Prohlem Set 2 'roblem 1 . this problem, we study the consequences of relative performance evaluation when agent: 'e altruistic, i.e., the}r care about other agents: payoffs. here is one principal and two agents. Agent. 1 chooses eort e1 to perform task 1 while gent 2 chooses effort tag to perform task 2. Both agents produce noisy= perfectlyicorrelatec 1tput: Fl 2 81 +5: ya '32 + 5? here [EH] = U and Var[5] = 1. gent 1 is riskaverse and maximizes Payoil Cq. Inle principal 15 Ink-leutId Up = Elyi] + E[y2] - E[wi] - E[w2]. simplify the problem, we assume that Agent 2's incentive scheme is fixed as W2 = y2/2. rther, Agent 2 will always work for the principal; there is no accept/reject decision for gent 2. the other hand, the principal chooses an incentive scheme for Agent 1 which includes eam component, w1 = a + By1 +by2, d Agent 1 chooses whether to accept or reject this offer.42 = e2 + E, here E[=] = 0 and Var[e] = 1. gent 1 is risk-averse and maximizes Ud = E[wi] - Var[w1]/2 - e1/2; gent 2 is risk-neutral but altruistic, and maximizes U2 = VE[U!] + E[w2] - e3/2. otice that v 2 0 captures how altruistic Agent 2 is, i.e., how much he cares about Agen s payoff Ua. The principal is risk-neutral and maximizes Up = Elyi] + E[yz] - E[wi] - E[w2]. o simplify the problem, we assume that Agent 2's incentive scheme is fixed as W2 = 92/2. urther, Agent 2 will always work for the principal; there is no accept/reject decision fo gent 2. n the other hand, the principal chooses an incentive scheme for Agent 1 which includelter: 1. Principal o'ers Agent 1 an incentive scheme on = o. + y] I byg. Step 2. Agent 1 aeeeIJt-s or rejects. If she I'CjUCt-S, the game ends and each agent receiv acre outside Option. Step 3. Agent 1 Chooses c] and Agent 2 chooses (i2- gtcp 4. The [,u'intupal pays ml and reg. First... assume for questions 1. to 3. that Agent. 2 is not altruistic, q- : 0. For the remaini] questions, 4. to 7. assume that Agent 2 is altruistic, 1r- : 1. Calculate non-altruistic agent 2's optimal choice of eort cg. Calculate Agent 1's optimal choice of effort cf: as a function of 55" Calculate the incentive strengths {3* and b* that. the principal optimally offers Agent Conrm that this corresponds to pure relative performance evaluation: If\" : _5*. (Hit because only Agent 1 chooses whether to accept, the usual trick applies only to Agent not to Agent 2.] Talculate assent. 2's optimal choice of effort a. as a function of .3 and f or b. In wort alculate Agent 1's optimal choice of effort et, as a function of 3. alculate the incentive strengths * and b* that the principal optimally offers Agent 1 onfirm that this corresponds to pure relative performance evaluation, b* = -3*. (Hint ecause only Agent 1 chooses whether to accept, the usual trick applies only to Agent 1 t to Agent 2.) alculate agent 2's optimal choice of effort es, as a function of 3 and / or b. In words hy does agent 2's effort choice depend on the incentive scheme offered to agent 1? alculate Agent 1's optimal choice of effort et, as a function of 3. alculate the incentive strengths S* and b* that the principal optimally offers Agen (Hint: you should find that the principal does not engage in relative performanc aluation.) xplain, in words, why the principal doesn't engage in relative performance evaluation hen agents are sufficiently altruistic