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 9 that may take one of the tow values 0 and El > 0. Assume that the high level of effort 8 = entails a cost to the agent of size ID, while the low level of effort 9 = 0 entails no cost. If the agent chooses effort 3 = 0 then output, y, is equal to 1 with probability p0, and y = 0 with probability 1 390. On the other hand, if the agent exerts effort 9 = til, then y = 1 with probability p1, and y = 0 with probability 1 pa. We assume p1 > 390. 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 w), 2 0, h = 0,1. Assume rst that the level of effort 9 is verifiable. (i) Solve for the rst-besr-optimal contract that the principal offers the agent to induce him to exert the high level of effort e = 9. (ii) Solve for the mt-best-optimai contract that the principal offers the agent to induce him to exert the low level of effort 8 = 0. (iii) Compare the principal's prots in (i) and (ii). Under which conditions would you expect the rst-best optimal contract to induce the agent to exert a high level of effort? Assume now that the effort level 9 is private information of the agent. While the amount of output y is veriable. (iv) Solve for the second-best-optimal contract that the principal offers the agent to induce him to exert the high effort 8 = El. (v) Solve for the second-bestoprimal contract that the principal offers the agent to induce him to exert the low level of effort e = 0. (vi) Compare the principal's prots 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)