Consider a principal-agent relationship in which more effort by the agent raises the likelihood of success. Does rewarding success, i.e., paying a bonus, increase effort in this case? I find that bonuses have not only an incentive but also an income effect. Overall, bonuses paid for success may well reduce effort and hence the probability of success. I also identify conditions under which the income effect dominates the incentive effect, and single out the hazard-rate of effort as a crucial determinant of this trade-off.bonus, premium, incentives, income effect, moral hazard

Wendelin Schnedler

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
University of Heidelberg 
Discussion Paper Series    No. 452 
Department of Economics 
You Don't Always Get What You 
Pay For 
Wendelin Schnedler
September 2007You Don't Always Get What You Pay For
Wendelin Schnedler
Department of Economics
University of Heidelberg
September 23, 2007
Abstract
Consider a principal-agent relationship in which more eort by the agent raises the
likelihood of success. Does rewarding success, i.e., paying a bonus, increase eort in
this case? I nd that bonuses have not only an incentive but also an income eect.
Overall, bonuses paid for success may well reduce eort and hence the probability
of success. I also identify conditions under which the income eect dominates the
incentive eect, and single out the hazard-rate of eort as a crucial determinant of
this trade-o.
Keywords: bonus, premium, incentives, income eect, moral hazard
JEL-Codes: D8, J3, M5
wendelin.schnedler@awi.uni-heidelberg.de
Department of Economics, University of Heidelberg, Grabengasse 14, D-69117 Heidelberg.1 Introduction
Managers often resort to premium payments in order to boost performance above ordinary
levels. Football players are promised a bonus if they win important matches, in some
universities professor's budgets directly increase with each article that they publish in a
reputed journal,1 and researchers receive pay raises for successful completion of a projects.
In all these cases, the outcome is not only determined by the eort of the agent but also
by factors beyond his inuence|that is, luck. Moreover, agents in these examples are
already motivated and exert eort. Under these circumstances, threatening to take money
in the case of failure rather than to give money in the case of success may induce adverse
reactions by the agent and crowd out rather than increase eort (see Bewley, 1995, Fehr
and Rockenbach, 2003, or Falk and Kosfeld, 2006). This is probably the reason why
bonuses rather than nes are often used when agents are already motivated. The logic
of bonus payments itself seems straight-forward: by paying a bonus on the condition of
success, the successful outcome becomes more attractive to the agent and he puts in more
eort to increase the likelihood of success. But do bonus payments necessarily increase
eort and the likelihood of success?
Here, I examine this question and nd that the answer is \no." There are plausible
situations in which paying a bonus reduces the incentive to exert eort. For example,
consider an agent who appreciates an increase of income more when he works less for
it. In other words, leisure is a normal good, and richer agents derive more pleasure from
exerting less eort. Then, the possibility of being lucky and receiving the bonus even
if no additional eort is exerted increases the agent's income and thereby reduces his
willingness to exert eort. This income eect has to be traded-o against the incentive
eect of the bonus. The income eect dominates if the marginal eect of eort on the
probability of success is small in relation to the probability of success when no additional
eort is exerted (Proposition 1).
1The university of Cologne in Germany has recently adopted such a system.
1The next section introduces the model, which is then analyzed in Section 3. Section 4
summarizes the ndings.
2 Model
Consider an agent who works for a principal on a project that can be successful (S = 1) or
not (S = 0). Success can be veried by courts and its probability p depends on the eort
e of the agent, where e is a real number. The agent likes receiving monetary transfers t.
In the absence of transfers, his utility initially increases in eort until some level, e0 > 0,
is reached and then decreases. In other words, there is an inner solution to the agent's
eort choice problem. The agent may, for example, be altruistically motivated, care about
reputation or there may be other incentives that are not explicitly modeled.2
Let the agent's utility function u(t;e) be twice continuously dierentiable in transfers
and eort. This specication is suciently exible to account for interaction between the
received transfers and eort. In particular, it may be that the agent prefers receiving
a transfer when he exerts less eort. This happens, for example, if the agent is too
exhausted from exerting eort to enjoy the transfer. If we regard exerting eort as the
opposite of leisure, the condition is fullled if leisure is a normal good. There is empirical
evidence that leisure is normal (see Ashenfelter and Heckman, 1974, or the survey article
by Blundell and MaCurdy 1999) so it is not implausible that at least some agents enjoy
transfers more when they exert less eort. In analogy to leisure being a normal good,
eort is called normal in this case. Formally, normal eort means that the cross-partial
derivative of the utility function of the agent is negative:
@@u(t;e)
@t@e < 0.
The principal only cares about the expected success of the project. Standardizing the
utility in case of failure to zero and in case of success to one, her utility is simply p(e):
She is endowed with sucient funds to transfer money to the agent but cannot take from
2Holmstr om and Milgrom (1991) employ a similar assumption to show that using performance mea-
sures may be detrimental in multiple-task models.
2him. In other words, she can only use carrots and not sticks. This assumption reects the
idea that in many contexts, it is not possible to adversely aect the agent without risking
a reduction of eort below e0. The principal has no opportunity costs of using funds for
incentives. While it may be more realistic to assume that there are opportunity costs,
this assumption does not aect our results about the eect of incentives on eort.
3 Analysis
The agent is not internalizing the externality of his eort choice on the principal. It
is hence straightforward that the choice of e0 is too small from a social point of view:
increased eort and an appropriate compensation of the agent could Pareto-improve the
situation. Moreover, it is in the principal's self-interest to employ a mechanism that
encourages the agent to exert more eort. Does such a mechanism exist?
Since eort itself cannot be contractually xed, the principal has to rely on the
project's success in order to provide incentives. More specically, any incentive scheme
for the given contractual environment is a pair of transfers (t0;t1), where t0  0 is the
transfer that the agent receives if the project is not successful and t1  0 the transfer
after success.3
In order to see whether contracts of this type can increase eort and hence yield Pareto-
improvements, consider the incentive problem of the agent. Given (t0;t1), the agent's
expected utility can be written as: u(t0;e)+p(e)(u(t1;e) u(t0;e)): By assumption, this
utility has an inner maximizer for t0 = t1 = 0. This maximizer is given by the rst-order
condition:
@u(t0;e)
@e
+ p
0(e)  (u(t1;e)   u(t0;e)) + p(e) 

@u(t1;e)
@e
 
@u(t0;e)
@e

= 0: (1)
Based on this condition, the implicit-function theorem can be employed to obtain the
following proposition.
3Transfers are non-negative because the principal cannot take from the agent.
3Proposition 1. Introducing a bonus marginally decreases eort if and only if the income
eect dominates the incentive eect:
p(e)  ( 1) 
@@u(t;e)
@e@t

  
t=0;e=e0
> p
0(e) 
@u(t;e)
@t

  
t=0;e=e0
: (2)
Proof. Applying the implicit-function theorem to (1) yields:
de
dt0
=  (SOC)
 1 
 
 p
0(e) 
@u(t;e)
@t
   
t=0;e=e0
+ (1   p(e)) 
@@u(t;e)
@e@t
   
t=0;e=e0
!
de
dt1
=  (SOC)
 1 
 
p
0(e) 
@u(t;e)
@t
   
t=0;e=e0
+ p(e) 
@@u(t;e)
@e@t
   
t=0;e=e0
!
;
where SOC is the second-order condition, which is negative at the maximum. If Inequal-
ity (2) holds, then eort must be normal,
@@u(t;e)
@e@t < 0, the derivative with respect to t0 is
negative, and the principal sets t0 to the lowest possible level t0 = 0: Likewise, Inequal-
ity (2) implies that the last term in parentheses in the second line is negative, so that
the derivative of eort with respect to t1 is also negative. Thus paying a marginal bonus,
t1 > 0, reduces eort below e0. Conversely, if Inequality (2) does not hold, the deriva-
tive of eort with respect to t1 is positive and paying a small transfer t1 > 0 increases
eort.
The important insight of this proposition is that a marginal increase in the bonus paid
in the case of success does not necessarily increase eort. Since success is an increasing
function of eort, paying for success may thus lower the probability of success.4
Rewriting Inequality (2) shows that the hazard-rate, p0(e)=p(e), determines whether
the income or the incentive eect dominates. If the change in the probability resulting
from the next unit of eort is small in relation to the current probability of success,
the income eect outweighs the incentive eect and eort decreases when a bonus is
introduced.
4The proposition describes the eect of the introduction of a small bonus on the provision of eort.
If Inequality (2) holds for arbitrary transfers and eorts, eort decreases irrespective of the bonus' size.
44 Conclusion
I have examined how the introduction of a success-dependent bonus aects agents' mo-
tivation. The main nding is that a bonus may well reduce the agent's eort and hence
the probability of success. Given that leisure is a normal good, paying a bonus has an
income eect that may be stronger than the incentive eect. The income eect dominates
whenever the hazard rate is small, that is if the eect of the next unit of eort on the
probability is small in relation to the probability of success when no additional eort is
exerted.
This trade-o between income and incentive eect has, to my knowledge, not yet been
studied in the agency literature, although utility functions that allow for income eects
have already been considered| see for example the famous analysis of the principal-agent
problem by Grossman and Hart (1983). Unlike the income eect from the labor supply
decision of households, the income eect here disappears if success is a perfect measure
of eort. Then, the agent has no choice but to exert eort if he wants to obtain the
bonus. The trade-o between income and incentives complements the well-known trade-
o between insurance and incentives (Holmstr om, 1979, and Shavell, 1979) and yields
similar predictions: a weaker link between eort and success|i.e., more noise|renders
bonus payments less eective, and empirically they should occur less often under these
circumstances. Notice, however, that the trade-o studied here applies even if the agent
is risk-neutral.
These ndings have the following practical implication: before trying to boost eort
by paying for success, it is sensible to ensure that the income eect does not dominate the
incentive eect. For example, one should verify that the eort of footballers, researchers,
or professors actually has a strong eect on the probability of success. If results largely
depend on luck, then paying bonuses may achieve the opposite of the desired eect and
decrease the probability of success.
5References
Ashenfelter, Orley and James Heckman, \The Estimation of Income and Substitu-
tion Eects in a Model of Family Labor Supply," Econometrica, January 1974, 42 (1),
73{85.
Bewley, Truman F., \A Depressed Labor Market as Explained by Participants," Amer-
ican Economic Review, May 1995, 85 (2), 250{254. Papers and Proceedings.
Blundell, Richard and Thomas Macurdy, \Labor supply: A review of alternative
approaches," in Orley Ashenfelter and David Card, eds., Handbook of Labor Economics,
Elsevier, 1999, chapter 27.
Falk, Armin and Michael Kosfeld, \The Hidden Costs of Control," American Eco-
nomic Review, 2006, 96 (5), 1611{1630.
Fehr, Ernst and Bettina Rockenbach, \Detrimental Eect of Sanctions on Human
Altruism," Nature, March 2003, 422, 137{140.
Grossman, Sanford J. and Oliver D. Hart, \An Analysis of the Principal-Agent
Problem," Econometrica, January 1983, 51 (1), 7{46.
Holmstr om, Bengt, \Moral Hazard and Observability," Bell Journal of Economics,
1979, 10 (1).
and Paul Milgrom, \Multitask Principal-Agent-Analysis: Incentive Contracts, Asset
Ownership, and Job Design," Journal of Law, Economics, and Organization, 1991, 7,
24{52. special issue.
Shavell, Steven, \Risk-Sharing and Incentives in the Principal-Agent Relationship,"
Bell Journal of Economics, 1979, 10, 55{73.
6

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