Consider the following risk-sharing problem. A Principal (owner of a firm) wants to hire an Agent to manage the firm. The Principal's utility-of-money function is U(m) = m while the Agent's utility-of-money function is: V(m) = ln(m) where ln denotes the natural logarithm. The profit of the firm will be $x1 with probability p and $x2 with probability (1−p). Consider the following contract: the Agent is paid $w1 if the outcome is x1 and is paid $w2 if the outcome is x2, where x1 = $2,500, x2 = $1,300, p = 1/3, w1 = $900, and w2 = $400.
(a) Calculate the Principal’s and Agent’s expected utility from this contract.
(b) Prove that this contract is not Pareto efficient.
(c) (c.1) Draw an Edgeworth box to represent all the possible contracts and identify the point that corresponds to the above contract. (c.2) Draw the indifference curve of the Principal that goes through that point and the indifference curve of the Agent that goes through that point. (c.3) State in words which of the two (if any) is a straight line.
(d) If you were to recommend a contract that is Pareto superior to the one given above, which of the following four suggestions would you make? (1) Increase both w1 and w2; (2) decrease both w1 and w2; (3) increase w1 and decrease w2; (4) decrease w1 and increase w2. Justify your suggestion using the Edgeworth box that you drew for part (c).