1. The owner of a pumpkin farm (the principal) hires a manager (the agent). The number of pumpkins produced by the end of the season is denoted by ỹ, where y is stochastic and depends on the effort level x chosen by the agent as the following:

The principal is risk-neutral but the agent is risk-averse. The preference of the agent is v(c, x) = √c-² where c is the consumption and x is the effort level.

For agents, there is no other source of income other than the payment received by the principal. The outside option of the agent is normalized to zero.

(a) Suppose the effort level z is observable. Solve for the optimal contract that maximizes the principal's profit.

(b) Prove that your answer in part (a) is not implementable if the effort x is not observable.

(c) Define the second-best problem of the principal and characterize the constrained-efficient contract.