08 Jun Question A principal offers an agent an enf
Question
A principal offers an agent an enforceable contract for work on a project. Under the contract, theprincipal 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 theagent’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. The principal and agent are bothrisk-neutral expected utility maximizers. The agent is willing to accept any contract with a nonnegativepayment function s(·). Suppose that the payment function s?is optimal for the principal among allcontracts with nonnegative 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 betweenthe payment functions s? and S.
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