The tenuous relationship between e¤ort and
performance pay
Ola Kvaløyyand Trond E. Olsenz
October 22, 2014
Abstract
When an agent is o¤ered performance related pay, the incentive effect is not only determined by the shape of the incentive contract, but
also by the probability of contract enforcement. We show that weaker
enforcement may reduce the agent’s e¤ort, but lead to higher-powered
incentive contracts. This creates a seemingly negative relationship between e¤ort and performance pay.
JEL codes: D86; M52
Keywords: Incentives; performance pay; contract enforcement
We thank three referees and the editor for constructive comments. We also thank
Hideshi Itoh, Anja Schottner, Gaute Torsvik, Joel Watson and seminar participants at
Paris School of Economics, University of Bonn, University of Minho, EARIE, EEA, and
the Law and Economics of contracts workshop in Bergen for helpful comments. Financial
support from the Norwegian Research Council (Grant # 196873) is gratefully appreciated.
y
University of Stavanger, 4036 Stavanger, Norway. ola.kvaloy@uis.no
z
Norwegian School of Economics, Helleveien 30, 5045 Bergen, Norway.
trond.olsen@nhh.no
1
1
Introduction
The last two decades have seen a strong growth in the use of performance
related pay. An increasing fraction of jobs explicitly pays workers for their
performance, using bonuses, commissions or some other kinds of merit pay
(see Lemieux et al., 2009). At the same time there seems to be an increase
in complaints and even lawsuits over unpaid bonuses.1 As a recent example,
104 bankers in London were suing Dresdner Kleinwort and Commerzbank
for $66 million worth of unpaid bonuses in the biggest case of its kind in
the UK.2 Also in the public sector, where so-called new public management
(NPM) reforms have introduced performance related pay in a wide set of
public service jobs, non-credible incentive pay is an issue. According to
OECD (2004) and Dahlström and Lapuente (2010), the lack of credibility is
an important obstacle to e¤ective incentive pay in the public sector, where
governments are tempted to modify a given incentive system ex post and
renege on promises in order to pursue other political goals.
These two trends - more use of performance related pay and complaints
about unpaid or non-credible bonuses - coincide with what seems to be an increasing skepticism over what performance related pay actually can achieve.
Standard economic models that predict a positive relationship between effort and performance pay are challenged by empirical work suggesting that
performance pay mitigates motivation and reduces e¤ort (see e.g. surveys
by Weibel et al, 2010 and Frey and Jegen, 2001).
1
See e.g. Alexandra Carn in Financial World, 2006-07, and Howard Meyers in New
York Law Journal, June 27, 2008.
2
See e.g. Financial Times, January 25, 2012
2
In this paper we show that these phenomena may be closely related. Uncertainty over bonus payments or weaker enforcement of bonus contracts,
may lead to higher bonuses and lower e¤ort, creating a negative equilibrium
relationship between performance pay and e¤ort. The relationship that we
propose contrasts with the standard explanation based on motivation crowding out, in which non-monetary intrinsic motivation is the essential factor.
Higher monetary rewards may there reduce intrinsic motivation to such an
extent that e¤ort is reduced.3 We show that variations in enforcement probability can have similar e¤ects as variations in intrinsic motivation, and that
the former can be an alternative explanation for a negative association between performance pay and e¤ort.
With "enforcement probability" we here mean the probability that an
employee who is entitled to a bonus actually receives the bonus. There are
a number of reasons why the employee may not be paid as promised. If the
incentive contract is incomplete, the employer may deliberately choose not
to honor the contract hoping that the court will not be able to enforce it.
The employer may also provide discretionary bonuses, where the bonus is
paid at the employer’s discretion and the employee is not fully protected by a
legally enforceable contract. Finally, there may be more or less unexpected
contingencies that arise during the employment relationship that make it
costly, or even impossible, for the employer to pay the bonus as promised.4
3
Recent papers show how the structure of monetary rewards may a¤ect agents’ perception of their tasks or own abilities, or undermine incentives for social esteem (see,
respectively, Benabou and Tirole, 2003, 2006, and Ellingsen and Johannesson, 2008). See
also Frey and Jegen (2001) for a review of previous literature on motivation crowding out.
4
In the Dresdner Kleinwort /Commerzbank case, the loss of 6.5 billion euros made
the bank unwilling to pay the bonuses. In the aftermath of the …nancial crisis one has
also seen examples where CEOs give up their bonuses after pressure from stakeholders
3
In order to capture some important implications of variations in enforcement probability in a simple way, we analyze a moral hazard model where
a principal must provide an agent with incentives to exert e¤ort, and where
the incentive contract is enforced with a probability v < 1. Our modeling
set-up can account for both legal and non-legal, or informal, enforcement
mechanisms. With legal enforcement, v is the probability that the court can
verify performance and thus enforce the contract. With informal enforcement, v can be seen as the probability that the principal will be socially or
politically committed to honor the contract.
It is natural to consider the probability of enforcement as a variable
rather than as an absolutely …xed parameter. Generally, the complexity of
a transaction, the strength of the enforcement institutions and the practice
of legal courts are factors that a¤ect legal enforcement (see e.g. Clague et
al. 1999, and Djankov et al. 2003).5 Also, informal contract enforcement,
such as the environments for social enforcement, may vary. Exogenous variations occur naturally across countries and industries, but can also a¤ect
a given contractual relationship via legal reforms, changes in legal practice,
standardization of industry contracts, changes in (labor) law or other institutional or organizational changes (see MacLeod, 2011, for a discussion).
In this paper, we show how exogenous variations in enforcement probaor politicians, e.g. the case of Royal Bank of Scotland, see Financial Times January 30,
2012.
5
For contracting parties these may constitute exogenous variations. But one can also
think of the enforcement probability as an endogenous variable, since the contracting
parties’e¤ort in writing a contract that describes a job’s tasks and operational performance
metrics may also a¤ect this probability (see Kvaløy and Olsen, 2009). In this paper,
however, we abstract from endogenous veri…ability, and treat enforcement as an exogenous
variable.
4
bility a¤ect both incentive design and e¤ort. Clearly, a weaker probability
will, all else equal, reduce the employee’s e¤ort, because the expected bonus
decreases. But weaker enforcement may also lead to higher-powered incentive contracts, although at the outset one might expect the opposite. No
incentive contract can be implemented in a situation where the principal
certainly won’t pay. And high-powered incentives can certainly be enforced
if the contract is honored for sure. Also, risk aversion on the part of the
agent can make it quite costly for the principal to o¤er incentives where very
high bonuses are paid with low probability, as the agent must be compensated for the high risk associated with such schemes. However, it turns out
that on the margin, the incentive intensity of the contract can be negatively
related to the probability of enforcement under quite standard assumptions.
The reason is that a reduction of the enforcement probability does not only
reduce e¤ort, but also reduces expected wage costs per unit of e¤ort, since
the probability that the principal actually has to pay as promised decreases.
This can make the principal o¤er higher-powered incentives which increase
e¤ort, but not necessarily up to the level it was prior to the change in enforcement probability.
We adopt the classical model on risk sharing vs. incentives, and show
that when enforcement is probabilistic, then under certain conditions contractual incentive intensity and e¤ort are (spuriously) negatively related.
We also point out that under risk neutrality and limited liability, e¤ort
may be (spuriously) completely independent of contractual incentives (due
to contractual incentives and enforcement then being perfect substitutes).
The negative relationship is a "false crowding out e¤ect" since total mon5
etary incentives, which is the product of the enforcement probability and
contractual incentives, is positively related to e¤ort. But since the enforcement probability does not show up in the incentive contract, it appears that
incentives and e¤ort are negatively related.
This result has an important empirical implication: When observing a
negative relationship between performance pay and e¤ort, one has to control for the probability that incentive contracts are actually honored. If not,
one may wrongfully infer that monetary incentives crowd out non-monetary
motivation. Controlling for enforcement probability is quite easy in experimental work. With …eld data, however, this is much more of a challenge.
Take the empirical work on New Public Management (NPM) as an example. NPM describes reforms in the public sector that are characterized by
an emphasis on output control, performance related pay and introduction of
market mechanisms. Scholars argue that NPM undermines - or crowds out
- intrinsic motivation and thus the e¤ort of public servants, see e.g. Weibel,
Rost and Osterloh (2010), and Perry, Engbers and Jun (2009). But if NPM
actually undermines e¤ort (which of course is debatable, see Stazyk, 2010),
would this necessarily come from crowding out of intrinsic motivation? Important aims of NPM include decentralization of management authority,
more discretion and ‡exibility, less bureaucracy and less rules. These institutional changes may a¤ect the enforcement environment. Indeed, OECD
(2004) argues that weak enforcement and implementation problems are one
of the key challenges for the introduction of pay for performance schemes in
the public sector.
6
6
Dahlström and Lapuente (2010) argue that lower enforcement in the public sector
6
Another example is the puzzling cross country relationship between wage
dispersion and productivity. Scandinavia, for instance, is known for a combination of high productivity and compressed wages (and thus lower powered
incentives, cet. par). There may be a number of institutional explanations
for this relationship (see Barth et al., 2014), our model o¤ers a complementary one: countries with high productivity and seemingly low-powered
incentives may have stronger enforcement institutions and thus more credible incentives.
The paper is organized as follows. In Section 2 we discuss related literature. In Section 3 we present the basic model and study variations in enforcement probability under limited liability and risk aversion, respectively.
Section 4 concludes.
2
Related literature
A contribution of the paper is to consider probabilistic enforcement in an
otherwise standard moral hazard model with risk aversion or limited liability.
In the classic moral hazard models (e.g. Holmström, 1979, Innes, 1990),
perfect enforcement is assumed, while in models of incomplete contracting,
it is commonly assumed that contracting is prohibitively costly so that legal
enforcement is impossible (starting with Grossman and Hart, 1986). A
literature has thus evolved investigating the feasibility of performance pay
leads to less use of performance pay, and provide some evidence that pay for performance
schemes are more frequently used in countries where the credibility of incentives presumably is higher. In contrast, we point out that the magnitude - or incentive intensity - of
a given performance pay scheme may actually increase in order to compensate for lower
enforcement.
7
schemes in situations where there is no scope for legal enforcement, see e.g.
Levin (2003) on relational contracts, and MacLeod (2003) on subjective
performance evaluation. However, imperfect legal enforcement (rather than
no legal enforcement) is increasingly recognized as an important ingredient
in models of contractual relationships. Some papers focus on the relationship
between ex post evidence disclosure and enforceability (Ishiguro, 2002; Bull
and Watson, 2004), while others focus on the relationship between ex ante
contracting and enforceability (Battigalli and Maggi, 2002, Schwartz and
Watson, 2004, Shavell 2006). There is also a growing literature on the
interaction between legal imperfect enforcement and informal (relational)
enforcement, see Sobel (2006), MacLeod (2007), Battigalli and Maggi (2008),
Kvaløy and Olsen (2009, 2012) and Dhillon and Rigolini (2011), and on the
role of imperfect enforcement in …nancial contracting, see Ellingsen and
Kristiansen (2011).
A central implication of our model is that enforcement and performance
pay - or what we will also call contractual incentives - may be substitutes.
In that sense our paper is related to models showing the substitutability
between explicit contracts and informal relational contracts (see Baker, Gibbons and Murphy, 1994, and Schmidt and Schnitzer, 1995). In these models, improved explicit contracts may reduce feasible incentive pay under
relational contracting, but e¤ort is still positively related to the sum of contractual incentives. In contrast, we …nd that e¤ort may be negatively related
to contractual incentives. Of particular interest in this respect is a recent
working paper by Demougin, Fabel and Thomann (2013). They present an
empirical analysis documenting a negative correlation between performance
8
pay and productivity in a large insurance company. Interestingly, they …nd
that a higher probability of premature contract termination increases the
explicit incentives, but reduces productivity. They also o¤er a theoretical
analysis that can capture this relationship. In a repeated game model, they
show that a higher probability of contract termination may reduce the effect of non-enforceable salary promises (implicit incentives) and thus create
a negative equilibrium relationship between productivity and explicit incentives. Our model suggests an alternative explanation: A higher probability
of premature contract termination may reduce expected enforcement probability v. This makes the …rm o¤er higher bonuses, but the product of higher
bonuses and lower v may lead to lower e¤ort.
Our result also bears similarities to Ehrlich and Becker (1972), who
contrary to the moral hazard intuition show that people may increase selfprotection activities in the presence of market insurance, which is a similar
kind of response as increasing e¤ort under lower-powered incentives.7 Our
argument also bears similarities to the type of argument Prendergast (2000
and 2002a,b) uses in a series of papers in order to explain a positive relationship between uncertainty and incentives. Prendergast shows that a
third variable, either delegation or a supervisor’s evaluation, may interact
with both uncertainty and incentives and thereby create a positive relationship between the two. Of particular interest is his model on the credibility of
subjective assessments, where third party investigations are used to generate
veri…able outcomes. Prendergast shows that when the outcome of investi7
Dionne and Eckhoudt (1985) and Konrad and Skaperdas (1993), among others, have
explained this by showing how more risk aversion actually may lead to less self-protection
activities.
9
gations is more uncertain, e¤ort incentives are reduced, and so the principal
must compensate by o¤ering higher-powered pay-for-performance schemes.8
In a di¤erent modeling set-up, we take this argument one step forward by
showing that the equilibrium relationship between e¤ort and performance
pay may be negative in situations where enforceability changes.9
3
Incentives and enforceability
We consider a relationship between a principal and an agent, where the agent
produces output x for the principal. Output is a random variable and the
agent’s e¤ort a a¤ects the probability distribution. We assume that output
is observable to both parties, but that the agent’s e¤ort level is unobservable
to the principal, so the parties must contract on output: The principal o¤ers
a wage w(x) = s + (x) where s is a non-contingent …xed salary and (x)
is a contingent bonus. The principal is assumed to be risk neutral, but we
allow the agent to be risk averse, with a utility function u(w).
We assume that contracts are not perfectly enforceable by the court of
law, but that there is a probability v 2 (0; 1) that the principal is committed
to honor the full contract, i.e. the …xed salary plus the discretionary bonus.
We invoke the standard assumption that participation from the agent is
8
Similar arguments are also present in the law and economics literature: If damages have
to be paid with a probability lower than 1, more severe punishments (‘damage multipliers’)
are needed for deterrence, see e.g. Cooter (1991).
9
Interestingly, Prendergast argues that third party investigations are more common
in jobs where it is important to identify mistakes. These are sometimes referred to as
guardian jobs (Baron and Kreps, 1999), and often apply to government bureaucrats.
Hence, performance related pay in the public sector may partly be used to compensate
for the risk of third party investigations, which in our model also may imply higher risk
of not being paid anticipated bonuses.
10
always veri…able by a court of law, and hence that the non-contingent …xed
salary is enforceable and must always be paid. The probability v is thus
the probability of legal or informal enforcement of the bonus contract (x).
Given that the agent accepts the contract, he is thus paid the …xed salary
s, then exerts (hidden) e¤ort a, after which the output x is realized and
observed by both parties. Finally the agent is paid the discretionary bonus
(x) with probability v. (So with probability 1
v he receives no bonus.)
We abstract here from strategic behavior that may a¤ect the probability
of enforcement. Instead the modeling approach allows for several kinds of
exogenous variations. One interpretation is that a court decides whether or
not the principal has to pay the bonus as promised. The parameter v is then
the ex ante probability that the contract will be legally enforced. A second
interpretation is that the parties learn about the contractual environment
ex post, for instance to what extent social or political concerns matter for
the given contractual relationship. The parameter v is then the ex ante
probability that the principal is committed to pay the discretionary bonus.
We assume that the probability v is common knowledge. However, in
many situations, one might think of v as being private information, for
instance that the principal knows more about the probability of enforcement
than the agent, or that parties have di¤erent beliefs about the enforcement
probability. An extension of the model in which v is private information
would thus be valuable. Still, our current set-up can account for situations
where the parties share the same belief about the enforcement environment.
We will now deduce the optimal contract and discuss variations in enforcement probability v.
11
We consider a simple setting with two outcomes, xh ; xl and xh > xl . The
probability distribution depends on (hidden) e¤ort, and to simplify notation
we will measure e¤ort a such that Pr( xh j a) = a. This e¤ort is costly with a
strictly convex cost function C(a), with C(0) = C 0 (0) = 0 and C(1) so large
that a = 1 is never pro…table, i.e. C(1) > xh .
After the agent has accepted a contract o¤er, he chooses e¤ort to maximize his expected utility, given by
U (a; w; v; s) = v(au(wh ) + (1
a)u(wl )) + (1
v)u(s)
C(a);
where u( ) is increasing and concave. For each outcome xj , the agent gets
the payment wj = s +
j
with probability v, and the payment (…xed salary)
s otherwise, and this gives expected utility as speci…ed. The expected utility
depends on the three payment levels wh ,wl and s, and we may thus consider
these as the principal’s instruments.
The agent’s optimal e¤ort is given by
Ua (a; w; v; s) = v(u(wh )
u(wl ))
(This ‘…rst-order approach’is valid since Uaa =
C 0 (a) = 0
(IC)
C 00 < 0.)
The principal then chooses wages (and e¤ort a) to maximize her payo¤,
subject to the agent’s choice, represented by IC, and the agent’s participation constraint:
U (a; w; v; s)
12
Uo
(IR)
The principal, assumed risk neutral, has payo¤
V (a; w; v; s) = a [xh
vwh ] + (1
a) [xl
vwl ]
(1
v)s
Consider …rst the maximization of V for given e¤ort a; i.e. the optimal
way to implement a given e¤ort level. Moreover, following Grossman and
Hart (1983), formulate the problem in terms of utilities (uh ; ul ; us ) and let
w(u) be the inverse of the utility function. The optimal implementation will
de…ne the utility levels (and corresponding wage payments) as functions of
a; v, i.e. ui = ui (a; v). Let V~ (a; v) be the optimal value function, i.e., for
given e¤ort a, V~ (a; v) is the principal’s value when this e¤ort is implemented
optimally (in a least costly manner). Consider then the choice of optimal
e¤ort. Given concavity (V~aa (a; v) < 0), this optimal e¤ort a = a(v) will be
given by the …rst-order condition (FOC) V~a (a; v) = 0.
Now consider an exogenous change of the enforcement probability (v).
To examine whether this change induces a negative relationship between
e¤ort and performance pay, we ask the following: i) will e¤ort increase when
the enforcement probability increases and ii) will contractual incentives at
the same time become weaker? Regarding the response of e¤ort it follows
from the FOC V~a (a; v) = 0 that we have
da
=
dv
1
V~av (a; v)
~
Vaa (a; v)
(1)
The sign of V~av (a; v) will thus determine whether e¤ort will increase or
decrease in response to variations in v. We will come back to this sign
13
below.
Consider next how contractual (monetary) incentives will vary with v.
From the agent’s IC constraint we see that contractual incentives for e¤ort
are given by uh
ul , i.e. the utility di¤erence between high and low output
that follows from the wages speci…ed in the contract. In equilibrium this
di¤erence will depend on v; it will be given by
m(v) = uh (a(v); v)
ul (a(v); v)
By the IC constraint, in equilibrium we will have vm(v) = C 0 (a(v)), and
da
00
thus, by di¤erentiation; m(v) + v dm
dv = C (a) dv , where a = a(v). Using
again IC, we then obtain
v dm
aC 00 (a) da v
=
m(v) dv
C 0 (a) dv a
1
(2)
Given that equilibrium e¤ort increases with higher v ( da
dv > 0), we see that
the sign of
dm
dv ,
i.e. whether contractual incentives increase or decrease
with higher enforceability, will be determined by the magnitudes of the two
elasticities on the RHS of (2), namely the elasticities of marginal costs and
equilibrium e¤ort, respectively. The former is exogenous (for given a), but
the latter is endogenous and must be calculated by usual comparative statics
methods (see formula (1)).
14
3.1
Risk neutrality and limited liability
Consider …rst the case of risk neutrality, represented by u(w) = w, and
limited liability for the agent, and assume the latter entails that all payments
must be non-negative (wh ; wl ; s
0). Assume also here for simplicity that
U0 = 0, so that the agent’s IR constraint will not bind.
By assumption, utilities and wages are now the same (ui = wi ), and
there is no need to distinguish between the two. The agent’s IC constraint
is here
v(wh
C 0 (a) = 0:
wl )
The principal’s payo¤ then takes the form
V (a; w; v; s) = axh + (1
a)xl
v(awh + (1
= axh + (1
a)xl
aC 0 (a)
where we have substituted for v(wh
vwl
a)wl )
(1
(1
v)s;
v)s
wl ) from the IC constraint.
Given the limited liability constraints of non-negative payments, it is
clearly optimal to set wl = s = 0. Hence we see that the principal’s optimal
value V~ (a; v) will in fact be independent of v in this case. The optimal
e¤ort, say a = a0 , is then also independent of v. But then we see from
the agent’s IC constraint that the monetary incentive, which in this case
is simply the payment (bonus) wh for a successful outcome, is given by
wh = C 0 (a0 )=v and is hence inversely proportional to v. Put di¤erently,
contractual incentives are here unitary elastic with respect to changes in v.
Any change in v is thus exactly o¤set by a change in monetary incentives
15
such that e¤ort is kept constant. This is of course in accordance with the
more general formula (2), where
v dm
unitary elastic ( m(v)
dv =
da
dv
= 0 when contractual incentives are
1).
This case illustrates in a simple way that enforcement and contractual
incentives may be substitutes. In this special case they are even perfect
substitutes, so variations in contractual incentives - due to variations in
enforcement probability - have no e¤ect on e¤ort at all. But monetary
incentives exactly o¤setting variations in v are special for this case.10 In
particular, perfect substitutability is not robust to changes in risk attitudes.
We consider the case of risk aversion next.
3.2
Risk aversion
Assume now that the utility function u(w) is strictly concave. Assume also
that there are no limited liability constraints, and consider the principal’s
optimization problem. Forming the Lagrangian L = V + (U
with multipliers
and
Uo ) + Ua ,
on the IR and IC constraints, respectively, one sees
that optimal utility levels uk (and associated payments w(uk )) for implementing a given e¤ort level a, satisfy
w0 (uk ) =
+
p0k (a)
; k = h; l;
pk (a)
10
w0 (us ) = ;
(3)
It can be shown that enforcement and contractual incentives are not neccesarily perfect
substitutes in a more general setting with continuos outcomes. The response on e¤ort will
then depend, among other things, on the shape of the cost function.
16
where ph (a) = a = 1
pl (a). Assume the utility function is such that
the IR constraint binds at the optimum.11 Noting that w0 (uk ) = 1=u0 (wk )
we see that the optimality conditions are standard (e.g. Holmström 1979),
and re‡ect the trade-o¤ between providing insurance and incentives for the
agent. This trade-o¤ is relevant for the performance dependent payments
wh ,wl , but not for the …xed payment s.
These optimality conditions together with the constraints now de…ne
(uh ; ul ; us ; ; ) and the principal’s optimal value V~ as functions of a; v.
Comparative statics for the optimal e¤ort level and contractual incentives
then follow from (1) and (2).
To illustrate the general points arising under risk aversion without getting expressions that are too messy we consider a speci…c utility function,
namely u(w) = w1=2 , implying that the agent exhibits constant relative risk
aversion with coe¢ cient 1=2. The appendix contains some comparative statics formulae for general u( ). In the remainder of the paper we thus invoke
the following assumption.
Assumption. u(w) = w1=2 and U0 > 0 is so large that (IR) binds.12
Consider now how optimal e¤ort responds to variations in v. Given
concavity (V~aa < 0), we saw above (formula (1)) that
da
dv
has the same sign
as V~av (a; v). For the speci…ed utility function we …nd
V~av (a; v) =
2(1
a)
C 00 (a)a
C 0 (a)
2a + 1 C 0 (a)2 =v 2
(4)
11
For u(w) de…ned for w > w, su¢ cient conditions are u(w+ ) = 1 and for every e¤ort
a, there is a wage that meets the participation constraint for that e¤ort (u(w) C(a) = U0 ).
12
We verify in the appendix that IR must bind if U0 is not too small.
17
Thus we may state the following
Lemma 1 Given concavity (V~aa < 0),
da
dv
n
has the same sign as 2(1
00
a) CC 0(a)a
(a)
This tells us that the elasticity of marginal cost is an important determinant for whether equilibrium e¤ort increases or decreases with higher
enforceability. If this elasticity is bounded above, say by
> 0, then we
1
2( +1) .
see that the expression in Lemma 1 will be negative for a > 1
This indicates that e¤ort may decrease with higher probability of contract
enforcement. Here we will focus on environments where this is not the case.
Note that, while the elasticity
C 00 (a)a
C 0 (a)
measures the percentage change
in marginal cost per percent increase in the probability of success (a), the
00
(a)
expression (1 a) CC 0 (a)
measures the percentage change in marginal cost per
percent reduction in the probability of failure. If the latter elasticity is no
smaller than 1, i.e. if marginal costs are elastic in this sense (with respect
to reductions in the probability of failure), we see that the expression in the
lemma is positive, and hence that e¤ort will then increase with higher v.13
We illustrate below that a relatively large class of cost functions has this
property.
Consider next whether we may have contractual incentives decreasing
and e¤ort increasing in v ( dm
dv < 0 and
da
dv
> 0). Recall that (1
00
(a)
a) CC 0 (a)
can
be interpreted as the elasticity of marginal costs with respect to reductions
in the probability of failure (1
a). As shown in the appendix we have the
following result.
Proposition 1 Suppose marginal costs are elastic in the sense that (a)
13
In fact an elasticity no smaller than 0.5 would be su¢ cient.
18
o
2a + 1 .
00
(a)
(1 a) CC 0 (a)
000
(a)
1, and moreover that (1 a) CC 00 (a)
1
2.
Then
da
dv
> 0 and
dm
dv
<
0, so e¤ ort and contractual incentives move in opposite directions when v
varies. Furthermore, contractual incentives in monetary terms (w(uh )
w(ul )) then also decrease in v, and hence move in the opposite direction of
e¤ ort.
The intuition for this kind of result is as follows. Improved enforceability
increases the agent’s incentives to exert e¤ort (other things equal), but also
increases the principal’s wage costs per unit of e¤ort (since the probability
that the principal actually has to pay as promised increases). Now, even
though the principal …nds it optimal to induce higher e¤ort when v increases,
she will make a trade-o¤ between the bene…ts from higher e¤ort and the
expected wage costs from higher v: She may thus reduce these wage costs by
providing lower-powered incentives. In other words, improved enforcement
may crowd out contractual incentives.
The intuition for responses to weaker enforcement is of course mirrored:
Weaker enforcement induces lower e¤ort since the probability that the agent
actually is paid decreases. But since the wage cost per unit of e¤ort is reduced, the principal may want to mitigate the reduction in e¤ort by providing higher-powered contractual incentives.
We now brie‡y examine what the conditions given in Proposition 1 may
entail. Recall that we have measured e¤ort a such that Pr( xh j a) = a. This
normalization has implications for the e¤ort cost function C(a). If e = q(a)
is the amount of ‘natural’e¤ort (e.g. measured in hours of work) required
to obtain probability of success a, and c(e) is the cost of this e¤ort, then
19
C(a) = c(q(a)), 0
a < 1. (For example, if a = Pr( xh j e) = e=(1 + e) then
e = q(a) = a=(1
a).) The cost function C( ) is convex when c( ) and q( )
are convex, which are both natural assumptions.
For such a cost function, i.e. a function of the form C(a) = c(q(a)),
where c(e) is the cost of ‘natural’ e¤ort, the elasticity of marginal costs
appearing in the proposition can be written as
(1
where "c (e) =
q
ec00 (e)
c0 (e)
a)
C 00 (a)
= "c (q(a)) q (a) + "q (a)
C 0 (a)
is the standard elasticity of marginal e¤ort costs, and
and "q are the elasticities of, respectively, q( ) and q 0 ( ) with respect to
reductions in the probability of failure;
q (a)
= (1
0
(a)
a) qq(a)
and "q (a) =
00
(a)
. We illustrate such calculations with an example.
a) qq0 (a)
(1
Example. Suppose q(a) = a=(1 a), re‡ecting that, in terms of ‘natural’
e¤ort e = q(a) we have Pr( xh j e) = q
q
1 (e)
= e=(1 + e). Then we …nd
= 1=a and "q = 2, so the elasticity of marginal costs appearing in the
proposition is (a) = (1
00
(a)
a) CC 0 (a)
= "c =a + 2, which certainly exceeds 1.
We moreover …nd here that
where14
c
=
ec000
c00
(a)(1
000
(a)
a) CC 00 (a)
= ( c =a + 6)"c =a + 6,
is the standard elasticity of c00 ( ). The second condition in
Proposition 1 will therefore hold if (2 c =a+11)"c =a+10 > 0. Both conditions
in the proposition will thus hold if
will also hold if
c
c
0; and if a is bounded below, they
is not too negative.15 A large class of reasonable cost
functions will have these properties.
14
Third order derivatives will necessarily appear since the FOC for e¤ort contains second
order derivatives.
15
In fact, from Lemma 2 in the appendix it follows that c
1 is a su¢ cient condition.
20
To illustrate the magnitudes of the e¤ects on e¤ort and contractual incentives, we have numerically solved the model as speci…ed in this example,
with cost of ‘natural’ e¤ort given by c(e) = e2 . For parameters U0 = 2,
xh
xl = 100, we …nd that when v is increased from v = 2=3 to v = 1, mon-
etary incentives decrease by 24%, while e¤ort increases by 5%. (Calculations
available from the authors.)
As a …nal remark, note (from (3)) that in the absence of limited liability,
the optimal contract has wl < s, which implies a negative bonus (a ‘stick’)
l
< 0 for low output. If the contract is enforced, the court will then
require the agent to pay the principal. We have (implicitly) assumed that
the probability of contract enforcement v is independent of who is in the
position to cheat on the contract. But one may also think of situations
where v depends on whether it is the principal or the agent who has to
pay. However, since negative bonuses are rarely observed in employment
contracts, it is also worthwhile to study the case with both limited liability
and risk aversion, i.e. impose wl
s in the current model. In this case a
bonus will only be paid for the high outcome (and enforced with probability
v), and it then turns out that e¤ort will unambigously increase with higher
v. Moreover, the conditions for e¤ort and contractual incentives to move in
opposite directions become di¤erent and in some sense stricter than those
given in Propositon 1. In particular, for the parametrization in our example
above, the conditions for
dm
dv
to be negative will only hold if equilibrium
e¤ort a is not too large.16
16
Proofs of theses assertions are available from the authors.
21
These modi…ed results follow under our maintained Assumption 1, which
includes a binding participation constraint for the agent. But in a setting
with limited liability (and risk aversion) it is of interest also to examine
the alternative case where IR doesn’t bind. In this case (IR not binding,
wl
s and u(w) = w1=2 ), it turns out that e¤ort unambigously increases
with higher v, while the conditions for
dm
dv
to be negative become much less
strict than those given in Proposition 1; in particular
C 000 (a)a
C 00 (a)
>
3
2
turns
out to be a su¢ cient condition. (See (11) in the appendix.) This illustrates
that the negative relationship between e¤ort and contractual incentives that
we have analysed, is not critically dependent on all the assumptions we have
made in the formal analysis leading to Proposition 1.
4
Concluding remarks
We o¤er a simple model where contractual monetary incentives and e¤ort
are negatively related even if there is no crowding out of non-monetary
motivation. The idea is that a change of contract enforcement can a¤ect
both e¤ort and contract design in ways that generate a negative equilibrium
relationship between e¤ort and bonuses. In particular, weaker enforcement
can induce lower e¤ort, but at the same time increase the incentive intensity
in the employment contract.
The model thus indicates that the combination of two recent trends; a
higher incidence of performance related pay, along with a skepticism over
whether performance pay actually increases e¤ort, may be associated with
weaker enforcement of bonus contracts.
22
One can also think of potential reinforcement mechanisms, which we
have not modeled in this paper. If there is a general increase in the use
of performance pay, this in itself may lower the probability of enforcement.
The reason is twofold: First, if the fraction of jobs with performance pay
increases, one might expect that newcomers, i.e. jobs/employment relationships that have newly introduced performance pay, on average have weaker
enforcement than those with experience. This may reduce the general enforcement expectations. Second, higher bonuses may increase the probability that the employer cannot a¤ord to pay, or may try to deviate from
paying. These issues will endogenize the enforcement probability, but in a
way that potentially reinforces the mechanisms we discuss in the paper.
23
Appendix
We …rst derive a comparative statics formula under risk aversion for
general utility function (when the IR constraint binds). Recall that (given
concavity V~aa < 0) the marginal e¤ect on e¤ort
da
dv
has the same sign as
V~av (a; v). Here we …nd
V~av (a; v) =
[w(uh )
w(ul )] + w0 (uh )(uh
ul ) + w00 (uh )(uh
ul )2 (1
a)
C 00 (a)a
C 0 (a)
(5)
v (w0 (uh )
w0 (ul )) + (w00 (uh )
w00 (ul ))(uh
ul )(1
a)
C 00 (a)a @ul
C 0 (a) @v
where ui = ui (a; v), and
w00 (us )(us ul ) + (1 v)aw00 (uh )(uh ul )=v
@ul
=
>0
@v
vw00 (us ) + (1 v) [aw00 (uh ) + (1 a)w00 (ul )]
The …rst line of (5) is positive due to convexity of w( ). If w000 ( )
(6)
0, then the
square bracket in the second line is positive, and the sign of V~av (a; v) cannot
be determined without comparing the magnitudes of the terms entering the
formula. Similarly, if w000 ( ) < 0, the sign of the square bracket, and hence
the sign of V~av (a; v) depends on the magnitudes of the terms in the formula.
To simplify these comparisons,we proceed in the paper with a speci…c utility
function (for which w000 ( ) = 0).
Veri…cation of (5) and (6)
Since V~ (a; v) is an optimal value function, we have V~a = La = Va +
Ua + Uaa . Since Ua = 0 by IC and Uaa =
From the conditions (3) we obtain
C 00 , we have V~a = Va
= (w0 (uh )
24
w0 (us ))a, and hence
C 00 .
V~a (a; v) = [xh
xl ]
v [w(uh )
w(ul )]
[w0 (uh )
w0 (us )] C 00 (a)a;
where the utilities ui are functions of (a; v), determined by conditions (3)
and the (binding) IR and IC constraints. From conditions (3) we moreover
get
w0 (us ) = aw0 (uh ) + (1
a)w0 (ul );
(7)
and thus
V~a (a; v) = [xh
xl ]
v [w(uh )
w(ul )]
w0 (uh )
w0 (ul ) (1
a)C 00 (a)a
(8)
where ui = ui (a; v), i = h; l; s are given by, say (7) and the IR and IC
constraints. Di¤erentiation then yields
h
h
V~av = [w(uh ) w(ul )] v w0 (uh ) @u
@v
i
h
l
h
w00 (ul ) @u
w00 (uh ) @u
@v
@v (1
V~av =
h
[w(uh )
(w00 (uh )
w(ul )]
a)C 00 (a)a
h
ul ) + v( @u
@v
From IC we have 0 = (uh
then yields
l
w0 (ul ) @u
@v
h
v (w0 (uh )
l
w00 (ul )) @u
@v
Collecting terms containing
@ul
@v ),
@ul
@v ,
and substituting for
l
w0 (ul )) @u
@v
w00 (uh )(uh
i
w0 (uh )(uh
i
ul )=v (1
ul )=v
a)C 00 (a)a
and substituting 1=v = (uh
@uh
@v
i
ul )=C 0 (a)
from IC then yields formula (5):
Consider …nally (6). This formula follows from straightforward di¤erentiation of the three equations (IC), (IR) and (7), and solving for
@ul
@v .
Proof of Lemma 1. We verify below that IR must bind if U0 is not
too small. For w(u) = u2 and hence w00 (u) = 2 we then have, from (6)
@ul
@v
=
(us ul )+(1 v)a(uh ul )=v
v+(1 v)[a+(1 a)]
= (us
25
ul ) + (1
v)a(uh
ul )=v
where from (7) we now have auh + (1
@ul
@v
= a(uh
a)ul = us , and hence
v)=v) = aC 0 (a)=v 2 ;
ul )(1 + (1
ul )v = C 0 (a). Then from
where the last equality follows from (IC); (uh
(5) we obtain
V~av (a; v) =
u2h
v [(2uh
u2l + 2uh (uh
ul ) + 2(uh
ul )2 (1
00
(a)
a)a CC 0 (a)
2ul ) + 0] (aC 0 (a)=v 2 )
So we have
V~av (a;v)
uh ul
=
[uh + ul ] + 2uh + 2(uh
ul )(1
00
(a)
a)a CC 0 (a)
2aC 0 (a)=v
and hence from IC
V~av (a;v)
C 0 (a)=v
00
(a)
= C 0 (a)=v + 2(C 0 (a)=v)(1 a)a CC 0 (a)
2aC 0 (a)=v
n
o
00 (a)
= 1 + 2(1 a)a CC 0 (a)
2a C 0 (a)=v
This proves formula (4). It remains to verify that IR must bind if U0 is
not too small.
If IR does not bind and thus
= 0, we have
@L @L
@ul ; @us
0, so ul = us =
0, and hence from IC vuh = C 0 (a). Then the expected wage payment is
vaw(uh ) = aC 0 (a)2 =v, and the principal’s value is axh +(1 a)xl aC 0 (a)2 =v.
This value is maximal (and positive) for some e¤ort an > 0, and the agent’s
expected utility is then an vuh
C(an ) = an C 0 (an )
C(an )
un . When the
agent’s outside value exceeds un , IR must bind. This completes the proof
of Lemma 1.
Proof of Proposition 1. We …rst state the following lemma, which is
proved below.
Lemma 2 Given concavity (V~aa < 0), a su¢ cient condition for contractual
26
incentives to be decreasing in v, ( dm
dv < 0) is that (a(v); v) < 0, where
(a; v) = 2(1
v)
(2
3a)
C 00 (a)
C 0 (a)
2(1
a)a
C 000 (a)
C 0 (a)
Moreover, concavity holds (V~aa < 0) if both (a; v) < 0 and V~av (a; v) > 0.
In this case we thus have
dm
dv
< 0 and
da
dv
> 0, so e¤ ort and contractual
incentives move in opposite directions. Furthermore, in this case contractual
incentives in monetary terms (w(uh )
w(ul )) also decrease in v, and hence
move in opposite direction of e¤ ort.
Given this lemma, we see that Proposition 1 follows if the conditions
stated there imply (a; v) < 0 and V~av (a; v) > 0. The latter follows from
00
(a)
a) CC 0 (a)
(a) = (1
(1
C 000 (a)
a) C 00 (a)
1
2
(a; v)(1
1 and formula (4). Now note that, for v 2 (0; 1) and
we have
a) = 2(1
< 2(1
v)(1
a)
(2
a)
(2
3a) (a)
3a) (a)
2a (a)(1
a (a) = (2
2a)(1
000
(a)
a) CC 00 (a)
(a))
0
Hence (a; v) < 0, and then Proposition 1 follows from the lemma.
Proof of Lemma 2. From (2) we see that
where
da
dv
=
1
V~ (a; v).
V~aa (a;v) av
(a; v) = v
Note that if V~aa < 0, then
dm
dv
De…ne
C 00 (a) ~
Vav (a; v)v + V~aa (a; v)
C 0 (a)
dm
dv
(9)
< 0 i¤ (a; v) < 0. On the other hand, if
V~av > 0 and (a; v) < 0, then we must have V~aa < 0 and
now show that (a; v)
00
(a(v)) da
< 0 i¤ CC 0 (a(v))
dv v 1 < 0
dm
dv
< 0. We will
C 0 (a)2 (a; v), where (a; v) is given in Lemma 2,
27
and hence that (a; v) < 0 implies (a; v) < 0. This will then prove the …rst
two statements in the lemma.
To this end, consider the terms in (a; v). From (8) we have
h
i h
i
0 (u ) @ul
00 (u ) @uh
00 (u ) @ul (1 a)C 00 (a)
h
V~aa = v w0 (uh ) @u
w
a
w
w
l @a
h @a
l @a
@a
[w0 (uh )
d
w0 (ul )] da
(a(1
From IC (v(uh ul ) = C 0 (a)) we have
@uh
@a
a)C 00 (a))
=
@ul
@a
+C 00 (a)=v. Substituting
this and w0 (u) = 2u in the expression for V~aa we obtain
V~aa =
v [2uh C 00 (a)=v]
[2uh
l
2ul ] @u
@a
v [2uh
a [2C 00 (a)=v)] (1
2a)C 00 (a) + a(1
2ul ] ((1
a)C 00 (a)
a)C 000 (a))
From IR, IC and (7) we obtain
ul = us
a(uh
= C 0 (a)(1
1=v)
ul ) = U0 + C(a)
aC 0 (a)=v
(10)
and hence
@ul
@a
aC 00 (a)=v
Using this and IC we then obtain
V~aa (a; v) 12 =
v [C 0 (a)=v + ul ] C 00 (a)=v v [C 0 (a)=v] (C 0 (a)(1
1
v)
aC 00 (a)=v)
a [C 00 (a)=v)] (1 a)C 00 (a) [C 0 (a)=v] ((1 2a)C 00 (a)+a(1 a)C 000 (a))
Now, collecting terms, and taking into account ul
V~aa (a;v)
v
C 0 (a)2
C 00 (a)
C 0 (a)
[5a
3]
2(v
1)
2(1
0 we have
h 00
(a) 2
a)a ( CC 0 (a)
) +
C 000 (a)
C 0 (a)
i
Next consider (a; v) de…ned in (9) above. From this de…nition, the last
inequality for V~aa and the formula (4) for V~av we have
(a;v)
C 0 (a)2
=v
C 00 (a) V~av (a;v)
C 0 (a) C 0 (a)2 v
+
V~aa (a;v)
C 0 (a)2
n
o
00 (a)
00 (a)
v CC 0 (a)
v 2a(1 a) CC 0 (a)
2a + 1 =v 2
n 00
h 00
(a)
(a) 2
+ CC 0 (a)
[5a 3] 2(v 1) 2(1 a)a ( CC 0 (a)
) +
28
C 000 (a)
C 0 (a)
io
Collecting terms we get
C 00 (a)
C 0 (a) (3a
(a;v)
C 0 (a)2
This veri…es that
2) + 2(1
(a;v)
C 0 (a)2
v)
2(1
000
(a)
a)a CC 0 (a)
(a; v), which was to be shown.
It remains to prove the last statement in Lemma 2, i.e. that contractual
incentives in monetary terms (w(uh )
w(ul )) also decrease in v. To show
this, note that di¤erentiation of w(uh )
d(uh ul )
0
l
w0 (ul ) du
+ (w0 (uh )
dv = w (uh )
dv
h
w0 (uh ) du
dv
where
duh
dv
=
ul = us
w(ul ) yields
d
dv uh (a(v); v).
a(uh
l
w0 (ul )) du
dv ;
From (10) and IC we have
ul ) = U0 + C(a)
a(uh
ul );
and hence
dul
dv
= (C 0 (a)
(uh
ul )) da
dv
a d(uhdv ul ) <
where the inequality follows from (uh
a d(uhdv ul )
ul ) = C 0 (a)=v and
da
dv
> 0. Thus
we have
d(uh ul )
0
l
h
w0 (ul ) du
+(w0 (uh ) w0 (ul ))( a d(uhdv ul ) )
w0 (uh ) du
dv
dv < w (uh )
dv
a) + aw0 (ul )) d(uhdv ul ) < 0
= (w0 (uh )(1
This demonstrates that when
v, then w(uh )
da
dv
> 0 and m = uh
ul is decreasing in
w(ul ) is also decreasing in v, as was to be shown. (This
property can be generalized to any (inverse utility) function with w000
0.)
This completes the proof.
Non-negative bonuses
Assume now that bonuses must be non-negtive, and hence wl
s, and
in addition that IR doesn’t bind. (An analysis for the case where IR binds is
available from the authors.) Then with
@L @L
@ul ; @us
= 0 in the Langrangean, we have
0, so ul = us = 0, which by assumption is the minimal utility for
29
the given utility function. (This implies that the constraint wl
s in fact
does not restrict the principal when IR does not bind.) From IC we then have
vuh = C 0 (a), and the expected wage payment for the principal is vaw(uh ) =
aC 0 (a)2 =v. The principal’s value is thus V~ = axh + (1
a)xl
aC 0 (a)2 =v.
Given that V~aa < 0, the optimal e¤ort is given by V~a = 0, and this e¤ort is
increasing in v i¤ V~av = (C 0 (a)2 + a2C 0 (a)C 00 (a))=v 2 > 0. This is satis…ed
for any strictly increasing and convex cost function.
From the proof of Lemma 2 we have that
dm
dv
< 0 if (a; v) < 0, where
(a; v) is de…ned in (9). By insertion we obtain here, after some algebra:
(a; v) = v
C 00 (a) ~
Vav (a; v)v + V~aa (a; v)
C 0 (a)
=
(3C 00 (a) + 2aC 000 (a))C 0 (a)
(11)
This shows that
dm
dv
< 0 if
C 000 (a)a
C 00 (a)
>
in the last paragraph of Section 3.2.
30
3
2,
and thus veri…es the assertions
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