8. optimal contract Ralph owns a production function. Randomness in the environment plus labor input from a manager combine to produce output. The output can be one of two quantities: x1 < x2. The manager’s input can be one of two quantities, L < H. Ralph is risk neutral. The probabilities are given below, and you should assume the higher output is sufficiently attractive that Ralph wants supply of input H in all that follows.

x1 x2 input H .1 .9 input L .8 .2 Ralph’s manager is risk averse and also incurs an unobservable personal cost in supplying the labor input. We model this in the usual way. The manager’s utility for wealth is as given in (13.2), with cH = 5, 000, cL = 0, and ρ = .0001. Also, the manager’s opportunity cost of working for Ralph is a certainty equivalent of M = 10, 000.

(a) Suppose the manager is trustworthy and will honor any agreement

(or, equivalently, serious penalties are feasible and the manager’s input can be observed.) What is the cost to Ralph of acquiring input H?

(b) Suppose the only observable for contracting purposes is the manager’s output. Determine the optimal pay-for-performance arrangement. What is the cost to Ralph of acquiring input H?
Draw the manager’s decision tree and verify the manager can do no better than accept Ralph’s terms and then supply input H. What is the manager’s certainty equivalent for the payment lottery that is faced?

(c) Why, in your solution to part

(b) above, is the manager paid more when the largest feasible output (i.e., x2) is observed?