Agent and principal utility

The utility of the agent is given by U(w, a) = √w − a, where w is wage and a is effort. The reservation utility of the agent is 1. There are two profit levels, πl = $0 and πh = $100. The principal is risk-neutral. The wages are restricted to be non-negative, w ≥ 0. Suppose there are three levels of effort for the agent, a1 = 1, a2 = 2 and a3 = 4. The probabilities of high profit after respective actions are given by p1,h = 0.40, p2,h = 0.60 and p3,h = 0.85. The principal cannot observe agent's effort but observes profit. The principal offers the agent a contract (wl, wh), where wl is the wage paid after low profit and wh is the wage paid after high profit. The agent can accept or reject the contract. If the agent accepts the contract, the agent chooses effort. Agent's payoff is the expected utility and principal's payoff is the expected profit. If the agent rejects, agent's payoff is the reservation utility 1 and principal's payoff is 0.

(a) If the effort were observable, what is the optimal contract for the principal?

(b) Suppose now that effort is not observable. What is the minimal cost to induce a2?

(c) Find the minimal cost to induce a3 when effort is not observable.

(d) What effort does the principal want to induce when effort is not observable? What is the optimal contract for the principal?