18. change of pace Ralph is studying principal-agent problems. The principal is risk neutral, while the agent is risk averse and also incurs a personal cost.

The agent can supply input L or input H. The agent’s preferences are given by √payment−c(input) where the agent’s input cost is 15 if input H is supplied and 0 if input L is supplied, i.e., c(H) = 15 and c(L) = 0. The principal wants the agent to supply input H. Output will be either x1 or x2. Probabilities are defined by π(x1|L) = .5 and

π(x1|H) = .2. In addition, the agent must face an expected utility of at least 50 "units" if this is to be an attractive option. Further suppose an additional measure or monitor is available for contracting purposes. The monitor will report y = g or y = b.

(a) Determine an optimal pay-for-performance arrangement for each of the following four cases:

case 1:

x1/g x1/b x2/g x2/b

π(x, y|H) .18 .02 .72 .08

π(x, y|L) .05 .45 .05 .45 case 2:

x1/g x1/b x2/g x2/b

π(x, y|H) .10 .10 .40 .40

π(x, y|L) .25 .25 .25 .25 case 3:

x1/g x1/b x2/g x2/b

π(x, y|H) .19 .01 .01 .79

π(x, y|L) .01 .49 .19 .31 case 4:

g/x1 x1/b x2/g x2/b

π(x, y|H) .16 .04 .16 .64

π(x, y|L) .40 .10 .10 .40

(b) Determine the conditional and unconditional likelihood ratios for the new measure in each of the four cases.

(c) In which of the four cases is the monitor useful? In which of the four cases is the monitor controllable by the agent? Carefully explain your findings.