Bengt Holmström Prize Lecture: Pay for Performance and Beyond

418 

The Nobel Prizes

1]. This corresponds to the agent choosing any uniform distribution [e, e + 1] at 

cost c(e). The density of a uniform distribution looks like a box. As e varies the 

box moves to the right. In first best, the agent should receive a constant payment 

if he chooses the first-best level of effort e

FB

. This can be implemented by paying 

the agent a fixed wage if the observed outcome x ≥ e

FB

 and something low enough 

(a punishment) if x < e

FB

. The scheme works, because two conditions hold: (a) the 

agent can be certain to avoid punishment by choosing the first-best effort level 

and (b) the moving support allows the principal to infer with certainty that an 

agent is slacking if x < e

FB

 and hence punish him severely enough to make him 

choose first best. In the general model, inferences are always imperfect, but will 

still play a central role in trading off risk versus incentives.

C. Second-best with Two Actions

I proceed to characterize the optimal incentive scheme in the special case where 

the agent chooses between just two distributions F

L

 and F

H

. This special case 

will reveal most of the insights from the basic agency model without having to 

deal with technical complications. Assume that F

H

 dominates the distribution 

F

L

 in the sense of first-order stochastic dominance: for any z, the probability 

that x > z is higher under F

H

 than F

L

. This is consistent with assuming that the 

high distribution is a more costly choice for the agent: c

L

 < c

H

. As an example, 

for x = e + ε and e

L

 < e

H

, F

H

 first-order stochastically dominates F

L

 regardless of 

how ε is distributed.

Assume that the principal wants to implement H, the other case is uninter-

esting since L is optimally implemented with a fixed payment. Let μ and λ be 

(non-negative) Lagrangian multipliers associated with the incentive compatibil-

ity constraint (2) and the participation constraint (3) in the principal’s program 

(1)–(3). The optimal second-best contract, denoted s

H

(x), is characterized by

 

u′(s

H

(x))

–1

 = λ + μ[1 − f

L

(x)/f

H

(x)], for every x.

4

 

(4)

Here f

L

(x) and f

H

(x) are the density functions of F

L

(x) and F

H

(x). It is easy 

to see that both constraints (2) and (3) are binding and therefore μ and λ are 

strictly positive.

5

The characterization is simple but informative. First, note that the optimal 

incentive scheme deviates from first-best, which pays the agent a fixed wage, 

because the right-hand side varies with x. The reason of course is that the prin-

cipal needs to provide an incentive to get the agent to put out high effort. Second, 

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Pay For Performance and Beyond 

419

the shape of the optimal incentive scheme only depends on the ratio f

H

(x)/f

L

(x). 

In statistics, this ratio is known as the likelihood ratio; denote it l(x). The likeli-

hood ratio at x tells how likely it is that the observed outcome x originated from 

the distribution H rather than the distribution L. A value higher than 1 speaks 

in favor of H and a value less than 1 speaks in favor of L.

Denote by s

λ

 the constant value of s(x) that satisfies (4) with μ = 0. It is the 

optimal risk sharing contract corresponding to λ.

6

 The second-best contract (μ

> 0) deviates from optimal risk sharing (the fixed payment s

λ

) in a very intui-

tive way. The agent is punished when l(x) is less than 1, because x is evidence 

against high effort. The agent is paid a bonus when l(x) is larger than 1 because 

the evidence is in favor of high effort. The deviations are bigger the stronger 

the evidence. So, the second-best scheme is designed as if the principal were 

making inferences about the agent’s choice, as in statistics. This is quite surpris-

ing because in the model the principal knows that the agent is choosing high 

effort given the contract she offers to the agent before the outcome x is observed. 

So, there is nothing to infer at the time the outcome is realized.

II. THE INFORMATIVENESS PRINCIPLE

The fact that the basic agency model thinks like a statistician is very helpful for 

understanding its behavior and predictions. An important case is the answer that 

the model gives to the question: When will an additional signal y be valuable, 

because it allows the principal to write a better contract?

A. Additional Signals

One might think that if y is sufficiently noisy this could swamp the value of any 

additional information embedded in y. That intuition is wrong. This is easily 

seen from a minor extension of (4). Let the optimal incentive scheme that imple-

ments H using both signals be s

H

(x,y). The characterization of this scheme fol-

lows exactly the same steps as that for s

H

(x). The only change we need to make 

in (4) is to write s

H

(x,y) in place of s

H

(x) and the joint density f

i

(x,y) in place of 

f

i

(x) for i = L,H. Of course, the Lagrange multipliers will not have the same values 

if y is valuable.

Considering this variant of (4) we see that if the likelihood ratio l(x,y) = 

f

H

(x,y)/f

L

(x,y) depends on x as well as y, then on the left-hand side the opti-

mal solution s

H

(x,y) must depend on both x and y. In this case y is valuable. 

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