A principal offers an agent an enforceable contract for work on a project.

Under the contract, the principal is to pay s(x) for x ∈ {1, 2, 3} units of publicly observable output.

The agent's effort level onthe job, e ∈ {0, 1}, cannot be observed by the principal.

The probability that the output is x given the agent's effort e is p(x|e), with p(x|1)/p(x|0) increasing in x.

Under the contract, if x units are produced,the principal's utility is x - s(x) and the agent's utility is s(x) - e. T

he principal and agent are both risk-neutral expected utility maximizers. The agent is willing to accept any contract with a non negative payment function s(·).

Suppose that the payment function s∗ is optimal for the principal among all contracts with non negative payments, and suppose that s∗ induces the agent to choose e = 1.

a. Is it possible that s∗(1) is positive? Is it possible that s∗(2) is positive? Explain.

b. Consider instead the payment function S that is optimal for the principal among non negative payment functions s satisfying the additional constraint [s(x) - s(y)]/(x - y) ≤ 1 for x 6= y. Why might it make economic sense to restrict attention to payment functions satisfying this constraint?

c. Assume that the payment function S in part b induces the agent to choose e = 1. Is it possible thatS(1) is positive?

d. Under the assumption of part c, suppose that S(2) > 0. What can be concluded about S(3) - S(2)? Beas specific as possible and justify your answer. Give an economic explanation for the difference between the payment functions s∗ and S.