everal empirical findings have challenged the traditional trade-off between risk and incentives. By combining risk aversion and limited liability in a standard principal-agent model the empirical puzzle on the positive relationship between risk and incentives can be explained.limited liability; piece rates; risk aversion

Matthias Kräkel

English

Research Papers in Economics

 
 
 
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
Discussion Paper 3/2007
Limited Liability and the Trade-off between Risk and
Incentives
by
Matthias Kr¨ akel
April 2007
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 

 
 
 
 
 

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
                                      The Bonn Graduate School of  Economics is
                                                             sponsored by theLimited Liability and the Trade-oﬀ between Risk and
Incentives*
Matthias Kräkel, University of Bonn**
Several empirical ﬁndings have challenged the traditional trade-oﬀ between
risk and incentives. By combining risk aversion and limited liability in a standard
principal-agent model the empirical puzzle on the positive relationship between
risk and incentives can be explained.
Key words: limited liability; piece rates; risk aversion; risk-incentives trade-oﬀ.
JEL classiﬁcation: D01; D82; J3; M5.
* Financial support by the Deutsche Forschungsgemeinschaft (DFG), grant SFB/TR
15 (”Governance and the Eﬃciency of Economic Systems”), is gratefully acknowl-
edged.
** Matthias Kräkel, Department of Economics, BWL II, University of Bonn, Ade-
nauerallee 24-42, D-53113 Bonn, Germany, e-mail: m.kraekel@uni-bonn.de, phone:
+49-228-739211, fax: +49-228-739210.
11I n t r o d u c t i o n
In the well-known principal-agent model with a risk neutral principal and
a risk averse agent we face the typical negative relationship between exoge-
nous risk and incentives: If risk is large and, hence, the risk premium (or
risk costs) from high-powered incentives would be high the principal will op-
timally choose low incentives for the agent. However, if risk is small so that
risk costs from induced incentives are rather low, the principal will prefer
high-powered incentives.
Prendergast (2002a) refers to several empirical studies which point to
a positive relation between risk and incentives and therefore challenge the
traditional view. Consequently, he oﬀers alternative explanations for this
empirical puzzle (Prendergast 2002a, 2002b). In his modeling, he assumes
that the agent is risk neutral in order to abstract from the classical trade-oﬀ.
However, this assumption may be crucial. For example, if in Prendergast
( 2 0 0 2 a )t h ea g e n tw e r er i s ka v e r s ea n dh i g hr i s km a k e st h ep r i n c i p a lp r e f e r
output-based to input-based contracts, the well-known traditional trade-oﬀ
would apply again. Another explanation for the empirical puzzle is intro-
duced by Raith (2003). There, the positive relationship between risk and
incentives comes as a by-product. Raith assumes that agents are risk averse
so that incentives indeed imply risk costs. However, since agents face a bind-
ing participation constraint and principals always realize zero proﬁts due to
market competition, neither the agent nor the principal (but society) has to
bear the risk costs from inducing incentives. Hence, less extreme competi-
2tion may lead to diﬀerent ﬁndings concerning the trade-oﬀ. Finally, Wright
(2004) and Serfes (2005) independently develop a matching approach that
can explain the puzzle. If competition makes less risk averse agents match
with more risky principals, the outcome may be a positive relation between
risks and incentives.
However, in this paper we oﬀer an alternative explanation for the empiri-
cal puzzle which does neither need risk neutral agents nor market competition
as a driving force. We only combine two standard contracting problems —
risk aversion and limited liability — which should be present in many circum-
stances so that this approach seems to be the most natural explanation for a
possibly positive relation between risk and incentives. The theoretical results
show that if the agent earns a positive rent in optimum and if his cost-of-
eﬀort function is not to steep, higher exogenous risk will lead to stronger
incentives. In this situation, the principal strictly gains from increased in-
centives leading to higher output since he has not to pay for the additional
incentives which only reduce the agent’s rent. However, if in optimum eﬀort
is not very sensitive to incentives the traditional trade-oﬀ will prevail.
Note that we restrict the set of possible contracts to linear incentive
schemes. This restriction is introduced for two reasons. First, most of the
empirical literature that challenges the traditional trade-oﬀ is based on linear
incentives schemes. Second, all of the cited papers that have dealt with
the given topic so far use a linear incentive scheme. Thus, adopting the
assumption on linear payment makes our ﬁndings directly comparable to the
3previous results.
2T h eM o d e l
We consider a typical principal-agent relationship between a risk neutral
principal P and a strictly risk averse agent A who has the utility function
u(I) with u0 (I) > 0, u00 (I) < 0 and I denoting A’s income. A produces
the output q = e + εr for P where e stands for A’s chosen eﬀort and εr for
exogenously given noise with subscript r indicating risk. Let εr be distributed
over [−¯ εr,¯ εr] according to the density function fr (εr) with mean E [εr]=
0 and variance σ2
r. In the following, we will distinguish two probability
distributions r = L,H with ¯ εL < ¯ εH and fL (εL) exhibiting second-order
stochastic dominance over fH (εH) which implies σ2
L <σ 2
H (Hadar and Russel
1971, Theorem 3) so that distribution fH (εH) is more risky than fL (εL).I n
other words, we can transform fL(εL) into fH (εH) by shifting probability
mass from the mean to the enlarged tails. There is the usual hidden-action
problem: P observes the veriﬁable output q but neither e nor εr. P also
knows A’s utility function and the probability distribution. A’s cost-of-eﬀort
function is given by c(e) with c0 (e),c 00 (e) > 0 and c(0) = 0.W er e s t r i c tt h e
set of possible contracts to linear payment schemes w = α + βq.H e n c e ,A’s
income is given by I = w − c(e)=α + βq − c(e). It is assumed that A is
protected by limited liability: P has to guarantee A a minimum payment so
that w ≥ wmin. A’s reservation utility is given by ¯ u with ¯ u = u(¯ w).T h e
timing of the model is the usual one: First, P oﬀers A ac o n t r a c t(α,β).
4Then, A accepts or rejects. Given that he has accepted P’s oﬀer, A then
chooses e. Finally, εr is realized and payments are made to P and A.
3 Optimal Linear Contract
P maximizes expected net proﬁts (1 − β)e−α subject to the incentive con-
straint, the individual-rationality constraint and the limited-liability con-
straint of the agent. A’s incentive constraint is given by
e
∗ ∈ argmax
e
½Z ¯ εr
−¯ εr
u(α + β (e + εr) − c(e))fr (εr)dεr
¾
which, by using Jensen’s Inequality, can be rewritten as
e
∗ ∈ argmax
e {u(α + βe− c(e) − ρr (β))}
with CE = α+βe−c(e)−ρr (β) being the certainty equivalent and ρr (β) > 0
the corresponding risk premium with ρ0
r (β) > 0 and r indicating the underly-
ing probability distribution. The ﬁrst-order condition u0 (CE)(β − c0 (e)) =
0 yields
e
∗ = h(β) (IC)
with h(·): =c0−1 (·) denoting the inverse of the marginal cost function which
is monotonically increasing because of the convex cost function.1
The individual-rationality constraint, which guarantees A’s participation,
Z ¯ εr
−¯ εr
u(α + β (e + εr) − c(e))fr (εr)dεr ≥ ¯ u,
1Note that the second-order condition u00 (CE)(β − c0 (e)) 2 − c00 (e)u0 (CE) < 0 is
always satisﬁed.
5can be simpliﬁed to
α + βe− c(e) − ρr (β) ≥ ¯ w. (IR)
Finally, the limited-liability constraint has to ensure A his minimum wage
wmin even in the worst case with the lowest realization of εr:
α + β (e −¯ εr) ≥ wmin. (LL)
To sum up, P maximizes expected net proﬁts subject to (IC), (IR) and (LL).
Inserting (IC) into P’s objective function, into (IR) and into (LL), the
principal’s optimization problem can be described by the Lagrangean
L(α,β)=( 1 − β)h(β) − α + λ1 [α + βh(β) − c(h(β)) − ρr (β) − ¯ w]
+λ2 [α + β (h(β) −¯ εr) − wmin]
with λ1,λ 2 ≥ 0 denoting the respective Lagrange multipliers. Diﬀerentiation
with respect to α and β yields that in optimum we must have
λ1 + λ2 =1 and (1)
−h(β)+( 1− β)h
0 (β)+λ1 [h(β)+βh
0 (β) − c
0 (h(β))h
0 (β) − ρ
0
r (β)] (2)
+λ2 [h(β)+βh
0 (β) −¯ εr]=0
According to condition (1), in optimum either the individual-rationality con-
straint is binding (λ1 =1and λ2 =0 ) or the limited-liability constraint
(λ1 =0and λ2 =1 ) which is also intuitively plausible since P does not want
to waste his money.
6First, consider the case of binding (IR). Substituting λ1 =1and λ2 =0
into (2) gives
h
0 (β)=ρ
0
r (β)+c
0 (h(β))h
0 (β).
Hence, if A does not earn a positive rent, P will choose optimal incentives
which maximize total welfare so that induced marginal returns equal the
sum of marginal risk costs and marginal costs of eﬀort. Second, if A earns a
positive rent and (LL) is binding, λ1 =0and λ2 =1imply
h
0 (β)=¯ εr (3)
which leads to the following result:
Proposition 1 If, in optimum, A earns a positive rent optimal incentives,
β
∗, will be described by h0 (β
∗)=¯ εr,o t h e r w i s eb yh0 (β
∗)=ρ0
r (β
∗)+c0 (h(β
∗))h0 (β
∗).
Proposition 1 shows that on the one hand we may have the well-known
solution for risk averse agents (who are not wealth-constrained) where the
principal maximizes total welfare so that the usual negative trade-oﬀ between
risk and incentives should apply. On the other hand, agents may earn posi-
tive rents so that the binding limited-liability constraint directly determines
the optimal level of incentives. Let β
∗ (σ2
r) denote the optimal piece-rate
dependent on the risk of the given probability distribution r = L,H.F r o m
(3), we immediately obtain the following corollary:
Corollary If A earns a positive rent in optimum and h(β) is convex, then
β
∗ (σ2
H) >β
∗ (σ2
L) so that risk and incentives will be positively related.
7The intuition for the interesting ﬁnding of the corollary is the following:
First, note that in case of a positive rent P does not have to pay for additional
eﬀort costs and additional risk costs implied by higher incentives. These costs
are fully borne by the agent since his positive rent is decreased. Hence, given
a positive rent the usual trade-oﬀ between risk and incentives is overridden.
However, as the corollary shows, P is not always interested in increasing
the piece rate when ¯ εr rises. P will only enhance incentives, if eﬀort strongly
reacts to an increased piece rate, i.e. if h(·) is convex or, in other words,
if the cost function is not too steep (for example, if c(e)=e1.5 in case of a
polynomial cost function). If risk is shifted to the tails so that ¯ εr increases, P
has to compensate A for this increase according to the (LL) condition. (LL)
shows that on the one hand P can decrease β in order to give the increase of
¯ εr al o w e rw e i g h t . 2 On the other hand, P can counterbalance the increase in
¯ εr by an increase of eﬀort e induced by a higher piece rate according to (IC).
Which of these alternatives is the better one from P’s perspective, crucially
depends on the shape of h(·).I f e∗ is very sensitive to incentives, P will
optimally choose a higher piece rate. If eﬀort does not react very much to
increased incentives in the optimum since the cost function is very steep, P
will prefer to decrease the piece rate when ¯ εr rises.
Finally, recall that plugging λ1 =0and λ2 =1into Eq. (2) yields
(1 − β)h
0 (β)+βh
0 (β) −¯ εr =0 ,
2Note that both α and wmin can be negative.
8which points to a further aspect. The ﬁrst term shows that increasing the
piece rate β also decreases P’s share in produced output. Hence, it will
only be optimal for him to increase the piece rate, if his decreased share is
overcompensated by the increase of eﬀort. To sum up, we need two conditions
for a positive relation between risk and incentives in a principal-agent model
— the agent must earn a positive rent so that the principal does not have to
pay for higher incentives, and eﬀort must be rather sensitive to incentives.
References
Hadar, J. and W.R. Russel, 1971, Stochastic dominance and diversiﬁcation,
Journal of Economic Theory 3, 288-305.
Prendergast, C., 2002a, The tenuous trade-oﬀ between risk and incentives,
Journal of Political Economy 110, 1071-1102.
Prendergast, C., 2002b, Uncertainty and incentives, Journal of Labor Eco-
nomics 20, S115-S137.
Raith, M., 2003, Competition, risk, and managerial incentives, American
Economic Review 93, 1425-1436.
Serfes, K., 2005, Risk sharing vs. incentives: contract design under two-
sided heterogeneity, Economics Letters 88, 343-349.
Wright, D.J., 2004, The risk and incentives trade-oﬀ in the presence of
heterogeneous managers, Journal of Economics 83, 209-223.
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Limited Liability and the Trade-off between Risk and Incentives

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