Problem Set 2 Problem 1 In this problem, we study the consequences of relative performance evaluation when agents are altruistic, i.e., they care about other agents' payoffs. There is one principal and two agents. Agent 1 chooses effort e, to perform task 1 while Agent 2 chooses effort ez to perform task 2. Both agents produce noisy, perfectly-correlated output: y1 = e1 + E, y2 = e2 + E, where E[e] = 0 and Var[e] = 1. Agent 1 is risk-averse and maximizes U! = E[wl] - Var[w1]/2 - e1/2; Agent 2 is risk-neutral but altruistic, and maximizes U2 = YE[Ud] + E[w2] - e3/2. Notice that y 2 0 captures how altruistic Agent 2 is, i.e., how much he cares about Agent I's payoff Ud. The principal is risk-neutral and maximizes Up = Elyi] + Ely2] - E[wi] - E[w2]. To simplify the problem, we assume that Agent 2's incentive scheme is fixed as W2 = 42/2. Further, Agent 2 will always work for the principal; there is no accept/reject decision for Agent 2. On the other hand, the principal chooses an incentive scheme for Agent 1 which includes a team component, w1 = a + Byl + by2, and Agent 1 chooses whether to accept or reject this offer. So, the timing is: Step 1. Principal offers Agent 1 an incentive scheme wj = o + By1 + by2. Step 2. Agent 1 accepts or rejects. If she rejects, the game ends and each agent receives zero outside option. Step 3. Agent 1 chooses e, and Agent 2 chooses ez. Step 4. The principal pays wj and w2- First, assume for questions 1. to 3. that Agent 2 is not altruistic, y = 0. For the remaining questions, 4. to 7. assume that Agent 2 is altruistic, y = 1. 1. Calculate non-altruistic agent 2's optimal choice of effort ez. 2. Calculate Agent 1's optimal choice of effort ei, as a function of B. 3. Calculate the incentive strengths B* and b" that the principal optimally offers Agent 1. Confirm that this corresponds to pure relative performance evaluation, b* = -B". (Hint: because only Agent 1 chooses whether to accept, the usual trick applies only to Agent 1, not to Agent 2.) 4. Calculate agent 2's optimal choice of effort ez, as a function of B and / or b. In words, why does agent 2's effort choice depend on the incentive scheme offered to agent 1? 5. Calculate Agent 1's optimal choice of effort ej, as a function of B. 6. Calculate the incentive strengths B" and b" that the principal optimally offers Agent 1. (Hint: you should find that the principal does not engage in relative performance evaluation.) 7. Explain, in words, why the principal doesn't engage in relative performance evaluation when agents are sufficiently altruistic