Suppose a risk neutral principal wants to hire an agent to do a project. The output .Y is either 1 (if successes) or 0 (if not). The probability of success depends on the agent's effort e ε [0, 1].
i.e., Pr(X = 1) = e.
The cost of effort is c(e) = 1/2e^2. Because the principal cannot directly observe e, he will write a contract on the outcome X, i.e., the principal pays the agent w_1 if it succeeds, and w_0 = 0 otherwise. The agent's monetary utility is u(w) = 3 w^1/3. The agent's outside reservation utility is assumed to be zero.
1. What's the first best effort level and contract?
2. Derive the optimal contract under moral hazard.