It has been argued in the multi-task agency literature that effort distortion can be mitigated by applying several performance measures in incentive contracts. This paper analyzes the efficient aggregation of multiple performance measures aimed at motivating non-distorted effort. It demonstrates that non-distorted effort can be induced by combining a sufficient quantity of informative performance measures. However, this is only optimal if the required aggregation concurrently maximizes the precision of the agent`s performance evaluation. This paper further illustrates how the optimal performance evaluation is affected by the ability of individual agents to perform relevant tasks.Multi-task agencies, effort distortion, performance measurement, incentives

Veikko Thiele

English

Research Papers in Economics

     
 
 
  
  
Volume 29, Issue 4 
  
Aggregating Performance Measures in Multi-Task Agencies 
  
 
 
Veikko Thiele  
Queen`s University 
Abstract 
It has been argued in the multi-task agency literature that effort distortion can be mitigated by applying several 
performance measures in incentive contracts. This paper analyzes the efficient aggregation of multiple performance 
measures aimed at motivating non-distorted effort. It demonstrates that non-distorted effort can be induced by 
combining a sufficient quantity of informative performance measures. However, this is only optimal if the required 
aggregation concurrently maximizes the precision of the agent`s performance evaluation. This paper further illustrates 
how the optimal performance evaluation is affected by the ability of individual agents to perform relevant tasks.
I like to thank Gerald Feltham, Sandra Chamberlain, Chia-Chun Hsieh, Anja Schöttner, Louis-Philippe Sirois, and Chee-Wee Tan for valuable 
comments and suggestions. 
Citation: Veikko Thiele, (2009) ''Aggregating Performance Measures in Multi-Task Agencies'', Economics Bulletin, Vol. 29 no.4 pp. 2693-
2699. 
Submitted: Aug 27 2008.   Published: October 28, 2009. 
 
     1. Introduction
In many employment relationships, ﬁrms utilize objective performance measures to provide their em-
ployees with incentives. Since effort is usually multidimensional, ﬁrms must not only induce a sufﬁcient
effort intensity, they must also motivate an efﬁcient effort allocation across tasks. However, if avail-
able performance measures do not reﬂect employees’ true contributions to ﬁrm value, the inclusion in
incentive contracts will motivate employees to choose inefﬁcient effort allocations across relevant tasks
(Feltham and Xie, 1994).
It has been argued in previous multi-task agency literature that effort distortion can be mitigated by
applying several performance measures in incentive contracts. In particular, Feltham and Xie (1994),
Datar, Kulp, and Lambert (2001), and Thiele (2007) have demonstrated that utilizing multiple measures
for evaluating an agent’s performance can improve the efﬁciency of his effort allocation. Nevertheless,
these papers stopped short of identifying the requirements of information systems for inducing non-
distorted effort. To close the existing knowledge gap, this paper analyzes the efﬁcient combination of
multiple performance measures aimed at motivating non-distorted effort.
This paper provides two important implications. First, inducing non-distorted effort necessitates
access to, at the very least, the same quantity of informative performance measures as the number of
tasks the agent has to perform. Motivating non-distorted effort however, is only optimal if the required
combination of performance measures concurrently maximizes the precision of the agent’s performance
evaluation. Second, the optimal aggregation of multiple performance measures depends on individual
agent’s ability to perform relevant tasks. Accordingly, mitigating potential effort distortion provokes
different performance evaluations for heterogenous agents even if their jobs are identical.
This paper proceeds as follows. In section 2, I give an overview of the model and derive the ﬁrst-best
effort allocation as a benchmark in section 3. The required aggregation of multiple performance measures
to induce non-distorted effort is derived in section 4 and analyzed in section 5. Section 6 concludes.
2. The Model
Consider a single-period agency relationship between a risk-neutral principal and a risk-averse agent.
The agent is employed to perform n  2 tasks which cannot be split and allocated to different agents.
Thus, the agent is in charge of implementing an effort vector e = (e1;:::;en)T, e 2 Rn+, where ei
denotes the agent’s non-veriﬁable effort allocated to task i.1 Implementing effort e imposes costs C(e) =
eT	e=2, where 	 is a symmetric and positive deﬁnite n  n matrix representing the agent’s marginal
effort costs. The agent’s preferences are represented by the negative exponential utility function
U(w;e) =  exp[ (w   C(e))]; (1)
where  denotes the measure of absolute risk-aversion and w his wage. His reservation utility is  U.
By implementing effort e, the agent contributes to the principal’s non-veriﬁable gross payoff V (e) =
Te, where  = (1;:::;n)T,  2 Rn+, characterizes the marginal effect of e on V (e). Since
V (e) is non-veriﬁable, it cannot be part of an explicit incentive contract. However, the principal re-
ceives an m-dimensional vector of veriﬁable and additively separable performance measures P(e) =
(P1(e);:::;Pm(e))T, P(e) 2 Rm. Let  = (!T
1;:::;!T
m)T denote the m  n matrix of the respective
performance measure sensitivities !i = (!i1;:::;!in)T 2 Rn+, i 2 f1;mg. Thus,
P(e) = e + "; (2)
1All vectors are column vectors where ‘T’ denotes the transpose.
1where " = ("1;:::;"m)T, " 2 Rm, is a normally distributed m-dimensional vector of random variables
with zero mean and covariance matrix . A performance measure Pi(e) is referred to be incongruent, if
there exists no constant  6= 0 satisfying  = !. Then, its exclusive application in an incentive contract
would motivate the agent to implement an inefﬁcient effort allocation across the relevant tasks (Feltham
and Xie, 1994; Baker, 2002).
In line with previous multi-task agency literature, I restrict my analysis to a linear compensation
scheme w:
w(e) =  + P(e); (3)
where  denotes the ﬁxed payment. Moreover,  = (1;:::;m)T,  2 Rm, is the vector of incentive
parameters representing the weight of each performance measure in the linear aggregation.2
3. The First-Best Effort Allocation
As a benchmark for the subsequent analysis, let us ﬁrst identify the ﬁrst-best (i.e. non-distorted)
effort allocation. Suppose the principal can contract over e. In this case, she would choose e aimed at
maximizing the difference between the gross payoff V (e) and costs C(e):
max
e (e) = 
Te  
1
2
e
T	e: (4)
Accordingly, the ﬁrst-best effort vector is characterized by
e
fb = 	
 1: (5)
For the remainder of this paper keep in mind that any implemented (second-best) effort vector e charac-
terizes a distorted effort allocation, if there exists no constant  6= 0 satisfying efb = e.
4. Aggregating Performance Measures
If the principal cannot directly contract over e, she faces an incentive problem for motivating an
appropriate effort intensity and effort allocation across the relevant tasks. Hence, the principal’s problem
is to design a contract (;
) that maximizes her expected proﬁt  = E[V (e)   w(e)] while ensuring
the agent’s participation. The optimal linear contract thus solves
max
;;e
  E[V (e)   w(e)] (6)
s.t.
e = argmax
~ e
E[U(w;~ e)] (7)
E[U(w;e)]   U; (8)
where (7) is the agent’s incentive, and (8) his participation constraint. Recall that w(e) is linear, U(w;e)
is exponential, and the error term " is normally distributed. Consequently, maximizing E[U(w;e)] is
analogous to maximizing the agent’s certainty equivalent
CE(e) =  + 
Te  
1
2
e
T	e  

2

T; (9)
2As shown by Banker and Datar (1989), a linear aggregation of performance measures is optimal whenever the noise term
is normally distributed.
2where 
T=2 describes the agent’s risk premium. To maximize his expected utility, the agent chooses
e = 	
 1
T. Apparently, if the principal receives at least two performance measures, she can inﬂu-
ence the relative effort allocation by adjusting the weights i, i = 1;:::;m, in the agent’s performance
evaluation.
Cost minimization requires setting  such that (8) binds. Solving CE(e) =  U for  and substi-
tuting this expression with e = 	
 1
T in the principal’s objective function yield an unconstrained
maximization problem:
max

  
T	
 1
T  
1
2

T	
 1
T  

2

T    U: (10)
The ﬁrst-order condition with respect to  leads to

 =

	
 1
T + 
 1
	
 1; (11)
where

	
 1
T + 
 1
is the inverse of an m  m matrix.
We can infer from 
 that the objective of aggregating performance measures is to balance three
effects: (i) the effort distortion characterized by 	
 1, (ii) the measure-cost efﬁciency described
by 	
 1
T; and (iii), the precision of the aggregated performance evaluation with the agent’s risk
aversion, characterized by . Since these three effects are determined by 	 and , we can conclude that
the optimal aggregation of performance measures is tailored to the agent’s speciﬁc characteristics. Thus,
the employment of heterogenous agents calls for different performance evaluations, inducing diverse
effort allocations across the relevant tasks.
5. Inducing the Efﬁcient Effort Allocation
As noted earlier, the principal can inﬂuence the agent’s effort allocation if she receives at least two
performance measures. The next proposition identiﬁes conditions which allow the principal to induce the
ﬁrst-best (i.e. non-distorted) effort allocation.
Proposition 1. If rank
T  n, the principal can aggregate the available performance measures to
induce e = efb, 0 <   1. However, this is only optimal, if and only if,
^  	
 1
T = ; ^  =
1   

: (12)
Proof See appendix.
The ﬁrst condition emphasizes that the principal needs access to an information system generating at
least the same quantity of performance measures as number of tasks the agent has to perform.3 Moreover,
their sensitivity vectors !i are required to be linearly independent, i.e. performance measures must differ
in their information content with respect to the implemented effort allocation. If these two requirements
are satisﬁed, the principal can combine the performance measures appropriately to induce the ﬁrst-best
effort allocation. However, as the second condition in Proposition 1 highlights, the aggregation of per-
formance measures with the purpose of motivating non-distorted effort is only optimal if the covariance
matrix  is a transformation of the measure-cost efﬁciency 	
 1
T. Intuitively, aggregating perfor-
mance measures to motivate non-distorted effort can only be optimal, if this concurrently maximizes the
3Note that this condition is sufﬁcient. For instance, the principal can also induce a ﬁrst-best effort allocation if one measure
is perfectly congruent.
3precision of the agent’s performance evaluation, and consequently, minimizes his risk premium. To see
this, consider the following example with two tasks and two performance measures:
 =

1
2

; 	 =

 1 0
0  2

;  =

!11 0
0 !22

;  =

2
1 0
0 2
2

:
In this example, performance measure P1(e1) captures only task 1, whereas task 2 is only measured by
P2(e2). For simplicity, assume that i =  i, i = 1;2, which implies efb = (1;1)T, i.e., the agent would
implement the same effort intensity for each task under ﬁrst-best. Using this example, condition (12)
from Proposition 1 simpliﬁes to
^ 
 
!2
11
 1 0
0
!2
22
 2
!
= 

2
1 0
0 2
2

: (13)
Let 2
1 > 2
2, i.e., performance measure P2(e2) is more precise than P1(e1). Then, for arbitrary ^  and
, motivating the ﬁrst-best effort allocation can only be optimal, if the less precise performance measure
P1(e1) is associated with a higher measure-cost ratio !2
11= 1 (i.e., !2
11= 1 > !2
22= 2). Clearly, to in-
duce the ﬁrst-best effort allocation, the principal can put a lower weight on the less precise performance
measure P1(e1) (i.e., 1 < 2), which in turn maximizes the precision of the aggregated performance
measure 
TP(e), and hence, curbs the risk imposed on the agent.
Finally observe that condition (12) (and for the above example, the simpliﬁed condition (13)) is tied
to the agent’s marginal effort costs parameterized by 	.4 Hence, depending on the characteristics of
the information system, inducing non-distorted effort can be optimal for a certain type of agent, but
inefﬁcient for other types. Consider again the above example with i =  i, i = 1;2. Clearly, for
arbitrary performance measure sensitivities !11 and !22, it can only be optimal to induce the ﬁrst-best
effort allocation, if the parameters  i, i = 1;2, of the agent’s marginal effort cost are such that the less
precise performance measure P1(e1) is associated with a higher measure-cost ratio !2
11= 1. Otherwise,
balancing effort incentives and insurance for the risk-averse agent requires the principal to combine both
performance measures differently, which in turn motivates the agent to implement distorted effort.
Put differently, for a given set of available performance measures satisfying rank
T  n (see Propo-
sition 1), personal characteristics of agents determine whether it is optimal for the principal to motivate
the efﬁcient (i.e., non-distorted) effort allocation by combining these measures appropriately. Moreover,
not only the respective informativeness of available performance measures, but also individual charac-
teristics of agents, dictate the relative importance of these measures for evaluating agents’ individual
contributions to ﬁrm value as basis for incentive payments.
The previous observations have two important implications. First, the principal has some latitude to
improve the efﬁciency of the induced effort allocation by employing ‘suitable’ agents for the relevant
jobs. The selection criteria, however, are not only determined by the potential contributions of tasks to
ﬁrm performance (captured by ), but also by the characteristics of the available information system
P(e). Second, instead of using standardized contracts, proﬁt maximization requires to tailor incentive
contracts to agents’ individual characteristics, even if their jobs are identical.
Even though it might be optimal to motivate the ﬁrst-best effort allocation for a certain type of agent,
it is not necessarily optimal to concurrently induce the ﬁrst-best effort intensity. To see this, recall that
non-distortion requires e = efb. Moreover, the agent implements the ﬁrst-best effort intensity only if
 = 1. This leads to the next Corollary to Proposition 1.
4In contrast, a change of the scalar  affects only the parameter ^  without violating (12).
4Corollary 1. Suppose that rank
T  n. Then, it is optimal to induce efb, if and only if  = 0 or
 = [0]ij, i;j = 1;:::;m.
Besides the conditions emphasized by Proposition 1, inducing the ﬁrst-best effort allocation and in-
tensity requires that either all performance measures are perfectly precise or the agent is risk-neutral.
For single-task agency relationships, it is well known that the latter criteria are sufﬁcient to achieve ﬁrst-
best if the agent is not ﬁnancially constrained. Multi-task principal-agent relationships however, impose
additional requirements on the information system with respect to the quantity and characteristics of con-
tractible performance measures. In particular, only if available measures can be combined such that the
agent’s performance evaluation reﬂects his true contribution to ﬁrm value, non-distorted effort can be
induced.
6. Conclusion
The application of performance measures in incentive contracts can motivate employees to implement
inefﬁcient effort allocations if their performance evaluations do not perfectly reﬂect their true contribu-
tions to ﬁrm value. This paper analyzes the aggregation of multiple performance measures as a means
of motivating a non-distorted effort allocation across relevant tasks. Two important observations are
noted. First, to induce non-distorted effort, the principal depends on a sufﬁcient quantity of informative
performance measures. However, motivating non-distorted effort is only optimal if the required aggrega-
tion of performance measures concurrently maximizes the precision of agents’ performance evaluations.
Second, the optimal aggregation of multiple performance measures is tied to individual agent’s ability
to carry out relevant tasks. Therefore, the intention to mitigate effort distortion can explain why the
performance of heterogeneous agents are evaluated differently even if their jobs are identical.
Appendix
Proof of Proposition 1.
The agent implements the ﬁrst-best effort allocation, if e = efb. Note that 0 <   1 since it cannot
be optimal to induce a higher effort intensity under second-best than under ﬁrst-best. Therefore,  needs
to solve 	
 1
T = 	
 1, which is equivalent to 
T = . If rank
T  n, there exists at least
one solution to this equation system. Particularly, h columns in 
T, n  h  m, must be linearly
independent.
Inducing the ﬁrst-best effort allocation is only optimal if e(
) = efb. This requires that 
T
 = ,
or equivalently, 
 = 


T 1
. Thus, e(
) = efb is equivalent to

	
 1
T + 
 1
	
 1 = 


T 1
; (14)
which can be transformed to
1   

	
 1
T = : (15)
2
5References
Baker, G. (2002) "Distortion and risk in optimal incentive contracts" Journal of Human Resources 37,
728-751.
Banker, R. D. and S. M. Datar (1989) "Sensitivity, precision, and linear aggregation of signals for perfor-
mance evaluation" Journal of Accounting Research 27, 21-39.
Datar, S., S. C. Kulp, and R. A. Lambert (2001) "Balancing performance measures" Journal of Account-
ing Research 39, 75-92.
Feltham, G. A. and J. Xie (1994) "Performance measure congruity and diversity in multi-task princi-
pal/agent relations" Accounting Review 69, 429-453.
Thiele, V. (2007) "Performance measurement in multi-task agencies" Research in Economics 61, 148-
163.
6

Aggregating Performance Measures in Multi-Task Agencies

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