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European Economic Review

European Economic Review 68 (2014) 93–105

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Bonuses and managerial misbehaviour

Caspar SiegertDepartment of Economics, University of Munich, Kaulbachstr. 45, 80539 Munich, Germany

a r t i c l e i n f o

Article history:Received 10 June 2013Accepted 19 February 2014Available online 12 March 2014

Keywords:ContractsIncentivesGovernanceExecutive compensation

x.doi.org/10.1016/j.euroecorev.2014.02.00721 & 2014 Elsevier B.V. All rights reserved.

ail address: caspar.siegert@lrz.uni-muenchenichael Peel and George Parker, “Brown atww.ft.com/intl/cms/s/0/d35908c6-8810-11dn alternative explanation would be that this blted in fines of several billion dollars and excholders' best interest.hn and John (1993) offer a theoretical expla

a b s t r a c t

Profit-based bonus payments have been criticised for encouraging managers to takeexcessively risky actions or to engage in other activities that are not in the firm's bestinterest. We show, however, that large bonuses may discourage managers from suchmisbehaviour, because they have more to lose in the event that misbehaviour is detected.Thus, large bonuses may be an optimal way for firms to control misbehaviour. Our findingsheds new light on recent proposals to regulate bonuses.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

During the recent financial crisis, excessive bonuses for bankers were frequently blamed as a source of irresponsiblebehaviour. In 2008, British Prime Minister Gordon Brown described the size and structure of bonuses in the financial sectoras “irresponsible” and the French minister of Economic Affairs Christine Lagarde judged incentive schemes for bankers as“perverse”.1 According to these arguments, bonus schemes induced bankers to take actions that were not in the long-terminterest of their employers, such as selling clients unsuitable products or engaging in risky trades. Bankers could reap sizablebenefits if everything went well, but their employer had little means of punishing them in case he found out that they hadbeen misbehaving. This line of argument suggests however that banks did collectively set sub-optimal incentive schemes, anidea which at least warrants some closer scrutiny.2

We define misbehaviour as an action that increases the contractible profit signal that the agent's bonus is based on, but isstill not in the principal's interest. We show that high-powered incentives are not only robust to potential misbehaviour, butthe cost of undesirable behaviour may even increase the optimal bonus an agent is offered. Offering large bonuses hencemay have been optimal even if banks were aware of their employees' ability to engage in misconduct. Our finding thatreducing pay-performance ratios may not be a suitable way to encourage compliance is consistent with the observation byJohn and Qian (2003) that controlling for leverage,3 banks offer incentives that are not significantly different from the onesgiven in firms where managerial misbehaviour is less costly. Comparisons of incentives in a number of different industriessuch as in Conyon and Murphy (2000), Murphy (1999) and Zhou (2000) paint a similar picture: Pay-performance ratios donot seem to be noticeably lower in sectors where compliance is key, such as the Financial Services or Natural Resources

.detacks ‘irresponsible’ City bonuses,” Financial Times, September 21, 2008, accessed May 31, 2012,d-b114-0000779fd18c.htmlehaviour was in the interest of shareholders. However, given that the misselling of financial productsessive risk-taking bankrupted other banks completely, these actions were probably at least ex post not

nation why pay-performance ratios should depend on the debt ratio of a company.

C. Siegert / European Economic Review 68 (2014) 93–10594

industries. Furthermore, our model is in line with the finding by Fahlenbrach and Stulz (2010) that the size of cash bonuses abank paid was not negatively correlated with performance during the recent financial crisis.

In this paper we propose a standard moral-hazard model where the agent is risk-neutral but protected by limitedliability. However, the contractible profit signal is not only affected by the agent's productive effort but also by misbehaviour.So by offering a bonus to alleviate the traditional effort supply problem, the principal inevitably creates some incentive formisconduct. This holds true even though the principal finds out about misbehaviour with positive probability and is able topunish the agent by not paying any bonus.

Whenever bonuses are small, an increase in the bonus raises the level of misbehaviour, since any action that positivelyaffects the profit signal now becomes more attractive. At the same time, the threat of losing the bonus in case of observedmisconduct is not an effective deterrent. But for large bonuses, the level of non-compliance is decreasing in the bonus. Thehigher the incentives for effort, the higher the expected bonus the agent is going to lose out on in case his misbehaviour isdiscovered, in which case he receives zero wage payments. Hence, with very high incentives he will be less prepared tojeopardise his expected earnings by taking undesirable actions and will be more likely to comply. Given that misbehaviour ismost pronounced for intermediate incentives, the principal optimally chooses “extreme” incentives. If effort is of littleimportance he will curtail incentives in order to reduce the level of misbehaviour. But if it is important to motivate effort,the principal will offer lavish bonuses in order to curb non-compliance. In both cases, the principal may find it optimal tocomplement bonuses with fixed wages that are paid regardless of the firm's profit as long as no evidence of misbehavioursurfaces in order to enhance compliance. Our model takes the somewhat extreme view that all wages the agent receives aregiven to ensure incentive compatibility. They are neither due to collusive practices in the determination of pay (Bebchuk andFried, 2006), nor do they reflect scarcity of prospective employees (Tervio, 2008; Gabaix and Landier, 2008). While this viewis unlikely to fully describe the reality of executive compensation, nevertheless it offers some interesting insights.In particular, fixed wages can not only be used in order to satisfy participation constraints, but they may also be anadditional disciplining device aimed against misbehaviour.

Our results imply that for top-management positions high incentives may in fact be a method to induce compliance,whereas lower ranks in a firm's hierarchy may receive very performance-inelastic pay in order to achieve the same goal.These effects play not only a role in the financial industry, but are always an issue if agents have the possibility to gameincentive schemes by engaging in actions that are harmful to their employer. Examples for undesirable behaviour in otherindustries may include illegal actions such as setting up a cartel or paying bribes in order to be awarded a contract. Whilesufficiently high corporate fines guarantee that these actions are often not in the principal's interest, a manager maynevertheless be tempted to engage in such behaviour in order to be awarded a bonus.4

Our results have a number of important policy implications. In particular, a legal cap on bonuses can be counter-productive and may lead to more misbehaviour. Moreover, if a policy maker wants to increase the level of compliance withinorganizations due to externalities created by misconduct, then it is always more efficient to impose (higher) fines on theprincipal in case misbehaviour is detected. This may even increase the bonus that a principal offers and can reduce thedistortions that are created due to the non-observability of effort.

In looking at a two-dimensional moral hazard model, our work is clearly related to the multi-tasking literature initiatedby Holmstrom and Milgrom (1991). Yet, by assuming that the manager is protected by limited liability, we reach quitedifferent conclusions. While in the traditional multi-tasking models the introduction of additional tasks typically reducesoptimal incentives, in our setting the opposite can be true: Incentive problems in the second dimension are mitigated byincreasing incentives in the first dimension. Precedents like the Enron case have shown that limited liability constraints areindeed an issue even for (usually rather wealthy) executives since courts are reluctant to enforce fines in excess of recentwage income.5

The use of large bonuses to induce compliance resembles the effect of efficiency wages as discussed for example byShapiro and Stiglitz (1984). The literature on efficiency wages argues that paying wages above the market clearing level maybe a way of discouraging misbehaviour. In our framework, the principal can achieve a similar effect by offering largebonuses. Interestingly, this holds true even though the prospect of earning a bonus is the very reason why employees maybe tempted to misbehave in the first place.

The idea that monetary incentives may trigger undesirable behaviour is not new and has for example been studied in theempirical literature on earnings management (Healy, 1985; Asch, 1989; Holthausen et al., 1995; Oyer, 1998; Larkin, 2007).Bergstresser and Philippon (2006) show that managers are more likely to inflate earnings if their equity-based incentives arelarge relative to total compensation. However, the evidence on the details of this relationship is rather inconclusive: Peng andRöell (2008) show that class-action law suits related to misleading financial statements become more likely if executives ownlarge amounts of stock options, but they do not find a significant effect of cash bonuses or stock holdings. Conversely, Johnsonet al. (2009) report a significant effect of unrestricted stock holdings on the likelihood of an “Accounting and Auditing

4 Our only key assumption is that the agent does not fully internalize the negative consequences of his actions. There are two reasons for this: (a) theagent's limited liability and (b) the imperfect observability of undesirable actions. Imperfect observability may arise because often, the negativeconsequences of undesirable effort only emerge in the distant future. Alternatively, it may be impossible to condition the agent's remuneration on certainoutcomes such as a drop in a firm's reputation, due to verifiability constraints. In this case the principal can only impose punishments in cases whereadditional evidence of misbehaviour is found.

5 See Bebchuk et al. (2006).

C. Siegert / European Economic Review 68 (2014) 93–105 95

Enforcement Release” by the SEC, but do not find any effect of stock option holdings. This mixed evidence suggests that therelationship between misbehaviour and monetary incentives may be more subtle than is typically assumed.

In two theoretical contributions, Spagnolo (2000, 2005) looks at the question of whether or not incentive contracts canmake collusive agreements harder to sustain. Fischer and Huddart (2008) consider spill-overs of social norms betweendivisions and derive implications of misbehaviour for the optimal scope of a firm. Milgrom and Roberts (1988) and Milgrom(1988) show that the prospect of attractive promotions may induce employees to spend an inefficient share of their time onlobbying for promotions and consider the implications for optimal promotion policies.

Another related strand of literature looks at the interplay between the incentives for effort, short-termism or risk-takingand a company's financial structure (Stein, 1988, 1989; John and John, 1993; Von Thadden, 1995; Biais and Casamatta, 1999;Bolton et al., 2006). More recent contributions on dynamic contracts have focused on the question how deferredcompensation (Edmans et al., 2012; Manso, 2011) or earnings- (rather than stock-) based compensation can mitigateshort-termism (Benmelech et al., 2010).6

The most closely related work is by Inderst and Ottaviani (2009) who look at optimal contracts if sales agents must beinduced to search for potential customers, but not to sell to unsuitable customers. When a sales agent is faced with anunsuitable customer, he is unable to earn any bonus by acting in the principal's interest and can only earn a bonus bymisbehaving. This implies that he can never lose out on any wage payments due to misbehaviour and unlike in our settinghigher incentives will never have a disciplining effect. Instead, they will always encourage misconduct.

The rest of the paper is organised as follows: Section 2 formulates the model and presents a simple example. Section 3analyses the agent's problem for a given contract, while in Section 4 we explore the properties of an optimal contract.Sections 5 derives testable predictions. In Section 6 we explore the policy implications of our model and discuss the results.

2. The model

We look at a one-shot game where a risk-neutral principal employs a risk-neutral agent to manage a firm. The firm canmake either high profits π or low profits π with Δ¼ π�π40. The agent's wealth is initially zero and has to remain non-negative in all states of the world since the agent is protected by limited liability. Hence, he can never receive a negativewage payment. The agent can exert unobservable effort which determines the probability aA ½0; a� that high profits arise.We will denote the agent's effort cost for working in the firm by C(a).

Furthermore, the agent has the possibility to unobservably increase the probability of high profits by uA ½0;u � at a privatecost K(u) if he engages in actions that are seen as undesirable by the principal. Since the overall probability of high profits cannot exceed one we assume that aþur1. Misbehaviour imposes an expected, non-verifiable cost of τðuÞ ¼ δγðuÞ on theprincipal where δ is some scalar, τð0Þ ¼ 0 and τ0ðuÞZΔ for all u. That is, the marginal cost of undesirable effort outweighs thebenefit of an increase in the likelihood of high profits from the principal's point of view. With a small probability p(u) theprincipal costlessly obtains hard information that the agent has been engaging in undesirable behaviour and can punish himby reducing his wage payments to any level that does not violate the limited liability constraint. Finally, we assume that theagent has an outside option of zero, so he will accept any contract he is offered and we can ignore his participation constraint.

2.1. A simple example

Let us start by considering a simple example to highlight the key effects of the model. We assume that C að Þ ¼ 13 a

3,K uð Þ ¼ 1

3u3 and pðuÞ ¼ 10u2. Moreover, let u ¼ 1ffiffiffiffi

10p and a ¼ 1� 1ffiffiffiffi

10p . Finally, for the purpose of this example we assume that

the agent only receives a wage in case the firm earns high profits and the principal has not obtained any evidenceof misbehaviour. We call this payment the bonus b. The agent's expected utility can be expressed as U ¼aþuð Þ 1�10u2

� �b� 1

3 a3� 1

3u3 and any interior optimum has to satisfy the following two first-order conditions:

∂U∂a

¼ 1�10u2� �b�a2 ¼ 0 ð1Þ

∂U∂u

¼ 1�10u2� �b�u2�20ub aþuð Þ ¼ 0: ð2Þ

We can check that the agent's choice of u will indeed always constitute an interior optimum. Moreover, we can restrictattention to bonuses bA ½0; b� for which the choice of a constitutes an interior optimum, too. If b4b, the principal could slightlyreduce the bonus bwhich would leave the agent's choice of a unaffected. Moreover, for any given level of a, the optimal choice ofu is increasing in b.7 Hence, reducing the bonus would leave a unchanged, lower the wage bill and reduce misbehaviour.

6 In a related paper Kwon and Yeo (2009) show that even if a manager's attempt to manipulate stock prices is self-defeating in equilibrium, his abilityto manipulate earnings still depresses incentives for (productive) effort.

7 To see this, we can take the partial derivative of the second first-order condition with respect to b, which gives us ð1�10u2Þ�20uðaþuÞ. From theF.O.C. we know that this must be positive for all u up to the optimal one, which implies that u is increasing in b.

Fig. 1. Misbehaviour as a function of bonus payments.

C. Siegert / European Economic Review 68 (2014) 93–10596

Solving for a and u gives us

a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

bð1þ20bÞ21þ50bþ400b2

s

u¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

b

1þ50bþ400b2

s:

Two key insights emerge from the expressions for a and u: (i) the level of effort a is monotonically increasing in b and (ii)the level of misbehaviour u is increasing in b if and only if bo0:05 where 0:05ob.8 Whenever b exceeds this threshold,misbehaviour is decreasing in the size of the bonus (see Fig. 1). To understand why this is the case, note that the bonusaffects the level of misbehaviour via two channels. First, an increase in the bonus increases the return to undetectedmisbehaviour, which is captured by the term ð1�10u2Þb in Eq. (2). This explains why u is initially increasing in b. But at thesame time, a larger bonus increases the expected payment that the agent loses out on in case his misbehaviour is detected,which is captured by the term 20ubðaþuÞ. This effect becomes stronger the larger the bonus. So for sufficiently large bonuses,any additional increase in the bonus reduces misbehaviour. The expected bonus the agent might lose out on is determined bytwo factors: The bonus itself and the probability with which the agent would obtain the bonus if he did not misbehave, a. Asthe bonus increases the agent will react by exerting more effort, which further increases the expected bonus that he might loseout on. We have discussed above that for a given level of effort a misbehaviour is always increasing in b. Hence, the fact thatmisbehaviour eventually decreases in the bonus crucially depends on the interaction between these two factors.

This simple example demonstrates that while bonuses may offer ways to game the bonus scheme by misbehaving, largebonuses may reduce the incentives to do so. We should understand this as a possibility result rather than a general property.Still, we will show below that this effect extends beyond our specific example and also arises under more general functionalforms and compensation schemes.

2.2. The general model

Let us assume that C(a), K(u) and p(u) are continuous, three times differentiable and satisfy the following three sets ofassumptions:

Assumption 1.

ðiÞ Kð0Þ ¼ 0; K 0ð0Þ ¼ 0; K″ðuÞ40;ðiiÞ Cð0Þ ¼ 0; C0ð0Þ ¼ 0; C″ðaÞ40; lim

a-aC0ðaÞ ¼1

Part (i) of the assumption says that the cost of undesirable behaviour is increasing and convex in u. Similarly, part (ii) saysthat the cost of effort is an increasing and convex function of a and ensures that the agent always finds it optimal to choosesome aoa. In contrast to our simple example the agent reacts to any increases in the rewards to effort by exerting moreeffort. This implies that the principal may now find it optimal to offer arbitrarily large bonuses if he values effort sufficientlystrongly.

Assumption 2.

p0ð0Þ ¼ 0; p″ðuÞ40; pðuÞ ¼ 1

8 The maximal bonus b that a principal might offer is implicitly defined by a2ð1þ50bþ400b2Þ ¼ bð1þ20bÞ2. Solving for b numerically yields

b ¼ 0:487.

C. Siegert / European Economic Review 68 (2014) 93–105 97

Assumption 2 guarantees that not only the explicit cost of undesirable behaviour but also the implicit cost, the risk ofbeing caught, is convex.9 The last part of the assumption makes sure that the agent always finds it optimal to choose aninterior level of u. Note that we do not assume that pð0Þ ¼ 0, so there may be a positive probability that an agent who hasbeen engaging in no misbehaviour at all is found guilty.

Assumption 3. C 0ðaÞ is convex and ðC‴ðaÞ=½C″ðaÞ�2ÞC0ðaÞo f for some f o2 .

This assumption ensures that the amount of effort an agent exerts is concave in the bonus he expects to get but is not tooconcave. Assumption 3 is for example satisfied by all power functions CðaÞ ¼ kar with k40 and rZ2. This class of functionsensures that all of the basic trade-offs considered in this paper hold. A function that satisfies all of our assumptions andallows us to abstract from potential corner solutions is CðaÞ ¼ a2=ða�aÞ.10

The model is fairly general, but it can be applied to excessive risk-taking in the financial industry: If the principal isunable to monitor the investment decisions of his banker, he has to incentivise the banker via bonus contracts. The bankercan exert effort looking for efficient projects in order to increase expected profits and hence the bonus that he can expect toearn. Additionally, he can spend some effort looking for excessively risky projects. There is a large probability ð1�pðuÞÞ thatthese risky investments have consequences indistinguishable from standard projects. However, with a small probability p(u)the principal obtains evidence on the misbehaviour (e.g. because the investments result in a large loss) and can punish theagent. Note that we abstract from the effect of excessive risk-taking on the uncertainty of the principal's profits. It would benatural to assume that the cost of misbehaviour is large in case of a crisis (in which case risk-taking results in a large loss)and zero otherwise. However, since the principal is risk neutral we are primarily interested in the expected cost of excessiverisk-taking τðuÞ.11

3. The agent's decision problem

Let us start by characterising the agent's decision problem. In general, an employment contract may specify differentwage payments depending on whether misbehaviour has been detected and whether the firm has been making high profits.However, without loss of generality we can restrict attention to the following contracts:

Lemma 1. An optimal contract can be fully characterized by a fixed wage w that the agent receives as long as no evidence ofmisbehaviour is found and an additional bonus bZ0 that he receives in case the firm earns high profits.

Proof. See the appendix.

An optimal contract does not pay any wages in case of observed misconduct, since any such payment would giveincentives to misbehave. While a bonus payment in case of observed misbehaviour and high profits might incentivise effort,the principal can implement the same level of effort and a strictly lower level of misbehaviour by increasing b instead. In ourinitial example we simplified the problem by assuming that w¼0. While this assumption did not affect our key results, wewill see that the principal may still find it optimal to pay a fixed wage in order to discourage misbehaviour. Finally, it is neveroptimal to punish the agent in the case of high profits: The same a and u could also be obtained by offering the agent nowages at all, strictly reducing the expected wage cost. This implies that bZ0.

The utility an agent receives for a given contract is

U ¼ ðaþuÞð1�pðuÞÞbþð1�pðuÞÞw�CðaÞ�KðuÞand any optimal choice of a and u has to satisfy the following two first order conditions:

∂U∂a

¼ 1�p uð Þð Þb�C0 að Þ ¼ 0 ð3Þ

∂U∂u

¼ 1�p uð Þð Þb�K 0 uð Þ�p0 uð Þb aþuð Þ�p0 uð Þw¼ 0: ð4Þ

Note that we have assumed the agent's effort cost to be additively separable in the two dimensions and hence there are notechnological complementarities between the two tasks. Nevertheless, we see that the two dimensions are stronglyintertwined: The bonus b will simultaneously encourage effort and influence the agent's choice of misbehaviour. Moreover,undesirable behaviour will itself reduce the probability with which a successful manager receives the bonus and willtherefore erode incentives for effort as can be seen in Eq. (3). Effort, on the other hand, increases the expected bonus theagent loses out on in case misbehaviour is detected and will increase the level of compliance as determined by (4).Nevertheless, we can show that there is always a unique optimum (see the appendix).

9 Weak convexity of p(u) is sufficient as long as p0ðuÞ40 for all u40.10 To see that this class of functions satisfies the second part of Assumption 3 note that the derivative of ðC000 ðaÞ=½C00ðaÞ�2ÞC0ðaÞ with respect to a is always

positive. So it is sufficient to show that ðC000ðaÞ=½C00ðaÞ�2ÞC0ðaÞ ¼ 3=2.11 The realized cost of misbehaviour only matters in that it may generate evidence that the banker has been misbehaving, which we assume to be the

case with probability p(u). Hence, the probability of obtaining evidence depends on the level of misbehaviour u.

C. Siegert / European Economic Review 68 (2014) 93–10598

Using conditions (3) and (4) we can implicitly define the optimum by

F � ð1�pðuÞÞb�K 0ðuÞ�p0ðuÞððGðð1�pðuÞÞbÞþuÞbþwÞ ¼ 0 ð5Þwhere the effort level is given by a¼ Gðð1�pðuÞÞbÞ and where G� C0�1 is a strictly increasing, concave function. In order todetermine the overall effect of an increase in the bonus b on the agent's choice of u we have to look at

dudb

¼ �∂F∂b∂F∂u

¼ ð1�pðuÞÞ�p0ðuÞððaþuÞþð1�pðuÞÞbG0ðð1�pðuÞÞbÞÞ�∂F=∂u

ð6Þ

with the denominator being positive by strict concavity of the agent's objective function in the optimum. It will be useful tostate some basic properties of the agent's choice of a and u:

Lemma 2. For any given w both the agent's effort and the probability of the firm making high profits are increasing in the bonusb: da=db40 and dðaþuÞ=db40.

Proof. See the appendix.

By Eq. (3) we know that the agent's effort is determined by the bonus he can expect to earn in case of high profits,ð1�pðuÞÞb. The first part of Lemma 2 corresponds to an upper bound on du=db and tells us that while an increase in b maylead to more undesirable behaviour and hence a larger p(u), the bonus an agent can expect to earn in case of high profits isstill strictly increasing in b. The second part of Lemma 2 establishes a lower bound on du=db. Even if larger bonuses lead toless undesirable behaviour, the overall probability of the firm making high profits (which is given by aþu) is still increasingin the bonus.

Proposition 1. For any given w there exists a strictly positive b such that

�

to G

If bo b an increase in the bonus leads to more misbehaviour: du=db40.

� If b4 b an increase in the bonus leads to less misbehaviour: du=dbo0.

Moreover, misbehaviour is always decreasing in the fixed wage w.

Proof. See the appendix.

Let us look in more detail at the numerator in Eq. (6). The first term, ð1�pðuÞÞ, captures the idea that an increase in b willraise the returns to undetected misbehaviour, which makes such actions more attractive. The second term corresponds tothe fact that an increase in the bonus increases the wage payments an agent can expect to earn if he is not caughtmisbehaving and that he may lose due to misconduct. This effect reduces the incentives for misbehaviour. As we havediscussed in the context of the initial example, this effect is driven by two factors. First, for any given a an increase in bincreases the expected wage payment by ðaþuÞ. Additionally, an increase in the bonus will increase the agent's effort a byð1�pðuÞÞG0ðð1�pðuÞÞbÞ, which also increases the payments the agent can expect to receive if he is not observed misbehaving.

Consider a situation in which b¼0 and the principal contemplates marginally increasing the bonus: In this case thesecond term vanishes, since there is no bonus the agent might miss out on. The marginal effect on the return tomisbehaviour on the other hand is still strictly positive, which implies that misbehaviour is increasing in the size ofincentives. However, for very large bonuses this logic no longer applies. Now the agent cares about losing his high(expected) compensation and any policy that increases this compensation will consequently reduce misconduct.To illustrate this, let us consider how the numerator in Eq. (6) changes as we increase b and hold u constant. The marginaleffect of an increase in b on the return to misbehaviour is constant. In contrast to this, the total wage payments an agent canexpect to earn (or lose, in case he is caught misbehaving) are convex in b. This implies that the second terms in thenumerator of Eq. (6) will become increasingly large and the effect an increase in the bonus has on the level of non-compliance will eventually become negative.12 The most intuitive way to think about this effect is to say that by setting ahigher bonus b the principal effectively relaxes the limited liability constraint of the agent. By using larger bonuses toaugment the agent's payoff if he is not caught misbehaving the principal increases the relative punishment that he canimpose on a demonstrably misbehaving agent, which discourages misconduct. An important observation is that this effectonly arises due to the interplay of higher bonuses and higher effort levels. For any exogenously given level of amisbehaviouris unambiguously increasing in b.13 Hence, an increase in the bonus can only have a negative effect on misconduct if theeffort level reacts sufficiently strongly to an increase in b. This is ensured by Assumption 3.

Employees that receive low bonuses are very amenable to the undesirable effects of bonus payments. An increase in theirrewards encourages misdeeds that increase the contractible profit signal. Highly incentivised employees on the other handcan expect to earn high bonuses even without misbehaving and are hence reluctant to breach their fiduciary duties. So any

12 To see this, note that for a given u the expected wage payments are convex in b if ðaþuÞþð1�pðuÞÞbG0ðð1�pðuÞÞbÞ is increasing in b. This is equivalent″ðβÞβþ2G0ðβÞ40 where β¼ ð1�pðuÞÞb, which is true by Assumption 3.13 The proof of this proceeds along the same lines as described in footnote 7.

C. Siegert / European Economic Review 68 (2014) 93–105 99

policy that increases wage payments will enhance compliance. Taking those two observations together, the function u(b)that determines the level of misbehaviour follows an inverted-U shape, as we have seen in our example in Fig. 1.

Finally, misbehaviour is globally decreasing in the fixed wage w. An increase in the fixed wage leaves the returns toundetected misbehaviour unchanged, but it increases the (relative) harm of being observed misbehaving. Hence, fixedwages can be an effective means of reducing misconduct. However, we will see that the principal will also use bonuspayments to influence the agent's choice of u.

4. The optimal contract

We can now turn to the optimal contract that the principal offers. From Lemma 1 we know that this contract can be fullycharacterised by a bonus bZ0 and a fixed wage wZ0. First, we will look at the case where a principal offers a contract withw¼0. This is optimal if misbehaviour is not too costly, since increasing w reduces u but comes at a strictly positive cost.Subsequently, we consider the case where w40 and discuss the determinants of whether the principal offers a fixed wageor not.

For notational simplicity we assume that instead of choosing b, the principal (equivalently) chooses the expected bonusβ¼ ð1�pðuÞÞb. The agent's choice of effort depends not only on b per se but also on the probability with which he believeshe will lose the bonus due to misbehaviour. So the effect any change in b has on the manager's choice of effort will beamplified or dampened by the effect it has on undesirable behaviour u. Assuming that the principal chooses the expectedbonus β directly allows us to ignore this issue and is without loss of generality. In this context, it will also be useful toexpress Proposition 1 in terms of β: there is a β ¼ ð1�pðuÞÞb such that whenever βo β we have du=dβ40 and vice versa.14

4.1. The case with no fixed wage

Let us assume that the optimal contract has w¼0. In this case the principal's objective function is given by

Πðβ;uðβÞÞ ¼ ðuþGðβÞÞðπ�βÞþð1�u�GðβÞÞπ�τðuÞ ð7Þwhere u is a function of β and a¼ GðβÞ. So the necessary condition for an optimum with β40 is given by

dΠðβ;uðβÞÞdβ

¼ ∂Π∂β

þ ∂Π∂u

dudβ

¼ 0: ð8Þ

where ∂Π=∂uo0.The first important observation is that the optimisation problem the principal faces is not necessarily concave. The cost of

misbehaviour makes very small and very large bonuses more attractive, since these bonuses guarantee high levels ofcompliance. Hence, for large costs of undesirable behaviour there are always multiple local maxima: (At least) one in whichthe principal pays small bonuses and du=dβ is positive, and (at least) one in which he offers lavish incentives and du=dβ isnegative.15 Still, we can show that the optimal bonus must be monotonically increasing in the value of effort Δ:

Lemma 3. The optimal bonus β is monotonically increasing in the value of effort Δ. Moreover, limΔ-0β¼ 0 and limΔ-1β¼1.

Proof. See the appendix.

We know from Lemma 2 that the probability of earning high profits ðaþuÞ is increasing in β. The principal hence offerslarger bonuses the more he values a high outcome. This holds true despite the potential multiplicity of optima: Any bonusthat is optimal for some Δ is larger than any bonus that is optimal for a smaller Δ. Henceforth, we will assume that theproblem has a unique optimum, abstracting from the non-generic case where two local optima generate exactly the samelevel of profits.

Let us now investigate how the optimal bonus changes as the damage that is caused by a misbehaving agent increases. Inorder to do so, let us recall that we have defined the cost of undesirable behaviour to the principal as τðuÞ ¼ δγðuÞ. So it isnatural to model a rise in the damage misbehaviour causes by an increase in the scaling parameter δ.

Proposition 2. There exists a strictly positive threshold Δ such that

�

the

β4bec

If Δ4 Δ the bonus is large ðβ4 βÞ and increasing in the cost of misbehaviour: dβ=dδ40.

� If Δo Δ the bonus is small ðβo βÞ and decreasing in the cost of misbehaviour: dβ=dδr0.

The last inequality is strict whenever β40.

14 Doing so is possible since Lemma 2 implies that β is a strictly increasing function of b and there is hence a one-to-one correspondence betweentwo.15 For sufficiently high costs of misbehaviour β¼ 0 is a local maximum since a marginally higher bonus would increase misbehaviour. Now, take someβ . Since du=dβ is strictly negative, for sufficiently large costs of misbehaviour we get dΠ=dβ40. However, as β-1 we must get dΠ=dβo0 since ∂Π=∂βomes strictly negative while du=dβ converges against zero. Hence, there must also be a local maximum with β4 β .

C. Siegert / European Economic Review 68 (2014) 93–105100

Proof. See the appendix.

At the heart of this proposition lies a straightforward intuition: Since compliance is highest for very small and very largebonuses, the principal will generally set more “extreme” incentives as his concern for compliance increases. If Δ is small, theprincipal does not value effort a very strongly and sets a low bonus regardless of the cost of misbehaviour. Consequently, wehave du=dβ40 and the principal reduces incentives even further if he wants to marginally increase the level of compliance.Conversely, whenever Δ is larger than some threshold Δ the opposite holds true. In this case effort is sufficiently importantand the principal offers seizable incentives in order to increase the probability of success. This implies that du=dβo0 andthe principal can increase compliance by marginally raising the bonus. So as his concern for compliance increases, heincreases a bonus that is large already.

While Proposition 2 describes how the optimal bonus changes with a marginal change in the cost of misbehaviour, thesame does not necessarily hold true for more radical changes in the damage a misbehaving agent causes. Assume that thecost of misbehaviour increases drastically. Even if a principal found it optimal to set high bonuses beforehand, after a largehike in the cost of non-compliance he may decide to scrap incentives altogether and offer very small bonuses, which resultsin negligible levels of misbehaviour. Since the principal's optimisation problem is not globally concave, any large change inthe cost of misbehaviour may render a different local optimum more attractive and may lead to a discontinuous change inthe optimal bonus.

4.2. The case with a strictly positive fixed wage

Let us now turn to the case where the principal offers a strictly positive fixed wage w40 in order to exploit the negativeeffect that w has on misbehaviour. Even though increasing w comes at a strictly positive marginal cost, doing so may beoptimal if the principal cares sufficiently about ensuring compliance. Now the principal's expected profit is given by

Πðβ;wÞ ¼ ðuþGðβÞÞðπ�βÞþð1�u�GðβÞÞπ�τðuÞþð1�pðuÞÞw: ð9Þ

By evaluating the first order conditions with respect to β and w we can characterize the level of misbehaviour that theprincipal implements. Unsurprisingly, this depends on the harm that a misbehaving agent causes. However, we will focus onhow a principal chooses to implement this level of misbehaviour. He can reduce misconduct either by adjusting the bonus orby increasing the fixed wage w. However, we can show that any optimal contract that has w40 satisfies the followingequality.

Lemma 4. If the principal chooses w40 and wants to implement a given level of compliance u, then the optimal bonus β satisfies

G0 βð ÞΔ¼ ð1�pðuÞÞp0ðuÞ : ð10Þ

Proof. See the appendix.

Whenever w40 the principal can offset the effect of a change of b on misbehaviour by simultaneously adjusting w. Thisimplies that we only have to check which of the two instruments is the least costly way to achieve a given level ofcompliance. We have seen that the principal may decide to pay the agent large bonuses in order to increase the wagepayments the agent loses out on if misbehaviour is detected. Alternatively, the principal could achieve the same by offeringa fixed wage component w which will be paid out whenever no undesirable behaviour is observed, irrespective of the firm'sprofits. Paying the agent only in the case of high profits has the advantage of increasing the expected punishment forobserved misbehaviour and motivating effort a at the same time; but it also increases the returns to undetectedmisbehaviour. This explains why it may be optimal to pay fixed wages in order to discourage non-compliance even ifdu=dbo0.

Let us define ~β as the bonus that a principal optimally offers in case the agent does not have any opportunity tomisbehave. This bonus is characterised by G0ð ~βÞΔ¼ ~βG0ð ~βÞþGð ~βÞ where the left hand side represents the benefit of increasedeffort. The right hand side represents the increase in the agent's expected compensation and is strictly increasing in ~β .

Proposition 3. Assume that the principal offers a contract with w40 and wants to implement a given level of compliance u.There exists a ~Δ40 such that

�

If Δ4 ~Δ the principal offers a larger bonus than if the agent was not able to misbehave: β4 ~β . � If Δo ~Δ the principal offers a smaller bonus than if the agent was not able to misbehave: βo ~β .

Proof. See text.

Whenever the value of effort Δ is large, ~β is large. Hence, the right hand side of Eq. (10) is smaller than ~βG0ð ~βÞþGð ~βÞ andthe principal chooses a bonus above the one that he would offer if the agent did not have the possibility to misbehave. Whilethe principal would offer a large bonus even if misbehaviour was not an issue, he offers even larger incentives if he cares

C. Siegert / European Economic Review 68 (2014) 93–105 101

about potential misconduct. Conversely, if Δ is small, the principal chooses a bonus that lies below the (small) bonus that hewould offer if the agent had no opportunity to misbehave.

Marginally distorting the bonus away from the size that would be optimal for a given level of misbehaviour is costless.Hence, the principal chooses to do so and will always exploit the positive effect a change in the bonus has on compliance.In particular, as long as effort is sufficiently important the principal will still set generous bonuses to reduce misbehaviour.This implies that the effects that we have shown in the last section will be present even if the principal offers a fixed wage asan additional way to encourage compliance.

However, the conditions under which it is optimal to pay a fixed wage are rather involved and cannot easily becharacterized without imposing specific functional forms. In general, the principal will only offer a fixed wage if the cost ofundesirable behaviour is sufficiently high, since increasing w comes at a strictly positive cost. However, if the cost ofundesirable behaviour becomes too large, the principal sets a bonus of β¼ 0. This is the only way to guarantee that the agentchooses u¼0 and renders fixed wages superfluous.

5. Industry implications

We would expect the cost of misbehaviour to be largely industry specific, while even within an industry firms exhibitconsiderable heterogeneity with respect to the importance of managerial effort due to different organisational choices.This allows us to characterise any industry by a distribution of bonuses that different agents are offered, H(b). In thefollowing, we assume that agents receive no fixed wages. This ensures that adjusting the bonus will always be the marginalinstrument to enhance compliance.

Consider an industry where the returns to effort Δ are distributed with strictly positive density in the interval ð0;1Þ andeach firm employs one manager. In order to ensure that even for arbitrarily large values of Δ misbehaviour is indeed not inthe principal's interest, let us redefine the cost of undesirable behaviour as τðuÞ ¼Δuþδγ ðuÞ, where δγ 0ðuÞ40. Thisspecification keeps the net cost of misbehaviour constant for different values of Δ.16 Moreover, for the purpose of the nextproposition we do not assume that the optimal bonus is unique for all values of Δ. Instead, we assume that whenever theprincipal is indifferent he chooses the smaller bonus.17

Proposition 4. The interquantile range Q ð1� ϵÞ �Q ϵ of the bonus distribution H(b) is increasing in δ for all small values of ϵ: Thelarger the harm caused by misbehaviour in a particular industry is, the more spread out the bonus distribution will be.

Proof. See the appendix.

We have already seen that the harm done by non-compliance will result in more “extreme” incentives. Those firms thatoffer very low pay-performance ratios will depress incentives further as the cost of undesirable behaviour increases, whilecompanies with high-powered incentives will increase bonuses. Hence, the bonus distribution will be more spread out in anindustry where misbehaviour is more costly.18 To illustrate the said properties, in Fig. 2 we consider an example wherepðuÞ ¼ ðu=0:25Þ2, CðaÞ ¼ a2=ð0:75�aÞ, KðuÞ ¼ u2=ð1�uÞ and τðuÞ ¼ΔuþTpðuÞ with TAf 0, 12 m, 25 m g.19

Proposition 4 relies on the fact that distorting incentives are always the marginal instrument to enhance compliance.This may be the case because increasing the fixed wage is either too expensive or too controversial, given that fixed wagesare typically more easily observable by shareholders ex ante than bonuses. However, we can also imagine situations whereindustries with higher costs of misbehaviour react by increasing fixed wages.

6. Discussion

In many cases, undesirable behaviour has negative externalities on society as a whole. Cartel agreements reduceconsumer surplus, bribes may undermine the rule of law, excessive risk-taking may create systemic risk and require publicbail-outs, etc. A natural question to ask is how a social planner may want to discourage misbehaviour. In our discussion, wewill again assume that the principal does not pay any fixed wages in order to focus on the interesting case where adjustingthe bonus is the marginal instrument to enhance compliance.

A measure that has received a lot of attention is a legal cap on bonuses. Our model shows that such caps may becounterproductive. By imposing caps that are close to the level of bonuses paid in an unregulated labour market, we mayincrease managerial misbehaviour. However, this does not hold for large interventions: Any cap that is sufficiently close tozero will result in negligible levels of misbehaviour. Moreover, given the non-concavity of the principal's objective function,

16 The principal loses any benefits that have accrued from misbehaviour and suffers an additional net cost δγ ðuÞ that is independent of Δ. While thisspecification is chosen mainly for technical convenience, it may represent a situation where the cost of misbehaviour consists of legal fines. Typically, suchfines are set as to claim back any benefits the principal may have had from misbehaviour plus an additional deterrent.

17 This guarantees that the bonus distribution is uniquely defined.18 We use the interquantile range (the difference in value between two given quantiles of a distribution) as a measure of how spread out a

distribution is.19 One interpretation of this cost function τðuÞ is that the principal only suffers a cost in those states of the world where he also receives hard evidence

on misbehaviour.

Fig. 2. The optimal bonus as a function of Δ.

C. Siegert / European Economic Review 68 (2014) 93–105102

the principal may voluntarily set a bonus that is strictly below the legal cap. However, such interventions erode incentivesfor managers to work hard and may not increase social welfare.

If the social planner can costlessly observe any evidence of misbehaviour, then he can also increase compliance byimposing fines on the principal if the agent has been observed misbehaving. From the principal's point of view, this is anincrease in the marginal cost of misbehaviour and the principal will react by adjusting bonuses. If effort is sufficientlyvaluable he will increase incentives in order to reduce misbehaviour. Since the principal always implements effort that is toolow from the point of view of a social planner, this is good news: By punishing the principal, the policy maker not onlyreduces unwanted behaviour, but also reduces the distortions created by the non-observability of effort. This explains whyimposing fines on the principal is always a weakly more efficient way to achieve a given level of misbehaviour than a bonuscap.20

The principal always chooses to punish the agent as fiercely as possible (given limited liability) if he observesmisbehaviour. However, a policy maker may dispose of additional options to penalise the manager, for example byimposing prison sentences. But even if the policy maker can impose arbitrarily large punishments, it is unclear whetherdoing so is optimal. While extremely harsh punishments can prevent misbehaviour, they also crowd out any compliance-enhancing incentives the principal might set. In particular, the principal may decide to pay smaller bonuses, whichaggravates the efficiency loss that is due to the non-observability of effort. This implies that the policy maker has someincentive to preserve the moral hazard problem in the second dimension in order to reduce the distortions in the firstdimension.

While our model may be able to explain high-powered incentives, it also suggests that shareholders should punish amanager fiercely in case he is caught misbehaving. The prevalence of generous severance package and other forms of“golden parachutes” suggests that this may not always be the case in reality. Understanding the reasons for sucharrangements is beyond the scope of this paper, but is undoubtedly an interesting question.

Appendix A. Mathematical appendix

Proof of Lemma 1. For the purpose of this proof we define the total wage that is paid in state ½i; j� as wi;j where iAfh; lg andjAfp;ng denote whether high profits (h) have been made or not (l) and whether misbehaviour has been detected (p) or not(n). The utility of the agent is then given by

U ¼ ðaþuÞðð1�pðuÞÞwh;nþpðuÞwh;pÞþð1�a�uÞðð1�pðuÞÞwl;nþpðuÞwl;pÞ�CðaÞ�KðuÞ

and the returns to a and u are

∂U∂a

¼ 1�p uð Þð Þwh;nþp uð Þwh;p� 1�p uð Þð Þwl;n�p uð Þwl;p�C0 að Þ∂U∂u

¼ 1�p uð Þð Þwh;nþp uð Þwh;p� 1�p uð Þð Þwl;n�p uð Þwl;p�K 0 uð Þ�p0 uð Þ aþuð Þ wh;n�wh;p� �þ 1�a�uð Þ wl;n�wl;p

� �� �

20 As usual, the first-best effort level would be implemented if the agent would reap the full benefits from effort, i.e. β¼ Δ. The principal always choosea bonus that is smaller than the first-best: Instead of choosing βZΔ, he can do strictly better by setting β¼ 0.

C. Siegert / European Economic Review 68 (2014) 93–105 103

Let us define the expected reward for high profits as β¼ ð1�pðuÞÞwh;nþpðuÞwh;p�ð1�pðuÞÞwl;n�pðuÞwl;p. It can never beoptimal to have βo0 since the same a and a weakly lower u can be implemented by offering a contract ð0;0;0;0Þ that has astrictly lower wage cost.Suppose a contract ðwl;n;wh;n;wl;p;wh;pÞ implements some aZ0 and has wl;p40 or wh;p40. Instead, we can choose a

contract ðwl;n; wh;n;0;0Þ that has

wl;n ¼wl;nþpðuÞ

1�pðuÞwl;p and wh;n ¼wh;nþpðuÞ

1�pðuÞwh;p

and would implement the same a if the agent were to choose the same level of u (which he is not). Now consider a contractð ~wl;n; ~wh;n;0;0Þ that does indeed have the same expected wage payments as the initial contract conditional on profits beingeither high or low. Clearly, this contract implements the same a. Furthermore, assume that it has a probability of detectingmisbehaviour ~pZpðuÞ. In this case ð1� ~pÞ ~wh;n ¼ ð1�pðuÞÞwh;n and ð1� ~pÞ ~wl;n ¼ ð1�pðuÞÞwl;n implies that ~wh;nZwh;n and~wl;nZwl;n. However, this contradicts the assumption that ~pZpðuÞ.So there exists a contract ð ~wl;n; ~wh;n;0;0Þ that has (i) thesame a, (ii) a weakly lower u and (iii) the same expected wage payments conditional on the realization of profits. Since βZ0we can express this contract as a fixed wage wZ0 that is paid whenever no evidence on misbehaviour is found and anadditional bonus bZ0 that is paid if the firm makes high profits. □

Proof of existence and uniqueness. Let us show that there always exists a unique optimum to the agent's problem. We canrestrict attention to contracts where b40, since for b¼0, the unique optimum has a;u¼ 0. Any optimum satisfiesð1�pðuÞÞb�C0ðaÞ ¼ 0 and there is a unique optimal a for any choice of u. This allows us to look at a one-dimensionaloptimization problem where the agent chooses u and a(u) is given by the above first order condition. We can easily checkthat a(u) is continuous in u, so the agent's utility is continuous on the closed interval ½0;u� and a maximum always exists.Moreover, any optimum must be interior. Now consider the largest optimal u ¼maxfarg maxufðaðuÞþuÞ

ð1�pðuÞÞbþwð1�pðuÞÞ�CðaðuÞÞ�KðuÞgg. By the necessary condition (4) we must have ð1�pðuÞÞb�bp0ðuÞaðuÞ40 at theoptimum since bp0ðuÞuþK 0ðuÞþwp0ðuÞ is positive at any interior maximum. Let us now show that for a given b the optimumis unique on the interval ½0; u�. In order to do so, we will need to find a lower bound for a0ðuÞ, which is strictly negative for allpositive bonuses. First, note that G� C0�1 is a concave function, since we can use the inverse function rule to show thatG″ðC 0ðaÞÞ ¼ �C‴ðaÞ=½C″ðaÞ�3 is negative for all a. As aðuÞ ¼ Gðð1�pðuÞÞbÞ this implies that a(u) is concave in u. So it suffices tolook at a0ðuÞ ¼ �bp0ðuÞG0ðð1�pðuÞÞbÞ. Using the fact that ð1�pðuÞÞb�bp0ðuÞaðuÞ40 and that Gðð1�pðuÞÞbÞZG0ðð1�pðuÞÞbÞð1�pðuÞÞb by concavity of G we get 14bp0ðuÞG0ðð1�pðuÞÞbÞ. It follows that a0ðuÞ4�1 for all uA ½0; u�.Since the agent's utility is given by UðuÞ ¼ ðaðuÞþuÞð1�pðuÞÞbþwð1�pðuÞÞ�CðaðuÞÞ�KðuÞ we get U″ðuÞ ¼

�p″ðuÞbðaðuÞþuÞ�ð2þa0ðuÞÞp0ðuÞb�p″ðuÞw�K″ðuÞ where several terms drop out because ð1�pðuÞÞb�C0ðaðuÞÞ ¼ 0 must holdfor any u. Given our lower bound for a0ðuÞ it is easy to see that U″ðuÞ is negative for all uA ½0; u� and the optimum isunique. □

For notational convenience we will henceforth write β¼ ð1�pðuÞÞbwherever possible. Moreover, we will no longer stressthat p¼ pðuÞ in the interest of brevity.

Proof of Lemma 2. First, let us show that da=db40 which is clearly the case if b¼0. If b40 the condition is equivalent todu=dboð1�pÞ=bp0 since da=db¼ ð1�pÞG0ðβÞ�bp0G0ðβÞdu=db. We want to show that

dudb

¼ ð1�pÞ�p0ðGðβÞþuÞ�p0βG0ðβÞ2bp0 þbðGðβÞþuÞp″þwp″þK″ðuÞ�½bp0�2G0ðβÞ o

ð1�pÞbp0

:

A sufficient condition for this inequality to hold is that

ð1�pÞ�p0βG0ðβÞ2bp0 þbðGðβÞþuÞp″þwp″þK″ðuÞ�½p0b�2G0ðβÞ o

ð1�pÞbp0

30obp0 þK″ðuÞþp″bðGðβÞþuÞþwp″

which is true by Assumptions 1 and 2.Now we need to show that dðaþuÞ=db40, which is equivalent to ð1�pÞG0ðβÞþð1�bp0G0ðβÞÞdu=db40. Again, it is sufficient

to consider situations where b40. From the proof of uniqueness we know that 1�bp0G0ðβÞ40, so the condition can only beviolated if du=dbo0. Let us hence look for a lower bound for du=db:

dudb

¼ ð1�pÞ�p0ðGðβÞþuÞ�p0βG0ðβÞ2bp0 þbðGðβÞþuÞp″þwp″þK″ðuÞ�½bp0�2G0ðβÞ

4� p0βG0ðβÞ2bp0 þbðGðβÞþuÞp″þwp″þK″ðuÞ�½bp0�2G0ðβÞ

4� ð1�pÞG0ðβÞ1�bp0G0ðβÞ ðA:1Þ

where the first inequality follows from the fact that ð1�pÞ4p0ðGðβÞþβÞ by Eq. (4) and the second inequality is implied by1�bp0G0ðβÞ40. Plugging (A.1) into our initial condition shows that indeed dðaþuÞ=db40. □

C. Siegert / European Economic Review 68 (2014) 93–105104

Proof of Proposition 1. We want to show that there is a bonus b such that b4 b3du=dbo0 and bo b3du=db40. It caneasily be seen that du=dbjb ¼ 040. Now, let us consider the case where b40 and show that du=db will become negative forvery large bonuses. First, note that 0ouou and 0o limb-1 uou. Since the denominator in (6) will always be positive, wejust have to show that the numerator will eventually turn negative. This in turn is equivalent to showing that bð1�αÞ∂F=∂bwill be strictly negative in the limit for some α. Using Eq. (4) we get that for all b40

∂F∂b

¼ K 0ðuÞþp0wb

�p0βG0 βð Þ:

Multiplying both sides by bð1�αÞ and taking the limit we get

limb-1

bð1�αÞ ∂F∂b

� �¼ lim

b-1K 0ðuÞþp0w

bα� lim

b-1p0

ð1�pÞ1�αlimb-1

βð2�αÞG0 βð Þ:

We can see that for α40 the first term is zero and that limb-1ðp0=ð1�pÞ1�αÞ is strictly positive. Finally, Assumption 3implies that fG0ðβÞþG″ðβÞβ40. This means that for 242�α4 f we get ð2�αÞG0ðβÞþG″ðβÞβ40 and βð2�αÞG0ðβÞ is increasing inb. So limb-1 βð2�αÞG0ðβÞ must be strictly positive and limb-1bð1�αÞ∂F=∂bo0 for all sufficiently small values of α. This meansthat for sufficiently large values of b we have du=dbo0.By continuity of du=db we know that du=db has at least one root. In order to show that it has exactly one root, we are now

going to show that du=db is strictly decreasing in b whenever du=db¼ 0. First, we can show that at any point wheredu=db¼ 0 we have

d2u

db2¼ �

d∂F∂b

� �db∂F∂u

:

Since ∂F=∂u is always negative we just need to show that

d∂F∂b

� �db

¼ �p0 1�pð ÞG0 βð Þ 2þ βG″ðβÞG0ðβÞ

� �

is strictly negative, which is true by Assumption 3. So du=db¼ 0 implies that d2u=db2o0.Finally, using Eq. (5) it is easy to show that u is decreasing in w. This concludes the proof. □

Proof of Lemma 3. The return to increasing the bonus β can be expressed as

dΠdβ

¼ dðaþuÞdβ

Δ�βð Þ� aþuð Þ�τ0 uð Þ dudβ

;

which is strictly increasing in Δ. Hence, the smallest bonus that is optimal for some Δ is larger than the largest bonus that isoptimal for any smaller Δ. It is strictly so whenever the principal chooses an interior bonus β40. Moreover, as Δ-1 wehave dΠ=dβ40 for all finite β, so limΔ-1 β¼1. Similarly, we get limΔ-0β¼ 0. □

Proof of Proposition 2. By assumption, the optimum of the principal's problem is unique. So a marginal change in the costundesirable behaviour imposes on the principal will never lead to a discontinuous change in the optimal bonus and we canrestrict attention to the neighbourhood around β where the principal's objective function is strictly concave.Whenever the principal chooses some β40 the optimal bonus is characterized by Eq. (8) and we can use the implicit

function theorem to get

dβdδ

¼ γ0ðuÞΘ

dudβ

where Θ¼ d2Π=dβ2o0 by local concavity. So dβ=dδ takes the sign of �du=dβ and we have dβ=dδ≷03�du=dβ≷03β≷β . Ifβ¼ 0 on the other hand, the optimum constitutes a corner solution and we may have dβ=dδ¼ 0.Using Lemma 3 we can express the relation above as dβ=dδ40 if Δ4 Δ and dβ=dδr0 if Δo Δ where Δ is defined by

Δ ¼ supfΔjβZmaxfarg maxβΠgg. Note that we have allowed for the fact that for some values of Δ the optimal incentive βmay not be unique. □

Proof of Lemma 4. The principal can either adjust the bonus b or the level of fixed wagesw in order to increase compliance.Using Eq. (4), we can show that the marginal rate of substitution between b and w which leaves u constant is

dwdb

¼ ð1�pÞp0

� G βð Þþuð Þ�βG0 βð Þ:

C. Siegert / European Economic Review 68 (2014) 93–105 105

A contract can only be optimal if the principal cannot increase profits by choosing a different combination of b and w thatimplements the same u. This implies that in an optimum with w40 we must have

∂Π∂b

þ ∂Π∂w

dwdb

¼ 0

3 1�pð ÞG0 βð Þ π�β�π� �� 1�pð Þ G βð Þþuð Þ� 1�pð Þ dw

db¼ 0

3G0 βð ÞΔ¼ ð1�pÞp0

: □

Proof of Proposition 4. For any given δ there exists some strictly positive threshold Δ such that whenever Δo Δ anyoptimal incentive must be smaller than b and vice versa. So the bonus distribution will always have strictly positive mass onboth sides of the threshold b.For two bonus distributions that result from different costs of misbehaviour, δ and δ4δ, we can find some ~ϵAð0;0:5Þ such

that for any ϵr ~ϵ the ϵ-quantile will have bo b and the (1�ϵ)-quantile will have b4 b for both δ and δ. An increase in themarginal cost of misbehaviour from δ to δ will leave the identity of the firm offering the (1�ϵ)-quantile bonus unchanged.Now consider the problem faced by that firm: Any bonus that was optimal for δ can no longer be optimal for δ4δ since itwould now pay to marginally increase the bonus. Moreover, no other bonus in ½b;bÞ can be optimal for δ since those bonusesare associated with a strictly larger level of misbehaviour. Finally, by design the optimal bonus cannot be less than b. So the(1�ϵ)-quantile bonus must be strictly increasing in δ. By asimilar argument we can show that the ϵ-quantile bonus is weaklydecreasing in δ, which concludes the proof. □

Appendix B. Supplementary data

Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.euroecorev.2014.02.007.

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