Question: A principal employs an agent to undertake a project that either succeeds and produces output \(y=V\), or fails and produces output \(y=0 .\) If the agent exerts effort \(e \in[0,1]\) at cost \(C(e)=e^{2} / 2\), the project succeeds with probability \(\operatorname{Pr}(y=V)=e\). Assume \(V<1\).

1. 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 \(W_{1}\) if the project succeeds and \(W_{0}\) if it fails. The principal's profit equals the output minus the wage, \(\Pi=y-W\). The agent is risk-neutral and has utility \(u=W-C(e)\). Assume the agent is protected by limited liability, so \(W_{1}, W_{0} \geq 0 .^{3}\)
2. Given contract \(\left(W_{1}, W_{0}\right)\), what effort will the agent choose?
   The principal seeks to choose wages and an effort recommendation, \(\left(W_{1}, W_{0}, e\right)\) to maximize her profits subject to the agent's first order condition (IC) and \(W_{1}, W_{0} \geq 0\) (LL).
3. Show that \(W_{0}=0\).
4. What is the optimal \(W_{1} ?\)
5. How does the optimal effort compare with the social planner's effort in (a)? What is the reason for this difference?

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