The contract derived in Example 15.1 has a negative wage of 20. What is the intuition for this…

The contract derived in Example 15.1 has a negative wage of 20. What is the intuition for this (seemingly strange) conclusion? Hint: check out note 5.

((Example 15.1 Specify the manager by a risk aversion measure of ρ = .1 and a personal cost of cH = 60, and define the inputs by H = 500 and L = 200. Also specify the contracting variable, (15.1), by α = .7 and σ 2 = 10, 000. Solving program (15.7), we find an optimal (linear) contract of ω ∗ = −20 and β = .20, a claim you should verify.6

Intuitively, the manager will supply input H only if incurring the personal cost has a compensating shift in the value of his compensation, i.e., only if β ≥ cH H−L = 60 500−200 = .20. In turn, the cost to the firm of this compensation package is, in total, the manager’s personal cost plus his outside opportunity (which we have normalized to M = 0) plus his risk premium. This risk premium, which totals 1 2 ρβ2 σ 2 , increases with slope β and is independent of the manager’s input. (This is evident in the certainty equivalent expressions (15.4) and (15.5).) Naturally, then, we keep the incentive intensity, the β, as small as possible, consistent with motivating input H. So we have β = .20.

The intercept, ω, is set to satisfy the individual rationality requirement: CEH = ω + βH − 1 2 ρβ2 σ 2 − cH = 0. We know β = .20, which provides a risk premium of 1 2 ρβ2 σ 2 = .5(.1)(.04)(10, 000) = 20. With H = 500 and cH = 60 we have CEH = ω + .2(500) − 20 − 60 = 0, or ω = −20. And we wind up with a compensation cost to the firm of C(H) = ω + βH = −20 + .2(500) = 80.

Notice, in both program (15.7) and Example 15.1, that we have not concerned ourselves with how the manager allocates his input. To see why, we return to Example 15.1.

((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