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 1 binary action (either X or Y ). There is a binary state of the world (either x or y) that is not known until after the agent makes his choices. The project is successful if and only if the action matches the state (x and X, or y and Y ), in which case the principal receives revenue π = Be, where B > 0.
Otherwise, if the project fails, then the principal receives π = 0, and he also incurs an additional cost of c = 1.
P can offer A an incentive scheme of the form w = βπ.
P and A disagree on how likely each state of the world is. P believes that x occurs with probability 1; A believes that x occurs with probability 1/3.
At the start of the game, P chooses one of two options.
• P can give A an order (either X or Y ). If A receives an order, he cannot disobey; however, he can choose any effort level he likes.
• Or, P can persuade A (either X or Y ). IF P chooses to persuade, then he is successful 50% of the time: A changes his mind, and believes (like P) that “x will occur with probability 1”. The other 50% of the time, persuasion fails: A continues to believe that “x will occur with probability 1/3”.
The game proceeds as follows:
1. The principal chooses whether to give the agent an order (either X or Y ), or to persuade the agent.
If the principal chooses to persuade the agent, then he learns whether persuasion was successful.
2. The principal offers the agent an incentive contract w = βπ.
3. The agent chooses an action (either X or Y ), and an effort level e ≥ 0. If the principal gave an order, the agent must obey.
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 w.1
We’ll work this out step-by-step:
a) Suppose the principal chooses to give A an order. What order does P choose? What effort level does the agent exert in step 3, given the principal’s choice of incentive strength β in step 2? What incentive strength β does the principal optimally choose in step 2?
b) Suppose the principal chooses to persuade A, and that persuasion succeeds. What effort level does the agent exert in step 3, given the principal’s choice of incentive strength β in step 2? What incentive strength β does the principal optimally choose in step 2?
c) Suppose the principal chooses to persuade A, and that persuasion fails. What effort level does the agent exert in step 3, given the principal’s choice of incentive strength β in step 2? What incentive strength 6 does the principal optimally choose in step 2?
d) For what values of B does the principal choose to persuade the agent, rather than order the agent?
e) Explain, in words, why the principal prefers to persuade the agent (rather than order the agent) when B is large.
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