principal agent problem

1. Consider a principal who hires an agent to run a very simple stochastic technology. The agent when hired may decide to exert a productive effort e that may take one of the tow values and > 0. Assume that the high level of effort e = entails a cost to the agent of size , while the low level of effort e = 0 entails no cost. If the agent chooses effort e = 0 then output, y, is equal to 1 with probability Po, and y = 0 with probability 1 - Po. On the other hand, if the agent exerts effort e = 0, then y = 1 with probability P2, and y = 0 with probability 1 - Po. We assume p. > po Finally, assume that both the principal and the agent are risk neutral, the agent has a reservation utility normalized to be 0, and that any contract offered to the agent needs to satisfy a limited liability constraint specifying that the agent cannot be paid a negative amount wh20, h = 0,1. Assume first that the level of effort e is verifiable. (1) Solve for the first-best-optimal contract that the principal offers the agent to induce him to exert the high level of effort e = . (ii) Solve for the first-best-optimal contract that the principal offers the agent to induce him to exert the low level of effort e = 0. (iii) Compare the principal's profits in (i) and (ii). Under which conditions would you expect the first-best optimal contract to induce the agent to exert a high level of effort? Assume now that the effort level e is private information of the agent. While the amount of output y is verifiable. (iv) Solve for the second-best-optimal contract that the principal offers the agent to induce him to exert the high effort e = 0. Solve for the second-best-optimal contract that the principal offers the agent to induce him to exert the low level of effort e = 0. (vi) Compare the principal's profits under (iv) and (v). Under which condition would you expect the second best optimal contract to induce the agent to exert a high level of effort? (vii) Compare your answer to (iii) with your answer to (vi). (v)