General Examination in Microeconomic Theory

Transcription

1 HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examination in Microeconomic Theory Fall 06 You have FOUR hour. Anwer all quetion Part A(Glaeer) Part B (Makin) Part C (Hart) Part D (Green) PLEASE USE A SEPARATE BLUE BOOK FOR EACH QUESTION AND WRITE THE QUESTION NUMBER ON THE FRONT OF THE BLUE BOOK. PLEASE PUT BOTH YOUR EXAM NUMBER AND PROFESSOR ON EACH BOOK. PLEASE DO NOT WRITE YOUR NAME ON YOUR BLUE BOOKS.

2 Glaeer Part A The town of Whoville i the perfect pot for the Grinch to build a factory that create Thnead and there i exce ocial benefit of B if the factory i built there (thi include the phyical cot of contruction), and that value goe to the Grinch. Aume that the ocial planner doen t know the value of B when deigning intitution, but doe know the ditribution of B. Unfortunately, there are alo Who down in Whoville and the factory cannot be built unle every Who move elewhere. The Who receive net benefit from reiding in their home relative to any other location. [Aume that thee benefit are obervable to the factory owner, but they cannot be verified in a court of law or ued a the bai for any political policy.] () If the factory owner bargain with the Who equentially [aume ome form of bargaining olution for each individual interaction], what will be the total cot of acquiring their home? Will thi lead to a ocial optimum? () If the factory owner bargain with the Who equentially, but only ign contingent contract in which the individual ale goe through if all Who agree to ell, then what will be the total cot of acquiring their home? Will thee lead to a ocial optimum? (3) Aume that a court can verify a market value for each home, which i the amount that each Who could ell their home on the open market (auming that there wa no factory in that home future). How doe thi price relate to the Who private valuation of the home? Aume further that the ocial planner can fix the premium over market price that the Grinch mut pay, and determine the premium rule that maximize expected ocial welfare. (4) Finally, conider the cae in which the Grinch can make a take-it-or-leave-it offer to all the Who for a total town ale price. The ale price i then ditributed to the Who baed on their hare of total market value of Whoville. If enough Who accept the offer then the Grinch get the whole town at that price. What will be the Grinch net revenue maximizing offer? What minimum approval rate hould the ocial planner require for an offer to go through? (For example, hould the planner allow any offer that i approved by a majority of the Who to go through or hould he require a uper-majority). (5) Reflect on the benefit and cot of thee different mean of handling the land ale. What other factor might influence the relative deirability of each one of the reule.

3 E. Makin General Quetion (Game Theory) Conider the game of Figure and anwer the following quetion about it. Give brief jutification for your anwer. (,3) T U U B U (-,4) D T D (3,-) B D (0,0) Figure A. What are the rationalizable trategie for each player? B. What are the Nah equilibria (including mixed trategie)? C. What are the ubgame perfect equilibria? D. Now uppoe the game of Figure i repeated infinitely many time and that player maximize the dicounted um of their payoff uing dicount factor, with 0. What i the et of payoff pair that can arie a the average payoff in ubgame perfect equilibria of the repeated for cloe to?

4 (,4) T U B U (-,3) T D (3,-) B D (0,0) Figure E. Conider the game of incomplete information in which the payoff can either be thoe of Figure or of Figure, with equal probabilitie. Player i privately informed of the actual payoff and player i not. Everything ele i common knowledge. What are the pure-trategy perfect Bayeian equilibria of thi game?

5 Oliver Hart Part C. (a) Conider a two date exchange economy with conumption at date 0 and. There i a ingle conumer, one good at each date, and there are S tate of the world (realized at date ). The conumer utility function i: S U u x0 u x, where x 0 i date 0 conumption, x i date conumption in tate, u >0, u <0, and 0. The conumer ha an initial endowment (ω 0, ω,, ω S ) 0. Write down the Arrow-Debreu (complete market) equilibrium for thi economy. Uing the conumer firt order condition and normalizing the price of date 0 conumption to be, derive a formula for the price q of each of the contingent commoditie, =,,S. Etablih the following, providing interpretation in each cae: () If t, then q > q t t. () If t, then q > q t t. (3) Each q i increaing in. (4) If =ω 0 for all, S q =. Finally, provide the formula for q in the cae that the conumer i rik neutral, and interpret it.. A eller S can provide a good to a buyer B at date. The partie meet and contract at date 0 and B invet i at date at peronal cot of i. B payoff i v + 4i if he receive the good, and zero if he doe not (and whether or not he receive the good B incur her invetment cot). There i no dicounting and there are no wealth contraint. B outide option at date 0 i zero. a. What i the firt-bet level of i? b. Suppoe that the partie write a long-term contract pecifying that they will trade the good at date at price p (and thi i enforceable). Can the firt-bet be

6 utained? Under the aumption that S ha all the bargaining power at date 0, what will p be? c. Suppoe that the partie cannot write a long-term contact but can write a hortterm contract pecifying i (aume that i i verifiable). The price p will be choen through bargaining at date, where again S ha all the bargaining power. S can make a lump um payment to B to peruade her to participate. Can the firt-bet be utained? What will p be? What will the lump um payment be? d. Aume now that neither a long-term contract on p nor a hort-term contract on i can be written (i i obervable but not verifiable). Again S ha all the bargaining power. Solve for the (econd-bet) equilibrium value of i and p. e. In a few word explain how ownerhip of an aet by B could improve the ituation in cae d.

7 Part D - Quetion Shortet Path Ranking (30 point) Conider a tournament T that repreent the reult of a majority vote on all pair of alternative choen from a finite et X with n element. I will give a method for creating a weak-ordering R of the element of X, that I will call Shortet Path Ranking, or SPR. The idea of the quetion i for you to examine the propertie of thi method and to criticize it or upport it a a olution concept for voting model baed on pairwie-majority information. The method i a follow:. For each ordered pair (x, y) X, compute the mallet number of tep (x, y) that are needed to go from x to y, via a equence of alternative in X, each tep of which i a majority victory. For example, if xt y, then (x, y) = ; if yt x but there exit z uch that xt zt y, then (x, y) =, etc. If there i no path from x to y, et (x, y) =.. For each x X, lit the number (x, y) in (weakly) increaing rank-order. Call the reulting n dimenional vector r(x) (for rank-ordered hortet path length). For example if X = {x, y, z, w} and (x, y) =, (x, z) = and (x, w) =, then r(x) = (,, ) 3. Finally, define the SPR a the weak order R on X that i equivalent to the lexically increaing order of the r(x) vector. For example, if r(x) = (,,,, 3) and r(y) = (,,,, ) then xry, and not yrx. If r(x) = r(y) then xry and yrx; that i x and y are in the ame equivalence cla of R. a) Show that every element of the Top Cycle of T, T C(T ), i trictly above every element of X\T C(T ) according to R. For part b,c,d,e of thi quetion, we are concerned with a particular example of a tournament, which we will call T, hown in the diagram on the next page (copied from Lecture and Lecture ). The relation T i hown in matrix form in the firt table below the diagram and the hortet path number (x, y) have been computed for your convenience and are hown in the econd table below the diagram.

8 x row\y column The Tournament T Entrie in the table above are if xt y and 0 otherwie x row\y column Entrie in the above table are (x, y) for the tournament T

9 b) Lit the hortet path from 3 to all the other element of X and thereby verify that the indicated number (3, x) in the table are correct. c) What i the SPR for the tournament T? d) Let T change to T which i identical to T except that for ome pair of alternative (w, v) we have wt v intead of vt w. Let R and R be their aociated Shortet Path Ranking. Show that if wrv then wr v. e) (i) Show that an alternative in the Uncovered Set of T can be ranked by R trictly below omething outide the Uncovered Set (ii) Explain why thi can happen. (iii) Do you think that thi phenomenon i a flaw in the idea of the SPR, or do you think that it might expre a good apect of thi ocial choice rule. Now, back to the general cae of the SPR. Regard the SPR a a ocial welfare functional, mapping profile of individual preference into weak ordering for the ociety, in the ene of Arrow. The SPR ocial welfare functional i computed by firt finding the tournament that thee preference generate, and the applying the definition above to thi tournament. f) Will the SPR atify Arrow Pareto Axiom? g) Will the SPR arify Arrow IIA Axiom? (You do not have to compute anything to anwer thi quetion. You may refer to anything preented in cla without proof.) 3

10 Part D - Quetion Shapley Value and Pre-nucleolu of a Quota Voting Game (0 point) There are four people on a committee. They are not equal in power. The rule of voting in the committee are that they each have "point" that weight their vote. The point are (4,3,,) repectively. It take 5 point to "win" a vote on any matter. A coalition that win get unit of payoff that they can hare in any way (maybe by tranferring money among themelve - they all have quai-linear preference). A loing coalition get nothing. a) What payoff allocation would the Shapley value recommend for thi ituation? b) What payoff allocation would the Pre-nucleolu recommend?

11 Part D - Quetion 3 Self-Driving Car (0 point) In the future we will have elf-driving car. Not only will the car drive by itelf, avoid colliion, reach the detination, and find a parking pot, it will alo chooe the bet route according to a pre-programmed algorithm in it computer. The car will communicate electronically with all other car, highway department and other agencie in order to decide on the route it will take. All the driver will need to do i to input a parameter that repreent the value of hi/her time, applicable to that day and that trip. At the end of the trip the car will credit or debit the owner bank account in a way that the owner undertand. All car owner have preference that depend on the time the trip take and on the amount of money that they pay. The preference are quai-linear in money. Thi quetion i concern a implified example of thi future world. There are two driver who need to go from the ame uburb to the ame city. There are two route Fat (F) and Slow(S). The driver can have one of two time-value, High, θ H, or Low, θ L. If there are either no car on the Fat route, or only one car on it, the time i take to ue the fat route i x F. The time it take to drive on the Slow route i x S. However, if they both drive on the Fat route on the ame day there will be a lot of congetion on the road. In that cae, both car on the Fat route would encounter a delay that make the trip take time x C even longer than the Slow route. Aume that the parameter atify: θ H > θ L and x F < x S < x C. The dollar-equivalent cot of driving time for a car whoe owner ha a given time-cot parameter i θ i x j, for i = H, L and j = F, S, C. The route choen by the computer for a given car can be a function of both reported θ. The monetary tranfer that the car implement can alo be a function of both reported θ. That i why the car computer may need to communicate with each other and with the bank. Finally, it i poible for the ytem to elect a randomized aignment of who drive on which road. In "mechanim deign language", the pace of all poible outcome function i the et of all random aignment of the two car to the two route, a a function of the true type. Both owner know therealized value of their own θ and that the two θ are tatitically independent of each other and are identically ditributed with the probability that θ = θ H equal to α. They do not have a direct obervation of the other driver θ. a) What are the effi cient way to aign car to route? What i the minimum expected total driving time cot? If you found in part a) that there are multiple effi cient outcome, take any one of them, hold it fixed, and ue that outcome function in anwering part b). b) I there a way to incentivize the driver to input their true θ in a Baye- Nah equilibrium and to have the car drive according to your candidate for an effi cient outcome?