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

a) Show that the agent's expected utility from a contract can be written as a + ße - 2ß² - 0.5e² .

b) Show that the principal's expected utility from a contract can be written as (1 - ß)e^-a

c) Write the principal's maximization problem when effort is verifiable.

d) When effort is verifiable, what are the optimal values for a and ß for the principal?

e) When effort is verifiable, what effort level does the principal contract the agent to exert?

f) Write the principal's maximization problem when effort is unverifiable. Why is it different from the maximization problem with verifiable effort?