Question 1 In this problem, we study the consequences of relative performance evaluation when agents are altruistic, i.e., they care about other agents' payo's. There is one principal and two agents. Agent 1 chooses effort 2, to perform task 1 while Agent 2 chooses e'ort 22 to perform task 2. Both agents produce noisy. perfectlycorrelated output: x1 : 21+ 5, x2 = e2 + s, where lE[e] = 0 and Var[e] = 1' To simplify the problem, we assume that the is no incentive scheme for Agent 2, 11 = 01 Further, Agent 2 will always work for the principal; there is no accept/reject decision for Agent 21 On the other hand, the principal chooses an incentive scheme for Agent 1 which includes a team component, 11: u+ bx, + 8362, and Agent 1 chooses whether to accept or reject this oen Agent 1 is risk-averse and maximizes 141: lE[rl] 7 VarlTll/Z , e?/2; Agent 2 is risk-neutral but altruistic, and maximizes u; : 7151141] em Notice that y 2 0 captures how altruistic Agent 2 is, i.e., how much he cares about Agent 1's payoff 141. The principal is risk-neutral and maximizes 7' : Elx1l+ Elle ' ElTll' So, the timing is: Step 1: Principal offers Agent 1 an incentive scheme 'rl : a + bx1 + 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 92. Step 4: The principal pays 11, First, assume for parts (a)-(c) that Agent 2 is moderately altruistic, y = 1/21 a) Calculate agent 2's optimal choice ofeort e; , as a function of b and B. In words, why does agent 2's e'ort choice depend on the incentive scheme o'ered to agent 1? b) Calculate Agent 1's optimal choice of effort 6f, as a function of b. c) Calculate the incentive strengths b' and B' that the principal optimally offers Agent 1. (Hint: because only Agent 1 chooses whether to accept, the usual trick for simplifying the principal's op- timization problem applies only to Agent 1, not to Agent 2.) d) You should have found from (c) that the principal does not engage in relative performance evalua- tion: 8' = 0. Explain, in words, why this is the case. (Hint: your answer should hinge on that fact that Agent 2 is altruistic.) For the remaining parts, assume that Agent 2 is highly altruistic, y = 1. e) Repeat parts (a)(c) for the case y : 1. What effort levels e; and e; do the agents choose given the principal's choice of b' and 13'? f) You should have found from (e) that agent 1 does not exert effort because the principal does not provide any incentives to motivate agent 1: 2f : O and b" : 0. Explain, in words, why the principal chooses not to motivate agent 1 when agent 2 is highly altruistic