9781118041581

Appendix to Chapter 14 A Principal-Agent Model 627

Note that the marginal cost of effort increases with e; expending more and

more extra effort becomes more and more costly. Thus, the agent faces a trade-

off. He or she benefits by increasing e and raising output (by which he or she

might increase the wage reward), but the extra effort comes at a personal cost.

The agent’s overall welfare or utility (call this UA) depends on the wage

received, net of the cost of effort: W CEW  .5ce^2.

Now consider the wage contract between principal and agent. The agent’s

wage is given by

[14A.2]

Here, the principal is able to observe and measure the agent’s output , but not

his effort e. Accordingly, the agent’s wage in Equation 14A.2 depends on out-

put, but not effort. Specifically, the agent receives a fixed-wage portion (a) plus

a profit-sharing portion (b) that depends directly on his output. We note two

extreme cases: b 0 corresponds to a fixed-wage contract, whereas b  1

means that the agent’s compensation varies dollar for dollar with output. For

b between 0 and 1, the agent’s wage involves partial profit sharing. Accordingly,

we speak of b as the agent’s profit-sharing rate.

Because the agent is risk averse, we need to focus on the agent’s certainty equiv-

alent wage. As noted in Chapter 12, the individual values a risky payment at an

amount below the payment’s expected value. This reduction in value is termed the

agent’s risk premium (RP).Thus, we write the agent’s certainty equivalent wage as

where E denotes expectation. (Here, we have substituted from Equations 14A.1

and 14A.2 and used the fact that the expected value of u is 0.) The agent’s risk

premium depends on the degree of risk inherent in the wage and on his or

her personal degree of risk aversion. The variance of his wage is exactly b^2 ^2 ,

and this we take as the appropriate summary measure of wage risk. Thus, we

can write the agent’s risk premium as RP  .5rb^2 ^2 , where the coefficient r

denotes the agent’s degree of risk aversion. The more risk averse is the agent,

then the greater is r. (If the agent were risk neutral, then r would be 0.)

We are now ready to write an expression for the agent’s overall level of

welfare

abke.5rb^2 ^2 .5ce^2. [14A.3]

[abkeRP].5ce^2

UAWCECE

abkeRP,

E[ab(keu)]RP

E[ab]RP

WCEE[W]RP

Wab.

c14AsymmetricInformationandOrganizationalDesign.qxd 9/26/11 11:03 AM Page 627