QUESTION A to G

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 81 to perform task 1 while Agent 2 chooses effort 32 to perform task 2. Both agents produce noisy, perfectly-correlated output: x1=el+e, where IE[] = O and Var[] = 1. Agent 1 is risk-averse and maximizes u1 = new Warm/2 ef/z; Agent 2 is risk-neutral but altruistic, and maximizes u2 = yIE[u1] + ]E[rz] 23/2. Notice that y 2 D captures how altruistic Agent 2 is, i.e., how much he cares about Agent 1's payoff MI. The principal is risk-neutral and maximizes 71 = IE[X1] + lElle - Elfll - lEszl- To simplify the problem, we assume that Agent 2's incentive scheme is xed as 1'2 = x2/2. 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, 1'1: 51 + bx1+ 3x2, and Agent 1 chooses whether to accept or reject this offer. So, the timing is: Step 1: Principal offers Agent 1 an incentive scheme 1'1 = a + bxl + 3362. 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 el and Agent 2 chooses 32. Step 4: The principal pays Tl and 1'2. First, assume for parts (a)-(c) that Agent 2 is not altruistic, y = D. a) Calculate non-altruistic agent 2's optimal choice of effort 3: . b) Calculate Agent 1's optimal choice of effort ef , as a function of b. c) Calculate the incentive strengths 5' and Be that the principal optimally offers Agent 1. Conrm that this corresponds to pure relative performance evaluation, 3' = 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. (1) 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? e) Calculate Agent 1's optimal choice of effort a? , as a function of b. f) Calculate the incentive strengths 19* and 3* that the principal optimally offers Agent 1. (Hint: you should nd that the principal does not engage in relative performance evaluation.) g) Explain, in words, why the principal doesn't engage in relative performance evaluation when agents are suiciently altruistic