This question asks you to think about the implications of moral hazard when both the prin- cipal and the agent are risk neutral, but the agent has limited liability. An agent is producing output a: for the principal. Output can take on two values: a," E {0, 1}. The probability that output is high depends on the agent's action 6 E [0, co), and is given by 10(6). Suppose that p is continuously di'erentiable, 10(0) = O, p'(0) > 1, and lim,.3_,DO p'(e) = 0. The effort cost of exerting e'ort level 6 is simply e. The agent's outside option gives her utility zero. Both the principal and the agent are risk-neutral. The principal can offer the agent a reward contract 8(53) as a function of output :L'. As a function of 9:, the principal's utility is then m 8(56). As a function of the effort level 6 and output :3, the agent's utility is 5(a) e. The timing of the contracting game is the following. First, the principal offers the agent a take-it-or-leave-it contract s(-). The agent accepts or rejects. If she accepts, she chooses her effort level. Finally, the output is realized and payments are made. (a) Suppose that there is no restriction placed on the function s(:r). Derive the properties of the Optimal contract and interpret. (b) Now suppose that the agent has limited liability: the principal's contract has to beat her outside option in every state of the world, not just on average. That is, 3(39) 2 0 for a: 6 {0,1}. Derive the prOperties of the Optimal contract, and compare it to those found in the previous part. Explain the intuition. (c) Based on these results, argue informally that one could have involuntary unemployment in the labor market: there could be workers out there willing to take jobs at below-market wages, yet rms are not hiring them nor lowering wages for existing workers