Exchange Agreements Facilitate Collusion

Abstract. A duopoly model with quantity competition is analyzed in which firmscollude in two markets. There is specialization in production in order to promoteefficiency. Firms may then either exclusively market one good each, or they mayagree to exchange goods and cross-supply a part of the production to the other firm. Itis shown that, compared to specialization in marketing, positive exchanges of goodsrelax the incentive constraints that limit the extent of collusion.

1. INTRODUCTION

Exchanges of goods between firms, each of which is capable of producing andmarketing the goods itself, have been a subject of concern to competitionauthorities. In 1988, for example, the Commission of the European Unionimposed fines summing up to ECU13 million on three Italian producers of flatglass. These producers had been engaged in reciprocal exchanges of goods. TheCommission showed that there was collusion in the market and alleged thatthe exchanges supported the collusion. The firms claimed that such exchangesimprove efficiency in production, but the Commission argued that theexchange agreement helped the firms in dividing up the market. The decisionwas overruled by the European Court of Justice in 1992. The Court denied thatthe Commission had proved that the exchanges had a systematic character, butsustained the accusations concerning collusion.1

A different policy was followed in the German power cable industry. TheGerman anti-trust authority, the Bundeskartellamt, allowed an exchangeagreement as a legal exception to the general cartel prohibition.2 From 1975on, firms were encouraged to specialize in several types of cables, while at the

ß Verein fuÈr Socialpolitik and Blackwell Publishers Ltd 2001, 108 Cowley Road, Oxford OX4 1JF, UKand 350 Main Street, Malden, MA 02148, USA.

German Economic Review 2(2): 113±125

Exchange Agreements FacilitateCollusion

Hans-Theo NormannHumboldt University, Berlin

1. See the Official Journal, No. L33/44; and the European Court of Justice, T68/89, T77/89 and T78/89.

2. Such an exception is possible if the conditions of §5 of the German competition law are met;that is, if the rationalization effects compensate for the restraint of competition.

same time committing themselves to cross-supply these cables to other firms inthe market upon demand. The purpose of this policy was to promote efficiencyby reducing the high fixed costs inherent in the production of power cables. In1980 the cartel contract was extended for another five years and then it wasstopped because no more efficiency gains could be expected. Firms had toreturn to `normal' competition. Twelve years later, in 1997, the Bundes-kartellamt imposed fines on these power cable producers because of a pricecartel and quota agreements. The fines totaled DM300 million (in an industrywith revenues of only DM1.1 billion), and they were the highest fines everimposed up to that time.3

These cases show that exchange agreements might be part of a largercollusive arrangement, but they also show that efficiency gains are possiblefrom such agreements. Spiegel (1993, p. 571) indicates the potential tradeofffrom an exchange agreement: Òn the one hand, it may enable firms to betterallocate production among themselves, thereby promoting efficiency. On theother hand, it may also facilitate collusion.'

Given that efficiency gains from specialization in production exist, thequestion is whether more competition results from an exchange agreement orfrom specialization in marketing the goods. In supporting the exchanges ofgoods, the aim of the policy of the Bundeskartellamt was to promotecompetition in marketing the products. By contrast, in the flat-glass case theCommission argued that specialization in marketing is more competitive. It isthe purpose of this paper to propose an analytical framework in which thesepolicies can be evaluated.

There are several theoretical papers that analyze the efficiency gains ofreciprocal exchanges. Kamien et al. (1989), Spiegel (1993), and Baake et al.(1999) have demonstrated such efficiency-enhancing consequences in a varietyof models. Exchange agreements4 allow firms to allocate production to themore efficient firm and fixed costs might also be saved.

Surprisingly little theoretical work has been done on the collusive effects ofexchange agreements. Since firms operate in more than just one market,Bernheim and Whinston's (1990) seminal paper on multi-market contactapplies. Bernheim and Whinston analyze the general effects that such multi-market contact has on the degree of collusion in these markets. They alsodiscuss the effect of reciprocal exchanges, but they come to a differentconclusion than this paper. Their results are discussed in the conclusion.

The scenario analyzed in this paper is as follows. Suppose that two firmsattempt to collude in two markets in an infinitely repeated game. Since firmshave different marginal cost, efficiency requires that they specialize inproduction. When exchange agreements are legal, firms choose how much to

3. See the annual reports of the Bundeskartellamt of 1976, 1986, and 1998, or Hahn and Normann(2000) for a detailed study of the case.

4. In the theoretical literature, `reciprocal dealing' and `cross-supplies' are used as synonymousterms for exchange agreements. A related phenomenon is `horizontal subcontracting'.

114 ß Verein fuÈr Socialpolitik and Blackwell Publishers Ltd 2001

H.-T. Normann

exchange in every period. They may decide to completely specialize inmarketing in which case they simply monopolize one market each.5 Otherwisethey decide to exchange a positive amount such that each firm has a positivemarket share in both markets. As it turns out, firms choose a positive transfer.The reason is that this facilitates collusion under fairly general modelingassumptions. Since the efficiency gains result from the specialization inproduction and not from the exchanges, the effect of exchange agreements onwelfare is unambiguously negative.

After the assumptions of the market model are stated in the next section,Section 3 introduces the exchange agreements. Section 4 contains the resultsand Section 5 is the conclusion.

2. THE MODEL

Consider two markets, a and b, and two firms, 1 and 2, which may operate inboth markets. Firms play a Cournot game in both markets. Their quantities aredenoted by qik, i � 1;2 and k � a; b. Inverse demand in market k is pk�Qk�, whereQk � q1k � q2k, k � a; b. Firm i's profit in market k is denoted by �ik�qik; qjk�,i � 1;2, i 6� j and k � a; b. The best reply of firm i to firm j's output in market k isdenoted by Rik�qjk�.

The following assumptions are employed in the paper:

Assumption A1. The inverse demand functions, pk�Qk�, k � a; b; are twicecontinuously differentiable and are downward sloping; that is, p0k�Qk� < 0.

Assumption A2. There are constant marginal costs for each firm in eachmarket, cik, i � 1;2 and k � a; b. Let c1a � c2a and c2b � c1b.

Assumption A3. Monopoly profits, �pk�qik� ÿ cik�qik, are concave in output;that is, p00k�qik�qik � 2p0k�qik� < 0:

Assumption A4. A monopolized market can profitably be invaded; that is,Rik�~qjk� > 0 for i; j � 1;2, i 6� j and k � a; b, where ~qjk is the unique maximizer of�pk�qjk� ÿ cjk�qjk.

Assumption A2 ensures that specialization in production makes sense. Withconstant marginal cost, all production may plausibly be shifted to one firm, themore efficient one. In that case, firm 1 produces in market a and firm 2produces in market b.6 If convex cost were assumed instead, there would beincentives not to allocate all production to one firm in each market. Efficiencywould require that both firms produce a positive amount of both goods.Constant marginal costs also simplify the analysis (see footnote 10 below).Further, no fixed costs are assumed though fixed costs are an important factor

5. In Edward's (1955) words, they develop `spheres of influence'.6. The assumption that each firm is more efficient in one market is without loss of generality.

The results below also hold if one firm is more efficient in both markets.

ß Verein fuÈr Socialpolitik and Blackwell Publishers Ltd 2001 115

Exchange Agreements Facilitate Collusion

suggesting the specialization in production. They can easily be added to themodel without altering the results. Assumption A3 ensures that there is aunique (joint) profit maximum. Assumption A4 is innocuous. If it was not met,a monopolized market would be a static Nash equilibrium and there would beno need to collude.

Firm i's profit in market k is �ik�qik; qjk� � �pk�Qk� ÿ cik�qik. The best reply offirm i, Rik�qjk�, is implicitly defined by @�ik=@qik � 0. Differentiating this termwith respect to qjk yields

R0ik � ÿ@2�ik

@qik@qjk

. @2�ik

@q2ik

as an expression for the slope of the best-reply function. From Assumption A2 itfollows that R0ik > ÿ1:

This stage game is repeated infinitely many times in periods t � 0; . . . ;1. Let�� denote the common weight with which firms discount future profits. Thediscount factor is commonly known.

3. EXCHANGE AGREEMENTS AND MULTI-MARKET CONTACT

The purpose of this paper is to analyze whether firms can use exchangeagreements to support collusion in the infinitely repeated game. In terms ofcontract theory, there are two types of contracts here. First, there is an implicitcontract about a collusive level of output. Second, there is an explicit contractabout the exchanges of goods.7 Final output decisions are non-cooperative andare assumed to occur after any possible exchanges. Whether or not firms adhereto the implicit contract is determined by the incentive constraint of theinfinitely repeated game. Therefore, firms choose outputs and exchanges inorder to maximize joint profits given the incentive constraint. In order toderive the incentive constraint, collusion, deviation and punishment profitsare analyzed in turn.

Consider collusion first. To produce the collusive level of output efficiently,specialization in production is required according to Assumption A2. Denotethe monopoly quantity produced by the efficient firm in market k by Qm

k . Thatis, Qm

k is the maximizer of �pk�Qk� ÿminfc1k; c2kg�Qk. Let pmk � pk�Qm

k �, k � a; b.Suppose that the firms attempt to implement collusive outputs ofQ�k � Qm

k ; k � a; b; which are produced efficiently.When exchanges are possible, firms still produce Q�a and Q�b but they are now

free to determine the market shares in a collusive equilibrium. In other words,given the pattern of specialization in production, firms decide whether or notto specialize in marketing the goods. It is useful to denote the quantitiesexchanged in proportions �a and �b of the collusive quantities. That is, �a � Q�a is

7. Schmidt and Schnitzer (1995) study the general interaction of explicit and implicit contractsin repeated principal±agent models.


H.-T. Normann

to be exchanged for �b � Q�b. In addition, a net payment T from firm 1 to firm 2is made. Collusive profits ��i consist of the gross profits made in both markets,denoted by ��i , and the payment T; that is, ��1 � ��1 ÿ T and ��2 � ��2 � T. Theexpressions for gross collusive profits are

��1 � �1ÿ �a�Q�ap�a � �bQ�bp�b ÿ c1aQ�a

��2 � �aQ�ap�a � �1ÿ �b�Q�bp�b ÿ c2bQ�b�1�

The payment T depends on �a and �b since T � ��1 ÿ��1 � ��2 ÿ ��2. Note thatfirms can always choose �a � �b � T � 0 such that no exchanges occur. Being alegal transaction, the exchanges are contractual. There is a new agreement inevery period.8

Some remarks about the exchange of quantities seem warranted. Note that itis the collusive level of output that determines the joint collusive profits,��1 ��2. The distribution of collusive profits results from some bargainingprocess not further specified. No matter what the market shares (as determinedby the exchanges) are, the distribution of collusive profits can always besustained by appropriately adjusting the payment T. Therefore, firms have noconflict in deciding about the exchanges since the agreement does not affectthe collusive profits. When deciding about exchanges of goods, firms have thecommon interest of preventing a deviation from collusion.

After any possible exchanges of goods, firms make their final outputdecisions for both goods. They may adhere to the implicit contract about thecollusive output, but they may also deviate. Formally, a deviation is as follows.Bernheim and Whinston (1990) show that, in a multi-market setting, deviationwill occur in all markets. The net deviation profits in both markets are �d

i , and�d

i denote the gross deviation profits, so one gets �d1 � �d

1 ÿ T and �d2 � �d

2 � T.Gross deviation profits depend on the degree of the exchange of goods

�d1 �

R1a��aQ�a� � pa��aQ�a � R1a��aQ�a�� ÿ ��aQ�a � R1a��aQ�a�� c1a

�R1b��1ÿ �b�Q�b� � pb��1ÿ �b�Q�b � R1b��1ÿ �b�Q�b��ÿ�R1b��1ÿ �b�Q�b� ÿ �bQ�b� � c1b if R1b��1ÿ �b�Q�b� > �bQ�b

�2�R1a��aQ�a� � pa��aQ�a � R1a��aQ�a�� ÿ ��aQ�a � R1a��aQ�a�� c1a

��bQ�bp�b if R1b��1ÿ �b�Q�b� � �bQ�b

8>>>>>><>>>>>>:

�d2 �

R2b��bQ�b� � pb��bQ�b � R2b��bQ�b�� ÿ ��bQ�b � R2b��bQ�b�� c2b

�R2a��1ÿ �a�Q�a� � pa��1ÿ �a�Q�a � R2a��1ÿ �a�Q�a��ÿ�R2a��1ÿ �a�Q�a� ÿ �aQ�a� � c2a if R2a��1ÿ �a�Q�a� > �aQ�a

�3�R2b��bQ�b� � pb��bQ�b � R2b��bQ�b�� ÿ ��bQ�b � R2b��bQ�b�� c2b

��aQ�ap�a if R2a��1ÿ �a�Q�a� � �aQ�a

8>>>>>><>>>>>>:8. If the exchanges and the payment T were non-contractual, it could be possible that a firm

deviates at this first stage in the sense that it receives a good but does not deliver and pay T.However, as in Bernheim and Whinston (1990), this possibility is not further investigated.



In the market in which a firm does not produce in a collusive equilibrium, thebest reply might be smaller than the quantity received as an exchange inequilibrium; that is, R1b��1ÿ �b�Q�b� � �bQ�b or R2a��1ÿ �a�Q�a� � �aQ�a. Forexample, if c2b < c1b and if �b � 1, R1b�0� < Qm

b � Q�b. In such cases, firms putthe collusive quantity on the market when deviating.9 In the market in which afirm produces in equilibrium, deviation might imply to produce less than Q�k .For example, when �a � 0; R1a�0� � Qm

a � Q�a. In such cases, firms produce thedeviation quantity.10

Deviation can only be prevented if it triggers some punishment. Supposethat a firm receives a present discounted value equal to PDV

pi after deviating.

Let �pi be the corresponding instantaneous profit; that is �p

i� PDV

pi �1ÿ ��:

The punishment can be reversion to a static Cournot±Nash equilibrium or theminmax strategy against the deviator (see, e.g., Fudenberg and Tirole, 1991).Optimal punishment paths (Abreu, 1986) are defined for symmetric firms only.

Collusion is supportable as a subgame-perfect equilibrium only if

�di �

��

1ÿ ��pi �

1

1ÿ ��i i � 1;2 �4�

Solving (4) for �� one finds

�� di ÿ��i

�di ÿ�

pi

� �i i � 1;2 �5�

Only if firms' actual discount factor, ��, satisfies �� maxf�1; �2g is collusionstable.

Generally, optimal collusion has firms maximizing joint profits given theincentive constraint, i.e.

maxQa; Qb; �a; �b

Xi�1;2

��i s.t. �� maxf�1; �2g �6�

where the joint collusive profit is distributed according to some bargainingrule.11 Note that changing the �k does not change collusive profits but, as willbe shown below, it does change the �i: It follows that, when the solutioninvolves Q�k > Qm

k ; k � a; b; the ��a and ��b solving (6) also solve

min�a;�b

fmaxf�1; �2gg �7�

9. The situation where firms want to destroy output when deviating does not occur whenproduction costs, c1a � c2a and c2b � c1b, are sufficiently high such that the best reply is greaterthan or equal to �kQ�k .

10. Note that, because marginal costs are constant, the reaction functions do not depend on thequantities exchanged. This would not be the case with convex cost functions where a firm'sreaction function depends on how much it has agreed to deliver to the other firm and howmuch of its sales arise from goods produced by the other firm.

11. Since �i � ��di ÿ ��i �=��d

i ÿ T�ÿ1�iÿ1 ÿ�pi �, the distribution of the collusive profits does affect

the �i. As a referee pointed out, firms may consider varying T in order to reach an even largerscope of collusive equilibria.

max

min


H.-T. Normann

The focus of the paper is on the effect of exchange agreements. Therefore,rather than the general objective (6), (7) is analyzed below for given collusiveprofits ��i and given Q�k � Qm

k ; k � a; b.12 From the firms' point of view, thesolution to (7) is optimal as it yields the largest scope for which collusion ispossible. From a policy perspective, this is the worst outcome for the samereason. The solution may be specialization in marketing, but it may alsoinvolve some �k 2 �0;1�; that is, a positive transfer of some good.

4. THE RESULTS

To see how exchange agreements affect collusion, it is useful to note how theminimum discount factors, �i, i � 1;2, react to changes in �k, k � a; b.

Lemma 1. For the partial derivatives @�i=@�a, i � 1;2,

sign@�1

@�a

� �� signfp�a ÿ pa��aQ�a � R1a��aQ�a��g

sign@�2

@�a

� �� signfminfpa��1ÿ �a�Q�a � R2a��1ÿ �a�Q�a�� ÿ p�a; 0gg

holds, and accordingly for @�i=@�b, i � 1;2:

The proof of the lemma, like all other proofs, can be found in the Appendix.Note that @�2=@�a � 0 and @�1=@�b � 0: Provided that Q�k � Qm

k ; k � a; b, holds,@�1=@�a � 0 and @�2=@�b � 0: This also holds for Q�k > Qm

k in the range of �k

where the solutions to (7) are found (see the proof of Lemma 2). Consider@�1=@�a to gain the intuition behind Lemma 1 in these cases. There are twoeffects: increasing �a reduces both gross deviation profits and gross collusiveprofits. The latter effect turns out to dominate the former. Therefore, increasing�a increases firm 1's deviation profit relatively and so �1 increases. It followsthat supplying more of firm i's good to firm j increases �i but reduces �j,i; j � 1;2; i 6� j.

There is a general property of all solutions which will also be helpful in theproofs below.

Lemma 2. Every solution of (7) involves �1 � �2.

The intuition behind Lemma 2 is straightforward. Over the relevant range ofparameters one can reduce �i at the expense of increasing �j: Therefore, �i > �j

cannot be a solution. Using the lemmas, the following proposition can beproved.

12. When �� is sufficiently high, firms get the unconstrained optimum with Q�k � Qmk . In such

cases, �a and �b other than those solving (7) may be sufficient to get �� maxf�1; �2g.



Proposition 1. Complete specialization in marketing never occurs; that is,(�a � �b � 0), (�a � �b � 1) and (�l � 1; �m � 0; l; m � a; b; l 6� m) do not solve(7). Moreover, ��k 2 �0;1�; k � a; b solve (7) provided that Q�k > Qm

k ; k � a; b.

Proposition 1 states that, in at least one market k, firms choose �k 2 �0;1�. Inother words, there is a positive (but not complete) transfer between firms of atleast one good. Under no circumstances is complete specialization in marketingchosen. When Q�k > Qm

k , the stronger statement can be made that each firm willhave a positive market share in both markets. The general conclusion is that, iffirms are free to agree on exchanges, they will always do so as this minimizesthe discount factor required to sustain collusion.

From Proposition 1, the welfare implications of exchange agreements aregenerally negative. Consider a collusive equilibrium with Q�k > Qm

k ; k � a; b;such that, with �a � �b � 0; �� maxf�1; �2g holds. By choosing the optimal��k, firms can reduce maxf�1; �2g. Consequently, firms may reduce outputfurther which implies lower welfare. An exception is the case when collusionwith Q�k � Qm

k ; k � a; b; is stable even without an exchange agreement. Thenthere is no negative impact on welfare from an exchange agreement but nopositive impact either.

Take, as an illustration, an example with linear demand. Assume thatQ�k � Qm

k and that punishments are reversions to the static Nash equilibrium.Suppose for simplicity symmetric markets such that demand in both markets ispk � 1:1ÿ Qk, k � a; b, and that c1a � c2b � 0:1 and c1b � c2a � 0:2. Each firmproduces in the market where it is more efficient. At the joint profitmaximizing equilibrium, firms realize profits of ��i � ��i � 1

4 (suggestingT � 0), both with specialization in marketing and with an exchangeagreement. In the Cournot equilibrium, firms earn �

pi � 1:12

9 � 0:82

9 � 37180. With

specialization in marketing, each firm is a monopolist in one market. In thatcase, when it deviates, a firm earns 1

25 by invading the other firm's market andstill earns 1

4 in its own market, so �di � 29

100. What is the result with an exchangeagreement? Solving for the optimal exchanges yields13 ��a � ��b � 4

10. With suchexchanges, when it deviates, a firm gets 14

100 in its own market and 13100 in the

market of the other firm, so �di � 27

100 <29100. The minimum discount factor

required to sustain collusion is �1 � �2 � 0:47 with specialization in marketingbut only �1 � �2 � 0:31 with the exchange agreement.

The above results are fairly general. Given the generality of the model, inProposition 1 only a negative statement about the optimal ��k can be made, but��k cannot be computed. Imposing symmetry, however, stronger conclusionscan be drawn. So assume that pa�Q� � pb�Q� for any Q and that cia � cib, i � 1;2and c1k � c2k, k � a; b. Symmetry also implies that some �

p1 � �

p2 exist. Further,

any reasonable bargaining rule yields ��1 � ��2.

13. This solution is obtained following the steps of the proof of Proposition 2. Given theparametrization of the example, this is a straightforward exercise.


H.-T. Normann

Proposition 2. With symmetric markets and symmetric firms, ��a � ��b � 12 is

the global minimum of (7).

The proposition shows that there is an incentive to exchange goods even withsymmetry. This is interesting as, because of the symmetry in costs, there are noefficiency gains from specialization.14 Firms nevertheless engage in exchangessince this enlarges the scope for collusion.

Consider again an example with linear demand and Nash reversion.Demand in both markets is pk � 1:1ÿ Qk, k � a; b; as above. Marginal costsare c1k � c2k � 0:1, k � a; b. At the joint profit maximizing equilibrium, firmsrealize profits of ��i � ��i � 1

4, both with specialization in marketing and with anexchange agreement. In the Cournot equilibrium firms earn �

pi � 2

9. Withspecialization in marketing, when it deviates, a firm earns 1

16 by invading theother firm's market and earns 1

4 in its own market, so �di � 5

16. According toProposition 2, optimal exchanges are ��a � ��b � 1

2. With such exchanges, whenit deviates, a firm gets 9

64 in either market, so �di � 18

64 <516. The minimum

discount factor required to sustain collusion is �1 � �2 � 0:69 withspecialization in marketing. With the exchange agreement the factor isreduced to �1 � �2 � 0:53.

5. CONCLUSION

In this paper it was shown that, if firms are free to engage in exchangeagreements, they will generally do so because there is a larger scope forcollusion with exchange agreements than where firms distribute one good eachas a monopolist. The results suggest that the policy in the German power-cablecase was wrong or, rather, that the Commission was right when it stopped theexchanges in the Italian flat-glass case. Markets are less prone to deviation fromcollusion when all firms have a positive market share in all segments.Conversely, with complete specialization, it is more attractive to defect fromsome collusive agreement. These findings support the Commission's point thatthe exchange agreement helped firms dividing up the market.

The result that exchange agreements have a collusive effect in addition tothat resulting from specialization in marketing is in contrast to a result inBernheim and Whinston (1990, Sec. 5). They show that a certain collusive priceis sustainable with exchange agreements under exactly the same condition asthat resulting from specialization in marketing. What is important for thedifferent results is that, in this paper, firms engage in Cournot competitionwhile, in Bernheim and Whinston, firms compete in prices. However, what iscrucial for the collusive effects of exchange agreements is not the strategicvariable, it is whether firms are committed to selling the quantities at their

14. In this case, both firms may well produce both goods and no exchanges would occur.However, infinitesimally small differences in marginal cost or fixed costs in production wouldagain suggest specialization in production, while the theoretical results would remain.



disposal. In Bernheim and Whinston, quantities received as part of theexchange are of no value when the rival defects and undercuts the price. Iffirms were actually committed to the quantities, the collusive effects ofexchange agreements would also occur, even though competition is in prices.15

There are papers related to the present one which show a similar collusiveeffect. For example, Martin (1995) shows that R&D joint ventures facilitatecollusion in the product market. Similarly, Schultz (1999) shows thatpromoting export cartels facilitates collusion in the domestic market. Thegeneral conclusion is that, allowing firms to collude in one market (or one stageof it) is likely to make collusion easier in the allegedly competitive segment ofthe industry.

APPENDIX

Proof of Lemma 1. Consider @�1=@�a. Rearrange �1 such that

�1 � �d1 ÿ��1

�d1 ÿ�

p1

� �d1 ÿ ��1

�d1 ÿ ��1 ��1 ÿ�

p1

�A:1�

and take the derivative with respect to �a

@�1

@�a�

@�d1

@�aÿ @�

�1

@�a

� ��1 ÿ�

p1�

��d1 ÿ ��1 ��1 ÿ�

p1�2

�A:2�

Omitting arguments and using the envelope theorem, one gets @�d1=@�a

� �R1ap0a ÿ c1a�Q�a. From @��1=@�a � ÿQ�ap�a it follows that

@�d1

@�aÿ @�

�1

@�a� �R1ap0a ÿ c1a�Q�a � p�aQ�a �A:3�� R1ap0a � pa��aQ�a � R1a� ÿ c1a� � Q�a � �p�a ÿ pa��aQ�a � R1a�� Q�a� �p�a ÿ pa��aQ�a � R1a��aQ�a�� Q�a �A:4�

Therefore, signfp�a ÿ pa��aQ�a � R1a��aQ�a��g is equal signf@�1=@�ag. Whenderiving @�2=@�a one has to additionally consider R2a��1ÿ �a�Q�a� � �aQ�a. It iseasy to see that in this case @�2=@�a � 0. Since @�2=@�a < 0 otherwise, it followsthat

sign@�2

@�a

� �� signfminfpa��1ÿ �a�Q�a � R2a��1ÿ �a�Q�a�� ÿ p�a;0gg �A:5�

The steps for deriving the signs of @�2=@�b and @�1=@�b are identical. ú

15. Such a possibility is considered, for example, in models with price and quantity competition(e.g., Allen, 1992), where firms are committed to the quantities they produced.


H.-T. Normann

Proof of Lemma 2. Consider @�1=@�a and note that @�1=@�aj�a�0 � 0 and@�1=@�aj�a�1 > 0: Since p�a ÿ pa��aQ�a � R1a��aQ�a�� monotonically increases in�a, a unique �a � f�aj@�1=@�a � 0g exists. Similarly, a unique �a � f�aj�aQ�aÿR2a��1ÿ �a�Q�a� � 0g exists for which @�2=@�aj�a��a

� 0: Note that �a < 0:5 < �a.A solution to (7) cannot involve �a < �a since @�1=@�aj�a<�a

< 0 and@�2=@�aj�a<�a

< 0. It cannot involve �a > �a either since @�1=@�aj�a>�a> 0 and

@�2=@�aj�a>�a� 0: Therefore, any solution involves �a 2 ��a; �a�. In this interval,

@�1=@�a � 0 and @�2=@�a � 0. The reverse inequalities hold for the derivativeswith respect to �b. It follows that no solution with �i < �j; i; j � 1;2; i 6� j, canexist since one can reduce �j at the expense of increasing �i, thereby reducingmaxf�1; �2g. Thus, in all solutions, �1 � �2. ú

Proof of Proposition 1. The first part of the proposition is only important whenQ�k � Qm

k as otherwise the stronger second part applies. If Q�k � Qmk , from (A.2),

(A.4) and (A.5), at �a � �b � 0, one has @�1=@�a � 0 and @�1=@�b < 0 for firm 1,and @�2=@�a < 0 and @�2=@�b � 0 for firm 2. By marginally increasing both �a

and �b one can reduce both �1 and �2. Similarly, if �a � �b � 1, one can reduceboth �1 and �2 by marginally decreasing both �a and �b. Hence, �a � �b � 0 and�a � �b � 1 cannot solve (7). With ��a � 0 and ��b � 1, one gets �1 � 0 and �2 > 0.From Lemma 2, this cannot be a solution. The same argument rejects, viceversa, ��a � 1 and ��b � 0 as a solution.

When Q�a > Qma , @�1=@�aj�a�0 < 0 and @�2=@�aj�a�0 < 0 and �a � 0 cannot be a

solution. Similarly, for �a > �a (where �a is defined as in the proof of Lemma 2),@�1=@�aj�a>�a

> 0 and @�2=@�aj�a>�a� 0. Since Q�a > Qm

a , �a < 1 and �a � 1cannot be a solution. Thus, �a 2 �0;1�. ú

Proof of Proposition 2. From (A.2), (A.4) and (A.5), differentiating ��1 � �2� withrespect to �a gives

@��1 � �2�@�a

� ��1 ÿ�

p1� � �p�a ÿ pa��aQ�a � Ra��aQ�a�� Q�a��d

1 ÿ ��1 ��1 ÿ�c1�2

� ��2 ÿ�

p2� � �pa��1ÿ �a�Q�a � Ra��1ÿ �a�Q�a�� ÿ p�a� � Q�a

��d2 ÿ ��2 ��2 ÿ�c

2�2�A:6�

and similarly for @��1 � �2�=@�b. (The firm index for the best reply is omittedsince R1k � R2k, k � a; b.) Consider �a � �b � 1

2 as a candidate solution. With�a � �b � 1

2, �d1 � �d

2, ��1 � ��2, and

pk��kQ�k � Rk��kQ

�k�� pk��1ÿ �k�Q�k � Rk��1ÿ �k�Q�k�� k � a; b;

holds. Thus both derivatives are zero and �a � �b � 12 yields an extremum of

��1 � �2�.Taking @2��1 � �2�=@�2

a gives a complicated second-order condition. At �a � 12,

sign@2��1 � �2�

@�2a

� �� sign

@K

@�a

� �



where

K � pa��1ÿ �a�Q�a � Ra��1ÿ �a�Q�a� ÿ pa��aQ�a � Ra��aQ�a��From

@K

@�a� ÿ p0a��1ÿ �a�Q�a � Ra��1� R0a�ÿ p0a��aQ�a � Ra��1� R0a� > 0 �A:7�

��a � 12 yields a minimum of �1 � �2 and accordingly for �b. With ��a � ��b � 1

2, itfollows that �1 � �2 and, thus, min�a; �b

f�1 � �2g is a solution to min�a; �b

maxf�1; �2g.To see that ��a � ��b � 1

2 is the global minimum of min�a; �bmaxf�1; �2g, note

that from Lemma 2, any optimum must satisfy �1 � �2. Now, �1 � �2 if and onlyif �d

1 ÿ ��1 � �d2 ÿ ��2. From symmetry, this holds if and only if ��a � ��b � 1

2. ú

ACKNOWLEDGEMENTS

I am grateful to Pio Baake, Rainald Borck, Dirk Engelmann, Ulrich Kamecke,JoÈrg Oechssler, and participants of the EARIE meeting in Lausanne and of theconference of the Verein fuÈr Socialpolitik in Berlin for helpful comments. Tworeferees, and the editor in charge of the paper, Urs Schweizer, provided usefulreports which led to significant improvements of the paper.

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