9781118041581

In turn, the principal’s level of welfare is

Finally, the total welfare of principal and agent together is

[14A.4]

The parties’ total welfare is the expected profit, ke (after adding together

each side’s respective profit share), net of the agent’s cost of effort and risk

premium (the cost of bearing risk).

Given that the agent seeks to maximize his overall level of welfare, what

level of effort will he or she expend? To maximize UAwith respect to e, we use

the marginal condition

Therefore, e* bk/c, where the asterisk denotes the optimal level of effort.

Note that b 0 implies e* 0. If the agent receives a fixed wage, W a, that

does not depend on performance, there is no point in expending costly effort.

The agent’s optimal level of effort increases with b and k. A greater profit-shar-

ing rate or greater productivity means a closer connection between the worker’s

remuneration and his effort; with more to gain in either case, he works harder.

Not surprisingly, effort varies inversely with its cost (c).

Let’s now characterize an efficient wage contract. The goal of efficiency (recall

the discussion in Chapter 7) is to maximize the total welfareof the parties. First, sub-

stitute the formula for the agent’s effort (e* bk/c) into Equation 14A.4.

[14A.5]

An efficient contract sets the sharing rate b to maximize total welfare in

Equation 14A.5. Note that total profit does not depend on the fixed wage (a).

Varying the fixed wage simply redistributes the split of the total profit. The

optimality condition with respect to b is

Therefore,

b*

k^2 /c

r^2 k^2 /c

dUT/dbk^2 /crb^2 bk^2 /c0.

bk^2 /c.5rb^2 ^2 .5b^2 k^2 /c

UTUPUAk(bk/c).5rb^2 ^2 .5c(bk/c)^2

dUA/debkce0.

UTUPUAke.5rb^2 ^2 .5ce^2.

(1b)kea.

E[(ab)]

UPE[W]

628 Appendix to Chapter 14 A Principal-Agent Model

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