A principal employs an agent to undertake a project that either succeeds and produces output y = x, or fails and produces output y = 0. If the agent exerts effort e [0,1] at cost C(e) = e2/2, the project succeeds with probability Pr(y = x) = e. Assume x < 1. a) Suppose a social planner can choose effort to maximize expected welfare, E[y] C(e). What is their choice of effort? Suppose the principal must incentivize the agent to choose effort. She pays the agent W1 if the project succeeds and W0 if it fails. The principal's profit equals the output minus the wage, = y W. The agent is risk-neutral and has utility u = W C(e). Assume the agent is protected by limited liability, so W1,W0 0.1 b) Given contract (W1,W0), what effort will the agent choose? The principal seeks to choose wages and an effort recommendation, (W1,W0,e) to maximize her profits subject to the agent's first order condition (IC) and W1,W0 0 (LL). c) Show that W0 = 0. d) What is the optimal W1? e) How does the optimal effort compare with the social planner's effort in (a)? What is the reason for this difference?