1. Consider a principal-agent relationship where agent’s effort, e, is not observable by the Principal, and hence it is not contractible. Agent’s output, z, is however contractible. Agent’s output is linked to agent’s effort by the following relationship:
z = e + x,
where x is a stochastic noise with E(x) = 0 and Var(x) = 2. (As in the lectures, E(.) and Var(.) denote the operators expected value and variance, respectively.)
The principal is risk neutral and has full bargaining power in designing the optimal linear contract , where w denotes the agent’s wage.
The preference of the risk averse agent is such that the certainty equivalent wealth of any risky income, I, is given by
CEWA(I) = E(I) – Var(I).
Given the contract designed by the principal, the agent chooses effort to maximise the certainty equivalent wealth of her income (wage) net of the cost of effort C(e) given by:
C(e) = .
Finally, the agent’s outside option (in terms of certainty equivalent wealth) is equal to 0.
a) Find the optimal contract designed by the principal, the total welfare (i.e., the total certainty equivalent wealth) generated by the optimal contract, and the expected utility (i.e., the certainty equivalent wealth) gained by each of the two parties. Provide detailed motivation and comments at all steps of your analytical answer.
