# Return to the setting of Example 13.5. Now assume the probability of output x1 under input L is…

Return to the setting of Example 13.5. Now assume the probability of output x1 under input L is .9 instead of 1.0. Determine the optimal pay-for-performance arrangement. Carefully explain the difference between this arrangement and that identified in Example 13.5.

Example 13.5((Example 13.5 Return to the illustrative payment functions in Example 13.4. It turns out case 4, with I1 = 5, 000 and I2 = 12, 305.66 is the optimal incentive arrangement in that setting where the only contractible variable is the output, x. The value of this arrangement to the manager is CEH = 3, 000, which equals his opportunity cost of M = 3, 000. The cost to the firm is E[I|H] = 8, 652.83. Exclusive of the personal cost, the manager’s certainty equivalent of this compensation arrangement is 8, 000 = CEH + cH. And 8,652.83 – 8,000 = 652.83 is the manager’s risk premium. This risk premium claim should, of course, be verified. Consider an individual with utility for wealth w given by U(w) = −exp(−.0001w). Our individual has no initial wealth and faces a lottery of 50−50 odds on 5,000 or 12,305.66. The expected value of this lottery is .5(5, 000)+.5(12, 305.66) = 8, 652.83. And if you check, you will see that its certainty equivalent is 8,000, implying a risk premium of 652.83.

As an aside, intuition guides us to the solution to program (13.5). Suppose we have a solution in which E[U|H, I] is strictly greater than U(M). We could then lower each payment a small amount, lowering the firm’s cost and not upsetting the other constraint. So anytime we have E[U|H, I] > U(M), we can find a less costly scheme. Therefore, the best scheme must have E[U|H, I] = U(M).

Similarly, suppose we have a scheme in which E[U|H, I] > E[U|L, I]. Now the incentive scheme is needlessly strong. Incentives, however, are not a free good. The manager’s pay is at risk, and the manager must be compensated for carrying this risk. So, if the incentives are too strong, they can be weakened in a way that lowers the cost to the firm. Hence, the best scheme must also have E[U|H, I] = E[U|L, I]. 16

We therefore have a constraint set of two equations in two unknowns:

And this can be readily solved for I2. 17

Regardless, several features of this exercise should be noted. First, we have I2 > I1. Notice in the above expression that M + cL − cH < M, as cH > cL. But this means I2−cH > M, or I2 > M+cH > M+cL = I1. This is no accident. We already know I1 = I2 (a flat wage) won’t work. What about I1 > I2? The manager would then face the prospect of switching to input L, incurring lower cost, and guaranteeing himself the larger prize. What a deal! Simply stated, incentive compatibility, expression (13.3), requires I2 > I1.

Second, with I2 > I1 the manager labors under an incentive arrangement. A bonus of I2 − I1 is paid if high output, x2, is produced. Of course, this means the manager’s wealth is at risk. This is contrary to efficient risk sharing, as the firm is risk neutral. In a sense, then, we trade off efficient risk sharing for incentive compatibility. Third, with the manager bearing risk, part of his compensation takes the form of a risk premium. We saw this in Example 13.5. To see it more generally, write out the individual rationality condition, (13.4), in a little more detail

Stare at this for awhile. We have a risky lottery of I1 or I2 that has a certainty equivalent of M + cH. This means it has a nontrivial risk premium of its expected value less that certainty equivalent, or

This risk premium is a deadweight loss of contracting for managerial action in such a setting.18 Fourth, a popular euphemism is that the manager is now paid for results, or “only results count.” This masks a subtle and important point. We want the manager to supply input H, but cannot directly observe whether input H is supplied. Output is observed, and we therefore use output to infer input. Casually, high output (i.e., x2) is consistent with supply of input H, while low output (i.e., x1) is more ambiguous. This is why the manager is paid more for high output. Output, then, is a source of value to the firm and a source of information in the contracting arrangement. Fifth, the overall exercise is one of engineering the manager’s decision tree, at minimum cost to the firm. Figure 13.1 was designed to convey this insight. At the time of contracting, the manager has three alternatives: reject the firm’s offer, accept the firm’s offer and supply L (be disobedient), or accept the firm’s offer and supply H (be obedient). Individual rationality requires E[U|H, I] ≥ U(M), and incentive compatibility requires E[U|H, I] ≥ E[U|L, I]. The constraints literally ensure the manager’s fully formed decision tree rolls back to the conclusion that supply of input H is desirable behavior from the manager’s perspective. Indeed, here we further assume that if indifferent the manager will honor the firm’s instruction, supply H in this case.19

Finally, our story sharply distinguishes the cases of observable and unobservable input. In the former, the cost to the firm of input H is simply the perfect market solution of IH = M + cH. In the latter, where only output is observed, the cost is this amount plus the above identified risk premium. Unobservable input raises the cost of managerial service. This occurs because output here is an imperfect indicator of input, and thus requires a risky payment to the manager; and we have grounded the model so the cost of the manager’s risk bearing is borne by the firm. In this way we readily see that the firm would pay up to this risk premium, expression (13.6), to be able to observe the manager’s input.))

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