Question 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 31 to perform task 1 while Agent 2 chooses effort 2 to perform task 2. Both agents produce noisy, perfectly-correlated output: xl :el+s, x2 : 82 + e, where lEle] = O and Var[e] = 1. Agent 1 is risk-averse and maximizes u, : 1am] warm/2 7 ef/z; Agent 2 is riskneutral but altruistic, and maximizes u: = y]E[u1] + IE[T2] 23/2. Notice that y 2 0 captures how altruistic Agent 2 is, i.e., how much he cares about Agent 1's payoff 1.11. The principal is riskaneutral and maximizes rt : 1E[x1]+ JE[x2] 7 IE[1'1]7 lE[1'2]. To simplify the problem, we assume that Agent 25 incentive scheme is xed as 12 = x212. 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, T1 = a + bx1+ Bxl, and Agent 1 chooses whether to accept or reject this offer. So, the timing is: Step 1: Principal offers Agent 1 an incentive scheme 1'] = a + bx] + sz. 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 e2. Step 4: The principal pays Tl and 7:. First, assume for parts (a)-(c) that Agent 2 is not altruistic, y : 0. a) Calculate nonaltruistic agent 2' s optimal choice of effort e; and caculate Agent 1' s optimal choice of effort 8:, as a lnction ofb. b) Calculate the incentive strengths 5' and B * that the principal optimally offers Agent 1. Conrm that this corresponds to pure relative performance evaluation, B3' = b* . (Hint: because only Agent 1 chooses whether to accept, the usual trick applies only to Agent 1, not to Agent 2.) For the remaining parts, assume that Agent 2 is altruistic, y = 1. c) Calculate agent 2' s optimal choice of effort 8;, 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? d) Calculate Agent 1' s optimal choice of effort 8:, as a function of b and calculate the incentive strengths If and 3* that the principal optimally offers Agent 1. (Hint: you should nd that the principal does not engage in relative performance evaluation.) e) Explain, in words, why the principal doesn't engage in relative performance evaluation when agents are sufciently altruistic