11. informativeness and usefulness Ralph, who is risk neutral, owns a production process. One of three feasible labor inputs, L < B < H, must be selected and H is desired.

The labor supplier’s preferences are modeled in the usual constant risk aversion fashion, with risk parameter ρ = .0001. The supplier’s outside opportunity offers a normalized certainty equivalent of M =

0; and his personal costs are given by cH = 4, 000, cB = 1, 000 and cL = 0. The contracting arrangement is limited to payment based on output, which can take on one of two possible values, x1 or x2.

Output probabilities are displayed below.

x1 x2

π(x|H) .10 .90

π(x|B) .70 .30

π(x|L) 1 0

(a) Determine an optimal pay-for-performance arrangement.

(b) Suppose it is possible to install an additional measure, or monitor.

This monitor will report bad news if input L is supplied and good news otherwise. Is this monitor useful?

(c) Is the monitor in

(b) controllable? Is it informative (conditionally controllable)? Carefully explain this case of a serious control problem, a monitor that is both controllable and conditionally controllable, and yet is not useful.