Question:

Suppose the principal chooses to persuade , and that persuasion fails. What effort level does the agent exert in step 3, given the principals choice of incentive strength in step 2? Show that the principal will always receive the same payoff, regardless of the choice of incentive strength.

In this question, we will consider a model of incentives and disagreement where the principal can either choose to give the agent an order, or persuade the worker. Unlike the model covered in class, orders are binding. There is a (P)rincipal and an (A)gent. A works on a project: he chooses effort e at cost 21e2 and a binary decision d (either 1 or 1 ). There is a binary state of the world (either 1 or 1 ) that is not known until after the agent makes his choices. The project is successful if and only if the action matches the state (d=), in which case the principal receives revenue v=Be, where B>0. Otherwise, if the project fails, then the principal receives v=0, and the principal also incurs an additional cost of c=1. P can offer A an incentive scheme of the form =bv. So, the principal's and agent's payoffs are =E[v]E[]{10iftheprojectfailsiftheprojectsucceedsu=E[]e2/2. P and A disagree on how likely each state of the world is. P believes that =1 occurs with probability 1 ; A believes that =1 occurs with probability 1/3. At the start of the game, P chooses one of two options. - P can give A an order i (either 1 or 1). If A receives an order, he cannot disobey, and must choose d=i; however, he can choose any effort level he likes. - Or, P can persuade A. If P chooses to persuade, then she is saccessful 50% of the time: A changes his mind, and believes (like P ) that =1 will occur with probability 1n. The other 50% of the time, persuasion fails: A continues to believe that " =1 will occur with probability 1/3 ". The game proceeds as follows: Step 1. The principal chooses whether to give the agent an order i, or to persuade the agent. If the principal chooses to persuade the agent, then she learns whether persuasion was successful. Step 2. The principal offers the agent an incentive contract =bv. 1 Step 3. The agent chooses a decision d, and an effort level e0. If the principal gave an order, the agent must obey. Step 4. The state of the world is revealed, and the project succeeds or fails (based on whether the agent chose the correct action). The principal pays the agent his wage . In this question, we will consider a model of incentives and disagreement where the principal can either choose to give the agent an order, or persuade the worker. Unlike the model covered in class, orders are binding. There is a (P)rincipal and an (A)gent. A works on a project: he chooses effort e at cost 21e2 and a binary decision d (either 1 or 1 ). There is a binary state of the world (either 1 or 1 ) that is not known until after the agent makes his choices. The project is successful if and only if the action matches the state (d=), in which case the principal receives revenue v=Be, where B>0. Otherwise, if the project fails, then the principal receives v=0, and the principal also incurs an additional cost of c=1. P can offer A an incentive scheme of the form =bv. So, the principal's and agent's payoffs are =E[v]E[]{10iftheprojectfailsiftheprojectsucceedsu=E[]e2/2. P and A disagree on how likely each state of the world is. P believes that =1 occurs with probability 1 ; A believes that =1 occurs with probability 1/3. At the start of the game, P chooses one of two options. - P can give A an order i (either 1 or 1). If A receives an order, he cannot disobey, and must choose d=i; however, he can choose any effort level he likes. - Or, P can persuade A. If P chooses to persuade, then she is saccessful 50% of the time: A changes his mind, and believes (like P ) that =1 will occur with probability 1n. The other 50% of the time, persuasion fails: A continues to believe that " =1 will occur with probability 1/3 ". The game proceeds as follows: Step 1. The principal chooses whether to give the agent an order i, or to persuade the agent. If the principal chooses to persuade the agent, then she learns whether persuasion was successful. Step 2. The principal offers the agent an incentive contract =bv. 1 Step 3. The agent chooses a decision d, and an effort level e0. If the principal gave an order, the agent must obey. Step 4. The state of the world is revealed, and the project succeeds or fails (based on whether the agent chose the correct action). The principal pays the agent his wage