Bengt Holmström Prize Lecture: Pay for Performance and Beyond

Pay For Performance and Beyond 

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So, assume as before that the agent can choose between low (L) or high (H) 

effort with a (utility) cost-differential equal to c > 0. Note that the likelihood 

ratio f

L

(x)/f

H

(x) with the normal distribution goes to infinity as x goes to negative 

infinity. Therefore, the right-hand side of (4), which is supposed to characterize 

the optimal solution, must be negative for sufficiently small values of x, because 

the values for λ and μ are strictly positive when high effort is implemented as 

discussed earlier. But that is inconsistent with the left-hand side being strictly 

positive for all x regardless of the function s

H

(x). How can that be? Mirrlees 

conjectured and proved that this must be because the first-best solution can be 

approximated arbitrarily closely, but never achieved.

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In particular, consider a scheme s(x) which pays the agent a fixed amount 

above a cutoff z, say, and punishes the agent by a fixed amount if x falls below 

z. If the utility function is unbounded below, Mirrlees shows that one can con-

struct a sequence of increasingly lenient standards z, each paired with an increas-

ingly harsh punishment that maintains the agent’s incentive to work hard and 

the agent’s willingness to accept all contracts in the sequence. Furthermore, this 

sequence can be chosen so that the principal’s expected utility approaches the 

first-best outcome.

One intuition for this paradoxical result is that while one cannot set μ = 0 in 

(4), one can approach this characterization by letting μ go to zero in a sequence 

of schemes that assures that the right-hand side stays positive.

The inference logic suggests an intuitive explanation. Despite the way the 

normal distribution appears to the eye, tail events are extremely informative 

when one tries to judge whether the observed outcome is consistent with the 

agent choosing a high rather than a low level of effort. The likelihood ratio f

L

(x)/

f

H

(x) goes to infinity in the lower tail, implying that an outcome far out in the 

tail is overwhelming evidence against the H distribution. Statistically, the normal 

distribution is more akin to the moving box example discussed in connection 

with first-best outcomes. Punishments in the tail are effective, because one can 

be increasingly certain that L was the source.

III. TOWARD A MORE REALISTIC MODEL

A. Why Are Incentive Schemes Linear?

One could brush aside Mirrlees’ example as extremely unrealistic (which it is) 

and proceed to study linear contracts because they are simple and widely used. 

But this would defeat the purpose of studying general incentives. An additive 

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The Nobel Prizes

production function with a normal error term is a very natural example to study. 

The unrealistic solution prods us to look for a more fundamental reason for lin-

earity than simplicity and to revisit the assumptions underlying the basic princi-

pal-agent model. This led us to the analysis in Holmström and Milgrom (1987).

The optimum in the basic model tends to be complex, because there is an 

imbalance between the agent’s one-dimensional effort space and the infinite-

dimensional control space available to the principal.

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 The principal is therefore 

able to fine-tune incentives by utilizing every minute piece of information about 

the agent’s performance resulting in complex schemes.

The bonus scheme that approaches first-best in Mirrlees’ example works for 

the same reason. It is finely tuned because the agent is assumed to control only 

the mean of the normal distribution. If we enrich the agent’s choice space by 

allowing the agent to observe some information in advance of choosing effort, 

the scheme will perform poorly. Think of a salesperson that is paid a discrete 

bonus at the end of the month if she exceeds a sales target. Suppose she can see 

her progress over time and adjust her sales effort accordingly. Then, if the month 

is about to end and she is still far from the sales target, she may try to move 

potential sales into the next month and get a head start towards next month’s 

bonus. Or, if she has already made the target, she may do the same. This sort 

of gaming is common (see Healy 1985 and Oyer 1998). To the extent that such 

gaming is dysfunctional one should avoid a bonus scheme.

Linear schemes are robust to gaming. Regardless of how much the salesper-

son has sold to date, the incentive to sell one more unit is the same. Following this 

logic, we built a discrete dynamic model where the agent sells one unit or none 

in each period and chooses effort in each period based on the full history up to 

that time. The principal pays the agent at the end of time, basing the payment on 

the full path of performance. Assuming that the agent has an exponential utility 

function and effort manifests itself as an opportunity cost (the cost is financial), it 

is optimal to pay the agent the same way in each period—a base pay and a bonus 

for a sale.

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 The agent’s optimal choice of effort is also the same in each period, 

regardless of past history. Therefore, the optimal incentive pay at the end of the 

term is linear in total sales, independently of when sales occurred.

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The discrete model has a continuous time analog where the agent’s instanta-

neous effort controls the drift of a Brownian motion with fixed variance over the 

time interval [0,1]. The Brownian model can be viewed as the limit of the discrete 

time model under suitable assumptions.

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 It is optimal for the principal to pay 

the agent a linear function of the final position of the process (the aggregate 

sales) at time 1 and for the agent to choose a constant drift rate (effort) at each 

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