Microeconomic Theory (501b) Comprehensive Exam

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1 Dirk Bergemann Department of Economics Yale University Microeconomic Theory (50b) Comprehensive Exam. (5) Consider a moral hazard model where a worker chooses an e ort level e [0; ]; and as a result, either a high or a low output y fy l ; y h g is realized with 0 = y l < y h < : Denote the probability that y = y h for e ort level e is given by p (e) = e: The employer s payo is y w; where w denotes the wage paid to the worker. The worker s payo is w e if she accepts the contract, and 0 if she rejects the contract. (a) Characterize the optimal contract in the case where w can depend on e: It is now su cient to compute the e cient solution which is simply ey h e and hence e = y h () and set the wage to w y h = y h and for all e 6= y h, we set w (e) = K for some su ciently large K. (b) Characterize the optimal contract when only y is observable and veri able. We observe that the agent is risk-neutral, there is no need to do risk sharing. The expected value of the project for the agent at the e cient e ort level () is given by and hence a wage y h y h h y = y h w (y) = y y h () leaves the agent as the residual claimant and with zero expected utility. Thus the principal receives all the surplus.

2 (c) How does your answer change if it is required that w 0? We observe from () that for y l the wage is negative and hence we need to change the solution. We thus set and need to determine w (y l ) = 0 w (y h ) = w optimally. The incentive compatibility leads us to or e ew e e (w) = w. We can solve the problem of the principal as w w (y h w) and we get as and hence (y h w) w = y h e = 4 y h. w = 0. Consider the standard job market signalling model. Nature chooses a worker s type (productivity) a f; g. The worker has productivity a with probability p (a) and for notational convenience de ne p, p (a = ) : The worker can choose an educational level e R + and the workers utility is u (w; e; a) = w e a if he is paid a wage w. The pro t of a rm to employ a worker with ability a at wage w is (a; w) = a w. In contrast to the standard signalling model, we consider a di erent timing of the event, thus a di erent extensive form. t = 0 : In period 0, the informed worked o ers a contingent wage schedule to rm fw(e)g where the wage schedule depends on the education level eventually chosen.

3 t = : In period, the uninformed rm accepts the wage schedule if the expected pro ts are larger or equal to zero. t = : In period, the informed worker chooses an education level and the utilities and pro ts are realized. (a) Formally de ne the pure strategies for the worker and rm and de ne the notion of a Perfect Bayesian Equilibrium for this game. Standard. (b) Characterize the equilibrium/a of this game. In particular is it unique or there are multiple equilibria. Does your characterization of the equilibrium depend on p and if so, how? It is su cient to consider the problem for the high productivity worker. Suppose he would like to make an o er by which he can separate himself from a low ability worker subject to and and fe;w(e)g n w (e) e o w (e ) e w (e ) e (IC ) w (e ) e w (e ) e (IC ) a w (e ) 0 (IR ) a w (e ) 0 (IR ) Thus to make incentive compatibility as easy as possible he suggests e = 0 and w (e = 0) =. As w (e ) =, it follows that after setting e =, he indeed imizes his payo. Suppose instead he would like to o er a pooling contract. Then he would suggest e;w n w e o + p w 0 (IR ) which would yield w = + p for all e. This suggest that there are two di erent cases to consider: (i) + p : a high productivity worker is better o in the least cost separating equilibrium than in the e cient pooling equilibrium;(ii) + p > : a high productivity worker is better o in the e cient pooling equilibrium. It is easy to verify that in the case where p, the high productivity worker cannot do better than o ering the separating contract, nor can the low productivity worker. More precisely, the high productivity worker strictly prefers this contract over any contract resulting in pooling or any contract with more costly separation. As for the low productivity worker, he has everything to lose by o ering another 3

4 contract which would identify himself. In the alternative case where p >, the unique equilibrium contract is the one where: w (e) = + p for all e 0: Again, if the rm accepts this contract, both types of workers choose an education level of zero. Thus, on average the rm breaks even by accepting this contract, provided that it is as (or more) likely to originate from a high productivity worker than a low productivity worker. Now, a high productivity worker strictly prefers to o er this contract over any other separating contract. Similarly, a low productivity worker has everything to lose from o ering another contract and thus identifying himself. Thus, in this case again this is the unique contract o er made by the workers. 3. Consider a rm that can invest an amount I in a project generating high observable cash ow C > 0 with probability and 0 otherwise: f L ; H g with H L > 0 and Pr[ = L ] =. The rm needs to raise I from competitive external investors who do not observe the value of. Assume that L C I > 0. Everybody is risk neutral, and there is no discounting. (a) Suppose now that the rms can only promise to repay an amount R chosen by the rm (with 0 R C) when case ow is C and 0 otherwise. Does there exist an equilibrium in which the high value rm receives funding? Does there exist a separating equilibrium in which the high value rm receives funding at terms di erent from the low value rm. If R is chosen such that I = ( L + ( ) H ) R, or R = I L + ( ) H, then the funding will be provided and the investor will at least break through on average. But this is a pooling equilibrium, and there is no separating equilibrium, because the low value rm would always want to mimic the high value rm. (b) Suppose now that the rm also has the possibility of pledging some assets as collateral for the loan: Should a default occur (the rm being unable to repay R), an asset of value K to the rm is transferred to the creditor. The size of the collateral K is a choice variable. Does there now exist a separating equilibrium in which the high value rm receives funding to di erent conditions than the low value rm? What is the minimal size of the collateral in any separating equilibrium. The high value rm can pledge a collateral such that the low value rm is indi erent: l R l = l R h + ( l ) K h 4

5 and since R l = I l we have which is paid with K h = I l h R h + ( h ) K h = I R l = I l and the solution is R h = h (I K h + h K h ) R l = l I and we get a solution l R l = l R h + ( l ) K h which is K h = R h = I. 4. Bilateral Trading. Consider the following trading game between a buyer and a seller. The buyer has a valuation v U [0; ] and seller has a cost c U [0; ] of producing the product. The valuation and the cost is private information to buyer and seller respectively and the utility function of buyer and seller is v p and p c if a trade occurs at p and is 0 and 0 if no trade occurs. Trade is voluntary. (a) Describe the ex post e cient trading rule in this environment. Compute the expected net value of the ex post e cient trading rule. (A diagram may help.) The e cient trading rule x is mapping x : [0; ] [0; ]! [0; ], describing the probability of trade as a function of value v and cost c. It is x (v; c) = if v c and x (v; c) = 0 if v < c. The expected net value is given by Z Z (v c) dv dc = 6. 0 c 5

6 (b) Consider now the following trading mechanism. Buyer and seller o er a bid and a ask price, p B and p S, respectively. A trade occurs if p B p S and the trading price is p = (p B + p S ) : (3) Formulate the trading game as a Bayesian game and de ne the notion of a Bayesian Nash equilibrium for this game. A pair of pure strategies is p B : [0; ]! [0; ] ; p S : [0; ]! [0; ] ; and a pair of mixed strategies is p B : [0; ]! [0; ] ; p S : [0; ]! [0; ] : A Bayes-Nash equilibrium is a pair (p B ; p S ) such that the buyer imizes for every v : Z p B 0 v p B + p s (c) I pb p S dc; and that the seller imizes for every c: Z p S 0 p B (v) + p s c I pb p S dv; (c) Consider now the following xed price strategy by each player 0 if v < p p B (v) = p if v p ; and p if c p p S (c) = if c > p ; for some p [0; ]. Does this strategy pair for a Bayesian Nash equilibrium for some p? For what values of p does it form a Bayesian Nash equilibrium? What is the expected net surplus in the xed trading rule and does/do the equilibria lead to the realization of the ex post e cient trading rule? Yes, because given the strategy of the other agent j, agent i cannot increase the probability of trade, and any other price would lead only to higher terms of trade. It thus forms a Bayes-Nash equilibrium for every p [0; ] and the expected surplus is given by Z p Z (v c) dv dc = p ( p), 0 p which reaches its imum at p = but only realizes a social surplus equal to =8. 6