Answers to Final Examination Economics 703 Spring 2016 1. (a) The characterization of incentive compatibility for linear environments applies here. So we have to have Z θi ȳi (s) ds Ui (θi ) = Ui (0) + 0 where ȳi (θi ) = Eθj y(θ1 , θ2 ). Also, Ui (θi ) = θi ȳi (θi ) + t̄i (θi ), where t̄i (θi ) = Eθj ti (θ1 , θ2 ). Note that if θi = 0, then the probability that θ1 + θ2 > 1 is 0. By assumption, if the sum of the θ’s is less than 1, the transfers are 0. Hence Ui (0) = 0. Rearranging the definitions and using Ui (0) = 0 gives −t̄i (θi ) = θi ȳi (θi ) − Z θi 0 ȳi (s) ds. Note that y(θ1 , θ2 ) is 1 iff θ1 + θ2 ≥ 1, so ȳi (s) is the probability that s + θj ≥ 1. Given the uniform distribution, this probability is s. So 1 1 −t̄i (θi ) = θi2 − θi2 = θi2 . 2 2 So −Eti (θ1 , θ2 ) = −Eθi t̄i (θi ) = Z 1 0 1 2 1 θ = . 2 i 6 (b) The answer to (a) tells us that the expected payment made by the two agents in total is 1/3. But the public good is being provided iff θ1 + θ2 > 1. The probability of this event is 1/2 (since this area is half the unit square). So the expected total transfers must equal 1/2, a contradiction. 1 2. (a) The principal chooses eH , wH , eL , and wL to maximize his expected profits subject to the individual rationality and incentive compatibility constraints: wj − ej ≥ 0, j = L, H θj eH eL ≥ wH − θL θL eH eL wH − ≥ wL − . θH θH As usual, the only binding constraints are the individual rationality constraint for type θL and the incentive compatibility constraint for type θH . Hence wL − wL = eL θL eL eL eH + − . θH θL θH Substituting these results into the objective function, the principal will choose eL and eH to maximize 1 √ eH eL eL eL 1 √ 2 eH − − + + 2 eL − 2 θH θL θH 2 θL wH = 2 The first–order condition for eH is the same as in the first–best (as usual), so eH = θH = 4. The first–order condition for eL is 1 2 or " # 1 1 1 1 1 = − √ − eL θL 2 θL θH 1 1 √ −1=1− . eL 2 Rearranging gives eL = 4/9. The wages are wL = 4/9 and wH = 2 + 4 2 20 − = . 9 9 9 (b) Suppose the principal offers only one wage–effort pair. The problem says the principal must hire the agent, so this must satisfy the individual rationality constraint for both types. Since the low type has higher costs, if (w, e) satisfies individual rationality for the low type, it must satisfy it for the high type. Incentive compatibility is no longer relevant since there’s no “lies” anyone can tell. So the √ best version of this option for the principal is to offer the w and e which maximize 2 e − w subject to w − (e/θL ) ≥ 0. This is the same problem as the first–best for the low type,√so we know that e = 1 and w = 1. If the principal follows this approach, his payoff is 2 e − w = 2 − 1 = 1. 2 If the principal offers two wage–effort pairs, then he has to choose the first–best efforts in each case. Otherwise, the contract cannot be ex post efficient since the total payoff to the principal and agent could be made larger. Hence we must have eL = 1 and eH = 4. We also have to satisfy incentive compatibility and individual rationality. So our constraints are wL − 1 ≥ 0 wH − 2 ≥ 0 1 eL = wL − wH − 2 ≥ wL − θH 2 eH wL − 1 ≥ wH − = wH − 4. θL Rewriting: wL ≥ 1 wH ≥ 2 3 2 wL ≥ wH − 3. wH ≥ wL + It’s easy to see that the first and third constraints imply the second. Also, if we ignore the fourth constraint, we get wL = 1 and wH = 5/2. Since this implies the fourth constraint, it can’t bind and these must be the wages. The principal’s profit is (1/2)(6 − 1 − 2.5) = 5/4 > 1. Hence this is the optimal contract. 3. Let ei , i = h, `, denote the education choice by type θi . For a separating equilibrium, we require incentive compatibility and also require that neither type wishes to deviate to some off equilibrium education level. In this context, incentive compatibility says √ √ eh e` ≥ 40 + 4 e` − 100 + 4 eh − 5 5 and √ √ 2eh 2e` ≤ 40 + 4 e` − . 100 + 4 eh − 5 5 Of course, the best way to prevent deviations to some off equilibrium education level is to suppose that the employers’ beliefs in response to such a deviation is to believe that the worker is type θ` . Suppose this is the belief in response to any devaition. Note that both types have nontrivial preferences regarding education levels, so we have to ensure that sticking to the equilibrium is better than the best deviation. √ For type θ` , the best deviation would be to that e which maximizes 4 e − 2e/5. The first–order condition for e is 2 2 √ − = 0, e 5 3 so the maximizer is 25. The only way to prevent the low type from wanting to deviate from e` to 25 is to set e` equal to 25. If we have a separating equilibrium where type √ θ` √ chooses any e` 6= 25, then his equilibrium payoff is 40 + 4 e` − 2e` /5 < 40 + 4 25 − (2)(25)/5 = 50. Because we’re assuming the employers believe he is the low type if he deviates, the right–hand side is the payoff type θ` would get if he deviated to 25. Hence he would want to deviate. So we must have e` = 25. Plugging this into the incentive compatibility conditions above, we can rewrite them as √ √ eh 25 ≥ 40 + 4 25 − = 55 100 + 4 eh − 5 5 and √ 2eh 100 + 4 eh − ≤ 50. 5 √ For the high type, the best deviation would be to the e which maximizes 4 e − e/5. The first–order condition is 1 2 √ − =0 e 5 so the √maximizing e is 100. If the high type deviates to e = 100, his payoff would be 40 + 4 100 − (100/5) = 60. So we have to add the constraint that √ eh 100 + 4 eh − ≥ 60. 5 Clearly, this implies the first of the two incentive compatibility conditions above. Finally, then, we see that eh must satisfy √ eh ≥ 60 100 + 4 eh − 5 √ 2eh 100 + 4 eh − ≤ 50. 5 So the set of separating equilibrium strategies for the worker consist of the pairs e` = 25 and any eh satisfying these two inequalities. You can simplify further and even prove that the set of such eh is non–empty, but this is sufficient. 4