Question 1 So, the timing is: In this problem, we study the consequences of relative performance evaluation when agents are Step 1: Principal offers A851\" 1 an incentive scheme 71 = a + bxl + sz. altruistic, 1'5" they care about other agents payoffs. Step 2: Agent 1 accepts or rejects. If she rejects, the game ends and each agent receives zero outside There is one principal and two agents. Agent 1 chooses effort e1 to perform task 1 while Agent 2 option. h rt t rf tk2.Bth t d ', rfctl- ltd t: c oosese o 52 ope arm as o agen 5 pro uce noisy pe e y corre a e outpu Step 3: Agent 1 chooses 61 an d Agent 2 chooses ez. "i = 91 + 5- Step 4: The principal pays 1'1. X1 = 22 + 8, where Elt] = 0 and Var[e] = 1 First, assume for parts (a)-(c) that Agent 2 is moderately altruistic, y = 1/2. To simplify the problem, we assume that the is no incentive scheme for Agent 2, 12 = 0. Further, a) Calculate agent 2's optimal choice ofeffort 62', as a function ofb and B. Agent 2 Will always work for the principal; there is no accept/reyect decision for Agent 2. In words, why does agent 2,5 effort choice depend on the incentive scheme offered to agent 1? On the other hand, the principal chooses an incentive scheme for Agent 1 which includes a team component, b) Calculate Agent 1's optimal choice of effort 21', as a function of b. Tl = a + bxl + BXZ' c) Calculate the incentive strengths b' and B' that the principal optimally offers Agent 1. (Hint: and Agent 1 chooses whether to accept or reject this offer. because only Agent 1 chooses whether to accept, the usual trick for simplifying the principal's 0p- , , , , timization problem applies only to Agent 1, not to Agent 2.) Agent 1 is riskaverse and maxim12es 2 d) You should have found from (c) that the principal does not engage in relative performance evalua- \"i = 1E["1] val'l'ril/2 ' e1/2; tion: 3" : 0. Explain, in words, why this is the case. (Hint: your answer should hinge on that fact that A ent 2 is altruistic. Agent 2 is risk-neutral but altruistic, and maximizes g ) \"z : VEluil 7 (22/2. For the remaining parts, assume that Agent 2 is highly altruistic, y = 1. Notice that y 2 0 captures how altruistic Agent 2 is, i.e., how much he cares about Agent 1's payoff e) Repeat parts (a)-(c) for the case 1' = 1_ What effort levels ei and e; do the agents choose given the ul. The principal is risk-neutral and maximizes principal's choice of b' and B"? 71 = ]E[x1] + ]E[x1] ]E[1l]. 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: e; : 0 and b' : 0. Explain, in words, why the principal chooses not to motivate agent 1 when agent 2 is highly altruistic