Return to Example 15.1, but now assume the manager’s outside certainty equivalent is M = 900….

Return to Example 15.1, but now assume the manager’s outside certainty equivalent is M = 900. What is the optimal contract? What does this tell you about our normalization of M = 0 in this setting.?

((Example 15.1 (continued) The manager’s performance measure x is given by x = a1 +.7a2 +ε and his compensation is given by I = −20 +.2x. If he supplies input H = 500, he can allocate his time and talent to the two tasks in any fashion consistent with a1, a2 ≥ 0 and a1+a2 ≤ 500. Any such feasible allocation provides an expected compensation of ω+ a1 + .7a2, a risk premium of 1 2 ρβ2 σ 2 = 20 and a personal cost of cH = 60. This implies a certainty equivalent of

So how do we maximize .2[a1 + .7a2] − 100 subject to a1, a2 ≥ 0 and a1 +a2 ≤ 500? The answer is simple. Each unit allocated to the first task produces one dollar of certainty equivalent, while each unit allocated to the second produces but 70 cents of certainty equivalent. We have an optimal allocation of a ∗ 1 = 500 and a ∗ 2 = 0, precisely as desired.

Now you know why we specified the performance measure in (15.1) to put less weight on the second than on the first task (i.e., 0 ≤ α < 1). This ensures the manager’s allocation of total input between the two tasks is a trivial exercise. He will allocate everything to the first task because doing so is more personally productive as long as explicit incentives are turned on, as long as β > 0. 7

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