7. scaling Ralph faces a recalcitrant manager. While Ralph wants input H, the manager prefers input L. (Yes, we have a prototypical contract story as studied in the chapter.) The manager is risk averse with preferences modeled in the usual fashion of expression (14.1). Assume

ρ = .0001, cH = 15, 000, cL = 10, 000 and M = 75, 000. The output probabilities are given below, where you will notice output is the only contracting variable and it can take on one of three possible values.

x1 x2 x3

π(x|H) .1 .2 .7

π(x|L) .5 .3 .2

(a) Determine an optimal contract.

(b) Now scale the setting, as we did in the chapter. What is the optimal contract in the scaled setting? Explain your reasoning.

8. shadow prices Return to the settings in Tables 14.1, 14.2 and 14.3 and focus on the optimal contract when the additional information variable is present, I∗xy. For each of the three cases determine the shadow prices on the individual rationality and incentive compatibility constraints and verify that the optimal contract satisfies the likelihood ratio condition in expression (14.6).