide to how long your answer should be. You may use the backs of the answer sheets as scratch paper, but do not write your answersthere.

2 WRITE YOUR EXAM CODE and Question Number at the top of each Qualifying ExaminationJune 5, 2006 Exam Code:_____page2 Problem 1.[60 minutes, total]Ma Ching-to consumes two goods, rice(r)and whiskey(x). Letpdenote the price of rice; whiskey is the numeraire with a price of1. Ma s utility function isu(r; x) =x+ logr:a)[15 minutes]What is the slope of Ma s indi erence curves for general values ofu; xandr?Make a rough drawing of Ma s indi erence map (show the shapes of the curves, but do notwaste time on a precise drawing). Label your axes. Do Ma s indi erence curves intersectthe axes?From the implicit function theorem we know that withuheld constantdxdr= @u@r@u@x= 1r:This slope is independent ofuandx, which means that all the indi erence curves are parallel, simplyshifting upwards the form of the utility function we see thatr=eu x, which means thatrcannot be0for any nitevalues ofuandx.

3 It follows that the indi erence curves do not intersect the whiskey axis. We also havethatx=u logr, so that intersections on the rice axis are the solutions ofu logr= 0, or atr= Qualifying ExaminationJune 5, 2006 Exam Code:_____page3b)[15 minutes]Find Ma s demand for ricer(p; y)and for whiskeyx(p; y), haveMUrp 1rpandMUx1 1:Ma consumes only rice so long asMUrp>MUx1;but if he consumes only rice then from his budget constraint we know thatr=y=p:Therefore, Ma consumes only rice if1rp 1y>1or, equivalently, ify <1:In that case,r=y=pandx= 0. Otherwise, Ma setsMUrp=MUx1;so thatr= 1=px=y 1:In sum, demand is given byr(p; y) =8<:ypfory <11=pfory 1andx(p) =(0 fory <1y 1 fory 1c)[5 minutes]Find the indirect utility demand into the utility function we getv(p; y) =8<:logypfory <1y 1 logpfory 1 Microeconomics Qualifying ExaminationJune 5, 2006 Exam Code:_____page4d)[15 minutes]Find the expenditure function and Hicksian compensated demand functionsfor rice and for expenditure functiony=e(p; u)is the solution ofu=v(p; y)fory.

4 Fory <1we haveu= logypsothaty=peu. Butpeu<1if and only ifp < e u. Fory 1, we haveu=y 1 logpory=u+ 1 + 1if and only iflogp uor, equivalently, if and only ifp e u. So we havee(p; u) =(peuforp < e uu+ 1 + logpforp e u:The Hicksian compensated demand for rice ishr(p; u) @e@p=(euforp < e u1=pforp e u:We cannot use this method to nd the compensated demand for whiskey, because the price of whiskey is thenumeraire and it is held constant. However, because at any price the utility derived from the compensateddemand for both goods must add up to the speci ed utility, it must be true thatu hx(p; u) + loghr(p; u)so thathx(p; u) u loghr(p; u):Thereforehx(p; u) =(0forp < e uu+ logpforp e u:e)[10 minutes]Supposey= 1and the price changes fromp= 1top= 2. Find the changes inthe quantities that Ma demands. Decompose the changes into the substitution and incomee 1all income is spent on rice, so thatr(p;1) = 1=pandx= 0.))

5 Whenpgoes from1to2,rgoesfrom1to1=2, butxstays at0. Atp= 1, we haveu=x+ logr= 0. Whenp= 2, compensated demandfor rice is1=2and that for whiskey islog 2. (Slutsky compensation y= 1, because we needy= 2forMa to be able to buy his original bundle. This means Slutsky compensated demand for rice would be1=2and for whiskey, it would be1.) Using the Hicksian de nition, then we have that the subsitution e ect is 1=2for rice and+ log 2for whiskey. The income e ect is zero for rice and log 2for wiskey it is theincome e ect that holds that actual demand for whiskey Qualifying ExaminationJune 5, 2006 Exam Code:_____page5 Problem 2.[60 minutes, total]An exchange economy has a single consumption good, two datest= 0;1, and agentsi= 1; : : : ; nwho do not own or consume anything att= 0. Att= 1thereare states of natures= 1; : : : ; S; agentihas endowment!isin states. The state of nature isunknown att= 0, all agents assign the same probability s>0to states.

6 Agenti s Bernoulliutility belongs to the linear risk-tolerance class, taking the formui(ci) =1b 1(ai+bci)1 1bwhereb >0;6= 1andai>0for )[30 minutes]Show thatex antePareto optimal allocations have the property thatcis=fi:!s+kiwherefi; kiare independent ofs, where!sdenotes the per capita endowment ofthe economy in Arrow-Borch PO conditions are: i su0i(cis) = , for alli; s, oru0i(cis)u0i(cir)= s rfor any pair of statess; rand any agenti. So the ratio of marginal utilities across any pair of states is equalized across all (ci) = (ai+bci) 1b, this implies thatai+bcisaj+bcjs=ai+bciraj+bcjracross any pair of agentsi; jand any pair ofstatesr; s. Henceai+bcisaj+bcjs=mij(1)for alls. Fix states; select an arbitrary agentj, and average (1) across allito obtain (using the budgetbalance condition which states that average consumption in any state equals the average endowment):a+b!

7 Saj+bcjs=mj(2)whereadenotes the average ofaandmjthe average ofmijacross alli. Hence (2) impliesaj+bcjs=1mj[a+b!s]and socjs=fj:!s+kj(3)wherefj 1mjandkj abmj Qualifying ExaminationJune 5, 2006 Exam Code:_____page6b)[30 minutes]Suppose that a full set of Arrow securities are traded att= 0. Derive anexpression for the equilibrium price of a statessecurity, as a function of the probabilityand per capita endowment associated with s net purchase of a state-ssecurity, andpsits price, theniselects a portfoliofzisgtomaximizePs sui(!is+zis)subject toPspszis= 0. The rst-order conditions for an optimal assetportfolio implypspr=u0i(cis)u0i(cir)= s r[ai+bcisai+b:cir] 1b;(4)orps s(ai+bcis) 1b=pr r(ai+bcir) 1b=Ki(5)independent of the state. (5) implies that[ps sKi] b=ai+bcis(6)and averaging acrossiwe obtain[ps sK] b=a+b!s(7)for some constantK, ,ps=K s[a+b:!s] 1b(8) Microeconomics Qualifying ExaminationJune 5, 2006 Exam Code:_____page7 Problem 3.

8 [60 minutes, total]Two male deer compete for the right to mate with a female male deer is strong (S)or weak(W)with an equal probability. Each male deer knows itsown type, but not its rival s. The two male deer make a simultaneous decision whether or not to ght with each other. If they don t ght, then the female deer will not mate with any of them; ifthey do ght, then the female deer will mate with the winner. (Notice that a male deer can mateonly ifbothagree to ght and it wins.) Suppose that a strong deer always wins a ght against aweak deer, and that if both deer have the same type, then each wins with probability12:Finally,suppose that ghting costs a deer 2, and, on top of that, a deer gets 0 if it doesn t mate, and 10if it does, regardless of whether it fought or )[5 minutes]Describe the set of pure strategies available to a male male deer has four pure strategies:G Give up in either case;F Fight in either case;C Clever, ght if strong, give up if weak;D Dumb, give up if strong, ght if F C DS:N F F NW:N F N Fb)[15 minutes]Is the situation in which strong deer ght and weak deer don t a Bayesian-Nashequilibrium?

9 The question is whether(C; C)is a BNE. Suppose that the other deer employs strategyC:In this case,not ghting gives both types of deer a payo of zero. Fighting gives a strong deer a payo of120@12 10 +12 0 2other is strong1A+12 0other is weak and therefore doesn t ght =32>0;and it gives a weak deer a payo of12 0 2other is strong +12 0other is weak = 1<0:That is, it is better to ght if and only if a deer is strong. It therefore follows that(C; C)is a Qualifying ExaminationJune 5, 2006 Exam Code:_____page8c)[30 minutes]Find all the pure strategy Bayesian-Nash equilibria. (Hint: there are three.)The best response correspondence for each deer is given by: AgainstC;as shown above, the best response isC: AgainstG;payo is zero regardless of what a deer does, so the best response isfG; F; C; Dg: AgainstF;not ghting gives both types of deer a payo of zero. Fighting gives a strong deer apayo of12(10 2) +12 12 10 +12 0 2 =112>0;and it gives a weak deer a payo of12(0 2) +12 12 10 +12 0 2 =12>0:So the best response againstFisF: AgainstD;not ghting gives both types of deer a payo of zero.

10 Fighting gives a strong deer apayo of12(0) +12(10 2) = 4>0;and it gives a weak deer a payo of12(0) +12 12 10 +12 0 2 =32>0:So the best response againstDisF: It follows that the BNE of the game are(C; C);(F; F);and(G; G):d)[10 minutes]Are all the Bayesian-Nash equilibria you found equally plausible?The BNE(G; G)is less plausible becauseGis weakly dominated byC(note that a strong deer alwaysweakly bene ts from ghting!). Microeconomics Qualifying ExaminationJune 5, 2006 Exam Code:_____page9 Problem 4.[60 minutes, total]A worker is equally likely to be either a Low or a High type the worker, nor the market can observe the worker s are twoperiods:t= 1andt= 2. In every period the worker exerts e orte2[0;1]. The worker s choiceof e ort is unobservable by anyone else. The utility cost of the e ort in every period is12e2forboth types of the worker. There are two possible output levels in every period:1and0.