MICROECONOMICS COMPREHENSIVE EXAM - UT Liberal Arts

1 MICROECONOMICS COMPREHENSIVE EXAMJUNE 2012 Instructions:(1) Please answer each of the four questions onseparatepieces of paper.(2) Please write only onone sideof a sheet of paper(3) When finished, please arrange your answersalphabetically(in the order in which theyappeared in the questions, 1 (a), 1 (b), etc.).As usual, answers without supporting arguments will receive little , the decision maker, can choose among three possible actions: going to a soccer game(S), going to a movie (M), or going for a walk (W). Her preferences over these alternativesdepend on the state of nature: rain or no rain. If it rains, she prefersMtoS,StoW,and, by transitivity,MtoW. If it does not rain, she prefersStoW,WtoM, and bytransitivity,StoM.

2 Alice has to decide which action to take before she finds out whether itwill rain or not. If prob(rain)=13, she is indifferent betweenMandW, and if prob(rain)>13,she strictly prefersMtoW. Further, if prob(rain)=35, she is indifferent betweenMandS,and if prob(rain)>35, she strictly prefersMtoS.(a) Without knowing the probability of rain, point out an action that Alice will definitelynot take if she is rational.(b) Give an example of a utility function that might be a von Neumann-Morgenstern utilityrepresentation of Alice s preferences in the sense that it does not contradict any of theabove statements.(c) Construct another utility function that represents the same preferences under certainty,but is not a von Neumann-Morgenstern representation of these preferences.

3 (d) Alice s brother, Bob, has the same preferences under certainty as Alice does. However,if prob(rain)=45, he prefersStoM; and if prob(rain)=14, he prefersMtoS. Are thesepreferences consistent with Bob being an expected utility maximizer? the following two good-two consumer pure exchange economy. Mr. 1 has en-dowmente1= (e11,e21) 0 and Ms. 2 has endowmente2= (e12,e22) 0. Their respectiveutility functions areu1(x1,x2) = 1(x1) +x2,andu2(x1,x2) = 2(x1) +x2,where, fori= 1,2, i(x)>0 forx >0, limx 0 i(x) = + , and i(x)<0 forx >0.(a) Show that any two Pareto optimal allocations in the interior of the Edgeworth box involvethe same consumption of the first good. In this case, what is the reason for differentlevels of the consumers utility in different Pareto optimal allocations?

4 (b) Suppose from now on that i(x) =ailnx,whereai>0,i= 1,2, anda1+a2= 1. Whatcan you say about the sharing rule for the first good for Pareto optimal allocations inthe interior of the Edgeworth box? Illustrate your answer with a diagram.(c) Let the second good be the numeraire. Find all competitive equilibria for this control a growing asset, perhaps trees growing in a patch of forest, or wool growingon a flock of sheep, or experience accumulating in an office full of software engineers learningtheir trade. If you liquidate the asset after it has been growing between time 0 and timet,it has a market valueP(t)> ( ) has the usual S shaped properties of a growth curve:P (t)>0 fort >0;P (t)>0 fort (0,T); andP (t)<0 fort (T, ). After liquidatingthe growing asset, you may, or may not, decide to sell, at a pricev, the infrastructurethat makes the asset productive (the acreage, flock, or office in the examples), and exit thebusiness.

5 (a) How does the solution to the problem maxt 0(P(t) +v)e rtdepend onr? Onv? Explainboth mathematically and intuitively. [Sometimes, but not always, it is easier to takelogarithms before going to work.](b) Now suppose that in addition to the sale value, you receive a flow utility,b(t) 0, fromthe asset (picnic time in the forest, amusement watching the sheep, amusement fromwatching the engineers swear when you make their programs input themselves). Howdoes the solution to the problem maxt 0[(P(t) +v)e rt+ t0b(s)e rsds]depend on thefunctionb( )? Onr?(c) Instead of selling the infrastructure, you may decide to keep it and regrow the so, the question is how often to liquidate the asset.

6 Show that the solution to theproblemmaxt 0P(t)[e rt+e r2t+e r3t+ ],(1)is smaller than the solution to the problem maxt 0P(t)e rt. What can you say aboutthevthat makes the solutions to maxt 0(P(t) +v)e rtand the problem in (1) equal?(d) If the asset starts growing at time 0, the probability that it is destroyed by the actionsof a malevolent universe at or before timetisF(t) (the malevolent universe takes theform of forest fires, wolves, and the ebola virus in the examples). The waiting time until destruction happens has a hazard rate,h(t), so thatF(t) = 1 e t0h(s) Show that if two waiting times,F1andF2, are hazard rate ordered, ( )> h2( ).Show thatF2first order stochastically dominatesF1, and that the decision maker isbetter off Suppose that you plan to sell the growing asset and liquidate the infrastructure atsome timetin the future, but that the asset may be destroyed at a random happens to the optimal planned time to sell if the random time suffers anupward move in the hazard rate?

7 44.[A fable about farmers and bandits] Group 1 are farmers, they live in a fixed location,and allocate their time budget of 1 betweenx1, farming activities, andy1defensive 2 are bandits, they roam about the forest, and allocate their time budget of 1 betweenx2, hunting and gathering in the forest, andz2, banditry. If the farmers allocatex1to farming,they havex 1to show for it, if the bandits allocatex2to hunting and gathering, they havex 2to show for it, (0,1). If (y1,z2) are the defensive-banditry allocations, then (y1,z2: )is the successful defense probability, that is, the probability that the farmers prevent thebandits from takingx 1from the farmers. We assume that ( , : ) is continuous, increasinginy1, decreasing inz2, and for each value of (0,1), ( , : ) has increasing differences iny1andz2.

8 If you prefer algebra, you can do this problem with (y1,z2: ) = (y 1)/(y 1+ z 2), (0,1). The utilities areu1=x 1 (y1,z2) andu2=x 2+x 1(1 (y1,z2)).(a) Suppose that (x1,y1) and (x2,z2) are picked simultaneously. Show that the game has apure strategy equilibrium.(b) How does the equilibrium set depend on ? [If you are doing the general functional form,your job is to give and interpret assumptions on and then justify your answer.](c) Suppose that the farmers allocate their efforts first, the bandits observe it, and thenchoose their effort levels. Under what conditions do the farmer s weakly prefer thisleader-follower equilibrium to the previous one? What about strictly prefer?(d) Give the efficient allocation(s) of activities.

9 Can they be supported as a subgame perfectequilibrium outcome of an infinitely repeated version of the game?5