13. optimal production plan Ralph, who is risk neutral, owns a production process. Production re quires input from a manager. This input can be one of three possible quantities: L < B < H. Output will be one of two possible quantities:
x1 < x2. The manager is risk averse and incurs a personal cost, precisely as specified in (13.2), with cL = 0, cB = 4, 000, cH = 10, 000 and ρ = .0001. The manager’s outside opportunity guarantees a wealth of M = 40, 000. The output probabilities are as follows.
x1 x2 input H 0 1 input B .1 .9 input L .9 .1

(a) Suppose the parties can contract on the output and the input supplied. Determine the best contract from Ralph’s perspective that will insure supply of input (i) H, (ii) B, and (iii) L.

(b) Suppose the parties can contract on the output, but not the input.
Determine the best contract from Ralph’s perspective that will insure supply of input (i) H, (ii) B, and (iii) L.

(c) Let x1 = 0 and x2 = 55, 000. Determine Ralph’s optimal plan under the contracting conditions in

(a) and under the contracting conditions in

(b) above.

(d) Let x1 = 0 and x2 = 59, 000. Determine Ralph’s optimal plan under the contracting conditions in

(a) and under the contracting conditions in

(b) above. Carefully explain your conclusions.

(e) Let x1 = 41, 000 and x2 = 46, 100. Determine Ralph’s optimal plan under the contracting conditions in

(a) and under the contracting conditions in

(b) above. Carefully explain your conclusions.