Managerial Bonus Systems in a Differentiated Duopoly: A Comment

MANAGERIAL AND DECISION ECONOMICS

Manage. Decis. Econ. 33: 61–70 (2012)

Published online 8 November 2011 in Wiley Online Library

Managerial Bonus Systems in aDifferentiated Duopoly: A Comment

Thijs Jansena,*, Arie van Lierb,d and Arjen van Witteloostuijnb,c,d

aDepartment of Quantitative Economics, University of Maastricht, Maastricht, The NetherlandsbUtrecht School of Economics, Utrecht University, Utrecht, The Netherlands

cDepartment of Management, University of Antwerp, Antwerp, BelgiumdDepartment of Organization and Strategy, Tilburg University, Tilburg, The Netherlands

(wileyonlinelibrary.com) DOI: 10.1002/mde.1562

*Correspondenand EconomBox 616,m.jansen@ma

Copyright ©

AdifferentiatedCournot duopoly is consideredwhere firm owners delegate the output decisionto a manager, who is rewarded on the basis of his performance. If this performance ismeasured in terms of (i) pure profits, (ii) a combination of profits and sales, (iii) a combinationof profits andmarket share or (iv) relative profits, the latter option strictly dominates the othersif the products are perfect substitutes. Recently it was claimed that this result does not hold forall levels of product substitutability. In this comment, we show however that this result isrobust against the introduction of product differentiation. Copyright © 2011 John Wiley &Sons, Ltd.

1. INTRODUCTION

Separation of ownership and control by CEOs is a wide-spread characteristic of modern (domestic and multina-tional) enterprises. This organizational authority structuregives an owner the possibility to hire a manager whoseremuneration scheme can be used as a strategicinstrument. This was already studied in the second halfof the 1980s in the seminal papers of Vickers (1985),Fershtman and Judd (1987) and Sklivas (1987)—VFJS,for short. In these papers, using the tool of so-called(strategic) delegation games, the compensation of a man-ager is only partially based on profits: owners direct theirmanagers to partially nonprofit-maximizing behavior soas to maximize their own profitability. This result isreferred to as the profit-maximization paradox (vanWitteloostuijn, 1998). In order to maximize her profits,

ce to: Maastricht University, School of Businessics, Department of Quantitative Economics, PO6200 MD Maastricht, The Netherlands. E-mail:astrichtuniversity.nl

2011 John Wiley & Sons, Ltd.

the firm’s owner should introduce performance-basedCEO remuneration that includes a nonprofit element.

In fact, the strategic use of partially nonprofit-basedmanagerial compensation can already be observed inthe early 17th century. Irwin (1991) studied the duopo-listic competition between the English East IndiaCompany and the Dutch United East India Company(‘Verenigde Oostindische Compagnie’) in trading withIndia and Southeast Asia. He observed that the Dutchemerged to dominate the trade. The Dutch companyowed its superior role—and assumed something similarto a Stackelberg leadership role against the English bythe 1620s—because the Dutch government used a man-agerial compensation scheme based on profit and salesvolume, whereas the English government stuck to amanagerial compensation based on pure profits alone.This historical anecdote relates to a crucial question inthe modern (strategic) delegation games literature:How does the use of managerial compensation byowners affect the performance of their firms, and doesthis impact upon competition in an oligopolistic

T. JANSEN ET AL.62

environment? Note that Basu (1995) first examined thisissue, by analyzing a duopoly in which both firms’owners have the choice between pure profits compensa-tion and a contract based on profits and sales. From hisanalysis, we learn that the owner who rewards her man-ager based on profits and sales emerges as a Stackelbergleader, with the rival becoming a Stackelberg follower.1

Vickers, Fershtman, Judd and Sklivas introduced thegame theoretical toolkit to provide an answer to this typeof questions, by using two-stage (strategic) delegationgames. In the first stage of such a game, the contractstage, owners decide on the constituent elements of themanagerial bonus system, which consists of a weightedcombination of pure profits and a nonprofit part. In theearly work in this tradition, this nonprofit bonus anchorwas assumed to be sales. In the second stage, the marketstage, knowing the composition of the remunerationscheme, each manager determines the strategy of hisfirm—that is, output if the market stage is modeled asCournot. In the VFJS analysis, the managerial bonus isproportional to a (linear) combination of profits andsales (denoted by S). Inspired by empirical research out-comes, the (strategic) delegation games were extendedto include yet another type of managerial bonus systemby Salas Fumas (1992) andMiller and Pazgal (2002). Inthis work, the bonus for the manager is based on ownprofits minus weighted rival profits, referred to as rela-tive profits (denoted by R). A more recent theoreticalcontribution comes from Jansen et al. (2007) and Ritz(2008). They analyze the case in which the manager isrewarded on the basis of a combination of profits andmarket share (denoted by M).

Jansen et al. (2009) comprehensively examined, for aduopoly market with product homogeneity, the strategicconsequences of the use of this set of three differentmanagerial bonus systems, as well as pure profit maxi-mization (denoted by P). Their main conclusions arethat, if owners can freely choose between these four re-muneration schemes, the relative profits option R turnsout to be the strictly dominant one. Manasakis et al.(2011) build upon the framework of Jansen et al.(2009), but assume that the two competing firmsproduce differentiated products. One of their mainresults is that the dominant managerial bonus systemdepends on the degree of product substitutability. Byrepairing a flaw in their paper, however, we correct thisclaim, that is, we show that the result of Jansen et al. isrobust against introducing the assumption that productsare differentiated. In other words, a remunerationscheme with a bonus based on relative profits is strictlydominant for any degree of product differentiation.Moreover, contrary to Manasakis et al. (2011), all our

Copyright © 2011 John Wiley & Sons, Ltd.

conclusions are based on analytical proofs rather thanon numerical calculations.

2. THE MODEL

In the market stage of the (strategic) delegation game,we consider a standard differentiated Cournot duopoly.Inverse demand is given by (see Singh and Vives,1984) pi=m� qi� gq3� i for i=1, 2. Here pi and qiare the price and quantity of the product of firm i, mequals market size and g2 (0, 1). We assume that thetwo firms possess the same constant marginal produc-tion costs c<m.

In line with the VFJS tradition, we assume that eachowner delegates the output decision to a manager whoreceives a fixed salary and a bonus. Similar to Jansenet al. (2009), we consider four types of bonuses. Thatis, the bonus of manager i may be proportional to

(1) pi: a pure profits contract (indicated by P);(2) pi+wiqi with wi≥ 0: a sales contract (indicated

by S);(3) pi þ wi

qiQ with Q = q1 + q2 and wi≥ 0: a market

share contract (indicated by M); and(4) pi�wip3� i with wi2 [0, 1]: a relative profits

contract (indicated by R).

Note that Manasakis et al. (2011) only examine the lastthree types of contracts, ignoring the standard Cournotbenchmark. This decision structure is modeled through atwo-stage delegation game, where in each stage the deci-sions are taken simultaneously and independently by bothrivals. In the first stage of the game, the contract stage,both owners—who want to maximize profits—write themanagerial bonus contract, which is publicly observable.More specifically, the owner of firm i chooses the type ofbonus and the weight wi (except for a P contract). In themarket stage of the game, the manager of firm i choosesthe output level of his firm in order to maximize his bonus.The managers determine these levels while knowing thecontracts concluded in the first stage.

In Sections 3 and 4, we analyze all 10 possible com-binations of bonus types, the symmetrical ones PP, SS,MM and RR, as well as the asymmetrical ones PS, PM,PR, MS, SR and MR. In Section 5, we prove that theresults of Jansen et al. (2009) are robust against intro-ducing the assumption that products are differentiated,contrary to the central claim in Manasakis et al.(2011). For the sake of readability, the main text focuseson the methodology and the results; mathematicaldetails are provided in the appendices.

Manage. Decis. Econ. 33: 61–70 (2012)DOI: 10.1002/mde

MANAGERIAL BONUS SYSTEMS 63

3. BONUS CONTRACT HOMOGENEITY

In this section, we consider all cases in which bothowners use the same type of bonus contract. First,we summarize the results that are already known inthe literature. As there was a flaw in Manasakis et al.(2011) for the case where both owners opt for a marketshare-related bonus, the outline of the corrected prooffor this case is given. Further details can be found inthe appendix.

Throughout this paper, we denote the productionlevel of a firm by qAB, where A2 {P, S,M,R} refers tothe bonus system used by this firm. Similarly, B2 {P, S,M,R} indicates the bonus system chosen by the rivalfirm. The notation for the corresponding profits isanalogous.

3.1. Results from the Literature

We have summarized all cases that have been reportedin the literature in Table 1.

The result for the standard profit-maximization case,where both managers maximize pure profits, wasobtained by Singh and Vives (1984). The sales casewas discussed in Nakamura (2008), where owner andmanager bargain over (the weight in) a sales-orientedcontract. If in this model the bargaining power of eachmanager equals zero, the result presented in the tableis obtained. The relative profits case was investigatedbyMiller and Pazgal (2005), who analyze strategic tradepolicies under Cournot and Bertrand competition. Intheir model, governments decide on subsidizing pro-ducts. If all governmental subsidies equal zero, the resultin Table 1 is obtained. The two nonprofit-maximizationresults were also provided by Manasakis et al. (2011).

3.2. Both Owners Choose a Market ShareContract

Next, we briefly introduce the methodology for thecase where each owner rewards her manager on thebasis of market share. As the two-stage delegationgame is solved by applying the principle of backward

Table 1. PP, SS and RR Cases

Type ofcontract Output level Profits

Profits qPP ¼ 12þg m� cð Þ pPP ¼ 1

2þgð Þ2 m� cð Þ2Sales qSS ¼ 2

4þ2g�g2 m� cð Þ pSS ¼ 2 2�g2ð Þ4þ2g�g2ð Þ2 m� cð Þ2Relative

profits qRR ¼ 2þg4 1þgð Þ m� cð Þ pRR ¼ 4�g2

16 1þgð Þ m� cð Þ2


induction, we start with the final stage of the game, themarket stage, where both managers are engaged inquantity competition.

As each manager wants to maximize his bonus—which is proportional topi þ wi

qiQ—by properly choosing

the output level, the first-order conditions for the marketstage lead to the system of equations

m� cð Þ � 2q1 � gq2 þ w1q2Q2

¼ 0

m� cð Þ � gq1 � 2q2 þ w2q1Q2

¼ 0:

8><>: (1)

Observe that this system cannot be solved explicitly.However, by adapting the proof of Jansen et al. (2007),we can show that, for all feasible weights w1 and w2, thesystem has a unique solution (q1(w1,w2),q2(w1,w2)).The partial derivatives of each output level with respectto the weights can be obtained by using the ImplicitFunction Theorem.

As, in the contract stage, each owner wants to max-imize her profits by strategically choosing the weightin the contract with her manager, the correspondingfirst-order conditions are

m� cð Þ � 2q1 � gq2½ � @q1@w1

� g@q2@w1

q1 ¼ 0

m� cð Þ � gq1 � 2q2½ � @q2@w2

� g@q1@w2

q2 ¼ 0:

8><>: (2)

In order to solve the two systems (1) and (2) simulta-neously, we exploit the problem’s symmetry; we takew1=w2=w and q1 =q2 =q. This leads to the following result.

2

Proposition 1If in a duopoly market both owners hire a managerwho is rewarded on the basis of profits and marketshare, then in equilibrium outputs and profits are

andqMM ¼ 3þ gþ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2

p8þ 6g

m� cð Þ

pMM ¼ 7þ5g�g2�g3þ 1�g�g2ð Þffiffiffiffiffiffiffiffiffiffiffi1þg2

p2 4þ 3gð Þ2 m� cð Þ2:

Using these results, we formulate3 a corrected versionof Proposition 1 in Manasakis et al. (2011).

Proposition 2If in a duopoly market both owners hire a manager andif the two owners reward their manager on the basis ofthe same type of contract, then for all 0< g≤ 1,

(a) qPP< qRR< qMM< qSS and pPP> pRR>pMM> pSS,and


T. JANSEN ET AL.64

(b) if products become more differentiated (that is, if gdecreases), all production and profit levels increase.

Part (a) of Proposition 2 shows that the result of Jansenet al. (2009) is robust against the introduction of productdifferentiation, which proves that the claim of the con-trary by Manasakis et al. (2011)in their Proposition 1 isincorrect. Obviously, the highest profits occur if bothcompetitors would agree to use bonuses based on pureprofits. Contracts with a nonprofit component, after all,lead to higher outputs and lower profits. Part (b) ofProposition 2 reveals a general and plausible result. Ifproducts become closer substitutes, the size of the marketsegment that each rival can exploit does shrink, implyingthat production levels decrease. Despite the fact thatbonuses direct managers to more aggressive behavior,the effect of the decreasing market segment sizedominates for all types of bonus anchors.

4. BONUS CONTRACT HETEROGENEITY

In this section, we first consider the case in which one ofthe owners adopts the contract with her manager beingrewarded on the basis of profits only. Jansen et al. (2009)show that, for a Cournot duopoly with homogeneous pro-ducts, this owner obtains the profits of a Stackelbergfollower. This property appears to be robust against theintroduction of a product differentiation assumption.Subsequently, we deal with the remaining three asymme-trical cases.

4.1. Endogenous Stackelberg Leadership: thePS,PM and PR Cases

We assume that one of the firms, say firm 1, chooses a bo-nus based on pure profits. So, we consider three cases: PS,PM andPR. In all these cases,manager 1maximizes his bo-nus, which is proportional to p1= [(m� c)� q1� gq2]q1,by strategically choosing the output level q1. This givesthe standard reaction curve with differentiated productsq1 ¼ 1

2 m� cð Þ � 12 gq2 . Given this reaction curve, the

profit for firm 2 is maximal if the iso-profit curve p2 =ltouches the above standard reaction curve. Obviously,this will be at the Stackelberg point Sg. For any type ofbonus (S,M or R), owner 2 can tune the contractualweight in such a way that the reaction curve of her man-ager passes4 through the Stackelberg point Sg. Hence, ifone of the owners uses a bonus based on pure profits,whereas the rival owner opts for another type of bonus,the latter owner acquires a Stackelberg leadership posi-tion. The corresponding outputs and profits are


and

qF ¼ 4� 2g� g2

4 2� g2ð Þ m� cð Þ; qL ¼ 2� g2 2� g2ð Þ m� cð Þ

pF¼ 4� 2g� g2ð Þ216 2� g2ð Þ2 m� cð Þ2; pL¼ 2� gð Þ2

8 2� g2ð Þ m� cð Þ2:

Therefore, the result of Basu (1995) and Jansen et al.(2009) for homogenous products holds, more generally,for differentiated products.

Proposition 3If in a duopoly market one of the owners rewards hermanager on the basis of pure profits and her rival usesanother type of bonus contract, then for all 0< g≤ 1,the second firm acquires an endogenous Stackelbergleadership position. If products become more differen-tiated, the profits of both firms increase.

4.2. The SR,MR and MS Cases

Aswe considered aforementioned duopolies involving onestandard pure profit-maximizing firm, three asymmetricalcases are left to be analyzed, namely SR,MR and MS.The first case is solved analytically by Manasakis et al.(2011) and the other cases numerically. We improve uponthis by offering analytical proofs for all three asymmetricalcases. In Appendices C and D, we derive formulas for theequilibrium quantities for theMR andMS cases. All resultsare summarized in Table 2.

The expressions for the equilibrium quantities for theMR andMS cases look rather horrifying, even to such anextent that one could have doubts whether such analysesare really needed. After all, equilibrium results can alsobe obtained numerically, as in Manasakis et al. (2011).However, to properly compare profits for highly differ-entiated products—for instance, in order to determinedominant strategies—we need exact expressions, in-stead of numerical approximations.

5. OPTIMAL CONTRACT DESIGN

Finally, we consider the case in which owners canfreely choose the type of contract for their managers.Hence, we now move to really strategic delegationgames. In order to prove that using a contract basedon relative profits is optimal for any degree of productdifferentiation, we summarize the profits for bothfirms for all possible options for the owners in Table 3.


Table 2. SR,MR and SM Cases

Type of contracts Output level

Sales/relative profitsqSR ¼ 8� 4g� 2g2 � g3

4 4� 3g2ð Þ m� cð Þ

qRS ¼ 4� 2g� g2

2 4� 3g2ð Þ m� cð Þ

Market share/relative profits qMR ¼ 8� 8gþ 6g2 � 4g3 � g4

2� gð Þ 2g3 � 8gþ 8½ � þ 2gffiffiffiffiD

p m� cð Þ

qRM ¼ �g g3 � 2g2 � 16gþ 16½ � þ 2þ gð Þ ffiffiffiffiD

p

2� gð Þ 4g3 � 16gþ 16½ � þ 4gffiffiffiffiD

p m� cð Þ

D ¼ 64� 64gþ 16g2 � 64g3 þ 40g4 þ 8g5 þ g6� �

Market share/salesqMS ¼ 32� 16g� 16g2 þ 5g4 � g

ffiffiffiffiE

p

64� 32g� 64g2 þ 8g3 þ 20g4 þ 8g5 � 2g6m� cð Þ

qSM ¼ �32gþ 4g3 þ 16g4 þ g5 þ 4� g2ð Þ ffiffiffiffiE

p

128� 64g� 128g2 þ 16g3 þ 40g4 þ 16g5 � 4g6m� cð Þ

E ¼ 256� 256gþ 64g2 � 96g3 þ 16g4 þ 16g5 þ 17g6� �


In order to compare the rows (columns) in this ta-ble, we represent—by using the expressions for theprofits that can be found in Appendix E—the graphsof the payoffs for firm 1 given the choice of the rivalin Figures 1–4.

According to these figures, for any g2 (0, 1], thefirst row—and for reasons of symmetry, also the firstcolumn—is strictly dominated by the fourth one. Fur-thermore, in the remaining 3� 3 table—for fixedg< 1—the third row (column) strictly dominates thesecond one, which in turn strictly dominates the firstone. Observe that the Table 1 in Jansen et al. (2009)reveals that this property does not hold if g = 1—thatis, if the products are homogeneous. Apparently,choosing a contract based on relative profits is the bestoption for the owners. In all, this gives a correctedversion of Proposition 2 in Manasakis et al. (2011).

Table 3. Four Bonus Systems Compared

Pureprofitsoriented

Salesoriented

Marketshare

oriented

Relativeprofitsoriented

Pureprofitsoriented

(pPP, pPP) (pPS, pSP) (pPM,pMP) (pPR,pRP)

Salesoriented

(pSP,pPS) (pSS, pSS) (pSM, pMS) (pSR,pRS)

Marketshareoriented

(pMP, pPM) (pMS, pSM) (pMM,pMM) (pMR, pRM)

Relativeprofitsoriented

(pRP, pPR) (pRS, pSR) (pRM,pMR) (pRR,pRR)


Proposition 4If in a duopoly market owners may freely choose betweenthe four types of bonus contracts introduced previously,then the dominant strategy for both owners for all degreesof product differentiation is to design amanagerial contractbased on relative profits evaluation.

This shows that the result of Jansen et al. (2009) isrobust against the introduction of the assumption ofproduct differentiation.

Next, we consider the case in which one firm, say firm2, is unable to change its type of managerial bonuscontract—that is, firm 2 is characterized by contractualinertia. Such inertia may be the consequence of a coun-try’s culture, for example, as indicated by the empiricalresearch of Borkowski (1999). If a fixed type of manage-rial contract belongs to the blueprint of firm 2, whereasthe rival firm is flexible concerning the choice of the typeof contract, the flexible firm can substantially increase itsprofits by making a profit-maximizing contractual shift.If, for instance, the inert firm 2 uses a contract based onpure profits, the flexible firm 1 can acquire a Stackelbergleadership position by using the S, M or R types of con-tract. Figures 2–4 reveal another regularity. Let us fixg2 (0, 1) and assume that the manager’s contract of theinert firm 2 is based on sales, market share or relativeprofits. Then, the profits of firm 1 increase if the type ofcontract shifts from P to S, from S to M and from M toR. Accordingly, the profits of firm 2 show a decreasingsequence. It can be proven that this pattern holds ingeneral, provided that products are no perfect substitutes,so for 0< g< 1. As mentioned previously, the flexiblefirm always maximizes its profits by adopting a contractbased on relative profits evaluation. The core message


× (m − c)2

SP = MP = RP

PP

0.16

0.14

0.12

0.10

0.08

0.06

0.5 0.6 0.7 0.8 0.9 1

Figure 1. Payoffs for a firm if the rival is pure profitsoriented.

× (m − c)2

SS

MS

RS

PS

0.16

0.14

0.12

0.10

0.08

0.06

0.5 0.6 0.7 0.8 0.9 1

Figure 2. Payoffs for a firm if the rival is sales oriented.

× (m − c)2

SR

MR

RR

PR

0.16

0.14

0.12

0.10

0.08

0.06

0.5 0.6 0.7 0.8 0.9 1

Figure 4. Payoffs for a firm if the rival is relative profitsoriented.

T. JANSEN ET AL.66

× (m − c)2

SM

MM

RM

PM

0.16

0.14

0.12

0.10

0.08

0.06

0.5 0.6 0.7 0.8 0.9 1

Figure 3. Payoffs for a firm if the rival is market shareoriented.

Copyright © 2011 John Wiley & Sons, Ltd. Manage. Decis. Econ. 33: 61–70 (2012)DOI: 10.1002/mde

of this argument is that flexibility in the choice of manage-rial remuneration contracts does pay off.

To conclude, we investigate those cases in which onecompany uses a relative profits-oriented bonus, whereasthe managerial bonus contract of the rival firm is basedon profits in combination with sales or market share.As Table 2 reveals, the output level of this firm differsfrom the Stackelberg quantity for both the SR and MRcases. In all, this provides our final proposition.

Proposition 5If one of the firms in a duopoly uses a managerial con-tract based on relative profits and the rival is sales ormarket share oriented, then the first firm obtains aStackelberg leadership position only if the productsare perfect substitutes.

6. CONCLUSION

Manasakis et al. (2011) develop a product differentia-tion version of the Cournot delegation game with prod-uct homogeneity of Jansen et al. (2009). They claim thatthe key finding of Jansen et al. (2009), that managerialremuneration based on relative profits dominates thepure profits, sales and market share alternatives, doesnot survive the introduction of product differentiation.In this comment, we correct the model of Manasakis etal. (2011) and prove that the original result of Jansenet al. (2009) is robust against relaxing the product ho-mogeneity assumption.Moreover, along the way, we in-troduce analytical proofs, instead of the numericalapproximations of Manasakis et al. (2011). In future


research, the robustness of the dominance of relativeprofits finding can be further explored—for instanceby relaxing the duopoly and Cournot assumptions.

Acknowledgement

Arjen van Witteloostuijn gratefully acknowledges the financialsupport through the Odysseus program of the Flemish ScienceFoundation (FWO).

NOTES

1. In what follows, for the sake of non-discriminatory con-venience, owners are considered to be female, whereasmanagers are male.

2. For details, we refer to Appendix A.3. For details, we refer to Appendix B.4. Because the second coordinate of Sg is less than

12 m� cð Þ for 0< g< 1, we can easily show that, ifmanager 2 is rewarded on the basis of relative profits,the contractual weight w2 is less than 1.

APPENDIX A: THE MM CASE

THE MARKET STAGE

Manager i maximizes his bonus by properly choosingthe production level qi. The corresponding first-orderconditions (the second-order conditions are fulfilled) are

m� cð Þ � 2q1 � gq2 þ w1q2Q2

¼ 0

m� cð Þ � gq1 � 2q2 þ w2q1Q2

¼ 0:

8><>: (A1)

By slightly adapting the proof of Lemma 1 in Jansenet al. (2007), we can show that for any feasible combina-tion of the weights w1 and w2, the two reaction curves,given by the two foregoing equations, intersect once.

If we denote the left-hand side of the ith equation of

system (A1) by Fi, then the Jacobian determinant JF ¼det @Fi

@qj

h iis strictly positive for allw1,w2≥0 and 0< g≤1.

According to the Implicit Function Theorem,

and

@q1@w1

¼ q2Q2JF

2þ 2w2q1Q3

� �@q2@w1

¼ q2Q2JF

�gþ w21Q3

q2 � q1ð Þ� �

:

As owner 1 wants to maximize her profits by strategi-cally choosing the weight w1, the first-order condition is


@p1@w1

¼ 0, @q1@w1

m� cð Þ � 2q1 � gq2½ � � gq1@q2@w1

¼ 0

, 2þ 2w2q1Q3

� �m� cð Þ � 2q1 � gq2½ � þ g2q1

�gw2q1Q3

q2 � q1ð Þ ¼ 0

,2 m� cð Þ � 4� g2� �

q1 � 2gq2 þ

w2q1Q3

2 m� cð Þ � 4� gð Þq1 � 3gq2½ � ¼ 0:

For owner 2, we obtain the first-order conditionby interchanging indices. So, the equilibrium has tosatisfy the two equations (A1) and the two first-orderconditions for both owners.

Solving the system

Exploiting the problem’s symmetry, we use q1 = q2 = qand w1 =w2 =w. Then we obtain

m� cð Þ � 2þ gð Þqþ w14q

¼ 0,w ¼ � m� cð Þ þ 2þ gð Þq½ �4qand

2 m� cð Þ � 4þ 2g� g2� �

qþ

w18q2

2 m� cð Þ � 4þ 2gð Þq½ � ¼ 0:

Substitution of the expression for w in the secondequation leads to the quadratic equation

8þ 6gð Þq2 � 6þ 2gð Þ m� cð Þqþ m� cð Þ2 ¼ 0:

Hence, in equilibrium, each firm produces

qMM ¼ 3þ gþffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2

p8þ 6g

m� cð Þ:

APPENDIX B. PROOF OF PROPOSITION 2

If both firms choose the same type of managerial bonuscontract, profits can be written as [(m� c)� (1 + g)q]q,where q is the output level of each firm in equilibrium.As, for any g, profits are decreasing for productionlevels beyond 1

2 1þgð Þ m� cð Þ, the statement pPP>pRRpMM>pSS follows from

12 1þ gð Þ m� cð Þ < qPP < qRR < qMM < qSS:

These inequalities can be easily checked. It isstraightforward to show that the derivatives of all outputand profit levels with respect to g are negative.


T. JANSEN ET AL.68

APPENDIX C. THE MR CASE

As owner 1 uses a market share-oriented managerialbonus contract, whereas owner 2 applies a contractbased on relative profits, the bonuses of manager 1 andmanager 2 are proportional top1 þ w1

q1Q and p2�w2p1,

respectively.

The market stage

As the managers want to maximize their bonus bystrategically choosing the output level, the first-orderconditions in the market stage (the second-order con-ditions are fulfilled) are

m� cð Þ � 2q1 � gq2 þ w1q2Q2

¼ 0

m� cð Þ � g 1� w2ð Þq1 � 2q2 ¼ 0:

((C1)

The contract stage

Using similar techniques as in Appendix A, we obtainthe first-order conditions in the contract stage

2 m� cð Þ � 4� g2 1� w2ð Þ½ �q1 � 2gq2 ¼ 02 m� cð Þ � 2gq1 � 4� g2ð Þq2þ

w1q2Q3

2 m� cð Þ � 3gq1 � 4� gð Þq2½ � ¼ 0:

8>><>>:

(C2)

We have to solve the four equations in systems(C1) and (C2) simultaneously.

Solving the system

According to the first equation of system (C1),w1q2Q2 ¼

2q1 þ gq2 � m� cð Þ. Substitution in the second equa-tion of system (C2) leads to

�8gq21 � 4þ 4g� 2g2� �

q22 � 12þ 2g2� �

q1q2þ 6þ 3gð Þ m� cð Þq1 þ 6þ gð Þ m� cð Þq2� 2 m�cÞ2 ¼ 0:

(C3)

By using the second equation of (C1) and the firstone of (C2), q1 and q2 can be expressed in terms of1�w2. We obtain

andq1 ¼ 2� g

4� 2g2 1� w2ð Þ m� cð Þ

q2 ¼ 4� g2 þ 2gð Þ 1� w2ð Þ8� 4g2 1� w2ð Þ m� cð Þ

Substitution of these formulas for q1 and q2 in (C3)gives

g[8� 8g +6g2� 4g3� g4](1�w2)2� 2[16� 8g� 8g2+

2g3� g4](1�w2)+8[4� 5g + g2] =0.


Solving this equation produces the formulas represented

in Table 2.

APPENDIX D: THE MS CASE

As owner 1 applies a market share-oriented managerialbonus contract, whereas owner 2 uses a contract basedon sales, the bonuses of manager 1 and manager 2 areproportional to p1 þ w1

q1Q and p2 +w2q2, respectively.

The market stage

As the managers want to maximize their bonus bystrategically choosing their output levels, the first-or-der conditions in the market stage (the second-orderconditions are fulfilled) are

m� cð Þ � 2q1 � gq2 þ w1q2Q2

¼ 0

m� cð Þ � gq1 � 2q2 þ w2 ¼ 0:

((D1)

The contract stage

Using similar techniques as in Appendix A, we obtainthe first-order conditions in the contract stage

2 m� cð Þ � 4� g2½ �q1 � 2gq2 ¼ 02 m� cð Þ � 2gq1 � 4� g2ð Þq2þ

w1q2Q3

2 m� cð Þ � 3gq1 � 4� gð Þq2½ � ¼ 0:

8>><>>:

(D2)

We have to solve the four equations in systems(D1) and (D2) simultaneously.

Solving the system

According to the first equation of system (D1),w1q2Q2 ¼

2q1 þ gq2 � m� cð Þ. Substitution in the second equa-tion of system (D2) leads to

�8gq21 � 4þ 4g� 2g2� �

q22 � 12þ 2g2� �

q1q2þ 6þ 3gð Þ m� cð Þq1 þ 6þ gð Þ m� cð Þq2� 2 m� cð Þ2¼ 0:

Substitution of q2 ¼ 12g 2 m� cð Þ � 4� g2ð Þq1½ �—

which follows from the first equation of (D2)—gives

32� 16g� 32g2 þ 4g3 þ 10g4 þ 4g5 � g6� �

q21 �32� 16g� 16g2 þ 5g4½ � m� cð Þq1 þ8� 4g� 2g2½ � m� cð Þ2 ¼ 0:Solving this equation generates the formulas repre-sented in Table 2.



APPENDIX E: THE PROFITS FOR ALL CASES

pPP ¼ 1

2þ gð Þ2 m� cð Þ2

pSP ¼ pMP ¼ pRP ¼ 2� gð Þ28 2� g2ð Þ m� cð Þ2

pPS ¼ pPM ¼ pPR ¼ 4� 2g� g2½ �216 2� g2ð Þ2 m� cð Þ2

pSS ¼ 4� 2g2

4þ 2g� g2½ �2 m� cð Þ2

pSM ¼�2048þ3072g2þ512g3 þ 128g4�1792g5 � 912g6þ688g7 þ 336g8þ20g9 � 55g10�2g11

8 32� 16g� 32g2 þ 4g3 þ 10g4 þ 4g5 � g6½ �2 m� cð Þ2

þ 256� 128g� 256g2 þ 80g3 þ 64g4 � 8g5 þ 4g6 � g7 þ 2g8� � ffiffiffi

Ep

8 32� 16g� 32g2 þ 4g3 þ 10g4 þ 4g5 � g6½ �2 m� cð Þ2

pSR ¼ 2� gð Þ 8� 4g� 2g2 � g3½ �16 4� 3g2ð Þ m� cð Þ2

pMS ¼ 2� g2� � 512� 512g� 256g2 þ 128g3 þ 320g4 � 128g5 � 72g6 þ 8g7 þ 21g8

64� 32g� 64g2 þ 8g3 þ 20g4 þ 8g5 � 2g6½ �2 m� cð Þ2

� 2� g2� � g 32� 16g� 16g2 þ 5g4½ � ffiffiffi

Ep

64� 32g� 64g2 þ 8g3 þ 20g4 þ 8g5 � 2g6½ �2 m� cð Þ2

pMM ¼ 7þ 5g� g2 � g3 þ 1� g� g2ð Þffiffiffiffiffiffiffiffiffiffiffiffiffi1þ g2

p2 4þ 3gð Þ2 m� cð Þ2

pMR ¼ 2� gð Þ �2g4 � 8g3 þ 12g2 � 16gþ 16½ �16 2� gð Þ g3 � 4gþ 4½ � þ g

ffiffiffiffiD

p� � m� cð Þ2

pRS ¼ 2� g2ð Þ 4� 2g� g2½ �28 4� 3g2ð Þ2 m� cð Þ2

pRM ¼ �2g9 þ 2g8 þ 76g7 þ 168g6 � 160g5 � 144g4 � 832g3 þ 1152g2 � 256

16 2� gð Þ g3 � 4gþ 4½ � þ gffiffiffiffiD

p� �2 m� cð Þ2

þ 2g6 þ 6g5 þ 12g4 þ 48g3 � 96g2 � 32gþ 64� � ffiffiffiffi

Dp

16 2� gð Þ g3 � 4gþ 4½ � þ gffiffiffiffiD

p� �2 m� cð Þ2

pRR ¼ 4� g2

16 1þ gð Þ m� cð Þ2

where

D ¼ g6 þ 8g5 þ 40g4 � 64g3 þ 16g2 � 64gþ 64

E ¼ 256� 256gþ 64g2 � 96g3 þ 16g4 þ 16g5 þ 17g6

Copyright © 2011 John Wiley & Sons, Ltd. Manage. Decis. Econ. 33: 61–70 (2012)DOI: 10.1002/mde

T. JANSEN ET AL.70

REFERENCES

Basu K. 1995. Stackelberg equilibrium in oligopoly: anexplanation based on managerial incentives. EconomicsLetters 49: 459–464.

Borkowski SC. 1999. International managerial performanceevaluation: a five-country study. Journal of InternationalBusiness Studies 30: 533–555.

Fershtman C, Judd K. 1987. Equilibrium incentives inoligopoly. American Economic Review 77: 927–940.

Irwin DA. 1991. Mercantilism as strategic trade policy: theAnglo-Dutch rivalry for the East India trade. Journal ofPolitical Economy 99: 1296–1314.

Jansen M, van Lier A, van Witteloostuijn A. 2007. A note onstrategic delegation: the market share case. InternationalJournal of Industrial Organization 25: 531–539.

Jansen M, van Lier A, van Witteloostuijn A. 2009. On theimpact of managerial bonus systems on firm profit andmarket competition: the cases of pure profit, sales, marketshare and relative profits compared. Managerial andDecision Economics 30: 141–153.

Manasakis C, Mitrokostas E, Petrakis E. 2011. Endogenousmanagerial incentive contracts in a differentiated duopoly,


with and without commitment. Managerial and DecisionEconomics 31: 531–543.

Miller N, Pazgal A. 2002. Relative performance as a strate-gic commitment mechanism. Managerial and DecisionEconomics 23: 51–68.

Miller N, Pazgal A. 2005. Strategic trade and delegated com-petition. Journal of International Economics 66: 215–231.

Nakamura Y. 2008. Bargaining over managerial contracts indelegation games: the differentiated goods case. Econom-ics Bulletin 12(6): 1–8.

Salas Fumas V. 1992. Relative performance evaluation ofmanagement. International Journal of Industrial Organi-zation 10: 473–489.

Singh N, Vives X. 1984. Price and quantity competition in adifferentiated duopoly. The Rand Journal of Economics15: 546–554.

Sklivas SD. 1987. The strategic choice of managerial incen-tives. The RAND Journal of Economics 18: 452–458.

Vickers J. 1985. Delegation and the theory of the firm. TheEconomic Journal 95: 138–147.

van Witteloostuijn A. 1998. Bridging behavioral and eco-nomic theories of decline: organizational inertia, strategiccompetition, and chronic failure. Management Science44: 501–519.


Documents

Managerial Bonus Systems in a Differentiated Duopoly: A Comment