9781118041581

Appendix to Chapter 14 A Principal-Agent Model 629

or

[14A.6]

(In the last step, we have simplified the expression by dividing numerator and

denominator by the common factor k^2 /c.)

Equation 14A.6 summarizes the factors that determine the optimal rate of

profit sharing. First, note that if the agent were risk neutral (r 0), the opti-

mal contract would set b* 1; the agent would be paid each extra dollar of

profit earned. As noted earlier in the chapter, this induces the agent to supply

the profit-maximizing level of effort (taking into account the cost of effort).

Similarly, if effort is very productive (k is very large), efficiency implies a high

sharing rate, again, to induce a high effort level. Conversely, if output is very

risky or the agent is very risk averse (high ^2 or high r), an efficient contract

sets a low profit-sharing rate, minimizing the agent’s exposure to profit risk.

Finally, if effort is very costly (high c), the efficient contract sets a low profit-

sharing rate and elicits a modest level of effort from the agent.

The bare-bones principal-agent model can be generalized in many useful

directions. We conclude by extending the setup to allow the principal to mon-

itor imperfectlythe agent’s effort. Suppose the principal observes:

where v is a random factor with mean 0 and standard deviation v. The prin-

cipal cannot measure and observe the agent’s true effort e. Rather, it observes

a proxy (H for “hustle”) that may overestimate or underestimate e. The prin-

cipal then offers a wage contract of the form

In this new contract, greater effort by the agent is (imperfectly) rewarded by

means of the positive coefficient d. For this contract, we can derive the agent’s

optimal level of effort: e* (bk d)/c. Although we will not write down their

formulas, the optimal coefficients b* and d*—in particular their relative mag-

nitudes—are of some interest. First, both coefficients are positive. This follows

from the informativenessprinciple (noted earlier in the chapter) that all per-

formance information should be used in setting compensation. Second, an

efficient wage contract places the greater weight on the performance measure

(or H) having the greater precision, that is, smaller variance. For instance, if

effort can be measured with relative precision (vis small relative to ), then

d* will be large and b* will be small. Conversely, if effort is very imprecisely

observed, profit sharing (b*) will carry the greater relative weight.

WabdH.

Hev,

b*

1

rc^2 /k^2  1

.

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