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Conversely, if l(x,y) does not depend on y the optimal scheme is of the form
s
H
(x). We can in that case write
f(x,y|e) = g (x|e)h(y|x), for every x, y and e = L, H.
(5)
On the left is the density function for the joint distribution of x and y given
e. On the right, the first term is the conditional density of x given e and the
second term the conditional density of y given x. The key is that the conditional
density of y does not depend on e and therefore y does not carry any additional
information about e (given x). In words we have the following:
Informativeness Principle (Holmström, 1979, Shavell, 1979). —An
additional signal y is valuable if and only if it carries additional
information about what the agent did given the signal x.
Stated this way the result sounds rather obvious, but it only underscores the
value of using the distribution function formulation.
7
One can derive a charac-
terization similar to (4) using the state-space formulation x(e,ε), but this char-
acterization is hard to interpret because it does not admit a statistical interpreta-
tion. The reason is that there are two ways in which y can be informative about e
given x. It could be that y provides another direct signal about e (y = e + δ, where
δ is noise) or y provides information about ε (y = δ, where δ and ε are correlated),
which is indirect information about e. Both channels are captured by the single
informativeness criterion.
As an illustration of the informativeness principle, consider the use of
deductibles in the following insurance setting. An insured can take an action to
prevent an accident (a break-in, say). But given that the accident happens, the
amount of damage it causes does not depend on the preventive measure taken by
the insured. In this case it is optimal to have the insured pay a deductible if the
accident happens as an incentive to take precautions, but the insurance company
should pay for all the damages, regardless of the amount. This is so because the
occurrence of the accident contains all the relevant information about the pre-
cautions the agent took. The amount of damage does not add any information
about precautions.
B. Implications of the Informativeness Principle
Several implications follow directly from the informativeness principle.
1. Randomization is suboptimal. Conditional on signal x, one does not
want to randomize the payment to the agent, because by eliminating the
Pay For Performance and Beyond
421
randomness without altering the agent’s utility at x gives the principal a
higher payoff. Randomization may be optimal if the utility function is
not separable.
8
2. Relative performance evaluation (Baiman and Demski, 1980, Holm-
ström, 1982) is valuable when the performance of other agents tells
something about the external factors affecting the agent’s performance,
since information about ε paired with measured performance x is infor-
mative about the agent’s action e. The important tournament literature
initiated by Lazear and Rosen (1981) studies relative performance evalu-
ation in great depth.
3. The controllability principle in accounting states that an agent’s incen-
tive should only depend on factors that the agent can control. The use of
relative performance evaluation seems to violate this principle, since the
agent does not control what other agents do. The proper interpretation
of the controllability principle says that an agent should be paid based
on the most informative performance measure available. Since x already
depends on outside factors (captured by ε) anything correlated with ε
can be used to filter out external risk, making the adjusted performance
measure more informative.
4. A sufficient statistic in the statistical sense is also sufficient for design-
ing optimal incentive contracts. For instance, the sample mean drawn
independently from a normal distribution with known variance is a suf-
ficient statistic for mean effort and can be used to evaluate performance
(Holmström 1982).
5. Optimal incentive pay will depend on lagged information if valuable
information comes in with delay.
Bebchuk and Fried (2004) among others have argued that CEOs should not
be allowed to enjoy windfall gains from favorable macroeconomic conditions.
Appealing to the informativeness principle, they advocate the use of relative per-
formance evaluation as a way to filter out luck. In many cases this is warranted,
but not without qualifications. In a multitasking context relative performance
evaluation may distort the agent’s allocation of time and effort. When agents that
work together are being compared against each other, cooperation is harmed or
collusion may result (Lazear 1989). Filtering out the effects of variations in oil
prices on the pay of oil executives will in general not be advisable (at least fully)
because this would distort investment and other critical decisions.
Longer vesting times, causing lagged information to affect pay, are unlikely
to negatively distort CEO behavior. On the contrary, too short or no vesting
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aggravate problems with short-termism and strategic information release. Allow-
ing executives to sell incentive options so quickly in the 1990s was clearly
unsound and the current vesting periods may still be too short.
C. Puzzles and Shortcomings
The informativeness principle captures the central logic of the basic one-dimen-
sional effort model. In doing so it helps explain some puzzling features of the
basic model and also its main shortcomings.
Surprisingly, the optimal incentive scheme need not be nondecreasing even
when x = e + ε. The reason is that higher output need not signal higher effort
despite first-order stochastic dominance. The characterization (4) shows that
the optimal incentive scheme is always monotone in the likelihood ratio l(x)
and therefore monotone in x if and only if the likelihood ratio is monotone, in x.
Suppose the density has two humps and the difference e
H
− e
L
is small enough so
that the two density functions will cross each other more than once (the humps
are interlocked). This creates a likelihood ratio that is non-monotone, implying
that there exist two values x such that the larger value has a likelihood ratio below
one, speaking in favor of low effort, while the lower value has a likelihood ratio
above one suggesting high effort. In line with inference, the agent is paid more
for the lower value than the higher value. One can think of the two humps as two
states of nature: bad times and good times. The higher outcome suggests that the
good state obtained, but conditional on a good state the evidence suggests that
the agent slacked.
9
One can get around this empirically implausible outcome by assuming that
the agent can destroy output, in which case only nondecreasing incentives are
relevant (Innes 1990). Or one can assume that the likelihood ratio is monotone, a
common property of many distribution functions and a frequently used assump-
tion in statistics as well as economics (Milgrom 1981). But the characterization
in (4) makes clear that the basic effort model cannot explain common incentive
shapes. The universal use of piece rates for instance cannot be due to similar
likelihood ratios.
The inference view also helps us understand a troubling example in Mirrlees
([1975]1999). Mirrlees studies the additive production function x = e + ε where
ε is normally distributed with mean zero and a constant variance. In other words,
the agent chooses the mean of a normal distribution with constant variance. He
analyzes the problem with a continuous choice of effort, but the main intuition
can be conveyed with just two actions.
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