A principal wants to hire an agent to run her firm. The agent is risk-averse and has utility U(w,e)= (50+w)^(0.5)v(e)where w is the wage she receives from the principal and v(e) are her costs from effort level e. There are only two possible effort levels, eL and eH, and v(eH) = 5 and v(eL) = 3. The agent is an expected utility maximizer and her reservation utility is U = 5.

There are only two possible outcomes for the firm's profit, x1 = 0 and x2 = 500. If the agent chooses e = eL, then the probability that the outcome is x1 = 0 is 1/2. If the agent chooses e = eH, then the probability that the outcome is x1 = 0 is 1/4.

The principal is risk-neutral and maximizes expected profit. Her profit is given by B(x, w) = xw where x is the outcome for the firm's profit. Assume the principal can offer negative wages, if she wants to punish the agent for a bad outcome.

Suppose that effort is unobservable. Which contract would the principal offer if she wants to induce e = eH ? Derive the contract with the Kuhn-Tucker method.