Question: Consider a situation in which a risk-neutral principal wishes to contract an agent to work on a project. The project produces output x = e^+e where e is the agent's effort and e is a normally distributed random variable with mean 0 and variance 1 (that is, e ~ N(0, 1)). The agent's utility of wealth function is u(w) = E(w)-2Var(w) where E is expectation and Var is variance. The agent's disutility of effort function is v(e) = 0.5e² and his reservation utility is u = 0. The principal can only offer contracts with the form w= a + ßx

From this we gather that the agent's expected utility from a contract can be written as a + ße - 2ß² - 0.5e². While the principal's expected utility from a contract can be written as (1 - ß)e^{- a}.

a) Given the contract w = a + ßx, what effort level will maximize the agent's expected utility?

b) Derive the optimal values for a and ß for the principal when effort is unverifiable.

c) How does the optimal ß in the verifiable effort contract differ from the optimal ß in the unverifiable effort contract? Discuss the implications for the efficiency of risk sharing and the efficiency of effort in the case of non-verifiable effort.

d) Suppose that in addition to observing output x, the principal observes the variable y = e + ? where ? ~ N(0, 1) is independent of e. If we assume that the principal's payoff does not depend directly on y, should he nevertheless include y in the contract he offers the agent? Why or why not?