This paper considers a principal-agent model in which an agent chooses the level of effort on two tasks that determine two separate outputs. The level of effort is private information to the agent. We examine the relationship between risk and incentives when the agent has a preference towards one of the two tasks. This is modelled by the assumption that the cost of effort is task specific. We show that when outputs are not perfectly measurable, even if the measurement errors have the same variance, the principal has to set different marginal incentives. This is because the agent tries to spread the risk, created by imperfect output measurement, by distributing more evenly his effort between the two tasks, thereby moving away from an efficient allocation of effort. If the two outputs are measured with different precision, it can be optimal for the principal to set higher marginal incentives on the output measured with less precision. Hence the standard result of the negative correlation between risk and incentives does not necessarily hold.uncertainty, risk spreading, efficiency, incentives

Marisa Ratto

English

Research Papers in Economics

 
     
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The Risk-Efficiency Trade-off With Task-
Specific  
 
Marisa Ratto 
 
November 2006 Updated from (March 2006) 
 
Working Paper No. 06/143 
 
 
 
ISSN 1473-625X 
 
  
 
CMPO Working Paper Series No. 06/143 
CMPO is jointly funded by the Leverhulme Trust and the ESRC 
 
 
The Risk-Efficiency Trade-off With Task-Specific  
Effort Costs * 
 
 
Marisa Ratto**
 
 
 
 
**Centre for Market and Public Organisation, University of Bristol 
 
 
 
November 2006 (Updated) 
 
 
 
 
Abstract 
This paper extends the standard approach of multitasking models by relaxing the assumption that effort 
on different activities are perfect substitutes. It is shown that, when the agent cares about how his total 
effort is allocated across different tasks, it can be optimal for the principal to set higher marginal 
incentives on the output measured with less precision. This result holds even if measurement errors  
are stochastically independent and activities are technologically independent. Hence the standard result 
of the negative correlation between risk and incentives does not necessarily hold. 
 
Keywords : uncertainty, risk spreading, efficiency, incentives. 
 
 
JEL Classification: D23, D81, M52 
 
 
Acknowledgements   
*I am grateful to Ian Jewitt, David de Meza, and Thibaud Vergé for very helpful discussions and their 
valuable comments.   
 
 
 
Address for Correspondence 
Department of Economics 
University of Bristol 
12 Priory  Road 
Bristol 
BS8 1TN 
marisa.ratto@virgin.net 
www.bris.ac.uk/Depts/CMPO/ 
 
 1 Introduction
A well-known and established result in the principal-agent literature is the trade-oﬀ be-
tween risk and incentives. In multi-tasking contexts, when outputs are measured with
diﬀerent precision, the prediction of the standard models is that the principal has to
weaken incentives to prevent the agent from diverting his of eﬀort away from the desired
allocation (Holmstr¨ om and Milgrom, 1991). In particular, if each outcome could be re-
warded in isolation, the principal would optimally set higher incentives on the output
that is a more accurate indicator of the agent’s underlying eﬀort. If, instead, one outcome
reﬂects multiple dimensions of the agent’s eﬀort, the prospect of the agent diverting his
eﬀort away from the less accurately measured tasks makes the principal weaken the overall
incentives. However, this theoretical prediction is not supported by empirical evidence,
which suggests a weak or even positive relationship between risks and incentives1. Recent
theoretical work has provided arguments to explain why incentives might be stronger in
more uncertain environments. Prendergast (2002) shows that delegation of responsibili-
ties and the use of output-related pay can be optimal and leads to a positive correlation
between uncertainty and incentives. This is because in a more uncertain environment the
principal may have little idea of what the right kinds of eﬀort are. She then optimally
delegates the choice of how to allocate eﬀort across tasks to the agent and sets output-
related pay to prevent the agent from choosing the action with the highest private beneﬁt.
Baker and Jorgensen (2003) show that an increase in output measurement noise always
reduces the optimal incentive strength, whereas an increase in volatility of the marginal
product of eﬀort leads to higher incentives. Therefore the type of uncertainty aﬀects the
1See Prendergast (2002) for an overview.
2correlation between risk and incentives.
This paper provides another argument for the positive correlation between risk and
incentives. The multitasking literature normally assumes that the cost of eﬀort is a
function of total eﬀort. An exception is Itoh (1991, 1992) who, however does not consider
the relationship between risk and incentives but examines the optimality of inducing team
production. Itoh (1991) makes a distinction between the case where the cost of eﬀort is a
function of total eﬀort exerted on diﬀerent tasks from the case where the cost of eﬀort is
additively separable across tasks. The author examines how the cost of inducing agents
to perform multiple tasks varies in these two contexts and when it is optimal to induce
team production, by rewarding agents on the same output measure. The optimality of
group rewards is further developed in Itoh (1992) who allows agents to have task speciﬁc
disutility. The author investigates the circumstances under which cooperation should be
fostered by linking the wage schedule of a risk averse agent to the outcome of the task
assigned to the other agent.
In our model we do not consider the possibility of assigning tasks to diﬀerent agents.
There is only one agent, as we focus on the implications of noise in output measurement
on the delivery of incentives. We abstract from job design and task assignment issues. We
extend the standard model by assuming that the cost of eﬀort is task-speciﬁc. The idea is
that the agent may have a preference towards some tasks, or may have greater ability on
some tasks, so that for equal total time spent on diﬀerent tasks, he ﬁnds it less costly to
devote relatively more time to the most preferred tasks. This assumption is more general
than the assumption typically made in the standard multi-tasking literature, where eﬀort
in diﬀerent activities are perfect substitutes, so that the agent only cares about total
eﬀort and not on how eﬀort is allocated across diﬀerent tasks. Contrary to Holmstr¨ om
3and Milgrom (1991) and to Baker and Jorgensen (2003), we show that the principal may
optimally set higher incentives for the eﬀort which contributes to the output measured
with less precision. This holds even if error terms are stocastically independent and
the activities are technologically independent. The result is merely explained by task-
preference. The agent’s decision on how to allocate his eﬀort across activities is now
aﬀected by an extra dimension: the relative attractiveness of the diﬀerent tasks.
2 The Model
We consider the interactions between an employer (the “principal”, she) and her employee
(the “agent”, he) following the model ﬁrst proposed by Holmstr¨ om and Milgrom (1991).
The agent chooses how much eﬀort to exert on two diﬀerent tasks and we denote these
levels by e1 and e2 . We assume the cost of eﬀort is quadratic and task-speciﬁc, i.e. it
depends on how the agent allocates his eﬀort. More speciﬁcally:
C (e1,e2) = c1e
2
1 + 2cce1e2 + c2e
2
2.
This is the way our model diﬀers from Holmstr¨ om and Milgrom (1991). In Holmstr¨ om
and Milgrom (1991) we would have c1 = c2 = cc. Here we allow c1 to diﬀer from c2.
The underlying idea is that the agent has some task preference or has greater ability on
one of the two tasks. We do not make any assumption on whether eﬀorts are substitutes
(cc ≥ 0) or complements (cc ≤ 0). In order to guarantee the existence of interior solutions
(ei > 0), we will assume that |cc| < min[c1,c2] .
The principal does not observe the amount of eﬀort exerted by the agent but only
4some noisy signals x1 and x2 :
xi = ei + εi,
where the random variables ε1 and ε2 are normally distributed with means zero and
covariance matrix Σ.
Given that eﬀort levels are not observed, the compensation contract speciﬁes a wage
W (x1,x2) which we assume to be linear:2
W (x1,x2) = w + α1x1 + α2x2.
The agent is risk-averse and we assume that his utility function takes the form u(W) = −e−rW ,
where r measures the agent’s risk aversion. The principal is risk-neutral and her expected
beneﬁt from task i is ei. We can therefore restrict our attention to the agent’s certainty
equivalent, which is:
CE (e1,e2) = W (e1,e2) − C (e1,e2) −
r
2
α
￿Σα, where α
￿ = {α1,α2}.
As common in this approach, the optimal linear wage is chosen so as to maximize the
total certainty equivalent (i.e. the joint surplus of the principal and the agent):
e
∗
1 (α) + e
∗
2 (α) − C (e
∗
1 (α),e
∗
2 (α)) −
r
2
α
￿Σα,
where the agent’s levels of eﬀort maximize his certainty equivalent:3
Ci (e
∗
1 (α),e
∗
2 (α)) = αi ⇔ e
∗
i (α) =
c−iαi − ccα−i
2(c1c2 − c2
c)
. (1)
Equation (1)does not apply in Holmstr¨ om and Milgrom (1991). If c1 = c2 = cc, we
would get e∗
i (α) + e∗
−i (α) = 1
2 max{αi,α−i},with e∗
i (α) = 0 whenever αi < α−i. The
2See Holmstr¨ om and Milgrom (1991, p. 31) for a discussion about the restriction to linear contracts.
3The subscripts on C denote partial derivatives and −i ￿= i ∈ {1,2}.
5agent would only care about total eﬀort and not about how eﬀort is allocated between
the two tasks.
From equation (1) it is clear that the way incentives on one eﬀort aﬀect the chosen
level of the other eﬀort depends on whether eﬀorts are substitutes or complements:
∂e∗
i
∂α−i
= −
cc
2(c1c2 − c2
c)
(2)
In fact, the sign of
∂e∗
i
∂α−idepends on cc: if ccis positive, an increase in the marginal incen-
tive on task e−i decreases eﬀort on task ei and viceversa. This is the same result as in
Holmstr¨ om and Milgrom (1991), although equation (2) does not apply to their model.
Solving for the optimal marginal incentives yields:
α
∗ = (I2 + r[Cij]Σ)
−1 .1
￿, (3)
where I2 is the identity matrix, [Cij] is the matrix of the second order derivatives of
the cost function and 1 ={1,1}. These conditions are (necessary and) suﬃcient when the
expression [Ci]
￿ Σ[Ci] is convex in (e1,e2), condition which is always met in the situations
we will analyze in this paper.
3 Identically Distributed Shocks
Suppose ﬁrst that the random variables ε1 and ε2 are i.i.d. with variance v, that is,
Σ = vI2 . Equation (3) thus rewrites as:
α
∗
i =
1 + 2rv(c−i − cc)
1 + 2rv(c1 + c2) + 4r2v2 (c1c2 − c2
c)
, (4)
and therefore α∗
i > α∗
−i if and only if ci < c−i.
6Proposition 1 When noises are i.i.d., the principal sets higher incentives on the least
costly eﬀort.
Noise in output measurement creates eﬀort distortion. The agent tries to distribute
more evenly his total eﬀort between the two tasks in order to spread the risk and moves
away from eﬃciency. Although she gets the same value from both eﬀorts and noises are
i.i.d., the principal has to intervene and set higher incentives for the least costly task to
reinforce eﬃciency in production.
Note also that, if cc > 0, the incentive contract is less powered than when cc < 0. This
is a general result in the literature: when tasks are complements the principal sets higher
incentives.4
If we compare our result to Holmstr¨ om and Milgrom (1991), in their context equation
(4) would become α∗
i = α∗
−i = 1
1+4rvc and the principal would optimally set equal incentives
on the two tasks.
The agent’s tendency to spread the risk should be maximized when outputs are per-
fectly positively correlated and should not exist when outputs are perfectly negatively
correlated (in this case the perfectly negative correlation between random shocks elimi-
nates any scope to spread the risk). This should be reﬂected in the optimal incentives
when noises are not independent. Suppose indeed that the two shocks have the same
variance but are now correlated, that is:
Σ = v.

 

1 ρ
ρ 1

 
, where |ρ| ≤ 1.
4See Dixit (2000) for an overview.
7In that case, equation (3) yields:
α
∗
i =
1 + 2rv[c−i − ciρ − cc(1 − ρ)]
1 + 2rv(c1 + c2 + 2ρcc) + 4r2v2(1 − ρ2)(c1c2 − c2
c)
.
Comparing the marginal incentives for the two tasks, we get that:
α
∗
i − α
∗
−i =
2(c−i − ci)(1 + ρ)rv
1 + 2rv(c1 + c2 + 2ρcc) + 4r2v2 (1 − ρ2)(c1c2 − c2
c)
As long as ρ ￿= −1, it remains the case that the principal sets higher incentives for the
least costly eﬀort. The intuition is the same as before: she needs to distort the marginal
incentives in order to compensate the tendency of the agent to over-invest in the more
costly eﬀort in order to spread the risk. Moreover,
∂
￿ ￿α∗
i − α∗
−i
￿ ￿
∂ρ
=
2r|c1 − c2|(1 + 2rv (c1 + c2 + 2cc) + 4r2v2 (1 + ρ2)(c1c2 − c2
c))
(1 + 2rv (c1 + c2 + 2ρcc) + 4r2v2 (1 − ρ2)(c1c2 − c2
c))
2 > 0.
Proposition 2 When the shocks are identically distributed but correlated, the principal
sets higher incentives on the least costly eﬀort. Moreover, the diﬀerence in the marginal
incentives increases with the correlation coeﬃcient.
If ρ = −1 then α∗
1 = α∗
2 = 1 and the ﬁrst-best outcome can be achieved. Because
shocks are perfectly negatively correlated, the agent no longer distorts his choice of eﬀort
when incentives are equal. Making the agent the residual claimant of production therefore
implements the ﬁrst best solution.
4 Independent Asymmetric Shocks
Suppose now that shocks are independent but measured with diﬀerent precisions, i.e.
variances might not be equal. The covariance matrix is then given by:
Σ =

 

v1 0
0 v2

 
.
8Equation (3) then yields:
α
∗
i =
1 + 2rv−i(c−i − cc)
1 + 2r(v1c1 + v2c2) + 4r2v1v2(c1c2 − c2
c)
.
Let us again establish the sign of the diﬀerence between marginal incentives set for the
two tasks. We have:
α
∗
i > α
∗
−i ⇔ v−i(c−i − cc) > vi(ci − cc) ⇔
1
vi
1
v−i
>
ci − cc
c−i − cc
. (5)
The relative strength of marginal incentives depends on the diﬀerence in error measure-
ment (v1 vs. v2) and on the slopes of the marginal costs of eﬀort (c1 vs. c2) as shown in
the following proposition:
Proposition 3 When shocks are independent and have diﬀerent precisions:
• the marginal incentives are always higher for the output that is more precisely mea-
sured when c1 = c2;
• the marginal incentives may be set higher for the least precisely measured output if
c1 ￿= c2. This can only be the case if the least precisely measured output is also the
least costly task for the agent.
Proof. When c1 = c2, equation (5) boils down to v−i > vi; hence the ﬁrst part of the
proposition. Suppose now that c1 ￿= c2 and vi > v−i. For equation (5) to hold, it must
then be the case that
ci − cc
c−i − cc
<
v−i
vi
< 1,
which requires ci < c−i.
9When there is no task-preference, i.e. c1 = c2, then the standard result holds.5 In this
case the agent’s decision on how to allocate his eﬀort across the two tasks only depends
on the relative precision of output measurement, as captured by the diﬀerence in optimal
marginal incentives set by the principal.
The case of task-preference (c1 ￿= c2) is however the most interesting. Now there is
an extra dimension to the agent’s decision of how to allocate his eﬀort across task: the
relative slope of the marginal costs. When setting optimal incentives the principal needs
not only to ensure that the agent exert more eﬀort on the more precisely measured output
but also that the agent does not move away from eﬃciency by exerting too much eﬀort
on the more costly task. Without loss of generality, suppose that v1 < v2.
If c1 < c2, then risk and eﬃciency considerations go in the same direction: the agent
is indeed more exposed to risk when he works on task 2 which is also more costly. The
principal will thus set lower incentives on the least precisely measured output.
If instead c1 > c2, then it is possible to have α∗
1 < α∗
2. This will indeed be the case
when c2 is suﬃciently smaller than c1. The intuition behind this result is that the agent
is exposed to more risk when he is working on output x2, but the eﬀort that determines
x2 is less costly than the eﬀort contributing to output x1. So risk and eﬃciency consid-
erations go in the opposite direction. If the diﬀerence in costs (relative attractiveness of
tasks) outweighs the diﬀerence in output measurement precision, it is possible to observe
higher incentives on the less precisely measured output. This reverses the result shown
by Holmstr¨ om and Milgrom (1991).
Note that the cost of inducing the agent to perform a given task is not independent
of the other task when c1 ￿= c2 even if activities are technologically independent (cc = 0)
5See Holmstr¨ om and Milgrom (1991) and Baker and Jorgensen (2003).
10and error terms are stochastically independent. This means that the principal cannot set
incentives on the two tasks separately even if tasks are technologically independent and
measurement errors are not correlated.
5 Conclusion
In this paper we consider the relationship between risk and incentives in a simple context
where one agent can perform two tasks and has a preference for one of the two tasks.
We model this by assuming that the cost of eﬀort is additively separable across tasks
and for initial equal levels of eﬀort, the marginal cost diﬀers across tasks. We show that,
if outputs are measured with diﬀerent precision, the standard result according to which
the principal sets higher incentives on the output measured with more precision may be
reversed. The diﬀerence in the marginal cost of eﬀorts may have an overwhelming eﬀect on
the diﬀerence in measurement precision and the principal optimally sets higher incentives
on the output measured with less precision.
11References
[1] Baker, G., Jorgensen, B., “Volatility, Noise and Incentives”, mimeo, 2003.
[2] Dixit A., “Incentives and organisations in the public sector: an interpretative review”,
Journal of Human Resources, 37(4), 2000, pp.696-727.
[3] Holmstr¨ om B., Milgrom P. , “Multi-task principal-agent analyses: Linear contracts,
asset ownership and job design”, Journal of Law, Economics and Organisation, 7
(Special Issue), 1991, pp.24-52.
[4] Itoh H., “Incentives to help in multi-agent situations”, Econometrica, 59, (3),1991,
611-636.
[5] Itoh H., “Cooperation in hierarchical organizations: an incentive perspective”, Journal
of Law, Economics and Organization, 8(2), 1992, pp. 321-345.
[6] Jensen, M., Murphy, K., “Performance pay and top-management incentives”, Journal
of Political Economy, 98 (2), 1990, pp. 225-264.
[7] Prendergast C., “The tenuous trade-oﬀ between risk and incentives”, Journal of Po-
litical Economy, 110 (5), 2002, pp.1071-1102.
12

Task-specific effort costs and the trade-off between risk and efficiency

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