Downstream Competition, Exclusive Dealing and UpstreamCollusion

Marie-Laure Allain∗, Claire Chambolle†and Clémence Christin‡

Abstract

This paper analyzes the impact of exclusive dealing contracts between upstreamand downstream firms of a vertical channel on the scope for upstream collusion. Weconsider a double duopoly framework and study upstream collusive strategies in a re-peated game, alternatively in the benchmark where firms agree on simple linear tariffcontracts, and when they may sign exclusivity contracts. We alternatively assumeperfect competition at one level (either upstream or downstream) while there is im-perfect competition at the other level. We show that when upstream competition isperfect, exclusivity contracts as no role on the scope for collusion. On the contrary,when downstream competition is perfect, exclusive dealing contracts may either facil-itate or deter upstream collusion in comparison with linear tariffs, depending on thestrength of competition at both levels. We show that when interbrand competition issoft, collusion is easier to sustain when producers can offer exclusive dealing contractsto retailers than when they cannot, whereas when interbrand competition is fierce,exclusive dealing contracts deter collusion.

Keywords: Collusion, Vertical Contracts.JEL Codes: L13, L41.

1 Introduction

Although the role of market structure on the scope for collusion has been widely studied,less attention has been paid to the influence of the demand side. Yet, as noted in the∗CNRS, Economics Department, Ecole Polytechnique (Route de Saclay, Palaiseau, France, email: marie-

laure.allain@polytechnique.edu) and CREST-LEI.†INRA (UR1303 ALISS, 94205 Ivry-sur-Seine, France; email: chamboll@ivry.inra.fr) and Economics

Department, Ecole Polytechnique.‡Economics Department, Ecole Polytechnique (Route de Saclay, Palaiseau, France, email:

clemence.christin@polytechnique.edu. We gratefully acknowledge support from the Ecole PolytechniqueBusiness Economics Chair and from the French-German cooperation project “Market Power in Verticallyrelated markets" funded by the Agence Nationale de la Recherche (ANR) and Deutsche Forschungsge-meinschaft (DFG) .

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European Commission’s Merger Guidelines, the stability of coordination in an industry isaffected by vertically related sectors. The European Commission Guidelines on VerticalRestraints (2000) also mention that the reduction of inter-brand competition betweencompanies operating on a market, including facilitation of collusion amongst suppliers orbuyers, is one of the negative effects that may result from vertical restraints and which ECcompetition law aims at preventing.

Two recent decisions by the European Commission put forwards the links betweenexclusive dealing contracts and collusive practices amongst suppliers. In these two cases,the European Commission fined brewers for collusive practices. In the beer sector, mostoperators of a drinks outlet sign a "beer tie" with a brewer, consisting of an exclusivedealing clause for the purchase of a certain type of beer granted to a brewer in exchange fora payment.1 In both cases, these "beer ties" have played an important role in the brewers’collusive agreements. In 2003, the European Commission has fined the two largest brewersin Belgium, Interbrew and Alken-Maes (a subsidiary of the French food group Danone)for collusion on the price of beers sold in supermarkets and in the hotels, restaurants andcafés ("horeca") industry. The brewers were condemned for having settled agreements toset prices and divide the market over the period 1993− 1998.2 In particular, in 1994 afterInterbrew had tried to break the exclusive dealing agreements existing between Alken-Maes and several drink outlets by selling them beer at a loss, they agreed explicitly torespect one another’s exclusive dealing contracts on the horeca market. In 2001, theEuropean Commission also fined five brewers in Luxembourg who had agreed on the mutualobservance and protection of their exclusive dealing clauses.3 The brewers had explicitlyplanned to pay one another penalties in case of a disrespect of this rule.

In these two cases, exclusive dealing contracts were part of the collusive agreements atthe upstream level of an industry. This article thus analyzes the role of exclusive dealingcontracts on the ability of upstream firms to collude both on prices and on a market sharingrule.

This paper is first related to the literature on collusion. Most of the theoretical literatureon collusion focuses on industries in which firms sell their products directly to consumers.However a few papers study the incentives to collude in vertically related industry. Snyder(1996) considers the effect of the size of buyers on upstream collusion and shows that largebuyers are more likely to deter collusion among their suppliers than small ones: his resultis consistent with the analysis of competition authorities.4 Nocke and White (2007) andNormann (2009) show that vertical mergers may facilitate upstream collusion. Finally,

1For instance, the brewer pays the lease for the drinks outlet or the drinks outlet licence.2Official Journal of the European Union, L200, 07/08/2003.3Official Journal of the European communities, L253, 21/09/2002.4The European Commission’s Merger Guidelines, for instance, state that “special consideration is given

to the possible impact of [countervailing power] on the stability of coordination”-Guidelines on the as-sessment of horizontal mergers under the Council Regulation on the control of concentrations betweenundertakings", Official Journal of the European Union, 2004/C 31/03, § 57.

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Jullien and Rey (2007) focus on the effect of Resale Price Maintenance contracts (RPM)on the stability of collusion among producers. They show that by reducing the volatilityof prices, the use of RPM facilitates the detection of deviation and thus favor upstreamcollusion.

To our knowledge, there is no paper focusing directly on the role of exclusive dealingcontracts on the upstream incentives to collude. On the other hand, the literature devotedto the analysis of exclusive dealing contracts rather points out the potential exclusionaryeffects of these contracts. For instance the seminal paper by Aghion and Bolton (1987)shows that an incumbent and its customer may deter the entry of a more efficient entrantthrough an exclusive dealing contract. Segal and Whinston (1996, 2000) also show thatwhen an efficient entrant needs a sufficient number of orders from buyers to cover its fixedcost of entry, exclusive dealing contracts between buyers and incumbent suppliers maydeter its entry. In their model, Fumagalli and Motta (2006) take explicitly into accountthe fact that buyers are firms, who may resell the input they bought from their suppliersto final consumers. They show that intense downstream competition may remove theincumbent’s incentive to deter entry through exclusive dealing contracts. These resultssuggest that exclusive dealing may have an indirect effect on upstream incumbent firms’incentives to collude. Indeed, a firm enters a market by cutting prices, which clearly pushesincumbent firms to react by lowering their prices thus destabilizing a potential collusion.By preventing entry, exclusive dealing contracts may thus indirectly facilitate upstreamcollusion. Finally, other works have focused on the dampening effect of theses exclusivecontracts on competition (Lin (1990), O’Brien & Shaffer (1993), Besanko & Perry (1994),Dobson & Waterson (1996)). But, still, an analysis of the direct effect of exclusive dealingcontracts on collusion in a dynamic setting is missing.

Our article studies a double duopoly vertical channel and alternatively considers thecase of perfect competition upstream and downstream. When there is perfect upstreamcompetition, i.e when goods are perfect substitutes, downstream outlets are differentiated.On the contrary, when considering perfect downstream competition, we assume that goodsare imperfect substitutes. Following the traditional analysis introduced by Friedman (1971)on tacit collusion, we thus determine horizontal collusive equilibria that can be sustained bytrigger strategies and study the effect of collusion on up- and downstream profits as well ason consumer’s surplus. We analyze the effect of exclusive dealing contracts on the stabilityof collusion by comparing collusion, deviation and punishment profits when upstream firmsmay or not offer an exclusive dealing contract to downstream firms. When exclusive dealingcontracts are not feasible, firms only offer linear tariff contracts to downstream firms. Whenexclusive dealing contracts can be offered, we assume that upstream firms may offer a menuof contracts to downstream firms consisting of linear tariff without exclusive dealing clauseand an exclusivity clause with a two-part tariff. The fixed part of the tariff is a priori paid

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by the upstream firm in return for exclusivity.5

In the richer set-up where upstream firms can offer exclusivity contracts to the down-stream firm, we first show that exclusive dealing contracts affect the collusive strategies.When exclusivity clauses are available, there is a unique collusive equilibrium where up-stream firms sign an exclusive contract with one of the downstream firms and respect oneanother’s agreement: upstream firms also collude by sharing the market. Moreover, be-cause of the fixed fee associated to the exclusivity clause, producers use efficient two-parttariff contracts which enables them to capture the whole industry profit. Exclusive dealingcontracts here enhance the profitability of collusion.

Our results are then derived in the two extreme cases of perfect competition at one level.In the case of perfect upstream competition, collusive profits are higher when exclusivityclauses are available. Punishment profits are zero with or without exclusive dealing andthe optimal deviation strategy consists in excluding the upstream rival and thus capturingthe whole collusive profits in the two cases. Thus, when competition is perfect at theupstream level, exclusive dealing contracts have no effect on the stability of collusion.

If competition is perfect at the downstream level, although collusive strategies are stilldifferent on equilibrium, there is no downstream margin and thus linear contracts beingefficient the collusive profits are identical with and without exclusive contracts. However,it is always costlier to deviate from collusion when exclusive dealing is available, for aproducer has to offer an exclusive contract to its rival’s retailer and cannot deviate by onlycutting his tariff. Punishment profits are always higher when exclusive dealing clauses areavailable. Indeed, without exclusive dealing, the four goods are sold and perfect down-stream competition ruins the producers profit. If exclusive dealing contracts are available,the competitive equilibrium is such that exclusive dealing contracts are signed and thus,retailers being asymmetric, interbrand competition preserves producer’s profits. Studyingthe collusion stability, we show that the loss in deviation profit due to exclusivity is reducedwhen interbrand competition increases: in the linear tariff case, it becomes optimal for adeviating firm to exclude his rival’s product from one or both outlets when interbrand com-petition increases. Thus we highlight that unless for fierce enough interbrand competitionwhere exclusive dealing hinders collusion, exclusive dealing generally facilitates collusion.

The paper is structured as follows. In Section 2, we present the general framework.Section 3 offers a benchmark analysis in the case of linear tariffs. Section 4 describes amore complex framework where producers may offer a menu of contracts to each retailer,allowing them to choose between an exclusive dealing contract associated to a two-parttariff and a non-exclusive linear tariff contract. Finally, section 5 concludes and offers adiscussion of potential extension of our results.

5In our model, downstream firms may also be ready to pay for an exclusive dealing contract.

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2 The model

Consider two firms A and B producing differentiated goods at the same marginal cost,normalized to 0. They cannot sell their products directly to final consumers, but reach themthrough two differentiated downstream firms, or retailers, called 1 and 2. A downstreamfirm’s initial marginal cost of retailing is normalized to 0.

As upstream and downstream firms are differentiated, there are potentially four goodsavailable for the consumers to purchase. The final consumers’ inverse demand function forgood K sold by firm i (denoted good Ki) is defined as:

pKi(qKi, qKj , qLi, qLj) ≡ 1− qKi − aqLi − bqKj − abqLj (1)

with i 6= j and K 6= L where qKi is the quantity of good K sold by downstream firm i

on the final market, and similarly for qKj , qLi, qLj . Parameter a ∈ [0, 1] represents thedegree of substitution between goods A and B, i.e. the level of inter-brand competition,whereas b ∈ [0, 1] represents the degree of substitution between downstream firms, i.e.the level of intra-brand competition. Note that if a = b = 0, all goods are independent,whereas a = b = 1 corresponds to a homogenous good. Besides, we assume for simplicitythat the substitution between one good sold in a given shop and the other good soldin the other shop is a combination of intra- and inter-brand competition.6 We denoteby DX

Ki(pA1, pA2, pB1, pB2) the final demand for good Ki under market structure X (forinstance, X = AB,AB means that both retailers can potentially sell both goods, X =A,AB means that retailer 1 has accepted A’s exclusive dealing contract whereas 2 hasaccepted no exclusive dealing contract, etc).7

The timing of the stage game is as follows:

Stage 1 Contract choices

(a) If exclusive dealing is available, each manufacturer K = A,B offers each retaileri = 1, 2:

- An exclusive dealing contract preventing retailer i from selling the competitor’sgood. This contract specifies a fixed fee FKi and a unit price wedKi;

- A unit price wirKi at which retailer i will be able to buy good K should she rejectboth exclusive offers.8

6For a discussion of these assumptions, see Allain and Chambolle (2010) or Dobson and Waterson(2007).

7Note that when there are only three goods offered on the final market, for instance if retailer 1 doesnot sell product A, the inverse demand function for each good is obtained by setting qA1 = 0. Invertingthe system yields the demand functions for the three goods.

8Note that non exclusive contracts may be discriminatory: We will show in the next section thatassuming non-discriminatory tariffs outside of exclusive dealing offers would not change our results.

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If exclusive dealing is not available, producers only offer each retailer a linear tariffwirKi.

Whether exclusive dealing is available or not, all contract offers are simultaneous andpublicly observable.

(b) If exclusive dealing contracts have been offered, retailers simultaneously acceptor reject the exclusive dealing offers and these acceptance decisions are public.

Stage 2 Retailers then compete in price on the final market: each retailer sets a price foreach good she sells and, given the price of the input, purchases the number of unitsshe requires.9

The stage game is infinitely repeated. Both upstream firms have the same discountfactor denoted δ ∈ [0, 1]. Each producer K maximizes the discounted sum of his presentand future profits ΠK =

∑∞t=0 δ

tπK(t), where πK(t) is K’s immediate profit at date t.Downstream firms are assumed to change at each period, so that they are neutral toupstream collusion: Downstream firms have no incentive to hinder collusion among theproducers in order to improve their own future profits.10

We focus on collusive strategies in which producers set collusive tariffs as long as bothof them have set collusive tariffs in the previous periods, and switch to competitive tariffsforever if an offer different from the collusive offer has been observed in the past. Then, itis a classic result that collusion is stable as long as:

πdK +δ

1− δπpK ≤

11− δ

πcK

where πdK , πpK and πcK respectively denote a producer’s deviation, punishment and collusiveprofits. Besides, we restrict our analysis to joint profit maximizing collusive strategies.

We analyze the scope for upstream collusion first when exclusive dealing contracts arenot allowed, then when they are available. We compare the profitability of and scope forcollusion in these two frameworks.

3 Benchmark: No exclusive dealing

As a first step, we consider in this section that exclusive dealing is not allowed. Producersoffer linear tariff contracts that may be discriminatory.

9Note that the demand for a given good at a given outlet may be zero.10This technical assumption allows us to focus on upstream collusion. It is used by Jullien and Rey

(2007) and Nocke and White (2007).

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3.1 Downstream competition stage

Given the four wholesale prices wKi, retailers set in stage 2 the four final prices. Note thatthe price of each good may be set high enough for the good to face zero demand. Given thefinal prices pA2 and pB2 set by its competitor, retailer 1’s best response prices maximize:

π1 = (pA1 − wA1)DA1(P ) + (pB1 − wB1)DB1(P )

where P ≡ (pA1, pB1, pA2, pB2) denotes the vector of final prices. The equilibriummarket structure depends on the set of wholesale prices. In particular, the case of perfectcompetition downstream is developed in Appendix A.1.

3.2 Joint-profit maximizing strategies

When colluding, upstream firms maximize the upstream industry’s aggregate profit.In other words, in stage 1 of the one-period game, each upstream firm sets the wholesale

price that maximizes∑K=A,B,i=1,2

wKiDAB,ABKi (p∗A1(w), p∗A2(w), p∗B1(w), p∗B2(w)),

where w is the vector of wholesale prices: w = (wKi)K∈{A,B},i∈{1,2}. The joint profitmaximizing tariffs are symmetric and equal to wirc = 1/2. Retailers accept all contractoffers and the four products are sold. Final price is thus pirc = 1 − 1

2(2−b) . The wholesaleprice wirc does not depend on a: producers are not competing on the upstream level andcan thus set the monopoly wholesale price, that is 1/2. Moreover, the final price pirc doesnot depend on a either: collusion at the upstream level is enough to annihilate the effectof inter-brand competition on downstream choices. Nevertheless, as tariffs are linear, theyare not efficient: double margin occurs here and influences final prices.We thus obtain thefollowing lemma:

Lemma 1. With linear tariffs, both producers set their wholesale price wirc = 1/2 andearn profit πircK = 1

2(1+a)(2−b)(1+b) in collusion.

3.3 Deviation from the collusive path

We now determine the optimal deviation strategy. In the following we consider that pro-ducer A deviates from the collusive path. We compute which tariff the deviating producerA should offer if he takes as given that his rival will offer wirc.

Lemma 2. When exclusive dealing is not available, a producer’s optimal deviation strategydepends on both upstream and downstream competition:

(i) When upstream competition is soft enough, the optimal deviation is such that allgoods are carried on the final market;

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(ii) When upstream and downstream competition levels are intermediary, the producer’soptimal deviation is to set asymmetric prices such that his rival’s good is carried by onlyone retailer;

(iii) When upstream competition is fierce enough, the optimal deviation is such that therival good is completely excluded from the market.

Proof. See Appendix A.

When interbrand competition is soft enough, A offers the same tariff to the two retailers.It has no incentive to cut prices so as to exclude B from the market. As a increases however,it becomes easier to exclude B because the increased competition means that for a givendrop in A’s tariffs, A’s market share increases more. As a consequence, there exists a valueof a above which it becomes profitable for A to offer a lower tariff to one of the retailers, inorder to exclude B from this outlet. Finally, when upstream competition is fierce enough,A sets symmetric tariffs again, such that B’s good is not sold in any outlet. As a increases,goods become closer substitutes and prices such that quantities of B1 and B2 are zeroincrease. It goes to wC when a goes to 1. We summarize this result in the following figure.

Figure 1: Goods sold on the final market when A deviates from collusion. Horizontal axisis for a and vertical axis is for b. Ex: AB,A means that 1 sells the two goods and 2 sellsonly good A.

Notice that when upstream competition is perfect (a = 1), the only deviation strategyfor A is to eliminate B from the two outlets. On the contrary, when there is perfectcompetition downstream (b = 1), a deviating producer may either choose to accommodate

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its rival’s product or to exclude it and the three market structures (AB,AB), (AB,A) and(A,A) may arise according to the level of a.

3.4 Punishment profits

Finally, we determine punishment profits. The stage game is solved with backward induc-tion in the Appendix.

Lemma 3. Assume perfect competition upstream (a = 1) or downstream (b = 1). Then,the stage game has a unique Nash equilibrium such that (i) the two producers offer the samewholesale tariff to the two retailers w∗ = 1−a

2−a , (ii) the two retailers set the same price foreach good, p∗ = 1− 1

(1−a)(2−a) , and (iii) the two goods are sold on the final market.

Proof. Consider first perfect competition upstream (a = 1). In that case, each retailer ionly sells one type of good, the price of which is denoted pi. The whole demand of retailer iis then supplied by the producer who offers the lowest wholesale price, and in equilibrium,it is immediate that wAi = wBi = 0.

Assume now perfect competition downstream (b = 1). The solution of the downstreamcompetition subgame is given in Appendix A.1. Producers now compete in prices, antici-pating the choices of retailers at the second stage. Consider first that wB1, wB2 and wA1

are given and A sets wA2 so as to maximize its profit. We denote by w∗A2(wA1, wB1, wB2)A’s optimal choice. For all wA1, it is possible to find w′A1 such that w∗A2(w′A1) = w′A1 andπ∗K(w′A1, w

′A1, wB1, wB2) > π∗K(w∗A2(wA1), wA1, wB1, wB2). In other words, for any pair of

tariffs such that wA1 6= wA2, A can always find a non-discriminatory tariff that gives hima higher profit than with the discriminatory tariff. Therefore, A’s best reply to any pair(wB1, wB2) is a non-discriminatory tariff. Similarly, B’s best reply to any offer by A is tooffer a non-discriminatory tariff. Besides, the best reply can always be expressed as follows(we only consider the non-discriminatory case as it is the only relevant case):

BRK(wL) =1− a(1− wL)

2.

At equilibrium, each retailer then sets w∗K = 1−a2−a .

The competitive wholesale tariff w∗ decreases with upstream competition (a). Besides,when a goes to 0, w∗ goes to the monopoly price (local monopolies), whereas it goes to0 when a goes to 1 (perfect competition). Moreover, the competitive final price p∗ isdecreasing when downstream competition increases. It goes to w∗ when b goes to 0.

Finally, the retailers’ equilibrium profit π∗i is increasing in a and decreasing in b: thefiercer upstream competition and the softer downstream competition, the larger the partof the total profit retailers can capture. Upstream profits π∗K decrease with a. Downstreamcompetition has however an ambiguous impact on upstream profits: due to the variationof downstream prices and thus of quantities sold, and although the wholesale tariff w∗ doesnot depend on b, π∗K decreases with b when b ≤ 1/2, and increases with b otherwise.

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3.5 Stable Collusion

We now determine the threshold discount factor above which collusion is stable. Withoutexclusive dealing, the condition such that the collusive strategy in which producers offer wC

as long as no deviation has been observed and w∗ otherwise can sustain a subgame-perfectequilibrium can be written as follows:

πDevK +δ

1− δπ∗K ≤

11− δ

πCK K ∈ {A,B} (2)

with πDevK K’s deviation profit. We denote by δ∗ the discount factor threshold above whichcollusion can be sustained.

Let us now analyze the upstream collusion strategies when exclusive dealing contractscan be signed by firms.

4 Non linear tariffs

In this section we assume that an exclusivity contract is both a linear wholesale pricecontract and an exclusivity clause which may be associated to a fixed fee transfer. Wealso consider here the possibility for upstream firms to offer a menu of contracts to eachretailer. As previously there is an infinite number of periods.

Henceforth, we will always assume that when each supplier signs an exclusive dealingcontract with one retailer, A signs a contract with 1 and B signs a contract with 2. Besides,analyzing deviation from collusion, we will always assume that B is the deviating producer.

4.1 Perfect competition upstream

We first consider that producers are perfect substitutes and show that in this case, theincentives for collusion are the same, whether producers can offer exclusive dealing clausesor not, although final prices and profits change when exclusive dealing offers are made. Forthis specific case, we use a general demand model. As producers are perfect substitutes,only two types of goods may be available on the final market, and their degree of differen-tiation is the degree of differentiation between retailers. We thus denote these goods by 1and 2, depending on the outlet in which they are sold.

We assume that the demand function for good i, Di(p1, p2), is differentiable, downward-sloping and satisfies the following properties:

∂Di

∂pi< 0,

∂Di

∂pj> 0,

∂Di

∂pi

∂Dj

∂pj− ∂Di

∂pj

∂Dj

∂pi≥ 0.

When exclusive dealing is not available, retailer i’s profit is given by: πi = (pi −wi)Di(p1, p2). When exclusive dealing is available, producer K can offer an exclusivedealing clause and a fixed fee FKi to retailer i. In that case, if i accepts K’s offer, her

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profit is given by πi = (pi − wi)Di(p1, p2) − FKi. We make the following assumptions onretailers’ profits:

∂2πi∂p2

i

< 0,∂2πi∂pi∂pj

> 0,∂2πi∂p2

i

∂2πj∂p2

j

− ∂2πi∂pi∂pj

∂2πj∂pj∂pi

≥ ∂Di

∂pj

∂2πj∂pj∂pi

.

These assumptions respectively ensure that there is an interior solution to the profit max-imization problem, that Bertrand reaction functions slope upward and finally that absentfixed-fee transfers, retailer i’s subgame equilibrium profit is decreasing in wi.11

Without exclusive dealing clauses, retailer i gets her supply from the producer whooffers the lower linear tariff, denoted by wi. The corresponding producer’s profit associatedwith retailer i is then wiDi(p1, p2). With exclusive dealing clauses, if i signs an exclusivedealing contract with K, then K’s profit associated with retailer i is wiDi(p1, p2) + FKi.

Consider the scope for collusion in the upstream market.

Proposition 1. When producers are perfect substitutes, allowing for exclusive dealing doesnot affect producers’ incentives to collude.

We prove this result by comparing producers’ collusive, deviation and punishmentprofits with and without exclusive dealing clauses.

Collusive profits and deviation from collusion. We first consider collusion anddeviation. When exclusive dealing is not available, producers set linear tariffs to maximizethe joint profit of the upstream industry. We denote these tariffs by (wc1, w

c2).12 We denote

a producer’s collusive profit by πirc. Consequently, a producer who deviates from collusioncan simply offer retailers a pair of tariffs (wc1 − ε, wc2 − ε) and capture the whole jointupstream profit, for producers are perfect substitutes: his deviation profit is thus equal to2πirc.

When exclusive dealing is available, collusive producers are able to capture the monopolyprofit of the whole industry by each offering a contract (wedc, F c) such that the joint profitof the industry is maximized. With efficient two-part tariff contracts the linear part max-imizes the joint profit of the industry while the fixed fee shares the profit amongst firms.We thus have the following programme:

wedc = arg maxwi

∑j=1,2

p∗j (w1, w2)Dj(p∗1(w1, w2), p∗2(w1, w2))

F ci = (p∗i (w1, w2)− wi)Di(p∗1(w1, w2), p∗2(w1, w2))

Parallel to this, producers offer a linear tariff without exclusive dealing such that no retailerhas any incentive to refuse the exclusive dealing offer: as a retailer’s profit in exclusive

11Further justification for these assumptions is provided by Shaffer (1991).12Classically, we assume that producers share the joint profit equally. This minimizes the threshold

discount factor above which collusion is stable.

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dealing is 0 with the collusive offer, this implies offering a linear tariff that is too high forthe retailer to sell any good in the third stage. If we denote the monopoly profit of theindustry by πm, then a producer’s collusive profit is πm

2 .Consequently, the optimal deviation for a producer consists in offering a slightly more

attractive exclusive dealing clause to each retailer, without changing the linear tariff offers(with or without exclusive dealing): producer A then deviates by offering each retailer acontract (wedc, F c−ε). By doing so he captures the whole monopoly profit of the industry,that is πm.

Finally, note that out-of-equilibrium tariffs do not enable producers to reduce the scopefor deviation by their competitor. Denote (w′, F ′) the outside equilibrium contract aproducer offers his rival’s exclusive retailer. By definition, we have the following condition:

πi(w′)− F ′ < πi(wedc, wedc)− F c = 0 (3)

Otherwise, the rival’s retailer would accept contract (w′, F ′). Imagine now that one of theproducers deviates from the collusive strategy by offering (wedc, F c − ε) to the retailers.The profit a retailer obtains by accepting the contract whether her rival accepts it or notis:

πi(wedc, wedc)− F c + ε > 0 (4)

That profit is positive, so that both retailers accept the deviation contract in equilibrium.As πi(wedc, wedc)−F c+ε > πi(w′)−F ′ the outside equilibrium contract offer cannot breakthe deviation.

Punishment profits. We determine the competitive equilibrium of the single periodgame. In the last stage of the game, retailer i sets price pi so as to maximize her profit,given that his unit retail cost is wi. The equilibrium price is given by p∗i (w1, w2). In thesecond stage of the game, two different cases occur, depending on whether exclusive dealingclauses are allowed or not.

When exclusive dealing clauses are not available, each retailer gets her supply from theproducer that offers her the lowest linear tariff. As a consequence, producers compete inprice to earn each retailer’s demand and set linear tariffs equal to their marginal costs: forall i producers offer the same linear tariff wirpi = 0. Indeed, on the one hand any producerwho offers a lower tariff than 0 earns a negative profit. On the other hand, if producer Koffers i a tariff wKi higher than 0, his rival can offer wKi − ε and earn a positive profitby capturing i’s total demand. Finally, in the competitive equilibrium, producers earn noprofit and retailer i earns a profit equal to πirp = p∗i (0, 0)Di(p∗1(0, 0), p∗2(0, 0)).

When exclusive dealing clauses are available, there exists an equilibrium in which eachproducer signs an exclusive dealing contract with one retailer and this contract involvesa negative fixed fee F p and a linear tariff that is higher than the producer’s marginalcost: wedp > 0. This equilibrium is such that a producer’s profit is 0. Indeed, if one

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producer offers a contract such that he earns a positive profit, his rival has an incentiveto offer a slightly more interesting contract to the concerned retailer. Finally, producersseek to maximise each retailer’s profit. As there is perfect competition amongst producers,the only equilibrium is such that a producer’s profit is 0.13 Besides, this equilibriumgives retailers a higher profit than the situation with interlocking relationships and lineartariffs equal to the producers’ marginal costs. Note that there is no equilibrium where noexclusive dealing contract is signed: a producer could deviate from such a situation byoffering each retailer an exclusive dealing contract (F − ε, wedp). Such a contract wouldbe accepted by each retailer and the deviating producer would thus earn a positive profit.As a consequence, when exclusive dealing clauses are available, each producer signs anexclusive dealing contract with one retailer and earns no profit.

Therefore, when producers are perfect substitutes, punishment profits are identicalwhether exclusive dealing is possible or not, and equal to 0.

Stable collusion. As πp = 0 whether exclusive dealing is available or not, we can simplifythe condition for the sustainability of collusion for both cases. Without exclusive dealing,the condition becomes: 2πirc ≤ 1

1−δπirc. Similarly, with exclusive dealing, the condition

becomes πm ≤ 11−δ

πm

2 . These two conditions are both equivalent to δ ≥ 12 . Therefore,

allowing for exclusive dealing has no impact on the incentives for collusion. However, it islikely that final prices will be lower and welfare higher with exclusive dealing than without,because in this specific case exclusive dealing enhances efficiency and does not reduce therange of products available.

4.2 Perfect competition downstream

We now consider that outlets are perfect substitutes. We solve the model with the lineardemand function defined in section 2 and assuming that b = 1. If producer K’s good issold by the two retailers, the retailer that sets the lowest price for this good captures thewhole demand for K. His demand is then:

DK(pK1, pK2, pL1, pL2) =1− a−min{pK1, pK2}+ amin{pL1, pL2}

1− a2

If both retailers set the same price for the same good, then the demand is shared equallybetween them.

Proposition 2. When outlets are perfect substitutes, exclusive dealing facilitates collusionwhen upstream competition is soft enough, and hinders collusion otherwise.

The following analysis proves this result by comparing the relevant profits.13This equilibrium is described in Shaffer (1991).

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4.2.1 Collusive profits

We first consider the producers’ collusive offers in the two cases.

Lemma 4. When outlets are perfect substitutes, producers’ collusive profits are the samewith or without exclusive dealing: producers share the industry-wide monopoly profit πm =

12(1+a) .

Without exclusive dealing, producer K’s optimal strategy in collusion is to offer eachretailer the wholesale tariff wK that maximizes the joint profit of the upstream sector, thatis wirc = 1/2. Then, as retailers are perfect substitutes, they both set prices equal to theirmarginal costs: pK = wirc, and thus share the demand equally for both products. Thejoint profit of the industry is thus completely captured by producers. Besides, as there isno double marginalization, this profit is equal to that of a two-good monopoly.

With exclusive dealing, producers can either use the same strategy or implement exclu-sive dealing. In the latter case, each producer chooses a retailer and offers her an exclusivedealing contract with a unit part wedc = a/2 and a fixed fee F c = 1−a

4(1+a) . This givesproducers the same profit as without exclusive dealing πedcK = πircK = 1

4(1+a) . Moreover,each retailer only sells one good but still has no profit.

Let us first assume that when exclusive dealing is available, producers always useexclusive dealing as a collusion strategy although exclusive dealing yields the same collusionprofit as interlocking relationships. Indeed, we will prove further that exclusive dealingcontracts are more robust to deviation, for producers can build out-of-equilibrium contractsthat are partially robust to deviation. Therefore, assume that collusion is such that retailer1 signs an exclusive dealing contract with A and retailer 2 with B. A thus sets his exclusivedealing offer to 2 in order to reduce the profitability of deviation for B. To ensure that 2will not accept A’s contract, (wedcA2 , F

cA2) is set so that:

πA,A2 (wedc, wedA2)− F cA2 ≤ πA,B2 (wedc, wedc)− F c. (5)

Besides, amongst the set of contracts that satisfy (5), A chooses the contract which mini-mizes B’s deviation profit. As the fixed fees set by A never affect his rival’s profit, A setswedcA2 in order to minimize B’s deviation profit, and the fixed part F cA2 to satisfy constraint(5). As a consequence, the fixed part is:

F cA2 = max{

(wedc − wedcA2 )(1− wedc), 0}. (6)

We determine the exact value of wedcA2 while analyzing the optimal deviation for B in thefollowing.

As producers offer a menu of contracts, it is also necessary to define the optimal in-terlocking relationship linear contract referred to as wirc. Recall that the collusive profitis the same whether producers sign exclusive dealing clauses or not with the retailers. In

14

particular, retailers earn 0 in both cases. As a consequence, producer A (resp. B) cannotoffer retailer 1 (resp. 2) wircA1 < wc as retailer 1 would earn a positive profit by accepting theinterlocking relationship contract wircA1. Similarly, wircA2 must be larger than wedc in order toprevent retailer 2 from accepting A’s non-exclusive contract and earning a positive marginon product A. We determine the exact values of wircA1 and wircA2 while analyzing the optimaldeviation for B in the next paragraph.

Finally, the collusive menu of contracts offered by A to 1 and 2 are respectively of theform (wedc, F c, wirc) and (wedcA2 , F

cA2, w

irc). Contracts offered by B are symmetric.

4.2.2 Deviation from collusion

In this section, we show that for most values of a, deviation from collusion is less profitablewhen producers can offer exclusive dealing contracts than when they cannot.

Lemma 5. Assuming that outlets are perfect substitutes, allowing for exclusive dealingreduces the deviation profit of a producer when goods are differentiated enough (a < 0.647)or close enough substitutes (a > 0.732).

The intuition for this result is as follows. When producers have access to exclusivedealing, they can choose between several collusive strategies that yield the same upstreamprofit but call for different deviation strategies. Producers thus choose the collusive strategythat makes deviation the least profitable. Obviously, for a given collusive strategy, adeviating producer also benefits from the richer set of contracts available to him andcan thus increase his deviation profit. However, the former effect offsets the latter. Inparticular, we show in the following that contrary to the interlocking relation case, aproducer cannot find a profitable deviation that keeps the market structure unchanged:the deviating producer is forced to steal his rival’s retailer, which may be a costly strategy.

In the following, we consider that the collusive strategy is such that producer A signsan exclusive dealing contract with retailer 1 and producer B signs an exclusive dealingclause with retailer 2. We focus on a deviation by producer B. Since we have alreadydetermined F cA2, we determine the optimal deviation for any pair (a,wedcA2 ). We assumethat the values of wircA1 and wircA2 that minimize B’s deviation profit are the lowest values Acan set for these two tariffs. This turns out to be true.

Deviation in market structure (A,B). Producer B is unable to offer retailer 2 anexclusive dealing contract so that the market structure is similar as in collusion and B’sfinal profit is higher than 1

4(1+a) . Indeed, assume that B sets 2’s exclusive dealing contract(weddB2 , F

dB2) to maximize his individual profit instead of the joint upstream industry’s profit

in market structure (A,B), given that A offers each retailer the corresponding collusivemenu of contracts (defined above).

15

In order to implement market structure (A,B) in deviation, B must then ensure that2 will choose his exclusive dealing contract over A’s exclusive dealing contract, A’s non-exclusive dealing contract, and finally over no contract at all, and equivalently that 1 willaccept A’s exclusive dealing clause. In other words, B must satisfy the following constraintsrespectively for retailers 1 and 2:

πA,B1 (wedcA1 , weddB2 )− F cA1 ≥ max{0, πAB,B1 (wircA1, w

irdB1 , w

eddB2 ), πB,B1 (weddB1 , w

eddB2 )− F dB1},(7)

πA,B2 (wedcA1 , weddB2 )− F dB2 ≥ max{0, πA,AB2 (wircA1, w

ircA2, w

irdB2), πA,A2 (wedcA1 , w

edcA2 )− F cA2}. (8)

First, by setting weddB1 = wirdB1 = +∞, B can ensure that constraint (7) is equivalentto πA,B1 (wedcA1 , w

eddB2 ) − F cA1 ≥ 0. Second, it is immediate that for any value of wedcA2 , given

the conditions on A’s collusive contracts offered to 2, πA,A2 (wedcA1 , wedcA2 )−F cA2 = 0. Besides,

since wircA2 > 12 > wedcA1 , it is obvious that if 2 refuses all exclusive dealing clauses, she

cannot sell good A and can thus only earn a positive profit by selling B. Then, B can setwirdB2 = +∞ to ensure that πA,AB2 (wircA1, w

ircA2, w

irdB2) = 0. As a consequence, constraint (8) is

equivalent to πA,B2 (wedcA1 , weddB2 )− F dB2 ≥ 0.

The optimal deviation tariff is such that constraint (8) is binding: F dB2 = πA,B2 (wedcA1 , weddB2 ).

B then sets weddB2 so that:

weddB2 = arg maxwed

B2

πA,BB (wedcA1 , wedB2) + F eddB2 s.t. πA,B1 (wedcA1 , w

eddB2 )− F cA1 ≥ 0.

The unconstrained optimal tariff would be weddB2 =a2(4−2a−a2)

8(2−a2). However, with this tariff,

1’s profit is negative or demand for good A is negative. As a consequence, producer B setsweddB2 so that πA,B1 (wedcA1 , w

eddB2 ) − F cA1 = 0, which implies weddB2 = a

2 . The resulting contractis thus exactly the collusive contract, and B then earns the collusive profit 1

4(1+a) .It is thus impossible for B to offer 2 a different contract than the collusive contract and

earn a higher profit than his collusive profit while keeping the market structure unchanged.As a consequence, in order to increase his profit in deviation, B must change the marketstructure, which implies stealing his rival’s retailer, 1: as he earned a profit 0 throughhis relationship with 1, he can now offer her a slightly more interesting contract than thecollusive contract offered by A and still earn a positive profit from this exclusive contract.Given this strategy, retailer 1 will only sell B’s good. Therefore, it may become profitablefor 2 to accept one of A’s contracts, especially when goods are differentiated enough. Asa result, it may be too costly for producer B to induce 2 to accept his exclusive dealingclause too.

Optimal deviation strategy. When producer B deviates from collusion, he can inducedifferent market structures. For example, he can set exclusive dealing clauses such thatthe two retailers accept these clauses, or that only one of them accepts the clause whilethe other retailer refuses all exclusive dealing offers and buys the two goods at their linear

16

tariff. We determine the optimal deviation as follows. First, B sets the optimal tariff inany possible contract structure: this gives us the profit B will earn if it maximizes hisindividual profit in a given market structure, given that A sets the collusive tariffs. Asstructure (A,B) is never profitable, we must determine B’s profit in all the following struc-tures: (B,A), (B,B), (B,AB), (AB,B), (AB,AB), (A,AB) and (AB,A). Second, B comparesthe profit he can earn in all the possible structures and chooses the structure that giveshim the highest profit for given values of a, wircA1, w

ircA2, w

edcA2 and F cA2.

We first determine the optimal deviation of B in each possible contract structure. Inthe following, we focus on the deviation in structure (B,AB), i.e. the deviation in whichretailer 1 signs an exclusive dealing clause with producer B and retailer refuses the twoexclusive dealing offers. We determine deviations in all other structures ((AB,AB), (AB,B),(B,A), (B,B), (A,AB) and (AB,A)) in the Appendix.

The menus of contracts offered to 1 and 2, respectively (wirdB1 , weddB1 , F

dB1) and (wirdB2 , w

eddB2 , F

dB2)

must therefore satisfy the participation constraints of the retailers:

πB,AB1 (weddB1 , wircA2, w

irdB2)− F dB1 ≥ max{πA,AB1 (wedc, wircA2, w

irdB2)− F c,

πAB,AB1 (wircA1, wirdB1 , w

ircA2, w

irdB2), 0}, (9)

πB,AB2 (weddB1 , wircA2, w

irdB2) ≥ max{πB,B2 (weddB1 , w

eddB2 )− F dB2,

πB,A2 (weddB1 , wedcA2 )− F cA2, 0}. (10)

First, we determine the value of the right-hand side of each of these inequalities. The fixedfee F dB1 is such that constraint (22) is binding. Second, we determine the optimal values ofweddB1 and wirdB2 , i.e. the values that maximize the profit of B πB,ABB (weddB1 , w

ircA2, w

irdB2) +F dB1.

We focus on the participation constraint of A, which enables us to find wirdB1 , weddB2 and

F dB2 for all cases. B wants’ 1’s profit to be as low as possible when it sells both goods while2 sells both goods. First, it should be noted that since wircA1 > wircA2, the profit of 1 on goodA in market structure (AB,AB) is 0. Then, 2 simply sets wirdB1 > wirdB2 so that 1 cannot sellgood B either in market structure (AB,AB). For instance, B can set wirdB1 = 1 and thusensure that πAB,AB1 (wircA1, w

irdB1 , w

ircA2, w

irdB2) = 0. Second, as wedc = wircA2, it is immediate

that πA,AB1 (wedc, wircA2, wirdB2)−F c = −F c < 0. As a result, the three constraints concerning

retailer 1 are summarized by πB,AB1 (weddB1 , wircA2, w

irdB2)−F dB1 ≥ 0. Therefore, we always have

F dB1 = πB,AB1 (weddB1 , wircA2, w

irdB2).

Consider now the constraints of retailer 2. In order to prevent 2 from acceptingB’s exclusive dealing offer, B has to offer 2 an exclusive dealing offer such that F dB2 ≥πB,B2 (weddB1 , w

eddB2 )−πB,AB2 (weddB1 , w

ircA2, w

irdB2). There is a set of such offers. For instance, B can

offer 2 a contract such that weddB2 = 1 and F dB2 = +∞. Not also that πB,AB2 (weddB1 , wircA2, w

irdB2) ≥

0 by definition, as retailer 2 earns always at least 0 when she does not sign any exclu-sive dealing contract. The participation constraint of 2 is thus πB,AB2 (weddB1 , w

ircA2, w

irdB2) ≥

πB,A2 (weddB1 , wedcA2 )− F cA2. The profit of retailer 2 in structure (B,A) has been defined previ-

17

ously.Finally, B can now set weddB1 and wirdB2 so as to maximize its total profit min{weddB1 , w

irdB2}D

B,ABB +

F dB1, provided that the participation constraint of retailer 2 is satisfied. The optimal tariffsare given by:

• If a < 0.841325, then weddB1 =a2(−4+2a+a2)

8(−2+a2)and wirdB2 = −4+2a+a2

4(−2+a2). Profits are πB,ABB =

(−4+2a+a2)2

32(2−3a2+a4)and πB,AB2 = − (8−8a−6a2+5a3)2

64(−2+a2)2(−1+a2);

• Otherwise, weddB1 = wirdB2 = 2 + a − 4+a3

2a . Profits are πB,ABB = −34 −

1a2 + 2

a andπB,AB2 = 0.

With such tariffs, the participation constraint of 2 is always strictly satisfied.

Depending on the values of a and wedcA2 , B can choose four different structures: (B,A),(B,B), (B,AB) and (AB,B): a deviation always implies that at least one retailer acceptsB’s exclusive dealing contract. Besides, if B chooses to sign an exclusive dealing clausewith his “historical” retailer 2, then B must sells his good through 1 too. In other words,the contract that is accepted by A’s retailer in deviation must always be different from thecollusive contract. It is not the case when exclusive dealing is not allowed: indeed, in thatcase, retailers may carry on buying good A at the collusive tariff wirc = 1/2 even when Bdeviates from collusion.

Note that in two of the four structures, (B,B) and (B,AB), the deviation profit ofproducer B does not depend on wedcA2 . They correspond to the yellow and pink areas in thefigure. Besides, from the figure, one can immediately see that it is possible for A to set wedcA2

so that the deviation structure is always one of these two. Choosing such a value of wedcA2 isoptimal for A. Indeed, consider for instance values of a in which the optimal deviation forB leads to either structure (B,A) or structure (B,AB). We denote by πB,AB (wedcA2 ) the profitof B in deviation (B,A) and by πB,ABB his profit in deviation (B,AB). In the relevant area,the profit of B in deviation is max{πB,ABB , πB,AB (wedcA2 )} ≥ πB,ABB . Then, the best A cando is thus to ensure that B’s profit is exactly πB,ABB , as any case in which B would choosestructure (B,A) instead would bring B a higher profit than πB,ABB . A similar reasoningimplies that for values of a where B’s optimal deviation is either (B,AB), (B,A) or (AB,B),A sets wedcA2 so that B’s deviation is (B,AB). Finally, in areas where the optimal deviationstructure is either (B,B) or (AB,B), depending on the value of wedcA2 , A sets wedcA2 so thatthe optimal deviation is (B,B). In particular, A can set wedcA2 = a

2 and F edcA2 = 0.Finally, the contracts accepted by the retailers when B deviates can be of three types.

First, when goods are differentiated enough, producer B signs an exclusive dealing clausewith 1, thus stealing A’s retailer, but allows retailer 2 to sell good A. When goods areclose enough substitutes, B manages to exclude A even though he only signs an exclusivedealing clause with 1 and not with 2. It is less costly for him not to sign an exclusive

18

dealing clause with 2 because he would have to pay to high a fixed fee to 2 otherwise.Finally, when the degree of substitution between goods is intermediary, B excludes A fromthe market by signing an exclusive dealing clause with the two retailers: the linear tariffthat would exclude A without exclusive dealing clauses is too low from the point of viewof producer B.

The tariffs offered by B in deviation and his corresponding profits are as follows:

• If a < 0.57, then the optimal contract structure in deviation is (B,AB) and tariffsare:

weddB1 = a2(4−2a−a2)8(2−a2)

, wirdB2 = 4−2a−a2

4(2−a2)and F dB1 = (4−2a−a2)2

64(1−a2)> 0.

and B’s deviation profit is:

πdB = (4−2a−a2)2

32(2−3a2+a4).

In the deviation, 1 sells good B at price wirdB2 and 2 sells A. It should be noted thatin that case, deviation (B,AB) is in fact equivalent to deviation (AB,AB). Therefore,as in the benchmark case, when a is low enough, B prefers to accommodate A indeviation rather than setting limit prices to exclude A.

• If a ∈ [0.57, 0.74], the optimal contract structure in deviation is (B,B). B sets thesame linear part weddB in both contracts and offers a fixed fee to each retailer suchthat her participation constraint is binding and we have:

weddB = 64−48a−32a2+24a3+8a4−2a5−a6

2(−2+a2)2(16−3a2),

F dB1 = −(4−a2)2(1−a2)(16−12a−6a2+3a3)2

4(16−3a2)2(2−a2)4+ 1−a

4(1+a) ,

F dB2 = − (2−a)2(1−a2)(16−8a−13a2+2a3+2a4)2

(2−a2)4(16−3a2)2.

and the deviation profit of B is:

πdB =a(96+64a−148a2−45a3+59a4+7a5−7a6)

4(1+a)(2−a2)2(16−3a2).

• If a ∈ [0.74, 0.96], the optimal contract structure in deviation is (B,B). B sets thesame linear part weddB = a

2 in both contracts and offers a fixed fee to each retailersuch that her participation constraint is binding. Fixed fees are then:

F dB1 = 0 and F dB2 = − (2−a)2(1−a)16(1+a) .

and the deviation profit of B is:

πdB =(2−a)(7a+3a2−2)

16(1+a) .

19

• Finally, if a > 0.96, then again the optimal contract structure is (B,AB). However,as in the two former cases, The linear tariff offered to 2 is equal to the linear part of1’s exclusive dealing contract. Tariffs are then given by:

weddB1 = wirdB2 = 2 + a− 4+a3

2a and F dB1 = 0.

Note that it is equivalent to the case where B offers wirdB1 = wirdB2 = 2 + a − 4+a3

2a

and no retailer accepts the exclusive dealing clause. In that case, although exclusivedealing clauses are not necessarily signed, the good A is not sold on the final market:as goods are close substitutes, B can exclude his rival without using exclusive dealingclause, and indeed has an incentive to do so. However, as wircA2 = a/2 < 1/2, thedeviation profit of B is lower than in the case without exclusive dealing, becauseexcluding A is more costly when exclusive dealing is possible.

Comparing the deviation profit of B with and without exclusive dealing, it is worthnoting that in the areas where the deviation market structure is the same with or withoutexclusive dealing (that is for low enough and high enough values of a), the deviation profitis always higher in the benchmark case than when exclusive dealing is available. As wehave pointed out, the reason it that when exclusive dealing is possible, out-of-equilibriumcontracts in the collusive strategy are set to be robust to collusion. Therefore, only incases where exclusive dealing enables the deviating producer to exclude his rival morethan in the benchmark case, i.e. when he signs an exclusive dealing with the two retailerswhereas he would have to partially accommodate A in the benchmark case, is it likely thatdeviation will be more profitable with exclusive dealing than without. This corresponds toa ∈ [0.647, 0.732].

4.2.3 Punishment profits.

Lemma 6. Assuming that outlets are perfect substitutes, when producers compete in prices,their equilibrium profit is always higher when exclusive dealing is available than when it isnot.

Consider first that exclusive dealing is not available. We have shown in the formersection that the equilibrium is symmetric and such that the four goods are sold on thefinal market. The equilibrium tariff is w∗ = 1−a

2−a and yields profit π∗ = 1−a(2−a)2(1+a)

.When exclusive dealing is available, this equilibrium does not hold, as one of the pro-

ducers always has an incentive to deviate from it and offer one retailer an exclusive dealingcontract. We now show that there exists an equilibrium where each retailer signs an ex-clusive dealing contract and both goods are sold on the final market. In the case of perfectcompetition downstream, exclusive dealing is equivalent to two-part tariffs. We assumefirst that each producer offers the same exclusive dealing contract to each retailer and eachretailer accepts the contract of a specific producer (for instance 1 accepts A’s contract and2 accepts B’s), and we show that this is an equilibrium.

20

Each producer sets his tariff so as to maximize his profit given that the two goods aresold by different retailers on the final market. Assume that A offers the same contract to thetwo retailers and 1 accepts A’s exclusive dealing contract. Then, if 2 accepts A’s exclusivedealing contract too, retailer i’s profit is −FAi. As a consequence, if B wants 2 to accepthis exclusive dealing contract in that case, he has to offer the latter a fixed fee such that 2’sprofit is −FAi. As a result, as long as FAi ≥ 0, B sets F pB2 = (pB(wedA1, w

edB2)−wedB2)DA,B

B2 .Fixed fees are determined in a similar way for all contracts. Each producer then setsthe unit part of his tariff so as to maximize the joint profit of the vertical structure:maxwK p

A,BK (wK)DK,L

Ki (wK , wL), with {K,L} = {A,B}. In this context, exclusive dealingis thus equivalent to two-part tariffs, since they enable producers to maximize the profit ofthe vertical structure instead of his own. The resulting equilibrium contract is thus givenby wedp = a2(1−a)

4−2a−a2 and F p = (1−a)(2−a2)2

(1+a)(4−2a−a2)2.

No producer has any incentive to deviate from this equilibrium. Indeed, assume thatB wants to deviate from this situation. As in the collusive case, he cannot earn a higherprofit than the profit he earns with tariff (wedp, F p) if the market structure remains (A,B).He thus has to exclude his rival from the market. However, he cannot do so either: forall values of wB2 and wB1, the profit earned by 1 if he accepts A’s offer is non-negative:1 thus never has an incentive to stop selling any good. The reasoning is symmetric if Bwants 2 to stop selling on the final market. Eventually, the fixed fee B would have to offera retailer in order to stop her from selling on the final market is too high to make such adeviation profitable.

With such a contract, the profit earned in exclusive dealing is 2(1−a)(2−a2)(1+a)(4−2a−a2)2

. It isalways higher than that earned without exclusive dealing contract. This effect tends tomake collusion more difficult for producer, as it increases the profit a producer can earnafter deviating.

4.2.4 Stable collusion.

The resulting threshold discount factor above which collusion is stable without exclusivedealing is given in the former section. With exclusive dealing, it is as follows:

δ∗ed = (−4+2a+a2)2

32−32a−12a2+12a3+a4 if a < 0.57

= (−4+2a+a2)2(64−96a−140a2+148a3+73a4−59a5−10a6+7a7)1024−2560a−1216a2+5504a3−720a4−3664a5+908a6+1060a7−251a8−139a9+21a10+7a11 if a ∈ [0.57, 0.74),

= (−4+2a+a2)2(8−16a+a2+3a3)128−384a+224a2+144a3−112a4−24a5+13a6+3a7 if a ∈ [0.74, 0.96),

= (−4+2a+a2)2(4−4a−4a2+3a3)64−128a−16a2+144a3−48a4−28a5+7a6+3a7 otherwise.

It is lower than δ∗ir as long as a < 0.575, as the following figure shows.

21

0.0 0.2 0.4 0.6 0.8 1.0a

0.55

0.60

0.65

0.70

0.75∆

*

Collusion only with ED

Collusion only

without ED

∆IR*

∆ED*

Figure 2: Stability of collusion with or without exclusive dealing.

In the case of perfect competition downstream, there is a trade-off between two effects:on the one hand, deviation is often less profitable with exclusive dealing than without,which tends to facilitate collusion; on the other hand, punishment is less efficient withexclusive dealing (punishment profits are higher then), which tends to hinder collusion. Asthe difference between the two punishment profits increases with a, the first effect tendsto become less and less important with regards to the second.

5 Conclusion

This article has shown that when downstream competition is perfect, exclusive dealingcontracts directly affect the stability of collusion at the upstream level of an industry, eitherthrough the collusive profit, the deviating strategies or the punishment profit. Moreover,if the first insight was that exclusive dealing practices would facilitate collusion, we havegiven arguments that corroborate this effect, and we have shown that it may also be thecase that exclusive dealing contracts prevent upstream collusion.

Several extensions would also be relevant to analyze the robustness of our results. Itwould first be interesting to analyze the intermediary framework where upstream firmscould offer a menu of contracts with two-part tariff contracts to downstream firms with orwithout the exclusivity clause. Even if, due to insistence issues, it would then be impossibleto determine the competitive equilibrium, we could analyze the case where upstream firmscould offer two-part tariff contracts when exclusive dealing is not available, and a menuof two-part tariff contracts with and without exclusivity when exclusive dealing contractsare available. These extensions would allow to separate the effects properly due to theexclusivity clause from the effect of a vertical restraint such as a two-part tariff, both of

22

these elements being interlocked in our model.Finally, a closely related issue would be to analyze the effect of another vertical restraint

such as "slotting allowances" on the stability of upstream collusion. Slotting allowancesare indeed a very current practice consisting of fees paid by upstream firms to be listed(and not necessarily bought) by a retailer. Considering that downstream firms are retailersconstrained in capacity who cannot offer more than one product in their shelves, it would beinteresting to compare the overall prospect for collusion among producers when only lineartariff contracts (resp. two-part tariff contracts) are authorized and when two part-tariffcontract (resp. three-part tariff contracts) with slotting allowances are allowed.

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[23] I. Segal and M.D. Whinston. Naked exclusion: Comment. American Economic Review,90 (1):296–309, 2000.

[24] C.M. Snyder. A dynamic theory of countervailing power. RAND Journal of Eco-nomics, 27:747–769, 1996.

24

Appendix

A Perfect competition downstream and no exclusive dealing

In this appendix, we determine the competitive, collusive and deviation profits when ex-clusive dealing is not available and retailers are perfect substitutes (b = 1).

A.1 Equilibrium of the downstream competition subgame

We assume that there is perfect competition downstream (b = 1) and that a ∈ [0, 1]. Thevector of wholesale prices is w ≡ (wA1, wA2, wB1, wB2). The demand for good K is givenby:

DK(pA1, pA2, pB1, pB2) = 1−a−min{pK1,pK2}+amin{pL1,pL2}1−a2

We determine the equilibrium of the downstream price competition subgame.

Best replies.Assume that j has set prices pAj and pBj . If pKj > 1+wKi

2 for all K ∈ {A,B} and alli, j ∈ {1, 2}, i 6= j, then both prices set by j are above the monopoly prices of i. Therefore,i sets the monopoly prices:

BRKi(pAj , pBj) = 1+wKi2 , BRLi(pAj , pBj) = 1+wLi

2 .

If wKi < pKj <1+wKi

2 and pLj >1−a+2apKj+wLi−awKi

2 , then i must set pKi = pKj − ε inorder to capture the whole demand for good A and then sets pLi to maximize her profitpKjDK(pKj − ε, pKj , pLi, pLj) + pLiDL(pKj − ε, pKj , pLi, pLj)}. This gives:

BRKi(pKj , pLj) = pKj − ε, BRLi(pKj , pLj) = 1−a+2apKj+wLi−awKi

2 .

If wKi < pKj <1−a+2apLj+wKi−awKi

2 for all K = A,B, then i cannot maximize herprofit with any good and has to set BRKi(pKj , pLj) = pLj − ε and BRLi(pKj , pLj) =pLj − ε. If pKj < wKi and pLj >

1−a+apKj+wKi

2 , then i cannot sell good K and thussets any price pKi ≥ wKi. She then sets pLi in order to maximize her profit (pLi −wLi)DL(wKi, pKj , pLi, pLj), which gives:

BRKi(pKj , pLj) ≥ wKi, BRLi(pKj , pLj) = 1−a+apKj+wLi

2 .

If pKj < wKi and wLi < pLj <1−a+apKj+wKi

2 , then i sets BRKi(pKi, pLi) ≥ wKi and onlysells good L at price BRLi(pKj , pLj) = pLj−ε. Finally, if pKj < wKi and pLj < wLi, Theni does not sell any good.

Note that it is impossible to have 1−a+2apB2+wA1−awB12 < pA2 <

1+wA12 and 1−a+2apA2+wB1−awA1

2 <

pB2 <1+wB1

2 at the same time.

25

Equilibrium.We now find the equilibrium of the downstream competition subgame.If wKj > 1+wKi

2 for all K = A,B, then it is immediate that p∗Ai = 1+wAi2 , p∗Bi = 1+wBi

2 ,p∗Aj = wAj and p∗Bj = wBj . Producer K’s profit is then π∗K = wKi(1−a−wKi+awLi)

2(1−a2).

If wKi >1+wKj

2 and wLj > 1+wLi2 , then:

• If wKj < 1 − a(1−wLi)2 and wLi < 1 − a(1−wKj)

2 , retailer i solves the problemmaxpLi(pLi−wLi)DL(pA1, pA2, pB1, pB2) and j solves maxpKj (pKj−wKj)DK(pA1, pA2, pB1, pB2).The resulting equilibrium prices are given by p∗Kj = (1−a)(2+a)+2wKj+awLi

4−a2 , p∗Ki =

wKi, p∗Li = (1−a)(2+a)+2wLi+awKj

4−a2 and p∗Lj = wLj . Producer K’s profit is then

π∗K = wKj((2−a2)(1−wKj)−a(1−wLi))(4−a2)(1−a2)

.

• If wKj > 1− a(1−wLi)2 , then the interior solution is impossible: Retailer j has to set

p∗Kj = wKj and i then solves maxpLi(pLi − wLi)DL(pLi, wLj , wKi, wKj), which givesp∗Li = 1

2(1− a+ awKj + wLi). The resulting profits are then:

π∗K = wKj((2−a2)(1−wKj)−a(1−wLi))2(1−a2)

, and π∗L = wLi(1−a−wLi+awKj)2(1−a2)

.

If wKi < wKj <1+wKi

2 and wLj > 1+wLi2 , then if j sets a price in the interval [wKj , 1+wKi

2 ],i can always set a price pKi = pKj − ε. As j cannot set a price lower than wKj , she setsp∗Kj = wKj . The best reply of retailer i is to set p∗Ki = wKj − ε. Retailer i then sets pLito solve the program:

maxpLi

(wKj − wKi)DK(wKj − ε, wKj , pLi, pLj) + (pLi − wLi)DL(wKj − ε, wKj , pLi, pLj),

which gives p∗Lj = wLj and p∗Li = max{

1−a+2awKj+wLi−awKi

2 , wLi

}. Note that p∗Li < wLj ,

which ensures that this vector of prices is an equilibrium. In that case, producers’ profitsare given by:

• If 1−a+2awKj+wLi−awKi

2 > wLi:

π∗K = wKi((1−a)(2+a)+a(wLi−awKi))2(1−a2)

− wKiwKj , π∗L = wLi(1−a+awKi−wLi)2(1−a2)

.

• Otherwise π∗K = wKi(1−a−wKj+awLi)1−a2 and π∗L = wLi(1−a+awKj−wLi)

1−a2 .

If wKi < wKj <1+wKi

2 and wLi >1+wLj

2 then retailer i sells good K and retailer j sellsgood L, and different cases may happen:

• If wKj >(1−a)(2+a)+2wKi+awLj

2 and wLi >(1−a)(2+a)+2wLj+awKi

2 , then:

26

– If wKi <(1−a)(2+a)+2wKi+awLj

2 and wLj <(1−a)(2+a)+2wLj+awKi

2 , then each re-tailer maximizes her profit and the resulting prices are p∗Ki = (1−a)(2+a)+2wKi+awLj

4−a2 ,

p∗Kj = wKj , p∗Lj = (1−a)(2+a)+2wLj+awKi

4−a2 and p∗Li = wLi. The profit of retailer

K is then π∗K = wKi((2−a2)(1−wKi)−a(1−wLj))(4−a2)(1−a2)

.

– If wKi >(1−a)(2+a)+2wKi+awLj

2 , then retailer i has to set price p∗Ki = wKi andretailer j maximizes his profit, which gives p∗Lj = 1

2(1− a+ wLj + awKi). Theresulting profits for producers are:

π∗K = wKi((2−a2)(1−wKi)−a(1−wLj))2(1−a2)

, π∗L = wLj(1−a+awKi−wLj)2(1−a2)

.

• If wKj <(1−a)(2+a)+2wKi+awLj

2 then p∗Ki = wKj − ε and retailer j sets p∗Lj to max-imize her profit, which implies that p∗Lj = max

{12(1− a+ awKj + wLj), wLj

}. The

resulting profits are:

– If 1−a+awKj+wLj

2 > wLj , then:

π∗K =wKi((1−a)(2+a)+awLj−(2−a2)wKj)

2(1−a2), π∗L = wLj(1−a−wLj+awKj)

2(1−a2).

– Otherwise π∗K = wKi(1−a+awLj−wKj)1−a2 and π∗L = wLj(1−a−wLj+awKj)

1−a2 .

Note that in the interval that we have specified here, we can never have wKj <(1−a)(2+a)+2wKi+awLj

2

and wLi <(1−a)(2+a)+2wLj+awKi

2 at the same time.If wKi < wKj <

1+wKi2 for K = A,B, then:

• If there exists L ∈ {A,B} such that wLj >1−a+2awKj+wLi−awKi

2 , the equilibriumprices are:

p∗Ki = wKj−ε, p∗Kj = wKj and p∗Lj = wLj , p∗Li = max{

1−a+2awKj+wLi−awKi

2 , wLi

}.

In that case, producers’ profits are given by:

– If 1−a+2awKj+wLi−awKi

2 > wLi:

π∗K = wKi((1−a)(2+a)+a(wLi−awKi))2(1−a2)

− wKiwKj , π∗L = wLi(1−a+awKi−wLi)2(1−a2)

.

– Otherwise π∗K = wKi(1−a−wKj+awLi)1−a2 and π∗L = wLi(1−a+awKj−wLi)

1−a2 .

• If for all K ∈ {A,B} we have wKj <1−a+2awLj+wKi−awLi

2 , then for all K the equi-librium prices are simply p∗Ki = wKj−ε and p∗Kj = wKj . Producer K’s profit is then

given by π∗K = wKi(1−a−wKi+awLj)1−a2 .

27

Finally, if wKj < wKi <1+wKj

2 and wLi < wLj <1+wLi

2 , then different cases mayoccur:

• If wKi >(2+a)(1−a)+2wKj+awLi

4−a2 and wLj >(2+a)(1−a)+2wLi+awKj

4−a2 , then:

– If wKj <(2+a)(1−a)+2wKj+awLi

4−a2 and wLi <(2+a)(1−a)+2wLi+awKj

4−a2 , then we havethe interior solution:

p∗Kj = (2+a)(1−a)+2wKj+awLi

4−a2 , p∗Li = (2+a)(1−a)+2wLi+awKj

4−a2 .

Upstream profits are given by π∗K = wKj((2−a2)(1−wKj)−a(1−wLi))(4−a2)(1−a2)

.

– If wKj >(2+a)(1−a)+2wKj+awLi

4−a2 , then p∗Kj = wKj and retailer i sets pLi tomaximize her profit: p∗Li = 1

2(1−a+wLi+awKj) and upstream profits are then

π∗K = wKj((2−a2)(1−wKj)−a(1−wLi)2(1−a2)

and π∗L = wLi(1−a−wLi+awKj)2(1−a2)

.

• If wKi <(2+a)(1−a)+2wKj+awLi

4−a2 and wLj >(2+a)(1−a)+2wLi+awKj

4−a2 , then p∗Kj = wKi− εand i sets pLi to maximize her profit, which gives p∗Li = max

{1−a+2awKj+wLi−awKi

2 , wLi

}.

The resulting upstream profits are:

– If 1−a+2awKj+wLi−awKi

2 > wLi:

π∗K = wKi((1−a)(2+a)+a(wLi−awKi))2(1−a2)

− wKiwKj , π∗L = wLi(1−a+awKi−wLi)2(1−a2)

.

– Otherwise π∗K = wKi(1−a−wKj+awLi)1−a2 and π∗L = wLi(1−a+awKj−wLi)

1−a2 .

• If wKi <(2+a)(1−a)+2wKj+awLi

4−a2 and wLj <(2+a)(1−a)+2wLi+awKj

4−a2 ,

– If wLj >1−a+2awKj+wLi−awKi

2 , then, as in the former case, p∗Kj = wKi− ε and isets pLi to maximize her profit, which gives p∗Li = max

{1−a+2awKj+wLi−awKi

2 , wLi

},

and the resulting profits are given by the former case.

– Otherwise, p∗Kj = wKi− ε and p∗Li = wLj − ε and the resulting upstream profits

are given by π∗K = wKj(1−a−wKj+awLi)1−a2 .

A.2 First stage: Upstream price setting

Collusion.In collusion, each producer maximizes the joint-profit of the industry. With the same

reasoning as in the competitive case, we obtain that in collusion, a producer prefers tooffer the same tariff to both retailers than a different tariff to each of them. Besides, thetariffs are then given by wC = 1

2 .

28

Deviation from the collusive path.Assume that A wants to deviate from the collusive path. He then takes as given that

B sets the tariff wC . He wants to maximize his individual profit. He can either set thesymmetric tariff such that B’s product is still sold, that is maximize:

πA =∑i=1,2

wAiDAB,ABAi (wA1, wA2, w

C , wC)

provided that all quantities are indeed positive. He can also offer asymmetric tariffs suchthat one of the retailers (say 2) decides not to sell good B, that is maximize:

πA =∑i=1,2

wAiDAB,AAi (wA1, wA2, w

C) (11)

provided that retailer 2 has indeed an incentive not to sell B and all other quantities arepositive. Finally, A can maximize his profit when the good B is never sold, provided thatboth retailers have an incentive not to sell B. In that case, we have:

πA =∑i=1,2

wAiDA,AAi (wA1, wA2)

Other possible strategies of A are not relevant.Producer A thus maximizes the three profits given above, respecting the specific con-

straints on quantities and on retailers’ profits in each case. The first deviation (such thatB sells his good to both retailers) leads A to set symmetric wholesale tariffs:

wA1 = wA2 = wAB,ABDev =2− a

4

This deviation is possible (that is all quantities are positive with this deviation) is andonly if a <

√3− 1. Otherwise, only one of the two other deviations can occur.

In order to completely exclude B from the market, A sets wholesale tariffs such thateach retailer is indifferent between selling both goods and selling only good A if her rivalsells only A. In other words, for the third deviation, A solves the equation:

(pA,AA1 − wA1)DA,AA1 (pA,AA1 , p

A,AA2 ) = (pAB,AA1 − wA1)DAB,A

A1 (pAB,AA1 , pAB,AA2 , pAB,AB1 )

+ (pAB,AB1 − wC)DAB,AB1 (pAB,AA1 , pAB,AA2 , pAB,AB1 )

where pA,AKi (respectively pAB,AKi ) is the equilibrium final price of good Ki when the marketstructure is (A,A) (resp. (AB,A)). This gives wA1, and similarly we find wA2 by solvingthe parallel equation for retailer 2. Finally, A sets symmetric wholesale tariffs:

wA1 = wA2 = wA,ADev = 1− 12a

Finally, we determine the deviation tariffs such that B sells only through one retailer,say 1. As the acceptation stage is not entirely committing since retailer 2 may still choose

29

to not sell B2 even if she has accepted the contract wB2 in the end of stage 1, two typesof deviations are possible.First, imagine that all firms believe that retailer 2 refuses a contract wB2 in the end ofstage 1 if she anticipates to not sell product B2 in stage 2. A must first make sure that 2 isindifferent between accepting both contracts wA2 and wB2 or accepting only contract wA2,taking as given that 1 accepts both contracts. A sets wA2 to solve the following equation:

(pAB,AA2 − wA2)DAB,AA2 (pAB,AA1 , pAB,AA2 , pAB,AB1 )

=∑

K=A,B

(pAB,ABK2 − wK2)DAB,ABK2 (pAB,ABA1 , pAB,ABA2 , pAB,ABB1 , pAB,ABB2 )

with wB2 = wC . We thus find an expression of wA2 as a function of wA1:

wDevA2 (wA1) =a(2− b− b2 + bwA1)− (2− b− b2)(1− wC)

a(2− b2)

If producer A offers a contract wDevA2 (wA1) − ε, the retailer 2 won’t sell product B2 andanticipating it refuses the contract wB2 in the end of stage 1. We replace wA2 by thisexpression in A’s profit function given by equation (11). The optimal wholesale tariff forA1 is the tariff that maximizes this profit. Finally, we then have an asymmetric deviationsuch that:

wAB,AA1 =4− 2a2b2 − a(2− b2)

8− 4a2b2

wAB,AA2 =a(2− b− b2 + bwAB,AA1 )− (2− b− b2)(1− wC)

a(2− b2)

which gives a profit ΠAB,ADev to producer A.

Second, imagine all firms believe that retailer 2 accepts both contracts wA2 wB2 in theend of stage one whether or not she anticipates to sell the good B2. A must make surethat 2 has no incentive to deviate from a situation where three products A1,B1 and A2are sold towards a situation where the four products are sold. Thus, A sets wA2 to solvethe following equation:

(pAB,AA2 − wA2)DAB,AA2 (pAB,AA1 , pAB,AA2 , pAB,AB1 )

=∑

K=A,B

(pAB,ABK2 − wK2)DAB,ABK2 (pAB,AA1 , pAB,ABA2 , pAB,AB1 , pAB,ABB2 )

where wB2 = wC . We thus find an expression of wA2 as a function of wA1:

wDev′

A2 (wA1) = 1− ab(1− wA1) + (2− b)(1− wC)2a

30

We replace wA2 by this expression in A’s profit function given by equation (11). Theoptimal wholesale tariff for A1 is the tariff that maximizes this profit. Finally, we thenhave an asymmetric deviation such that:

wAB,A′

A1 =4a(2− a) + ab2(5a− 4a2 − 6)− 2b2(1− a)2(1 + a) + b4(1− 2a)(1− 2a2)

2a(4− 3b2)(2− a2b2)− b4

wAB,A′

A2 = 1−ab(1− wAB,AA1 ) + (2− b)(1− wC)

2a

which gives a profit ΠAB,A′

Dev to producer A. This deviation is only possible if quantities ofgoods A1, A2 and B1 are positive, which is true only for a ∈ [2 −

√2, a], with a ≈ 0, 76,

and b ∈ [b(a), b(a)], where b and b are two functions from (0, 1) to (0, 1).

Eventually, we compare the profits obtained by A with these four deviations and finda decreasing function of a b : a 7→ b(a) and an increasing function of a b : a 7→ b(a) suchthat A’s optimal deviation leads to the market structure (AB,AB) when:

• a ≤ 0, 67 or

• a ∈ (0, 67,√

3− 1] and b < b(a) or b > b(a)

We find two increasing functions of a bl : a 7→ bl(a) and br : a 7→ bl(a) and a witha ≈ 0, 7545 separating the two types of deviations towards the market structure (AB,A)and a profit ΠAB,A′

Dev when

• a ∈ (0, 67,√

3− 1] and b ∈ [b(a), b(a)] or,

• a ∈ (√

3− 1, a] and b > bl(a).

and the market structure (AB,A) and a profit ΠAB,ADev when

• a ∈ (√

3− 1, (√

17−1)4 ] and b ∈ [br(a), bl(a)].

It leads to the market structure (A,A) otherwise.

B Exclusive dealing: Perfect competition downstream

B.1 Punishment profits

Candidate equilibrium with market structure (A,B). We consider the punishmentstrategies when exclusive dealing is available and retailers are perfectly competitive (b = 1).

The candidate equilibrium is such that the market structure is (A,B) and each produceroffers each retailer the same contract (wedp, F edp):

wedp =a2(1− a)

4− 2a− a2and F p =

(1− a)(2− a2)2

(1 + a)(4− 2a− a2)2

31

We will prove below that it is indeed the case that all contracts are symmetric. Thecorresponding profit for each producer is denoted by:

πedp =2(2− 2a− a2 + a3

)(1 + a) (4− 2a− a2)2

We analyze the incentives for producer B to deviate towards market structure (B,B).As A’s contracts are identical, B’s optimal deviation is given by a contract (wB, FB) thatmaximizes B’s profit, provided that both retailers have an incentive to accept the contract.The most binding constraint is that in case one of the retailers has accepted B’s contract,the other could prefer A’s initial contract. B’s programme is given by:

maxwB

wB(1− wB) + 2FB

s.t.− FB ≥πA,B1 (wedp, wB)− F edp (12)

1’s profit if she accepts A’s contract instead of B’s is:

πA,B1 (wedp, wB)− F edp =a(2− a2

) (a3 + 4wB − 2awB − a2(1 + wB)

)(2 + a− a2) (4− 2a− a2)2

Replacing in (12), we solve the maximization programme, which gives:

wB =(1− a)

(8 + 4a− 4a2 − a3

)2(2− a)(1 + a) (4− 2a− a2)

FB = −(1− a)a

(2− a2

) (8 + 4a− 8a2 − 3a3 + 2a4

)2(2− a)2(1 + a)2 (4− 2a− a2)2

B’s resulting profit is always lower than πedp: this deviation is thus not profitable for B.It is straightforward to see each producer’s best out-of-equilibrium exclusive dealing

contract offer is the identical to his equilibrium offer, in order to limit potential monopo-lization of the market by the rival.

Candidate equilibrium with market structure (B,B). We now show that it isimpossible to have an equilibrium where one producer monopolizes the downstream market.Consider the following candidate equilibrium: B sets wB1 = wB2 = w in the interval [0, 1/2]and FB1 = FB2 = F = w(1−w)

2 . If it is an equilibrium, then it is Pareto-dominated bythe equilibrium we determined in the previous paragraph. However, as B transfers all hisprofit to the retailers, this candidate is also the most robust to a deviation by A.

We determine A’s optimal deviation from this candidate equilibrium. We assume thatA only tries to capture one retailer, say 1, by offering her an exclusive dealing contract(wA1, FA1). This contract solves the following program:

maxwA1

wA1DA,BA1 (wA1, w) + FA1

s.t.(pA − wA1)DA,BA1 (wA1, w)− FA1 ≥ −F

32

As the constraint is binding, we find that wA1 = aw2 , which yields profit:

πA =4(1− a)2 − 2(1− a)2(4 + (2− a)a)w + (8− a(8 + a(5− 2(4− a)a)))w2

4(2− a)2 (1− a2)

which is positive. A thus always has an incentive to deviate from this candidate equilibrium.As a consequence, there is no equilibrium of that type.

Candidate equilibrium (AB,AB). We now show that it is impossible to have an equi-librium where both producers sell both retailers. Consider that all producers set the samelinear tariff wA1 = wA2 = wB1 = wB2 = w. Then all retailers set the same price p = w

and each retailer sells a quantity q = 1−w2(1+a) of each good. Then, each producer earns a

profit equal to w 1−w1+a .

If A wants to deviate from this candidate equilibrium by offering 1 an exclusive dealingcontract, he sets wA1 and FA1 to solve the program:

maxwA1

wA1DA,BA1 (wA1, w) + FA1

s.t.(pA − wA1)DA,BA1 (wA1, w)− FA1 ≥ 0

As in the former paragraph, he then sets wA1 = aw2 and FA1 = (pA − wA1)DA,B

A1 (wA1, w).This yields a profit − (2+a(−2+wb))2

4(−2+a)2(−1+a2), which is always higher than w 1−w

1+a .

B.2 Deviation profits

B.2.1 Variable profits

We first determine the profits of the two retailers and of producer B in the differentmarket structures that may occur in the deviation from collusion. These profits will allowus to determine the participation constraints of the retailers in deviation. We considera deviation by B. In the following, we always assume that wircA1 and wircA2 are as lowas possible, provided that in collusion retailer 1 accepts A’s exclusive dealing offer andretailer 2 accepts B’s exclusive dealing offer. We thus have wircA1 = 1 − a(2−a)

2(2−a2)> a

2 andwircA2 = wedc = a

2 . Recall also that F cA2 = max{0, (wedc−wedcA2 )(1−wedc)} and F c = 1−a4(1+a) .

Finally, we determine variable profits, i.e. profits net of potential fixed fees.

Case (A,A). We denote by (A,A) the case where both 1 and 2 have accepted A’s exclu-sive dealing offer, regardless of which goods are actually sold on the final market. Retailer1 was offered by A the exclusive dealing contract wedc = a

2 and F c = 1−a4(1+a) and retailer 2

was offered the exclusive dealing contract wedcA2 ≥a2 and F cA2 = 0 or wedcA2 <

a2 and F cA2 = (a2

−wedcA2 )(1 − a2 ). We thus need to analyze the two cases with respect to the value of wedcA2 .

If wedcA2 < a2 , as there is perfect competition between the two retailers if they both sign

33

an exclusive dealing contract with retailer A, it is obvious that 2 wins the competitionand 1 realizes a strictly negative profit as it incurs a positive fixed fee F c = 1−a

4(1+a) . More

precsiely, 2 sets pA,AA = a2 as the monopoly price 1+wedc

A22 can never be lower than a

2 . IfwedcA2 ≥

a2 , 1 sets a price that is the minimum between the optimal price set by an uncon-

strained monopolist which buys A at the price a2 , i.e

2+a4 , and the price of its rival wedcA2 :

pA,AA = Min(2+a4 , wedcA2 ). 1 thus realizes a profit which rewrites as:

πA,A1 (wedc, wedcA2 )− F c = − 1−a4(1+a) if wedcA2 ∈

[0, a2),

= (1− wedcA2 )(wedcA2 − a

2

)− 1−a

4(1+a) if wedcA2 ∈[a2 ,

2+a4

),

=a(4−3a+a2)

16(1+a) otherwise.

This profit is strictly positive whenever:

wedcA2 >2+a4 −

14

√4a− 3a2 + a3

1 + a(13)

The profit of 2 in this framework is always πA,A2 (wedc, wedcA2 )− F cA2 = 0.

Case (A,B). We denote by (A,B) the case where 1 has accepted A’s exclusive dealingoffer and 2 has accepted B’s exclusive dealing offer, regardless of which goods are actuallysold on the final market. In that case, retailers’ profits are given by the following expres-sions. Retailer 1 sets pA to maximize her profit and retailer 2 sets pB to maximize herprofit.

1. If weddB2 < 2 + a − 4+a3

2a , then pA = wedc = a2 and pB = a−1+wedc

a = 3a−22a . The price

pB is a limit-price such that DA = 0. Note that it is impossible for B to set theunconstrained monopoly price, for wedc is too low. In that case, retailers’ variableprofits are:

πA,B1 (wedc, weddB2 ) = 0 and πA,B2 (wedc, weddB2 ) =(

3a−22a − w

eddB2

)2−a2a .

Note that when a < 0.80606, 2+a− 4+a3

2a < 0 and there is no value of weddB2 for whichDA = 0.

2. If weddB2 ∈[2 + a− 4+a3

2a , 1− a(2−a)2(2−a2)

), then pA,BA = 2−a2+awedd

B2(2−a)(2+a) and pA,BB = 4(1+wedd

B2 )−a(2+a)

2(2−a)(2+a) ,which corresponds to the interior solution. The resulting profits are:

πA,B1 (wedc, weddB2 ) =a(2wedd

B2 −a)(8(1−a)−3a2+2a(a2+weddB2 ))

4(4−a2)2(1−a2)and πA,B2 (wedc, weddB2 ) = (2(2−a2)(1−wedd

B2 )−(2−a)a)2

4(4−a2)2(1−a2).

34

3. If weddB2 ∈[1− a(2−a)

2(2−a2), 1− a(2−a)

4

], then pA,BA = 1 − 1−wedd

B2a and pA,BB = weddB2 , which

corresponds to the case where 1 sets a limit price so that DB = 0. Profits of retailersare then:

πA,B1 (wedc, weddB2 ) = a2(1−a)(1−2weddB2 )−4(1−wedd

B2 )(1−(1+a)weddB2 )

4a2(1+a)and πA,B2 (wedc, weddB2 ) = 0.

4. Otherwise, 1 can set the monopoly price pA,BA = 2+a4 when selling A (regardless of

the price set by B for good 2), and resulting profits are:

πA,B1 (wedc, weddB2 ) =a(4−3a+a2)

16(1+a) and πA,B2 (wedc, weddB2 ) = 0.

Case (B,A). We denote by (B,A) the case where 1 has accepted B’s exclusive dealingoffer and 2 has accepted A’s exclusive dealing offer, regardless of which goods are actuallysold on the final market. In that case, retailer 1 sets pB to maximize her profit and retailer2 sets pA to maximize her profit. We get the following equilibrium profits.

1. If weddB1 < 1− 2(1−wedcA2 )

a , then 1 can set her monopoly price pB,AB = 1+weddB1

2 and variable

profits are πB,A1 (weddB1 , wedcA2 ) = (1−wedd

B1 )2

4 , πB,A2 (weddB1 , wedcA2 ) = 0 and πB,AB (weddB1 , w

edcA2 ) =

weddB1 (1−wedd

B1 )2 .

2. If weddB1 ∈[1− 2(1−wedc

A2 )a , 1− (2−a2)(1−wedc

A2 )a

), then 1 sets a limit price so that DA = 0:

pB,AB = 1 − 1−wedca2a . Again, we have πB,A2 (weddB1 , w

edcA2 ) = 0. Profits of 1 and B are

given by:

πB,A1 (weddB1 , wedcA2 ) = (wedc

A2 −1)(1−wedcA2 −a(1−w

eddB1 ))

a2 and πB,AB (weddB1 , wedcA2 ) = (1−wedc

A2 )weddB1

a .

3. If weddB1 ∈[1− (2−a2)(1−wedc

A2 )a , 1− a(1−wedc

A2 )

2−a2

), then the interior solution applies. Prices

are pB,AA = (1−a)(2+a)+2wedcA2 +awedd

B14−a2 and pB,AB = (1−a)(2+a)+2wedd

B1 +awedcA2

4−a2 and profits aregiven by:

πB,A1 (weddB1 , wedcA2 ) = ((1−a)(2+a)+awedc

A2 −(2−a2)weddB1 )2

(4−a2)2(1−a2),

πB,A2 (weddB1 , wedcA2 ) = ((1−a)(2+a)+awedd

B1 −(2−a2)wedcA2 )2

(4−a2)2(1−a2),

πB,AB (weddB1 , wedcA2 ) =

weddB1 ((1−a)(2+a)+awedc

A2 −(2−a2)weddB1 )

(4−a2)(1−a2).

4. If weddB1 ∈[1− a(1−wedc

A2 )

2−a2 , 1− a(1−wedcA2 )

2

), 2 sets a limit price such that DB = 0:

pA = 1 − 1−weddB1a . Then we have πB,A1 (weddB1 , w

edcA2 ) = πB,AB (weddB1 , w

edcA2 ) = 0 and

πB,A2 (weddB1 , wedcA2 ) = (1−wedd

B1 )−a(1−wedcA2 )(wedd

B1 −1)

a2 .

35

5. Finally, if weddB1 ∈[1− a(1−wedc

A2 )2 , 1

), 2 can set the monopoly price pA = 1+wedc

A22

and profits are πB,A1 (weddB1 , wedcA2 ) = πB,AB (weddB1 , w

edcA2 ) = 0 and πB,A2 (weddB1 , w

edcA2 ) =

(1−wedcA2 )2

4 .

Case (B,B). We denote by (B,B) the case where both 1 and 2 have acceptedB’s exclusivedealing offer, regardless of which goods are actually sold on the final market.

Case (AB,B). We denote by (AB,B) the case where 1 has refused both exclusive dealingoffers and 2 has accepted B’s exclusive dealing offer, regardless of which goods are actuallysold on the final market. In that case, profits are given by the following expressions.

Case wirdB1 < weddB2

1. If wirdB1 +1

2 < weddB2 , then 1 can set the monopoly price for B 1+wirdB1

2 and we have threecases depending on the relative values of wirdB1 and wircA1:

• If wirdB1 < 1 − 2−a2(2−a2)

, then 1 sells B only at price 1+wirdB1

2 and earns a profit(1−wird

B12

)2, whereas 2 earns 0.

• If wirdB1 ∈[1− 2−a

2(2−a2), 1− a2(2−a)

2(2−a2)

], then 1 sells A and B respectively at price

1+wircA1

2 and 1+wirdB1

2 . Demand for good K sold by 1 is DK1 = 1−a+awL1−wK12(1−a2)

, andprofits are π1 = (pA1 − wircA1)DA1 + (pB1 − wirdB1)DB1 and π2 = 0.

• Finally, if wirdB1 > 1 − a2(2−a)2(2−a2)

, then 1 sets the same prices as before but sells

only good A. Demand for A is 1−wircA1

2 and 1’s profit is(

1−wircA1

2

)2=(a(2−a)4(2−a2)

)2,

whereas 2 earns 0.

2. If wirdB1 < weddB2 <wird

B1 +12 , then 1 cannot set her monopoly price for good B anymore.

Instead, she must set pB = weddB2 and then sets pA1 to maximize her profit. Then, wehave again three cases:

• If wirdB1 <2−2a2+a2(2−a2)

, then DA1 = 0: 1 sets pB = weddB2 and only good B is sold onthe final market. Demand for good B is 1− weddB2 and profits are:

π1 = (weddB2 − wirdB1)(1− weddB2 ),

π2 = 0.

• If wirdB1 ≥2−2a2+a2(2−a2)

and weddB2 <(2+a)(1−a)+awirc

A1−a2wird

B12(1−a2)

then we have the interiorsolution where:

pA1 = 1−a+a(2weddB2 −w

irdB1 )+wirc

A12 ,

pB1 = weddB2 .

36

As before, 1 sells the two goods and demand for each is: DA1 = 1−a+weddB2 −w

ircA1

2(1−a2)

and DB1 = (2+a)(1−a)−2(1−a2)weddB2 −a

2wirdB1 +awirc

A12(1−a2)

. 2 earns 0 and 1 earns π1 =(pA1 − wircA1)DA1 + (pB1 − wirdB1)DB1.

• Finally, if wirdB1 ≥2−2a2+a2(2−a2)

and weddB2 ≥(2+a)(1−a)+awirc

A1−a2wird

B12(1−a2)

, then demand forB is 0. Two cases occur:

(a) If weddB2 ≥ 1 − a2 + a

wircA12 , then 1 can set its unconstrained monopoly price

pA = 1+wircA1

2 and earn profit(

1−wircA1

2

)2while 2 earns 0.

(b) If weddB2 < 1 − a2 + a

wircA12 , then A cannot set the unconstrained monopoly

price for A. She must set pA so that DB = 0 in market structure (A,B):pA = 1

a(a− 1 + weddB2 ). Then, we have DA1 = 1−weddB2

2 , and DB1 = DB2 = 0.Profits are π1 = (pA1 − wircA1)DA1 and π2 = 0.

Case weddB2 ≤ wirdB1

1. If weddB2 ≤a(1−a)2−a2 (≤ a

2 )

• If weddB2 ≤ 2wirdB1 − 1, product A is not sold and retailer 2 charges the monopoly

price on product B: pB2 = 1+weddB2

2 . Profits are π1 = 0 and π2 = (1−weddB2

2 )2.

• If weddB2 ≥ 2wirdB1−1, product A is still not sold but retailer 2 cannot charge a pricehigher than wirdB1 on product B, otherwise retailer 1 would enter the market forB. Thus pB2 = wirdB1(−ε). Profits are π1 = 0 and π2 = (wirdB1 − weddB2 )(1− wirdB1).

2. If a(1−a)2−a2 ≤ weddB2 ≤

a2

Retailer 2 maximizes its profit:

Max(pB2 − weddB2 )(1− peddB2 )

s.t. pB2 ≤ wirdB1

s.t. pB2 ≤−1+a+wirc

A1a

Thus the optimal price for retailer 2 is pB2 = min{−1+a+wircA1

a , wirdB1}. Profits areπ1 = 0 and π2 = (pB2 − weddB2 )(1− pB2).

3. If a2 ≤ weddB2 ≤ wirdB1

• If weddB2 ≤Min{awircA1+(2+a)(1−a)

2−a2 ,(4−a2)wird

B1−awircA1−(2+a)(1−a)2 }, there is an interior

equilibrium where 1 sells A and 2 sells B at prices:

p∗A1 = (2+a)(1−a)+2wircA1+awedd

B24−a2

p∗B2 = (2+a)(1−a)+2weddB2 +awirc

A14−a2

37

Profits are:

π∗1 = (p∗A1 − wircA1)1−a−p∗A1+ap∗B21−a2

π∗2 = (p∗B2 − weddB2 )1−a−p∗B2+ap∗A11−a2

• If weddB2 ≥Min{awircA1+(2+a)(1−a)

2−a2 ,(4−a2)wird

B1−awircA1−(2+a)(1−a)2 },

(a) If (4−a2)wirdB1−awA1−(2+a)(1−a)

2 ≤ weddB2 ≤ wirdB1(≤ awircA1+(2+a)(1−a)

2−a2 : always in that case)Then pB2 = wirdB1 and DB1 = 0.

– If wirdB1 ≥wirc

A1−1+aa , then retailer 1 sells A and 2 sells B; profits are

π1 = (1−a+awirdB1−w

ircA1 )2

4(1−a2)

π2 = (wirdB1 − weddB2 ) (1−a2)(1−wirdB1 )+awirc

A11−a2

– If wirdB1 ≤wirc

A1−1+aa , then retailer 1 exists the market and retailer 2 sells

B; profits are

π1 = 0

π2 = (wirdB1 − weddB2 )(1− wirdB1)

(b) If awircA1+(2+a)(1−a)

2−a2 ≤ weddB2 ≤(4−a2)wird

B1−awircA1−(2+a)(1−a)2

Then pB2 = weddB2 and π2 = 0. The value of pA1 depends on the relativevalues of weddB2 and wircA1:

– If weddB2 > 1 − a + a(1+wircA1 )

2 = 1 − a2(2−a)4(2−a2)

then retailer 1 sells A at the

monopoly price pA = 1+wircA1

2 . Profits are:

π1 =(

1− wircA1

2

)2

,

π2 = 0.

– If weddB2 ≤ 1 − a + a(1+wircA1 )

2 , then 1 sets pA so that DB2 = 0: pA =

1 − 1−weddB2a . Then, demands are DA = 1−wedd

B22 and DB2 = DB1 = 0.

Profits are:

π1 =(

1−1− weddB2

a− wircA1

)1− weddB2

a, π2 = 0.

Case (B,AB). We denote by (B,AB) the case where 1 has accepted B’s exclusive dealingoffer and 2 has refused both exclusive dealing offers, regardless of which goods are actuallysold on the final market. The final outputs and prices depend on the relative values of wirdB2

38

and weddB1 . Indeed, different cases may occur in market structure (B,AB): 1 sells B while 2sells A, both retailers set the same price for B and 2 sells A, 2 sells A and B, A cannot besold and either one of the retailer sells B. Here, we summarize these different cases.a <

√23 .

1. If weddB1 < 2− 2a + a− a2

2 , then:

• If wirdB2 < −1 + 2weddB1 , then 2 sells B only at the monopoly price pMB = 1+wirdB2

2

and profits are πB,ABB = wirdB2 (1−wird

B2 )2 and πB,AB2 = (1−wird

B2 )2

4 ;

• If wirdB2 ∈ [−1 + 2weddB1 , weddB1 ], then 2 sells B only at price pB = weddB1 and profits

are πB,ABB = wirdB2(1− weddB1 ) and πB,AB2 = (1− weddB1 )(weddB1 − wirdB2);

• If wirdB2 ∈ [weddB1 ,32 −

1a ], then 1 sells B at price pB = wirdB2 and demand for A

is negative for all values of pA. Then, profits are πB,ABB = wirdB2(1 − wirdB2) andπB,AB2 = 0;

• Finally, if wirdB2 > 32 −

1a , then 1 sells B at price pB = 1 − 1−wirc

A2a so as to

prevent 2 from selling good A; A’s price is pA = wircA2 = a2 and profits are

πB,ABB = − (1−wircA2 )(1−a−wirc

A2 )

a2 and πB,AB2 = 0.

2. If 2− 2a + a− a2

2 < weddB1 < 32 −

1a , then:

• If wirdB2 <32 −

1a , see the previous case;

• If wirdB2 ∈ [32 −1a ,

12 −

a−2weddB1

4−a2 ], then 1 sells B at price wirdB2 and 2 sells A at

price pA = 2−a+2awirdB2

2 . Profits are πB,ABB =wird

B2 (2(2−a2)(1−wirdB2 )−a(2−a))

4(1−a2)and

πB,A2 = (2−a−2a(1−wirdB2 ))2

16(1−a2).

• Finally, if wirdB2 >12−

a−2weddB1

4−a2 , then 1 sells B at price 12−

a−2weddB1

4−a2 and 2 sells A at

price −2+a2−aweddB1

4−a2 . Profits are πB,ABB = − (4(1−weddB1 )−(2−a)(4+a))(2(2−a2)(1−wedd

B1 )−a(2−a))4(4−a2)2(1−a2)

and πB,AB2 = ((2−a)(2−a2)−2a(1−weddB1 ))2

4(4−a2)2(1−a2).

3. If 32 −

1a ≤ w

eddB1 < 5

4 −12a , then:

• If wirdB2 < −1 + 2weddB1 , then see previous case;

• If wirdB2 ∈ [−1 + 2weddB1 ,23 −

1a ], see previous case (wirdB2 < weddB1 ).

• If wirdB2 ∈ (23 −

1a , w

eddB1 ], then 2 sells B at price weddB1 and A at price aweddB1 +

1−awirdB2

2 − a4 . Profits are πB,ABB = wirdB2

(1− weddB1 −

a(2−a)−2a2(1−wirdB2 )

4(1−a2)

)and

πB,AB2 =(

(2+a)2−8(2−a)(a+wirdB2 )+4a2wird

B2 (1+wirdB2 )

16(1−a2)+ weddB1 (1− weddB1 + wirdB2)

);

• If wirdB2 ∈ [weddB1 ,12 −

a−2weddB1

4−a2 ], see previous case (wirdB2 >32 −

1a);

39

• If wirdB2 >12 −

a−2weddB1

4−a2 , see previous case.

4. If 54 −

12a ≤ w

eddB1 <

−2+a+a2

2−2+a2 , then:

• If wirdB2 <32 −

1a , see previous case (wirdB2 < −1 + wirdB2);

• If wirdB2 ∈ [32 −1a ,−1 + 2weddB1 ], then 2 sells both goods at two-product monopoly

prices: pA = 1+wedcA2

2 and pB = 1+wedcB1

2 . Profits are πB,ABB = (2(1−wirdB2 )−a(2−a))wird

B24(1−a2)

and πB,AB2 =4(2−2a+a2)(1−wird

B2 )+(2−a)2−4(1−(wirdB2 )2)

16(1−a2).

• If wirdB2 ∈ [−1 + 2weddB1 , weddB1 ], then 2 sells the two goods, B at price weddB1

and A at price pA = 14(2 − a + 4aweddB1 − 2awirdB2) and profits are πB,ABB =

wirdB2 (4(1−a2)(1−wedd

B1 )−a(2−a)+2a2(1−wirdB2 ))

4(1−a2)and πB,AB2 = (1 − weddB1 + wirdB2)weddB1 +

(2−3a)2−4wirdB2 (2(2−a)−a2(1+wird

B2 ))16(1−a2)

• If wirdB2 > weddB1 , see previous case.

5. If −2+a+a2

2−2+a2 ≤ weddB1 < 1

4

(4− 2a+ a2

), then:

• If wirdB2 < 2weddB1 + 2(1−weddB1 )

a2 − 1a −

12 , see previous case (wirdB2 < weddB1 );

• If wirdB2 ∈ 2weddB1 + 2(1−weddB1 )

a2 − 1a −

12 , w

eddB1 ], then 2 sells A only at price pA =

1 − 1−weddB1a and B is not sold on the final market. Profits are πB,ABB = 0 and

πB,AB2 = (2−2a+a2−2weddB1 )(−1+wedd

B1 )

2a2 ;

• If wirdB2 > weddB1 , then 2 sells only A at the limit price pA = 1− 1−weddB1a such that

DB = 0. Final profits are πB,ABB = 0 and πB,AB2 = (2−2a+a2−2weddB1 )(−1+wedd

B1 )

2a2 .

6. Finally, if 14

(4− 2a+ a2

)< weddB1 < 1, then:

• If wirdB2 <32 −

1a , see previous case;

• If wirdB2 ∈ [32 −1a ,

12

(2− 2a+ a2

)], then 2 sells the two goods at the two-product

monopoly prices and profits are πB,ABB = (2(1−wirdB2 )−a(2−a))wird

B24(1−a2)

and πB,AB2 =

(1−a2 )2−2a(1−a2 )(1−wirdB2 )+(1−wird

B2 )2

4(1−a2).

• Finally, if wirdB2 >12

(2− 2a+ a2

), 2 sells only good A at the monopoly price.

B’s profit is thus πB,ABB = 0, and 2’s profit is πB,AB2 = (2−a)216 .

a ≥√

23 .

When a ≥√

23 , we find the same prices, outputs and profits as in the previous case

as long as weddB1 < 32 −

1a and weddB1 > 1

4

(4− 2a+ a2

). In between, we have the following

results:

40

1. If weddB1 ∈ [32 −1a ,−2+a+a2

2−2+a2 ], see previous case (weddB1 < 5

4 −12a).

2. If weddB1 in(−2+a+a2

2−2+a2 , 5

4 −12a ], then:

• If wirdB2 < −1 + 2weddB1 , then 2 sells the two goods at their monopoly price and

profits are πB,ABB = wirdB2 (1−wird

B2 )2 and πB,AB2 = (1−wird

B2 )2

2 ;

• If wirdB2 ∈ [−1 + 2weddB1 ,32 −

1a ], then 2 sells good B at price weddB1 but cannot sell

good A, and profits are πB,ABB = wirdB2(1−weddB1 ) and πB,AB2 = (1−weddB1 )(weddB1 −wirdB2);

• If wirdB2 ∈ (32 −

1a ,

4−2a−4weddB1 +a2(−1+4wedd

B1 )

2a2 ], then 2 sells good B at price weddB1

and good A at price 14(2 − a + 4aweddB1 − 2awirdB2), and profits are πB,ABB =

wirdB2 (−4+2a+a2+4wedd

B1 −4a2weddB1 +2a2wird

B2 )4(−1+a2)

and πB,AB2 = weddB1 (1 − weddB1 + wirdB2) +−4+a(12−8wird

B2 )+16wirdB2−a

2(9+4wirdB2 +4(wird

B2 )2)16(−1+a2)

;

3. If weddB1 in(54 −

12a ,

14

(4− 2a+ a2

)], see previous case (weddB1 >

−2+a+a2

2−2+a2 ).

Case (A,AB). We denote by (A,AB) the case where 1 has accepted A’s exclusive dealingoffer and 2 has refused both exclusive dealing offers, regardless of which goods are actuallysold on the final market.

Case (AB,A). We denote by (AB,A) the case where 1 has refused both exclusive dealingoffers and 2 has accepted A’s exclusive dealing offer, regardless of which goods are actuallysold on the final market.

In market structure (AB,A), demand functions are (when the two goods are sold):

DAB,ABA =

1− a−min{pA1, pA2}+ apB1

1− a2,

DAB,ABB =

1− a+ amin{pA1, pA2} − pB1

1− a2.

Given that 2 has an exclusive dealing contract with A and wedcA2 is not yet determined, itis not obvious which retailer will sell good A: it depends on the value of wedcA2 relative towircA1. Then, as in the previous case, several corner solutions may occur.

First, it is immediate that if wedcA2 > wircA1, then 2 does not sell any good and thereforeearns profit 0. We thus focus on the case wedcA2 < wircA1.

Assume first that we always have pA < wircA1. Then, the different cases are as follows.

1. If wedcA2 <2(1−a)2−a2 and wirdB1 > 1− a(1−wedc

A2 )2 , then 2 charges the monopoly price 1+wedc

A22

for good A and 1 cannot sell good B. The profit of retailer 2 is then:

π2 =(

1−wedcA2

2

)2− F cA2.

41

2. If wirdB1 ∈[

(2+a)(1+a)+awedcA2

2−a2 , 1− a(1−wedcA2 )

2

]and wirdB1 < 1 − a2(2−a)

2(2−a2), then 2 sets pA so

that DB(pA, wirdB1) = 0, which implies pA = 1 − 1−wirdB1a . Again, 2 sells A and 1 does

not sell anything. The profit of 2 is then:

π2 = (a(1−wedcA2 )−(1−wird

B1 ))(1−wirdB1 )

a2 − F cA2.

3. If wirdB1 ∈[

(2−a2)−(2+a)(1−a)a < wirdB1 ≤

(2+a)(1+a)+awedcA2

2−a2

]and wirdB1 <

(2+a)(4(1−a)+a2(2−a))2a(2−a2)

−2wedc

A2a , then we have the interior solution where 2 sells goodA at price pA = (2+a)(1−a)+2wedc

A2 +awirdB1

4−a2

and 1 sells good B at price pB = (2+a)(1−a)+2wirdB1 +awedc

A24−a2 . The profit of retailer 2 is

then:π2 = ((2+a)(1−a)−(2−a2)wedc

A2 +awirdB1 )2

(4−a2)2(1−a2)− F cA2.

4. If wirdB1 <((2+a)(1−a)−(2−a2)wedc

A2 +awirdB1 )2

(4−a2)2(1−a2), then 1 sells B and 2 does not sell any good.

1 then either charges the unconstrained monopoly price for B or the limit-price suchthat DA = 0, depending on the value of wirdB1 . However, which of the two happens isunimportant as the profit of 2 is always 0 in that case.

5. If wedcA2 ≤2(1−a)2−a2 and wirdB1 > 1− a2(2−a)

2(2−a2), then 2 sells A and 1 does not sell any good.

However, A sets a constrained price: pA = wircA1(−ε) so that 1 cannot sell good A.The profit of retailer 2 is then:

π2 = (wircA1 − wedcA2 )(1− wedcA2 ).

6. Finally, if wirdB1 ∈[

(2+a)(4(1−a)+a2(2−a))2a(2−a2)

− 2wedcA2a , 1− a2(2−a)

2(2−a2)

], then we have again a

duopoly where 2 sells A and 1 sells B, but 2 charges pA = wircA1(−ε) to ensure that 1cannot sell A. The profit of retailer 2 is then:

π2 =a(2(2−a2)(1−wedc

A2 )−(2−a)a)4(1−a2)(2−a2)

(wirdB1 − a

2

)− F cA2.

Case (AB,AB). We denote by (AB,AB) the case where no retailer has accepted anyexclusive dealing offer, regardless of which goods are actually sold on the final market.

In market structure (AB,AB), demand functions are (when the two goods are sold):

DAB,ABA =

1− a−min{pA1, pA2}+ amin{pB1, pB2}1− a2

,

DAB,ABB =

1− a+ amin{pA1, pA2} −min{pB1, pB2}1− a2

.

As wircA1 > wircA2, good A is sold by 2 in equilibrium. Then, B can ensure that 2’s profitselling good B is 0 and 2’s profit selling good A is minimized by setting wirdB2 = wirdB1 . On

42

the one hand, if wirdB2 < wirdB1 , then 2 can sell good B at price wirdB1 and earns a positive profit

selling B while its demand for good A is 1−a−pA2+awirdB1

1−a2 . On the other hand, if wirdB2 ≥ wirdB1 ,

then B cannot sell good B, while its demand for good A is 1−a−pA2+awirdB2

1−a2 . Therefore, fora given price pA2, demand for A increases with wirdB2 when wirdB2 ≥ wirdB1 . 2’s profit sellinggood A is thus minimized when wirdB2 = wirdB1 . Finally, 2’s total profit is minimized forwirdB2 = wirdB1 .

In the equilibrium of the downstream subgame, 1 never sells good A and, if sold, Bis always sold at cost: pB = wirdB1 . Then, different cases occur depending on the value ofwirdB1 . We only determine the profit of 2 in these various cases.

1. If wirdB1 <3a−22a , then good B is sold at price wirdB1 while demand for A is 0. The profit

of firm 2 is then 0.

2. If wirdB1 ∈[

3a−22a , wircA1

], then we find the interior solution: firm 2 sets pA = arg maxpA(pA−

wircA2)DAB,ABA and sells good A at price pA = 2−a−awird

B14 . The profit of firm 2 is then

(2−3a+2awirdB1 )2

16(1−a2).

3. If wirdB1 ∈(wircA1,min

{4−2a+a2

4 , 1− a2(2−a3)2(2−a2)

}], then the interior solution is not pos-

sible because it gives DB < 0. However, the monopoly solution is not possible either,because it would yield DB > 0. Therefore, 2 sets pA so that DB = 0, which impliespA = 1− 1−wird

B1a . The resulting profit of 2 is (a(2−a)−2(1−wird

B1 ))(1−wirdB1 )

2a2 .

4. If wirdB1 ∈(

min{

4−2a+a2

4 , 1− a2(2−a3)2(2−a2)

}, 4−2a+a2

4

], then 2 would like to implement

the same strategy as above, but cannot do so because then pA2 > wircA1 and 1 canthen set pA1 ∈ (wircA1, pA2) and get the demand for good A. Thus, 2 sets pA2 = wircA1

and earns a profit (wircA1 − wircA2)(1− wircA1) = a(2−a)2(2+a)(1−a)4(2−a2)2

.

5. Finally, if wirdB1 >4−2a+a2

4 , two cases may occur:

• If a <√

3− 1, 2 can set the monopoly price for good A 1+wircA2

2 = 2+a4 . Indeed,

we find that DAB,ABB (wirdB1 ,

1+wircA2

2 ) < 0. In that case, only good A has a positivedemand on the final market: DAB,AB

A = 2−a4 . 2’s profit is

(2−a4

)2.• Otherwise, again, the former strategy is such that pA2 > wircA1. Then, as in case

(4), 2 must set pA = wircA1 and earns profit a(2−a)2(2+a)(1−a)4(2−a2)2

.

B.2.2 Deviation in structure (B,B)

We focus on the deviation in structure (B,B), i.e. the deviation in which producer Bmanages to have the two retailers accept his exclusive dealing offer. The menus of contracts

43

offered to 1 and 2, respectively (wirdB1 , weddB1 , F

dB1) and (wirdB2 , w

eddB2 , F

dB2) must therefore satisfy

the participation constraints of the retailers:

πB,B1 (weddB1 , weddB2 )− F dB1 ≥ max{πA,B1 (wedc, weddB2 )− F c, πAB,B1 (wircA1, w

irdB1 , w

eddB2 ), 0},(14)

πB,B2 (weddB1 , weddB2 )− F dB2 ≥ max{πB,A2 (weddB1 , w

edcA2 )− F cA2, π

B,AB2 (weddB1 , w

ircA2, , w

irdB2), 0}.(15)

First, we determine the value of the right-hand side of each of these inequalities. Fixed feesF dB1 and F

dB2 are such that these constraints are binding. Second, we determine the optimal

values of weddB1 and weddB2 , i.e. the values that maximize the profit of B πB,BB (weddB1 , weddB2 ) +

F dB1 + F dB2.

Participation constraint of retailer 1. Comparing πA,B1 (wedc, weddB2 )−F c, πAB,B1 (wircA1, wirdB1 , w

eddB2 )

and 0, it is first immediate that πAB,B1 (wircA1, wirdB1 , w

eddB2 ) ≥ 0 for all values of the linear tar-

iffs. Indeed, if 1 refuses both exclusive dealing offers, she can at worse decide not to sellanything and then earn 0.

Second, it should be noted that in this case, producer B’s optimal strategy involvessetting wirdB1 so that 1’s profit in πAB,B1 (wircA1, w

irdB1 , w

eddB2 ) is as low as possible. This happens

when wirdB1 = weddB2 . First, when wirdB1 = weddB2 , 1 earns no profit by selling good B. Second,1’s profit through the selling of A is then minimized. Indeed, it is straightforward that theprofit associated with selling A, increases with the price of B. Then, if B sets wirdB1 > weddB2 ,the final price of good B is pB > weddB2 . If on the contrary B sets wirdB1 < weddB2 , then thefinal price of good B is always pB = weddB2

14, but 1 earns a positive margin when she sellsgood B. Finally, if wirdB1 = weddB2 , then we have again that pB = weddB2 : the profit that 1earns by selling A is minimized and 1 earns no margin by selling B. We thus always havethat wirdB1 = weddB2 .

Similarly, producer B has an incentive to offer the same linear part to the two retailersin the exclusive dealing offer: weddB1 = weddB2 . The argument is very similar to the former.As retailers are perfect substitutes, only the retailer with the lowest linear tariff will sellgood B. If they are offered different linear tariffs, the latter retailer, say i thus earns apositive margin on every unit of good sold by charging the price pB = weddBj . It is efficientfor B to offer the same tariff to both retailers and have them set the final price equal to thewholesale tariff, in order to maximize and capture the joint profit of the industry. If oneretailer needs to earn a positive profit due to its participation constraint, then the choiceof the fixed fee solves this issue. We denote by weddB this common linear part. We can thustremendously simplify the expressions of profits specified previously.

Replacing wirdB1 , weddB1 and weddB2 by weddB in the expressions of the profits given previously,

we find that πA,B1 (wedc, weddB ) − F c > πAB,B1 (wircA1, weddB , weddB ) if and only if weddB2 > a

2 . Fi-nally, as we set F dB1 so that constraint (14) is binding, we find that F dB1 = πB,B1 (weddB , weddB )−

14In order to have p∗B < weddB2 , B would have to set wird

Bi < 0, which is impossible.

44

πdev1 = −πdev1 , since there is perfect competition on the market for good B and both re-tailers have the same marginal cost. The profit πdev1 that retailer 1 would earn if he wasto refuse B’s exclusive dealing offer is given by:

πdev1 = 0 if weddB ≤ a2 ,

= ((2−a)(2−a2)−2a(1−weddB ))2

4(4−a2)2(1−a2)− 1−a

4(1+a) , if weddB ∈(a2 , w

ircA1

],

= (a(2−a)−2(1−weddB ))(1−wedd

B )

2a2 − 1−a4(1+a) , if weddB ∈

(wircA1, 1−

a(2−a)4

],

=(

2−a4

)2 − 1−a4(1+a) , if weddB ∈

(1− a(2−a)

4 , 1].

Participation constraint of retailer 2. As for retailer 1, it is first immediate thatπB,AB2 (weddB1 , w

ircA2, w

irdB2) ≥ 0. Second, from a symmetric reasoning to that made for retailer

1 , we conclude that the best strategy for producer B is to set wirdB2 = weddB .Finally, we find that if 2 refuses B’s exclusive dealing offer and chooses the best alter-

native option, then the fixed part offered to 2 is F dB2 = πB,B2 (weddB , weddB ) − πdev2 = −πdev2 ,where πdev2 is defined as follows:

1. If weddB < 3a−22a , then:

• If the following conditions are satisfied:

weddB > 1 +(a− 2

a

) (a2 − w

edcA2

),

weddB > 1 + (4−a2)√

(2−a)(1−a2)(a−2wedcA2 )−2(2−a2)(1−wedc

A2 )

2a ,

weddB >(2−a2)wedc

A2 −(1−a)(2+a)

a ,

then the profit of 2 is:

πdev2 = πB,A2 − F cA2 = ((2−a2)(1−wedcA2 )−a(1−wedd

B ))2

(4−a2)2(1−a2)− F cA2.

• Otherwise the profit of 2 is πdev2 = πB,AB2 = 0.

2. If weddB ∈[

3a−22a , wircA1

], then:

• If the following conditions are satisfied:

weddB > 1 +(a− 2

a

) (a2 − w

edcA2

),

weddB < 32 + −1+a

−2+a2 −2wedc

A2a ,

45

and if wedcA2 > 1− (2−a)(2−a2)(8−7a2+a4)4(4−4a2+a4)

+ (4−a2)(1−a2)√

(6−a2)(2−a2)

4(4−4a2+a4)

weddB <(40−24a2−2a3+3a4)

2(6−a2)(2−a2)+ 4(1+wedc

A2 )

a(6−a2)

− (4−a2)q

4−16a4+36a3−20a+5a2−(9−a2)(2−a)a5+2(2−a2)(2(4−a2)(1−a2)−a(8−7a2+a4))wedcA2 +4(2−a2)2(wedc

A2 )2

a(6−a2)(2−a2),

weddB >(40−24a2−2a3+3a4)

2(6−a2)(2−a2)− 4(1+wedc

A2 )

a(6−a2)

+(4−a2)

q4−16a4+36a3−20a+5a2−(9−a2)(2−a)a5+2(2−a2)(2(4−a2)(1−a2)−a(8−7a2+a4))wedc

A2 +4(2−a2)2(wedcA2 )2

a(6−a2)(2−a2),

then the profit of 2 is again:

πdev2 = πB,A2 − F cA2 = ((2−a2)(1−wedcA2 )−a(1−wedd

B ))2

(4−a2)2(1−a2)− F cA2.

• Otherwise, the profit of 2 is πdev2 = πB,AB2 = (2−3a+2aweddB )2

16(1−a2).

3. If weddB ∈[wircA1,

4−2a+a2

4

], then the profit of 2 is:

πdev2 = πB,AB2 = ((1−a)(2+a)−2(1−weddB ))(1−wedd

B )

2a2 .

4. Finally, if weddB > 4−2a+a2

4 , then the profit of 2 is πdev2 = πB,AB2 =(

2−a4

)2.Optimal deviation tariff and profit of B in (B,B). The profit of the deviatingproducer B in structure (B,B) is given by πB,BB (weddB , weddB ) + F dB1 + F dB2 = weddB (1 −weddB )− πdev1 − πdev2 . B sets weddB to maximize his profit. The resulting tariff is:

weddB = max{a2 ,

64−48a−32a2+24a3+8a4−2a5−a6

2(2−a2)2(16−3a2)

}.

The profit of B is then:

πB,BB =a(96+64a−148a2−45a3+59a4+7a5−7a6)

4(1+a)(2−a2)2(16−3a2)if a < 0.742486,

=(2−a)(7a+3a2−2)

16(1+a) otherwise.

It is lower than B’s collusive profit for all a < 0.520024. In other words, this deviation isonly profitable for high enough values of a.

B.2.3 Deviation in structure (B,A)

We focus on the deviation in structure (B,A), i.e. the deviation in which retailer 1 signsan exclusive dealing clause with producer B and retailer 2 signs one with producer A. The

46

menus of contracts offered to 1 and 2, respectively (wirdB1 , weddB1 , F

dB1) and (wirdB2 , w

eddB2 , F

dB2)

must therefore satisfy the participation constraints of the retailers:

πB,A1 (weddB1 , wedcA2 )− F dB1 ≥ max{πA,A1 (wedc, wedcA2 )− F c, πAB,A1 (wircA1, w

irdB1 , w

edcA2 ), 0},(16)

πB,A2 (weddB1 , wedcA2 )− F cA2 ≥ max{πB,B2 (weddB1 , w

eddB2 )− F dB2, π

B,AB2 (weddB1 , w

ircA2, w

irdB2), 0}.(17)

First, we determine the value of the right-hand side of each of these inequalities. The fixedfee F dB1 is such that constraint (22) is binding. Second, we determine the optimal values ofweddB1 and weddB2 , i.e. the values that maximize the profit of B πB,BB (weddB1 , w

eddB2 )+F dB1 +F dB2.

Participation constraint of retailer 1. Comparing πA,A1 (wedc, wedcA2 )−F c ,πAB,A1 (wircA1, wirdB1 , w

edcA2 )

and 0, it is first immediate that πAB,A1 (wircA1, wirdB1 , w

edcA2 ) ≥ 0. Indeed, if 1 refuses both ex-

clusive dealing offers, she can at worse decide not to sell anything and then earn 0.Second, as in the previous case, B wants to set wirdB1 so that 1’s profit in πAB,A1 (wircA1, w

irdB1 , w

edcA2 )

is as low as possible. Two cases arise. First, when wedcA2 ≤ wircA1 , 1 earns no profit by sellinggood A, and thus 1’s profit on product B is minimized when wirdB1 = 1.When wedcA2 > wircA1,2 has no profit and 1 can earn a positive profit either on both goods or on one of the twogoods A and B . Clearly, for any given wedcA2 and wirdB1 , as long as for a price pB > wirdB1

there is a positive demand for good B, the profit of 1 is higher by selling both A and B.Thus for any wedcA2 and wirdB1 , the total profit of 1 is minimized for wirdB1 = 1. We thus alwaysset that wirdB1 = 1.

Replacing wirdB1 by this value in the expressions of the profits given previously, we find

that πA,A1 (wedc, wedcA2 )−F c > πAB,A1 (wircA1, 1, wedcA2 ) if and only if wedcA2 >

2+a4 −

14

√4a−3a2+a3

1+a .

Finally, as we set F dB1 so that constraint (22) is binding, we find that F dB1 = πB,A1 (weddB1 , wedcA2 )−

πdev1 , where πB,A1 (weddB1 , wedcA2 ) is the variable profit of 1 in structure (B,A) and will be de-

fined later, and πdev1 is the profit that retailer 1 would earn if he was to refuse B’s exclusivedealing offer and is given by:

πdev1 = 0 if wedcA2 ≤ 2+a4 −

14

√4a− 3a2 + a3

1 + a,

= (1− wedcA2 )(wedcA2 − a

2

)− 1−a

4(1+a) if wedcA2 ∈

(2+a4 −

14

√4a− 3a2 + a3

1 + a, 2+a

4

],

=(2− a)2

16− 1− a

4(1 + a)if wedcA2 ∈

(2+a4 , 1

].

Participation constraint of retailer 2. First, as in the (B,B) case, B sets wirdB2 so as tominimize πB,AB2 (weddB1 , w

edcA2 , w

irdB2). From the same reasoning as before, we conclude again

that the optimal tariff is wirdB2 = weddB1 . Second, it is immediate that it is optimal for B to setF dB2 very high to ensure that πB,B2 (weddB1 , w

eddB2 )− F dB2 < 0. In particular, B can set F dB2 =

+∞. The choice of weddB2 is irrelevant: we can for instance set weddB2 = wirdB1 = 1. Finally,

47

the profit of 2 if she refuses A’s exclusive dealing offer is given by πB,AB2 (weddB1 , wedcA2 , w

irdB2),

that is:

πdev2 = (2−a(3−2weddB1 ))2

16(1−a2)if weddB1 ∈

[min

{0, 3a−2

2a

}, wircA1

),

= (2−2a+a2−2weddB1 )(wedd

B1 −1)

2a2 if weddB1 ∈[wircA1,

14

(4− 2a+ a2

)),

= (2−a)216 if weddB1 ∈

[4−2a+a2

4 , 1],

= 0 otherwise.

In order to ensure that 2 will indeed accept A’s exclusive dealing offer, B must set weddB1

such that πB,A2 −F edcA2 ≥ πdev2 . Depending on the value of wedcA2 , this constraint may or maynot be binding.

Optimal deviation tariff and profit of B in (B,A). The profit of B in deviation(B,A) is πB,AB (weddB1 , w

edcA2 ) + F dB1 = pB,AB DB,A

B − πdev1 , where pB,AB and DB,AB have been

previously determined.If we do not take into account the participation constraint of retailer 2 for now, the

optimal tariff is:

weddB1 = a2((1−a)(2+a)+awedcA2 )

4(2−a2)if wedcA2 < 1− a(2−a2)

4−3a2 ,

≤ 1− (2−a2)(1−wedcA2 )

a if wedcA2 ∈[1− a(2−a2)

4−3a2 , 2−a2

),

= 0 otherwise.

This solution satisfies the participation constraint of retailer 2 if the following conditionsare satisfied:

• If a ≤ 0.70656, then:

wedcA2 ∈[

4√

64−64a−144a2+160a3+156a4−196a5−75a6+112a7+12a8−24a9−a10+a12

32−32a2+2a4+a6

+16(2−a−4a2)+2a3(1+a)(18−5a)−a6(1+a)

(4−a2)(−8+6a2+a4), a(

1− a4−a2

)],

• If a ∈ [0.70656, 0.714953] then:

wedcA2 ∈[

2(2−a2)√

(2−a)(1−a2)a(16−16a−12a2+22a3−5a4−8a5+4a6)

(4−3a2)2

− (1−a)(16+8a−32a2−10a3+21a4+4a5−4a6)(4−3a2)2

, a(

1− a4−a2

)],

• or if a ∈ [23 , 0.714953], then wedcA2 > 1− a(2−a2)4−3a2 ;

48

• If a ∈ [0.714953, 0.841325], then:

wedcA2 ≥2(2−a2)

√(2−a)(1−a2)a(16−16a−12a2+22a3−5a4−8a5+4a6)

(4−3a2)2− (1−a)(16+8a−32a2−10a3+21a4+4a5−4a6)

(4−3a2)2,

• Finally, if a ∈ [0.841325, 1], then wedcA2 ∈[a2 , 1].

The resulting variable profit of B is then:

πB,AB (weddB1 , wedcA2 ) = ((1−a)(2+a)+awedc

A2 )2

8(1−a2)(2−a2)if wedcA2 < 1− a(2−a2)

4−3a2 ,

= (1−wedcA2 )(a−1+wedc

A2 )

a2 if wedcA2 ∈[1− a(2−a2)

4−3a2 , 2−a2

),

= 14 otherwise.

Otherwise, we set weddB1 so that πB,A2 (weddB1 , wedcA2 ) − F edcA2 = πdev2 . There is not always

a solution in the relevant area. The areas where there exists a solution are given in thefollowing along with the deviation tariff and the profit of producer B.

• If wedcA2 ∈[a(

1− a4−a2

),min

{a(2+a)(4−3a)

4(2−a2), 1− 2−a

2(2−a2)

}], then weddB1 = (2+a)(4−3a)

2(2−a2)−

2wedcA2a and πB,AB (weddB1 , w

edcA2 ) =

(a−2wedcA2 )(−3a+a2+a3+2wedc

A2 −a2wedc

A2 )2(−1+a)a2(1+a)(−2+a2)

.

• If a > 23 and wedcA2 ∈

[1− 2−a

2(2−a2), 1− a(2−a2)

4−3a2

], then weddB1 = 3a−2

2a and again πB,AB (weddB1 , wedcA2 ) =

(1−wedcA2 )(a−1+wedc

A2 )

a2 .

• If a ≥ 1√2, and wedcA2 > 1− (2−a)((20−26a2+9a4−a6)−(4−a2)(1−a2)

√8−4a2+a4)

4(2−a2)2, then :

weddB1 = 1− 2(2−a2)(1−wedcA2 )−(4−a2)

√(2−a)(1−a2)(a−2wedc

A2 )

2a ,

and the variable profit of B is:

πB,AB (weddB1 , wedcA2 ) =

((1−a)(2+a)+awedcA2 +2wedd

B1 )((1−a)(2+a)+awedcA2 −(2−a2)wedd

B1 )(4−a2)2(1−a2)

.

• If a > 0.467215 and:

wedcA2 ∈[1− (2−a)(8−7a2+a4)

4(2−a2)+ (1−a2)(2−a)

4

√6−a2

2−a2 , 1−(2−a)((20−26a2+9a4−a6)−(4−a2)(1−a2)

√8−4a2+a4)

4(2−a2)2

],

then B sets:

weddB1 = −4(1+wedcA2 )

a(6−a2)+ 40−24a2−2a3+3a4

2(6−a2)(2−a2)

− (4−a2)q

4−20a+5a2+36a3−16a4−18a5+9a6+2a7−a8+2(2−a2)(8−8a−10a2+7a3+2a4−a5)wedcA2 +4(2−a2)2(wedc

A2 )2

a(6−a2)(2−a2),

and again the variable profit of B is:

πB,AB (weddB1 , wedcA2 ) =

((1−a)(2+a)+awedcA2 +2wedd

B1 )((1−a)(2+a)+awedcA2 −(2−a2)wedd

B1 )(4−a2)2(1−a2)

.

49

B.2.4 Deviation in structure (B,AB)

We focus on the deviation in structure (B,AB), i.e. the deviation in which retailer 1signs an exclusive dealing clause with producer B and retailer refuses the two exclusivedealing offers. The menus of contracts offered to 1 and 2, respectively (wirdB1 , w

eddB1 , F

dB1)

and (wirdB2 , weddB2 , F

dB2) must therefore satisfy the participation constraints of the retailers:

πB,AB1 (weddB1 , wircA2, w

irdB2)− F dB1 ≥ max{πA,AB1 (wedc, wircA2, w

irdB2)− F c,

πAB,AB1 (wircA1, wirdB1 , w

ircA2, w

irdB2), 0}, (18)

πB,AB2 (weddB1 , wircA2, w

irdB2) ≥ max{πB,B2 (weddB1 , w

eddB2 )− F dB2, π

B,A2 (weddB1 , w

edcA2 )− F cA2, 0}.(19)

First, we determine the value of the right-hand side of each of these inequalities. The fixedfee F dB1 is such that constraint (22) is binding. Second, we determine the optimal values ofweddB1 and wirdB2 , i.e. the values that maximize the profit of B πB,ABB (weddB1 , w

ircA2, w

irdB2) +F dB1.

Participation constraint of retailer 1. We first determine wirdB1 , weddB2 and F dB2 for all

cases. B wants’ 1’s profit to be as low as possible when it sells both goods while 2 sellsboth goods. First, it should be noted that since wircA1 > wircA2, the profit of 1 on good A

in market structure (AB,AB) is 0. Then, 2 simply sets wirdB1 > wirdB2 so that 1 cannot sellgood B either in market structure (AB,AB). For instance, B can set wirdB1 = 1 and thusensure that πAB,AB1 (wircA1, w

irdB1 , w

ircA2, w

irdB2) = 0. Second, as wedc = wircA2, it is immediate

that πA,AB1 (wedc, wircA2, wirdB2)−F c = −F c < 0. As a result, the three constraints concerning

retailer 1 are summarized by πB,AB1 (weddB1 , wircA2, w

irdB2)−F dB1 ≥ 0. Therefore, we always have

F dB1 = πB,AB1 (weddB1 , wircA2, w

irdB2).

Participation constraint of retailer 2. Consider now the constraints of retailer 2. Inorder to prevent 2 from accepting B’s exclusive dealing offer, B has to offer 2 an exclusivedealing offer such that F dB2 ≥ πB,B2 (weddB1 , w

eddB2 ) − πB,AB2 (weddB1 , w

ircA2, w

irdB2). There is a set

of such offers. For instance, B can offer 2 a contract such that weddB2 = 1 and F dB2 = +∞.Not also that πB,AB2 (weddB1 , w

ircA2, w

irdB2) ≥ 0 by definition, as retailer 2 earns always at least 0

when she does not sign any exclusive dealing contract. The participation constraint of 2 isthus πB,AB2 (weddB1 , w

ircA2, w

irdB2) ≥ πB,A2 (weddB1 , w

edcA2 )−F cA2. The profit of retailer 2 in structure

(B,A) has been defined previously.

Optimal deviation tariff and profit of B in (B,AB). Producer B sets weddB1 andwirdB2 so as to maximize its total profit min{weddB1 , w

irdB2}D

B,ABB + F dB1, provided that the

participation constraint of retailer 2 is satisfied. The optimal tariffs are given by:

• If a < 0.841325, then weddB1 =a2(−4+2a+a2)

8(−2+a2)and wirdB2 = −4+2a+a2

4(−2+a2). Profits are πB,ABB =

(−4+2a+a2)2

32(2−3a2+a4)and πB,AB2 = − (8−8a−6a2+5a3)2

64(−2+a2)2(−1+a2);

50

• Otherwise, weddB1 = wirdB2 = 2 + a − 4+a3

2a . Profits are πB,ABB = −34 −

1a2 + 2

a andπB,AB2 = 0.

With such tariffs, the participation constraint of 2 is always strictly satisfied.

B.2.5 Deviation in structure (AB,B)

We focus on the deviation in structure (AB,B), i.e. the deviation in which retailer 1refuses the two exclusive dealing offers and retailer 2 signs an exclusive dealing clause withproducer B. The menus of contracts offered to 1 and 2, respectively (wirdB1 , w

eddB1 , F

dB1) and

(wirdB2 , weddB2 , F

dB2) must therefore satisfy the participation constraints of the retailers:

πAB,B1 (wircA1, wirdB1 , w

eddB2 ) ≥ max{πA,B1 (wedc, weddB2 )− F c, πB,B1 (weddB1 , w

eddB2 )− F dB1, 0}, (20)

πAB,B2 (wircA1, wirdB1 , w

eddB2 )− F dB2 ≥ max{πAB,A2 (wircA1, w

irdB1 , w

edcA2 )− F cA2, π

AB,AB2 (wircA1, w

irdB1 , w

ircA2, , w

irdB2), 0}.(21)

First, we determine the value of the right-hand side of each of these inequalities. The fixedfee F dB2 is such that constraint (21) is binding. Second, we determine the optimal values ofwirdB1 and weddB2 , i.e. the values that maximize the profit of B πAB,BB (wircA1, w

irdB1 , w

eddB2 ) +F dB2.

Participation constraint of retailer 1. B sets (weddB1 , FdB1) to ensure that πB,B1 (weddB1 , w

eddB2 )−

F dB1 < 0, which can be achieved by setting F dB1 = +∞. Then, we simply need to compareπA,B1 (wedc, weddB2 )− F c to 0 and find that 1’s profit when she accepts A’s exclusive dealingcontract or does not take part in the market, which we denote by πdev1 , is:

1. If weddB2 < a/2, then πdev1 = 0: accepting A’s collusive exclusive dealing offer wouldyield a profit πA,B1 − F c < 0 and 1 therefore prefers to exit the market.

2. If weddB2 ∈[a2 , w

ircA1

], then 1 accepts A’s collusive exclusive dealing offer. We then have

the interior solution where 1 sells A and 2 sells B. The profit of 1 is:

πdev1 =a(8−8a−3a2+2a3+2awedd

B2 )2(4−a2)2(1−a2)

(weddB2 − a

2

).

3. If weddB2 ∈[wircA1,

4−2a+a2

4

], then 1 accepts A’s exclusive dealing contract and sets pA

so that DB(pA, weddB2 ) = 0. This implies that pA = a−1+weddB2

a and the profit of retailer1 is then:

πdev1 = (a(2−a)−2(1−weddB2 ))(1−wedd

B2 )

2a2 − 1−a4(1+a) .

4. Finally, if weddB2 ≥4−2a+a2

4 , then 1 accepts A’s exclusive dealing clause and chargesthe monopoly price for good A: pA = 1+wc

2 . The profit of retailer 1 is then:

πdev1 =(

2−a4

)2 − 1−a4(1+a) .

51

Participation constraint of retailer 2. Note first that πAB,AB2 (wircA1, wirdB1 , w

ircA2, , w

irdB2) >

0 by definition. Then, we must compare πAB,AB2 (wircA1, wirdB1 , w

ircA2, , w

irdB2) and πAB,A2 (wircA1, w

irdB1 , w

edcA2 )−

F cA2 in order to determine 2’s participation constraint. We denote by πdev2 the profit of 2when she refuses B’s exclusive dealing offer. B then sets F dB2 = πAB,B2 (wircA1, w

irdB1 , w

eddB2 )−

πdev2 , where we have:

• If wirdB1 ∈[

3a−22a , wircA1

], then if the following three conditions are satisfied:

wirdB1 < 32 + −1+a

−2+a2 −2wedc

A2a ,

wirdB1 <(40−24a2−2a3+3a4)

2(6−a2)(2−a2)+ 4(1+wedc

A2 )

a(6−a2)

− (4−a2)q

4−16a4+36a3−20a+5a2−(9−a2)(2−a)a5+2(2−a2)(2(4−a2)(1−a2)−a(8−7a2+a4))wedcA2 +4(2−a2)2(wedc

A2 )2

a(6−a2)(2−a2),

wirdB1 >(40−24a2−2a3+3a4)

2(6−a2)(2−a2)− 4(1+wa2edc)

a(6−a2)

+(4−a2)

q4−16a4+36a3−20a+5a2−(9−a2)(2−a)a5+2(2−a2)(2(4−a2)(1−a2)−a(8−7a2+a4))wedc

A2 +4(2−a2)2(wedcA2 )2

a(6−a2)(2−a2).

Then, the binding constraint is given by:

πdev2 = πAB,A2 − F cA2 = ((2+a)(1−a)−(2−a2)wedcA2 +awird

B1 )2

(4−a2)2(1−a2)− F cA2.

• If wirdB1 <3a−22a , and if the following two conditions are satisfied:

wirdB1 > 1 + (4−a2)√

(2−a)(1−a2)(a−2wedcA2 )−2(2−a2)(1−wedc

A2 )

2a ,

wirdB1 >(2−a2)wedc

A2 −(1−a)(2+a)

a ,

then (AB,A) is the binding constraint. The profit of 2 is again:

πdev2 = ((2+a)(1−a)−(2−a2)wedcA2 +awird

B1 )2

(4−a2)2(1−a2)− F cA2.

• Otherwise, the binding constraint is (AB,AB), and we have:

πdev2 = 0 if wirdB1 <3a−22a ,

= (2−3a+2awirdB1 )2

16(1−a2)if wirdB1 ∈

[3a−22a , wircA1

],

= (a(2−a)−2(1−wirdB1 ))(1−wird

B1 )

2a2 if wirdB1 ∈(wircA1,min

{4−2a+a2

4 , 1− a2(2−a3)2(2−a2)

}],

= (wircA1 − wircA2)(1− wircA1) if wirdB1 ∈(

min{

4−2a+a2

4 , 1− a2(2−a3)2(2−a2)

}, 4−2a+a2

4

],

=(

2−a4

)2 if wirdB1 >4−2a+a2

4 and a <√

3− 1,

= (wircA1 − wircA2)(1− wircA1) if wirdB1 >4−2a+a2

4 and a ≥√

3− 1.

52

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Π2AB,AB

=0 Π2AB,AB

=0

Π2AB,AB

=

I2 - a I3 - 2 wB1irdMM2

16 I1 - a2MΠ2

AB,AB=

I2 - a I3 - 2 wB1irdMM2

16 I1 - a2MΠ2

AB,A=

IH2 + aL H1 - aL - I2 - a2M wA2edc

+ a wB1irdM2

I4 - a2M2 I1 - a2M-FA2

c

Π2AB,AB

=

I2 wB1ird

- 2 + 2 a - a2M I1 - wB1irdM

2 a2

Π2AB,AB

=HwA1irc

-wA2edcL H1-wA2

edcL

Figure 3: Areas and profit of 2 (binding constraint) when a = 3/4 (wedcA2 as horizontal axisand wirdB1 as vertical axis).

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Π2AB,AB

=

I2 - a I3 - 2 wB1irdMM2

16 I1 - a2MΠ2

AB,A=

IH2 + aL H1 - aL - I2 - a2M wA2edc

+ a wB1irdM2

I4 - a2M2 I1 - a2M-FA2

c

Π2AB,AB

=

I2 wB1ird

- 2 + 2 a - a2M I1 - wB1irdM

2 a2

Π2AB,AB

=

I1 - wA2irc M2

4=H

2 - a

4L2

Figure 4: Areas and profit of 2 (binding constraint) when a = 1/2 (wedcA2 as horizontal axisand wirdB1 as vertical axis).

53

Optimal deviation tariff and profit of B in (AB,B). Producer B sets wirdB1 andweddB2 so as to maximize its total profit min{wirdB1 , w

eddB2 }D

AB,BB + F dB2, provided that the

participation constraint of retailer 1 is satisfied. The optimal tariffs are given by:

B.2.6 Deviation in structure (AB,AB)

In this section, we show that producer B cannot earn a higher profit in a deviation whereno retailer signs any exclusive dealing clause than in a deviation where 1 signs B’s exclusivedealing clause.

The participation constraints of retailers are given by:

πAB,AB1 (wircA1, wirdB1 , w

ircA2, w

irdB2) ≥ max{πA,AB1 (wedc, wircA2, w

irdB2)− F c,

πB,AB1 (weddB1 , wircA2, w

irdB2)− F dB1, 0}, (22)

πAB,AB2 (wircA1, wirdB1 , w

ircA2, w

irdB2) ≥ max{πAB,B2 (wircA1, w

irdB1 , w

eddB2 )− F dB2,

πAB,A2 (wircA1, wirdB1 , w

edcA2 )− F cA2, 0}. (23)

Then, it is immediate that by setting F dB1 = F dB2 = +∞, B minimizes πAB,B1 (wircA1, wirdB1 , w

eddB2 )−

F dB1 and πAB,B2 (wircA1, wirdB1 , w

eddB2 ) − F dB2. Then as wedc = wircA2 = a

2 , it is immediate that 1earns no profit in market structure (A,AB). Finally, the participation constraints of retail-ers summarize to πAB,AB1 (wircA1, w

irdB1 , w

ircA2, w

irdB2) ≥ 0 and πAB,AB2 (wircA1, w

irdB1 , w

ircA2, w

irdB2) ≥

max{πAB,A2 (wircA1, wirdB1 , w

edcA2 )− F cA2, 0}.

Consider first the unconstrained optimum in structure (AB,AB). Profits of retailersand of B in this structure are given previously. Besides, in the deviation, it is optimal forB to set wirdB1 = wirdB2 so as to capture all the profit resulting from the selling of good B.In particular, this implies that pAB,ABB = wirdB1 . The profit of B in this structure is thussimply:

• If wirdB1 <−2+3a

2a , then DA = 0 and πAB,ABB = wirdB1(1− wirdB1);

• If wirdB1 ∈[−2+3a

2a , wircA1

), then the interior solution applies and πAB,ABB =

wirdB1 (4−2a−a2−4wird

B1 +2a2wirdB1 )

4(1−a2);

• Otherwise 2 charges either the monopoly price for A or a limit price such thatDB = 0,and the profit of B is thus 0.

The optimal solution is then given by:

• If a < 0.841325, then weddB1 =a2(−4+2a+a2)

8(−2+a2)and wirdB2 = −4+2a+a2

4(−2+a2). Profits are πB,ABB =

(−4+2a+a2)2

32(2−3a2+a4)and πB,AB2 = − (8−8a−6a2+5a3)2

64(−2+a2)2(−1+a2);

• Otherwise, weddB1 = wirdB2 = 2 + a − 4+a3

2a . Profits are πB,ABB = −34 −

1a2 + 2

a andπB,AB2 = 0.

54

These are the same tariff and profit as in structure (B,AB). As in addition, B must takeinto account the participation constraint of retailer 2, which may further reduce this profit,we find that the deviation profit of B in structure (AB,AB) is always at best equal to hisprofit in structure (B,AB).

55