DISCUSSION
PAPER SERIES
IN ECONOMICS
AND MANAGEMENT
Performance Measure Congruence and Efficient
Allocation Across Tasks
Wendelin Schnedler
Discussion Paper No. 05-36
GERMAN ECONOMIC ASSOCIATION OF BUSINESS
ADMINISTRATION – GEABA
Performance Measure Congruence
and Efficient Allocation Across Tasks
Wendelin Schnedler∗
May 27, 2005
Abstract
Performance measures may be aligned with the performance to be measured
(performance measure congruence) and agents can agree with the principal on the relative importance of different tasks (goal congruence). Here,
we show that performance measure congruence is equivalent to goal congruence. This means that the details of how the performance measure is
used are irrelevant for the relative assessment of tasks. Moreover, we find
that goal congruence ensures that the agent efficiently allocates his activity
across tasks. We thus prove the long standing conjecture that congruent performance measures lead to a desirable allocation of activity between tasks.
Keywords: incentive scheme, multitasking, goal congruence
JEL-Codes: M52, D82, M41
Introduction
Incentive schemes are designed to overcome the externality problem of somebody
who carries out an activity (a worker, employee, manager; shortly called agent, he)
∗ University
of Heidelberg, Department of Economics, Grabengasse 14, 69117 Heidelberg
([email protected]). The author wishes to thank Jörg Oechssler and Dirk
Sliwka for helpful discussions. The paper also profited from the contributions of participants at the
Kolloquium für Personalökonomik in Konstanz, and seminars in Heidelberg, Lyon, and Cologne.
All errors remain my own.
1
which affects somebody else (a firm, employer, share holder; shortly called principal, she). Depending on the details of the incentive scheme the implemented
activity may be more or less close to the efficient activity. Roughly speaking,
there may be two problems: First, the agent displays too little activity. Second,
the agent focuses on wrong aspects. While the first problem has been the centre
of attention in the early moral-hazard models (Holmström 1979, Shavell 1979),
the second problem is at heart of the multi-tasking literature starting with Holmström & Milgrom (1991). The deviation of the agent from the efficient solution
can be assessed by computing the agency costs. In a single task setting, we know
these costs to be entirely due to the first problem because there is no opportunity
to focus on the wrong aspect. In a multiple task setting, however, it is difficult to
identify how much of the inefficiency is caused by too little activity and how much
by concentrating on the wrong tasks. Accordingly, it is hard to assess whether the
agent’s input is optimally allocated across tasks.
Here, we suggest a way to disentangle overall activity level and allocation across
tasks by measuring activity in terms of the activity costs incurred by the agent.
Too little activity then means that the agent’s costs are below the efficient level,
while a wrong focus means that the agent spends too much on one task and too
little on another. This idea leads to a natural definition of an optimal allocation
across task: an allocation is optimal if the agent employs the available cost budget in an efficient manner. In other words, an optimal allocation is constrained
efficient, where the constraint is that the agent’s costs should not exceed a certain
level. Incentive schemes which achieve the constrained efficient allocation are optimal in the sense that they minimise agency costs amongst all incentive schemes
which implement the same activity level at the same price.
An important ingredient to any incentive scheme is some measure of the agent’s
performance. The effect of the agent’s activity at one task on this measure in relation to another task may be the same as the respective relative effect on the principal’s benefit. Then, the performance measure is called congruent. Congruent
performance measure have attracted wide-spread interest in the literature dealing
with the allocation of activity across various tasks (Feltham and Xie 1994, Baker
2000, Banker and Thevaranjan 2000, Feltham and Wu 2000, Datar, Kulp, and
Lambert 2001, Baker 2002). Presumably this interest is founded on the idea that
congruent performance measures lead to some type of optimal allocation across
tasks. There are, however, surprisingly few results which show congruent performance measures to be optimal. Datar et al. (2001) prove that congruent per2
formance measures minimise agency costs in a noise-free environment where the
agent can be made full residual claimant. However, often the agent cannot be
made residual claimant and the first-best activity level is not achieved. For example, the agent may be risk-averse and desire protection against changes in income
based on a noisy measure (Holmström 1979 and Shavell 1979). Also, he could
be credit constrained and incapable of “buying the shop” from the principal. Finally, investment considerations may require a particular ownership structure (see
e.g. Grossman and Hart 1986 or Hart and Moore 1999). Schnedler (2004) deals
with the problem that the first-best activity level is not achieved and finds that
congruent performance measures are still optimal, if uncertainty is measured in a
particular way.
The result of Schnedler (2004) is obtained under a specific set of assumptions
which Feltham & Xie (1994) introduced to the multi-tasking setting by Holmström & Milgrom (1991) and which are evoked by most of the literature on multitasking (see e.g. Feltham and Xie 1994, Baker 2000, Banker and Thevaranjan
2000, Feltham and Wu 2000, Datar, Kulp, and Lambert 2001, Baker 2002). For
example, the benefit to the principal, the wage function and the performance measure are assumed to be linear, while costs are quadratic. These assumptions are
conducive for the optimality of congruent performance measures. Linear benefit
and quadratic costs imply that the efficient allocation across tasks does not change
in the activity level. A linear wage based on a linear performance measure ensures
that the allocation of activity across tasks chosen by the agent is also independent
of the activity level. As congruent performance measure induce the efficient allocation at the first-best activity level, they also produces the efficent allocation at
any other level and are hence optimal. But what happens if the benefit is not linear
or costs are not quadratic so that the efficient allocation across tasks changes with
the activity level? What are the consequences of performance measures where the
marginal effect of behaviour and thus the induced allocation across tasks is different for different activity levels? Will congruent performance measures still lead
to an efficient allocation across tasks? There is a simple intuition which suggests
that congruent performance measures are not generally optimal from an allocative point of view. If a congruent performance measure is used and the agent is
not residual claimant, he only partially internalises the effect on the principal’s
benefit. Thus, he might neglect costly but beneficial tasks. Then, a performance
measure which overemphasises these tasks leads to a better allocation than a congruent measure. In other words, it seems relevant how the observed performance
translates into payments and ultimately into gains for the agent. The effect on the
3
gains of the agent’s behaviour on one task relative to another may be the same as
the relative effect on the principal’s benefit. This idea of goal congruence appears
more relevant for the allocation of activity across tasks than performance measure
congruence. While goal congruence may be latent in many multiple task models, it is rarely mentioned explicitly (for an exception see Banker and Thevaranjan
2000) and has not been formalised hitherto.
Here, we examine the relationship between performance measure congruence,
goal congruence and constrained efficiency. We provide a general definition of
performance measure congruence. We show that this definition is in line with
various congruity concepts (Feltham and Xie 1994, Baker 2000 and 2002, Datar,
Kulp, and Lambert 2001). We formalise the verbal definition for goal congruence by Anthony & Govindarajan (1995) and show that performance measure
congruence is a necessary and sufficient condition for goal congruence. Finally,
we relate these two concepts which are mere descriptions of incentive schemes to
the results of the scheme: we prove that goal congruent incentive schemes lead
to a constrained efficient allocation and that this allocation can only be achieved
when the scheme exhibits some type of goal congruence. Overall, the result of an
incentive scheme is thus desirable from an allocative standpoint if and only if the
employed performance measure is congruent.
This result is remarkable in at least four ways. First, it also holds when the agent
does not fully internalise the principal’s benefit. Thus, the idea that partial internalisation might adversely affect goal congruence is wrong. Second, the result is
derived in a rather general setting. In particular, it does not rely on the specific
assumptions normally evoked in the literature on multi-tasking. Third, it is the
congruence of the performance measure and not the nature of the wage scheme
which is important for the efficiency of the allocation. Given a performance measure, the attention that the agent devotes to different aspects of the work is robust
to the details of how this measure translates into pay for the agent. Fourth, the result establishes a strong theoretical tie between performance measure congruence
and efficient allocation across tasks. This tie justifies the widespread interest in
congruent performance measures in the literature.
The rest of the paper is organised as follows. In the next section the model is introduced, constrained efficiency, goal congruence and performance measure congruence are defined, and we show that the definition of performance measure congruence agrees with various notions of congruity in the literature. In the third section,
4
we establish the main results linking performance measure congruence, goal congruence and efficient allocation across tasks. The fourth section concludes.
1
The model
In this section, we introduce a rather general multiple task hidden-action model.
There is an agent (he), who carries out n tasks a = (a1 , . . . , an ), which entail (quasi
convex) costs C(a) but produce a benefit B̃ to a principal (she); the choices of the
agent ai are real numbers. Besides the systematic effect of the agent’s activity on
the principal’s benefit which we model using a (strictly quasi-concave) function
B(a), the benefit of the principal is also influenced by factors beyond the control
of the agent. These factors are represented by teh random variable η. Overall,
the benefit is a continuously differentiable and increasing function of controllable
and non-controllable influences: B̃(B(a), η). The principal’s utility increases in
the benefit B̃ and decreases in any wage payment W to the agent: uP (B̃,W ). The
agents utility decreases in the costs and increases in the wage uA (C,W ). Both
functions are assumed to be continuously differentiable in both arguments.
In absence of any agreement between principal and agent, the choice of the agent
disregards the effect on the principal: aA ∈ AA := argmaxa uA (C(a), 0). Denote
the respective utility by uA := uA (C(aA ), 0). In order to find the pareto-efficient
activity choice, suppose that the activity a can be stipulated in a contract. For simplicity, we make the principal the residual claimant and let her reimburse any extra
costs, which are incurred by the agent. Thus, the agent receives a reimbursement
wage wc which is implicitly defined by uA (c, wc ) = uA . Then, the principal selects
an activity aFB and a reimbursement wage wFB which maximises her utility while
ensuring that the agent is not worse off than before:
max
Eη uP B̃(a), w
a,w
such that
uA (C(a), w) ≥ uA .
Because the utility of the principal is strictly quasi-convex and costs are quasiconcave, this maximisation problem has a unique solution.
In the first-best situation, which we have just described, the principal has full
power over the choice of a. What happens if this choice is limited to activities
which cost at most c? For example, the principal may have limited funds to spend
5
on the reimbursement wage. Then, the principal will allocate the cost “budget”
c efficiently over the n tasks. The resulting constrained efficient allocation ac
and reimbursement wage wc can be obtained from the following maximisation
programme:
max
Eη uP B̃(a), w
a,w
such that
and
uA (C(a), w) ≥ uA
C(a) ≤ c.
Again, strict quasi-concavity of benefit and quasi-convexity of costs ensure a
unique solution.
The starting point for contractual analysis is that a contract which stipulates the
efficient or constrained efficient activity is not feasible because this activity cannot be (perfectly) observed at court. Accordingly, the contract cannot be enforced.
Still, the situation can be potentially improved by using any available verifiable
information about the principal’s benefit, the agent’s behaviour or his costs.. We
suppose that this information is summarised in an overall performance measure P̃.
Of course, the information may be combined in different ways and thus result in
different performance measures. Here, we want to abstract from the origins of the
performance measure, take it as given and examine its properties. A performance
measure generally depends on factors, which are under the control of the agent
and which we summarise by a continuously differentiable function P(a). Again,
it is well possible that the performance measure also hinges on factors beyond the
agent’s control, which we represent by a random variable ε. Overall, the performance measure is the following differentiable real-valued function: P̃(P(a), ε),
which we standardise to be increasing in the systematic influence of the agent P.
By linking the agent’s
wage to the performance measure, we obtain an incentive
1
scheme W P̃(a) . Via this link it may be possible to make the agent aware of the
consequences of his doings. Thus, the choice of a suitable performance measure
and wage function may help to reduce or even eliminate the externality problem.
1 We
require W (·) to be continuously differentiable almost everywhere (with respect to the
probability measure of ε). This assumption is not particularly restrictive and it is essential to be
able to compute the marginal effect of the agent’s behaviour on his gains from the scheme. In
other words, without this assumption the agent himself would have problems to assess the gains
from his behaviour.
6
1.1
Incentive schemes with consequences
The usefulness of incentive schemes is limited for at least two reasons. First, it
may be cheaper to stick with the initial activity of the agent aA (for an example
see the home contractor model in Holmström and Migrom 1991). Second, it may
be favourable that the agent “sells” the
the right to determine a. The
Pprincipal
P
principal then picks a ∈ argmaxa Eη u B̃(a), 0 and the agent will ask for a
compensation which covers at least his costs C(aP ). This puts an upper bound
on the costs of an activity which is implemented by an incentive scheme. Taken
together, incentive schemes are only used if they lead to some behaviour different
from aA which does not cost more than C(aP ).
Definition 1 (Normal incentive schemes). An incentive scheme is called normal
if it implements a behaviour aIC which the agent would not choose otherwise and
where the costs to the agent are not too high:
aIC 6∈ AA and aIC ∈ a|C(a) < C(aP ) .
An incentive scheme can only influence the agent’s behaviour if his gains change
with his activity. We thus expect from a normal incentive scheme that the agent is
able to affect his gains from the scheme.
Proposition 1. For a normal incentive scheme, there is a task i where the agent’s
choice ai affects his gains at the induced activity aIC :
h
i
d
Eε uA (W (P̃(P(a), ε)), c) 6= 0 for a = aIC .
dai
Proof. Since aIC is implemented by the incentive scheme, it must be an optimal
choice for the agent and hence fulfil the necessary (first-order) conditions.
h
i
d
A
0 =
Eε u (W (P̃(P(a), ε)),C(a))
dai
h
i
d
d A
=
Eε uA (W (P̃(P(a), ε)), c) +
u (w,C(a)), for some w and c.
dai
dai
∂
∂
A
∂C(a) u (w,C(a)) ∂ai C(a). Because the incentive scheme is normal, there is at least one task i such that ∂a∂ i C(a)|a=aIC 6= 0. In
∂
addition, the utility of the agent drops in the costs, so ∂C(a)
uA (w,C(a)) < 0. But
A
this implies that Eε u (W (P̃(P(a), ε)), c) 6= 0 for a = aIC .
Computing the second term, we get:
7
Proposition 1 assures us that there is at least one task i, which has an effect on the
gains of the agent. This will later enable us to express effects in relation to this
task. For convenience and without loss of generality, we assume this task to be
the first task: i = 1.
1.2
Constrained efficient schemes and agency costs
Any incentive scheme W (P̃) can be decomposed into a component which is constant and a component which is variable in the performance measure: W (P̃(a)) =
w0 + w1 (P̃(a)), where w1 (0) = 0. The base wage w0 only affects how the gains
from the interaction between principal and agent are distributed but has no effect
on the agents activity choice. It is convenient to choose w0 so that either the principal’s or the agent’s utility remains constant. As customary, we suppose that the
agent’s utility is constant. Then, the value of an incentive scheme is entirely reflected in the utility of the principal because she is the only one who benefits from
the scheme. Next, we want to eliminate the noise related to the wage scheme by
using the certainty equivalent w instead of the random wage2 :
h i
h i
Eηε uP B̃ B(aIC ), η ,W (P̃ P(aIC ), ε ) = Eη uP B̃ B(aIC ), η , w ,
where aIC denotes the activity implemented by the scheme. By comparing this
utility with the utility when it is possible to stipulate the activity in a contract, we
obtain the loss from operating the incentive scheme. These agency costs amount
to:
h i
Eη uP B̃ B(aFB ), η , wFB − Eη uP B̃ B(aIC ), η , w .
This representation of agency costs does not reveal whether they were caused by
a wrong focus on tasks or an insufficient activity level. What is the activity level
induced by the scheme? The implementation of activity aIC leads to direct costs
c = C(aIC ) for the agent. These costs can be regarded as an indicator of the agent’s
activity level. The respective reimbursement wage to the agent for the desired
activity level is wc . However, the perceived costs of implementing the activity
level (in terms of the certainty equivalent w) may exceed the reimbursement wage;
the difference w − wc is a premium which the principal has to pay because she
cannot stipulate the activity level c directly in a contract. With this notation and
2 The
certainty equivalent w exists because we assumed the utility of the principal to be continuously differentiable in money transfers
8
terminology in place, we can now decompose agency costs into three meaningful
parts:
Eη uP B̃ B(aFB ), η , wFB − Eη uP B̃ (B(ac ), η) , wc
{z
}
|
costs of operating at c
+ Eη uP B̃ (B(ac ), η) , wc − Eη uP B̃ (B(ac ), η) , w
{z
}
|
costs of implementing c
h i
+ Eη uP B̃ (B(ac ), η) , w − Eη uP B̃ B(aIC ), η , w .
{z
}
|
costs of allocating c
The first term describes any losses which result from the fact that the activity level
c is different (usually below) the first-best level: the efficient allocation under this
constraint is ac and the respective reimbursement wage wc . The next loss term
takes into account the premium required to purchase c. The final term then reflects
the loss because activity is wrongly allocated across tasks: the agent chooses aIC
instead of ac . As we are particularly interested in this last allocative aspect, we
define an “ideal” incentive scheme from an allocative point of view.
Definition 2 (Constrained Efficient Scheme). An incentive scheme W (P̃(·)) is
called constrained efficient at c if and only if the induced activity aIC is constrained efficient, formally
aIC = ac for c = C(aIC ).
If an incentive scheme is constrained efficient at all a, it is called constrained
efficient.
The following proposition substantiates the claim that constrained efficient incentive schemes are “ideal” from an allocative point of view.
Proposition 2 (Optimality of constrained efficient schemes). Given all schemes
which purchase activity level c at price w, it is optimal to use a scheme which is
constrained efficient (at c).
Proof. If w and c are fixed, the costs of operating at c and implementing c are
likewise fixed. The only costs which are still influenced by the incentive scheme
are the costs of allocating c. These costs are minimised for the allocation ac , which
is implemented if and only if the incentive scheme is constrained efficient.
This proposition asserts that constrained efficient incentive schemes minimise
agency costs if we abstract from the costs of obtaining activity level c and concentrate on the allocation of activity across tasks.
9
1.3
Goal congruence
According to Anthony & Govindarajan (1995), goal congruence reflects the extent
to which the relative importance of alternative goals to the agent is similar to the
principal. For the context of the multiple task model, we interpret the goals to
mean the expected gains arising to principal and agent from a particular task i. In
order to identify the gains, we fix the costs to the principal (by holding the wage
w constant in the utility of the principal) and the agent (by keeping labour costs in
his utility at the level c). The relative importance of various activities can then be
represented by the marginal effect of the activity on task i on the expected gain in
relation to the activity on task one.
Definition 3 (Goal congruence). A normal incentive scheme is called goal congruent if and only if the relative marginal effect on the expected gains of principal
and agent are identical:
P
A
d
d
dai Eη u B̃ (B(a), η) , w
dai Eε u c,W (P̃(P(a), ε))
= d
∂
A c,W (P̃(P(a), ε)),
P B̃ (B(a), η) , w
E
u
E
u
ε
η
da
∂a
1
1
for all tasks i, all activities a and all costs c and wages w.
Normality ensures that there is at least one task, which we can use as a “numeraire” because the agent affects his gains and those of the principal with this
task. If there is another task j which likewise affects both, goal congruence implies that the effect of an arbitrary task i relative to j on the gains of the agent is
the same as on the benefits of the principal. Formally, this can be seen by dividing
the right-hand and left-hand side of the equation for task i by those of task j. The
definition of goal congruence can be interpreted to mean that the marginal rate of
substitution induced by the incentive scheme is identical to the marginal rate of
substitution in the production of benefit.
1.4
Performance measure congruence
Neither constrained efficiency nor goal congruence has received formal treatment
in the literature on effort allocation across tasks. This literature is chiefly concerned with the relationship between the performance measure P̃ and the quantity
to be measured, the benefit B̃. There are various approaches to describe this relationship formally (Feltham and Xie 1994, Baker 2000, Datar, Kulp, and Lambert
10
2001, Baker 2002). However, these concepts are all set in the analytically convenient but specific setting of a linear benefit function, linear performance measures
and linear performance wage contracts and are hence of little use for our general
setting. We thus develop our own more general definition. As similar relative
effects on performance measure and benefit seem to be important in all these concepts, we define a performance measure to be congruent if these relative effects
are identical to those on the benefit.
Definition 4 (Congruent Performance Measure). A performance measure is called
congruent (with the benefit) for activity a, if and only if the marginal effects of ai
relative to a1 on the benefit are identical to the respective relative marginal effects
on the performance measure:
d P̃(P(a),ε)
dai
d P̃(P(a),ε)
da1
=
d B̃(B(a),η)
dai
d B̃(B(a),η)
da1
for all tasks i. If a performance measure is congruent for all activities a, it is
called congruent (with the benefit).
Again, the first task is used as a numeraire. Like before for goal congruence, we
can deduce from the definition that all well-defined relative marginal effects are
identical:
d P̃(P(a),ε)
dai
d P̃(P(a),ε)
da j
=
d B̃(B(a),η)
dai
d B̃(B(a),η)
da j
A measure is thus congruent if the marginal rates of substitution of performance
measure and benefit production are identical. The most generic congruent performance measure is the benefit itself: P̃ = B̃.
The definition of performance measure congruence was set in terms of the measure and benefit, while the agent can only influence the controlled performance
and controlled benefit. Now, we want to define a congruence concept which deals
with the systematic effect on performance measures.
Definition 5. The systematic component of a performance measure P(a) is congruent with the systematic component of the benefit B(a) at a if and only if the
marginal effects of ai relative to a1 on these components are identical:
∂P(a)
∂ai
∂P(a)
∂a1
=
11
∂B(a)
∂ai
.
∂B(a)
∂a1
for all tasks i. If the systmatic components are congruent at all a, they are called
congruent.
Fortunately, it does not matter whether the systematic component of performance
measure and benefit are congruent or the measure and the benefit itself because
the two definitions are equivalent.
Proposition 3. A performance measure is congruent with the benefit (at a), if
and only if the systematic components of performance measure and benefit are
congruent (at a); P(a) is congruent with B(a) (at a) if and only if P̃(a) is congruent
with B̃(a) (at a).
Proof. By the definition of P(a) and ε, we get:
∂P̃(P(a),ε) ∂ε
dε
dai ,
where
∂ε
dai
d P̃(P(a),ε)
dai
=
∂P̃(P(a),ε) ∂P(a)
dai
dP(a)
+
= 0 by definition. Now, use this information to comd P̃(P(a),ε) d P̃(P(a),ε)
/ da1 = ∂P̃(P(a),ε)
/ ∂P̃(P(a),ε)
. Analdai
∂ai
∂a1
d B̃(B(a),η) d B̃(B(a),η)
∂B̃(B(a),η) ∂B̃(B(a),η)
/ ∂a1 .
benefit that
/ da1
=
dai
∂ai
pute the relative marginal effect:
ogously, we derive for the
Taken together, we obtain the desired result for any a. It thus holds for a performance measure which is congruent at a as well as for measures which are
congruent, generally.
This proposition essentially means that factors beyond the control of the agent are
irrelevant for performance measure congruence. This irrelevance stems precisely
from the fact that the agent has no influence on these factors.
What happens if we re-scale a congruent performance measure, will it remain
congruent? The answer is given in the following proposition.
Proposition 4 (Inheritance of performance measure congruence). Any non-constant
continuously differentiable transformations t(·) of a congruent performance measure is congruent with the benefit.
Proof. Compute the derivative of the transformed performance measure with re
d t (P̃(P(a),ε))
spect to an arbitrary dimension i:
= t 0 P̃(P(a), ε) d P̃(P(a),ε)
. Thus,
dai
dai
the ratio of the derivative for the i-th and first component is:
d t (P̃(P(a),ε))
dai
d t (P̃(P(a),ε))
da1
t 0 P̃(P(a), ε)
= 0
t P̃(P(a), ε)
12
d P̃(P(a),ε)
dai
d P̃(P(a),ε)
da1
=
d B̃(B(a),η)
dai
,
d B̃(B(a),η)
da1
where the last equality results from the congruence of P. Hence, the transformation is congruent with the benefit.
This proposition allows us to identify congruent performance measures. For example all non-constant differentiable transformation of the benefit t(B̃) are congruent performance measures. If benefit function and performance measure are
linear in activities, the set of congruent measures takes a particularly simple form:
performance measure are congruent if their coefficient vector b = (b1 , . . . , bn ) has
the same direction as the coefficient vector of the benefit function β = (β1 , . . . , βn )
(see Proposition 1 in Schnedler 2004). In other words, a performance measure
is congruent if and only if its coefficient vector is a multiple of the benefit coefficient vector: b = γβ, with γ 6= 0.3 This linear setting has been examined in
various articles on multitasking (Feltham and Xie 1994, Baker 2000, Banker and
Thevaranjan 2000, Feltham and Wu 2000, Datar, Kulp, and Lambert 2001, Baker
2002). Some of these articles define concepts to describe the “proximity” of performance measures and benefit. We can thus check whether the definition of congruent performance measures yields performance measures which are considered
“close” to the benefit in the literature. Feltham & Xie (1994) proposed to use the
discongruity,
n−1 n
2
δFX := ∑ ∑ β j bk − βk b j .
j=1 k= j+1
If we apply the incongruity concept of Datar et al. (2001) to a single performance
measure, we get:
n
δDKL := ∑ (βi − γbi )2 .
i=1
Baker (2002) suggests to measure congruence using the cosine of the angle α
between the two coefficient vectors:
δB := cos(α).
The following proposition links all these concepts to the definition of performance
measure congruence given here.
Proposition 5. Congruent performance measures minimise the discongruity of
Feltham & Xie (1994), δFX , and the incongruity of Datar et al. (2001), δDKL .
3 Note,
that the geometrical term “congruence” is slightly abused because geometrical congruence requires γ = 1.
13
Congruent performance measures also maximise the congruence of Baker (2002),
δB .
Proof. By definition, δFX and δDKL have to be non-negative. If a congruent measure b = γβ is plugged in, both attain their minimum of zero. The maximum of δB
is one. The angle between β and a congruent b = γβ is zero, the respective cosine
is one and the maximum is attained.
Proposition 5 shows that our general definition of performance measure congruence is in accordance with various ideas in the more specific linear setting (this
also includes the idea of decomposing the coefficient vector into a congruent and
incongruent component by Banker and Thevaranjan 2000). While the given definition of performance measure congruence may be appealing and related to other
congruence concepts, it is similarly ad-hoc as these concepts. In order to make the
definition relevant, we will have to link it to constrained efficiency or some other
form of optimality.
2
Congruence concepts and constrained efficiency
We have now defined three concepts: constrained efficiency gives us a benchmark
for a desirable allocation across tasks; goal-congruence is a property of incentive
schemes which relates the gains of principal and agent; finally, performance measure congruence links the performance measure to the principal’s benefit, which
should be internalised by the agent. How are these three concepts related?
Before we deal with the relationship between congruent performance measures
and constrained efficient allocations, we want to link the two congruence concepts.
Theorem 1. A normal incentive scheme based on a performance measure P̃ is
goal congruent (at a) if and only if this performance measure is congruent (at a).
Proof. Let us start with the effect of the agent on his expected gains with respect
to some activity i. It can be shown that these gains are a multiple of the marginal
effect on the performance measure: k ∂a∂ i P(a), where k is independent of i (see
Lemma 1 in the appendix). If we now set this effect in relation to the effect on the
first task, we get:
A
∂P(a)
d
dai Eε u (c),W (P̃(P(a), ε))
∂ai
= ∂P(a)
(1)
d
A
da1 Eε u (c),W (P̃(P(a), ε))
∂a1
14
Hence, the incentive scheme is goal congruent if and only if
∂P(a)
∂ai
∂P(a)
∂a1
=
∂B(a)
∂ai
,
∂B(a)
∂a1
which by Proposition 3 holds if and only if the performance measure is congruent
with the benefit. The result was derived for any a, so it is true for performance
measure and goal congruence at a as well as for performance measure and goal
congruence, generally.
The intuition for this result is simple. The shape of the wage function influences
the gains of the agent from different tasks in the same way. If these gains are expressed in relation to each other, the effect of the wage function washes out. The
agent perceives the relative importance of the tasks independent from the wage
and only based on the performance measure. Performance measure congruence
can thus ensure that principal and agent assess gains similarly.
Theorem 1 is important for three reasons. First, it formally links two prevailing
ideas in the literature that of congruent “goals” and of congruent measures. Second, this link has a quite general character as it is remarkably unaffected by the
shape of the wage function. In other words, the agent agrees with the principal
on the relative importance of tasks irrespective of how the performance measure
translates into his wage. Third, the fact that the agent perceives the relative gains
independent from the wage function suggests that he may also allocate activity
independent from the wage.
While goal congruence describes the concurrence of the relative assessment of
tasks by principal and agent, such a concurrence is only meaningful as an instrument to achieve a desired allocation across tasks. The next theorem asserts that
goal congruence achieves precisely this.
Theorem 2. Take a normal incentive scheme. If this scheme is goal congruent, it
is also constrained efficient. If it is constrained efficient, it is goal-congruent at
the implemented activity.
Proof. Let aIC be an activity which the agent takes when faced with the incentive scheme. Then, the the available budget is c = C(aIC ). For the constrained
efficient allocation, we need to maximise Eη uP (B̃(B(a), η), w) under the constraint C(a) ≤ c. Since the utility of the principal uP (B̃(B(a), η), w) is a strict
15
quasi-concave function Eη uP (B̃(B(a), η), w) is also strict quasi-concave. Because C(a) is quasi-convex, the set {a|C(a) ≤ c} is convex. Overall, we are
maximising a strict quasi-concave function on a convex set. Consequently, the
(constrained) efficient allocation ac is uniquely
determinedby the first-order con
ditions of the Lagrangian L(a, λ) := Eη uP (B̃(B(a), η), w) − λ(C(a) − c). These
first-order conditions are:
∂
d
Eη uP (B̃(B(ac ), η), w) = λ C(ac ) for
dai
∂ai
c
C(a ) = c
i = 1, . . . , n
(2)
As the incentive scheme is normal, c < C(aP ). Consequently, ac 6= aP and the
derivative of the principal’s utility is different from zero with respect to some
component dad i Eη uP (B̃(B(ac ), η), w) 6= 0 and neither is the right-hand side of the
first equation in the system. This enables us to rewrite the first-order conditions:
∂
d
Eη uP (B̃(B(ac ), η), w) = λ
C(ac )
da1
∂a1
P
∂
d
c
c ), η), w)
E
u
(
B̃(B(a
η
dai
∂ai C(a )
=
for i = 2, . . . , n
d
∂
P (B̃(B(ac ), η), w)
c)
E
u
C(a
η
da1
∂a
(3)
1
c
C(a ) = c
The first equation in this system determines the Lagrange-multiplier λ. Since the
right-hand side is positive the mulitplier is non-zero and the cost constraint binds.
Summarising, an allocation ac is the constrained efficient if and only if
)
( d
P
∂
C(a)
E
u
(
B̃(B(a),
η),
w)
η
da
∂a
= ∂i
i = 2, . . . , n ∧ C(a) = c .
ac ∈ Ac := a d i P (B̃(B(a), η), w)
E
u
C(a)
da1 η
∂a1
Let us turn to the agent’s problem. As the space of activities is unbounded, the
allocation aIC chosen by the agent must fulfil the first-order conditions to be a
maximiser. We now derive these first-order conditions. The effect of the agent on
his utility unfolds via the costs and the wage:
h
i
h
i
d
d
Eε uA (C(a),W (P̃(P(a), ε))) =
Eε uA (C(a), w)
dai
dai
h
i
d
A
+
Eε u (c,W (P̃(P(a), ε)) ,
dai
16
where w and c are suitably chosen. The first term is:
h
i
d
d A
∂C(a)
∂
A
A
Eε u (C(a), w) = Eε
u (C(a), w) = Eε
u (C(a), w)
dai
dai
∂C(a)
∂ai
Using this and Lemma 1 from the appendix, we get the following first order conditions:
h
i
∂C(a)
d
∂
A
A
Eε u (W (P̃(P(a), ε)), c) = −Eε
u (w̃,C(a)) ·
dai
∂C(a)
∂ai
for i = 1, . . . , n. Because the incentive scheme is normal and because the agent’s
utility falls in costs, the left-hand side as well as the right-hand side is different
from zero at least for one task. We can then use this task as the “numeraire” and
express the conditions i = 2, . . . n by the conditions
A
∂C(a)
d
dai Eε u (c,W (P̃(P(a), ε))
∂ai
= ∂C(a)
.
(4)
d
A
da1 Eε u (c,W (P̃(P(a), ε)))
∂a1
Because c := C(aIC ), it holds that aIC ∈ {a|C(a) = c}. So, overall:


∂C(a)
 d Eε uA (c,W (P̃(P(a), ε)))

dai
∂ai
aIC ∈ AIC :=
=
i
=
2,
.
.
.
,
n
∧
C(a)
=
c
.
∂C(a)
 dad Eε uA (c,W (P̃(P(a), ε)))

1
∂a1
Using goal congruence, we get that AIC = Ac and any maximiser aIC is also constrained efficient. Conversely, given that there is an induced allocation aIC ∈ AIC ,
which is also constrained efficient, aIC ∈ Ac , it must hold that aIC ∈ AIC ∩ Ac . But
this implies goal congruence at aIC .
The intuition why congruent performance measures are required for constrained
efficiency can be seen in Figure 1: amongst all allocations with the same costs, the
constrained efficient allocation is identified by the point where the marginal rate
of substitution in costs and benefit are identical. A non-congruent performance
measure has a marginal rate of substitution different from the benefit and hence
only congruent performance measures can lead to the constrained efficient allocation.
This result finally links the speculative “ad-hoc” ideas about measure and goal
congruence to an actual problem faced by a mechanism designer: minimising
17
activity task j
benefit isoquant=isoquant of congruent measure
measure
congruent
at a ic a jc
a jc
measure not
congruent
at a ic a jc
cost isoquant
a ic
activity task i
Figure 1: Necessity of performance measure congruence
The activity of the agent can only be constrained efficient if the marginal rate of subIC
stitution of the benefit is identical with that of the performance measure at (aIC
i , a j ).
agency costs. Once an incentive scheme is goal congruent, the resulting allocation of activity across tasks is constrained efficient. Note that the scheme is not
only constrained efficient for a particular activity level c but for all levels. So, it
does not matter what activity level the principal wants to implement, with a goal
congruent scheme, she always gets constrained efficiency.
By combining this result with Theorem 1, we immediately get the following corollary.
Corollary 1 (Performance measure congruence and constrained efficiency). Take
a normal incentive scheme. This scheme is constrained efficient if the performance
measure is congruent. If the scheme is constrained efficient, the performance
measure is congruent at the implemented activity.
Proof. Iterative application of Theorems 1 and 2.
This corollary justifies why congruent performance measures are attractive from
an allocative point of view. They ensure that no higher benefit can be obtained
by re-allocating activity across tasks. So, if a non-congruent measure is found to
be better (for an example see Schnedler 2004), this superiority cannot be due to
allocative properties of the measure but must be due to other measure properties
– for example the measure may allow to impose less uncertainty on the agent.
18
It is interesting to note that the wage function plays a surprisingly unimportant
role for the results. This means that irrespective of the details on how a congruent performance measure is used in an incentive scheme, constrained efficiency is
guaranteed. On the other hand, the design of the wage function of course influences on how much (or how little) surplus is produced via the overall activity level.
Earlier, we presented the intuition that congruent performance measures may possibly only lead to good allocation for specific activity levels (for example the firstbest activity level c = C(aFB ). Contrary to this intuition, Corollary 1 asserts that
using a congruent performance measure achieves constrained efficiency irrespective of the activity level that the principal aspires.
3
Conclusion
A naive approach to incentive scheme design may start with a performance measure that reflects the relative importance of different activities to the principal. In
other words, the designer tries to create a performance measure which is congruent with the benefit. For example, revenue may be used as a performance measure
if the principal wants to maximise revenue. It is not always that obvious whether a
performance measure is congruent and we provided a result which allows to identify congruent measures. However, this result is only a starting point and future
research should address the question how congruent performance measures can be
constructed. Here, we focused on the question: why should they be constructed?
Knowing that it is impossible or very costly to obtain the desired level of the activity from the agent, this activity should at least be focused on the right tasks. Put
differently, incentive schemes should lead to constrained efficiency. Intuitively,
measures which are congruent with the benefit are plausible candidates to induce
constrained efficiency. On the other hand, it needs to be taken into account that
performance measure variation rarely maps directly into variation of the agent’s
wage. The ensuing partial internalisation may prevent constrained efficiency.
Here, we have shown that the first intuition is nevertheless founded: constrained
efficiency can be achieved if and only if the performance measure is congruent.
This result holds in a general framework and even if the agent does not fully internalise the consequences of his actions on the principal. Moreover, the result
is valid independently from the details of how wage is tied to the performance
19
measure.
This is good news for practitioners because it means that the naive approach of
using a congruent performance measure is sensible – irrespective of the details of
how wage relates to performance. From a purely allocative standpoint, there is no
loss from using –for example– a linear scheme. This may explain the pervasive
nature of such schemes in reality. The result may also elucidate why shares are
handed to workers although the effect in terms of increased activity is minute. The
shares are not intended to stimulate activity but to ensure that workers focus on
the right tasks.
References
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Grossman, S. J. & Hart, O. D. (1986). The costs and benefits of ownership: A
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Lemma 1. The gains from task i to the agents can be written as a multiple of of
the marginal effect on the systematic component of the performance measure:
h
i
∂P(a)
d
Eε uA (W (P̃(P(a), ε)), c) = K ·
,
dai
∂ai
with K being independent of i.
Proof. We decompose
the state space of ε into two sets: the set of ε for which
U W (P̃(P(a), ε)), c is not differentiable and the set for which it is differentiable.
Because W is almost everywhere differentiable any points where the function is
not differentiable have no probability mass and can be ignored. The second set
can be subdivided into sets of those values for which the function is constant and
˙ k ACk and sets for which the function is non-constant ∪
˙ k ANC
takes the values wk , ∪
k .
21
Summarising, we can express the expected utility given the agent has selected a
as:
Z
Eε U(W (P̃(P(a), ε)), c) =
U(W (P̃(P(a), ε)), c) f (ε)dε
˙ Ck
∪A
Z
+
U(W (P̃(P(a), ε)), c) f (ε)dε
˙ NC
∪A
k
(5)
Z
=
U(W (P̃(P(a), ε)), c) f (ε)dε
˙ Ck
∪A
+ ∑ U(wk , c) · Prob ε ∈ W (P̃(P(a), ε)) = wk .
k
Next, we take the derivative of this expected utility with respect to the activity on
task i, ai , and get (recall that we ignore any effects of the activity on costs):
d
Eε U(W (P̃(P(a), ε)), c)
dai
Z
∂W ∂P̃ ∂P
∂
U(W (P̃(P(a), ε)), c) ·
·
·
f (ε)dε
=
∂W
˙ NC
∂P̃ ∂P ∂ai
∪A
k
∂P(a)
∂
+ ∑ U(wk , c) ·
Prob ε ∈ W (P̃(P(a), ε)) = wk ·
∂P(a)
∂ai
k
Z
∂
∂W ∂P̃
=
U(W (P̃(P(a), ε)), c) ·
·
f (ε)dε
∂W
˙ NC
∂P̃ ∂P
∪A
k
#
∂P(a)
∂
+ ∑ U(wk , c) ·
Prob ε ∈ W (P̃(P(a), ε)) = wk ·
, (6)
∂P(a)
∂ai
k
where the last equality is obtained by factoring out
22
∂P(a)
∂ai .