Discussion Paper No. 1021

NAKED EXCLUSIONUNDER
EXCLUSIVE-OFFER COMPETITION

Hiroshi Kitamura
Noriaki Matsushima
Misato Sato

March 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Naked Exclusion under Exclusive-offer Competition∗
Hiroshi Kitamura†

Noriaki Matsushima‡

Misato Sato§

March 2, 2018

Abstract
This study constructs a model of anticompetitive exclusive-offer competition between two existing upstream firms. Under exclusive-offer competition, the upstream
firm’s profit depends on the rival’s exclusive offer. If the rival makes an exclusive offer
acceptable for the downstream firm, the upstream firm is excluded unless it succeeds in
exclusion. Consequently, the upper bound of exclusive offers becomes higher than when
one of the upstream firms is a potential entrant that cannot make any exclusive offer.
Thus, the exclusion of the existing upstream firm can be an equilibrium outcome even in
the case where the potential entrant is never excluded.

JEL classification codes: L12, L41, L42.
Keywords: Antitrust policy; Exclusive dealing; Exclusive-offer competition; Imperfect competition.
∗

We thank Takanori Adachi, Masaki Aoyagi, Jay Pil Choi, Susumu Imai, Akifumi Ishihara, Atsushi Kajii, Simon Loertscher, Toshihiro Matsumura, Stuart McDonald, Masao Ogaki, Johannes Paha, Jérôme Pouyet,
Takeharu Sogo, and Zhiyong Yao as well as the conference participants at the 2nd Asia-Pacific Industrial Organisation Conference (University of Auckland), 44th European Association for Research in Industrial Economics
Annual Conference (Maastricht University), 15th Annual International Industrial Organization Conference (Renaissance Boston Waterfront Hotel), and XXXII Jornadas de Economı́a Industrial (Universidad de Navarra)and seminar participants at Keio University, Kyoto Sangyo University, Osaka University, Sapporo Gakuin University, and Tohoku University. We gratefully acknowledge the financial support from JSPS KAKENHI grant
numbers JP15H03349, JP15H05728, JP15K17060, JP17H00984, JP17J03400, and JP17K13729. The usual
disclaimer applies.
†
Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, Kyoto 603-8555,
Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp‡
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. E-mail: nmatsush@iser.osaka-u.ac.jp§
Postdoctoral Research Fellow of the Japan Society for the Promotion of Science (JSPS), Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, Kyoto 603-8555, Japan. E-mail:smisato@gwmail.gwu.edu

1 Introduction
Exclusive contracts have been a controversial issue in competition policy because they are
seemingly anticompetitive due to the exclusion of rival firms. However, by taking into account all members’ participation constraints for such exclusive dealing in the contract party
under a one-seller-one-buyer framework, Posner (1976) and Bork (1978) show that suchcontracts do not exist and conclude that rational economic agents do not engage in anticompetitive exclusive dealing.1 In rebuttal to the Chicago School argument, post-Chicago
economists indicate specific circumstances under which anticompetitive exclusive dealing
occurs (Aghion and Bolton, 1987; Rasmusen, Ramseyer, and Wiley, 1991; Segal and Whinston, 2000; Simpson and Wickelgren, 2007; Abito and Wright, 2008).
The common feature of these studies by the post-Chicago economists is that the entrant
is a potential entrant, which cannot make an exclusive offer. However, in real business situations, exclusive contracts can be signed to deter existing firms. For example, in the Intel
antitrust case, AMD and Transmeta, which were already in the market, were excluded.2 Furthermore, in the case of Virgin Atlantic Airways vs. British Airways, the former charged
that it was excluded through British Airways’ exclusive dealing with corporate customers
and travel agents.3 In these cases, the excluded firms might also be able to make exclusive
offers.4 More importantly, in the “cola wars” between PepsiCo and Coca-Cola, both firms ac1

For the analysis of the impact of this argument on antitrust policies, see Motta (2004), Whinston (2006),
and Fumagalli, Motta, and Calcagno (2018).
2Intel was accused of awarding rebates and making various other payments to major original equipment
manufacturers (e.g., Dell and HP). See Gans (2013) for an excellent case study of the Intel case.
3

Virgin Atlantic Airways charged that British Airways granted rebates to travel agents or corporate customers only if they purchase all or a certain percentage of their travel requirements from British Airways. See“Virgin Atlantic Airways v. British Airways, 872 F. Supp. 52 (S.D.N.Y. 1994)” JUSTIA US LAW, December
30, 1994 (link).
4

The other example of excluding existing rivals is found in aviation industry. The Boeing Company andAirbus sometimes award exclusivity to one or two jet engine makers over the others. For example, The Boeing
Company selects General Electric as the exclusive engine supplier for Boeing 777X in 1999. Airbus grants an
exclusive contract to Rolls-Royce for the A330neo in 2014. See “GE Unit Lands Exclusive Boeing Pact For
Developing Commercial Jet Engine” The Wall Street Journal, July 8, 1999 (link) and “Airbus selects Rolls-

1tually make exclusive offers and retailers, restaurants, cinemas, and universities choose one
exclusive offer over the other to obtain a large monetary transfer from either supplier.5 Thus,
this study aims to ascertain how exclusive-offer competition affects anticompetitive exclusive
dealing.
In this study, we construct a model of anticompetitive exclusive contracts that deter anexisting upstream firm. Although most previous studies assume that upstream firms produce
perfectly homogeneous products, we assume that upstream firms produce horizontally differentiated products so that they earn positive profits under upstream duopoly. This modelingstrategy is close to that of Wright (2008). Although Wright (2008) assumes that multiple
downstream firms play an essential role in exclusion, we assume the presence of only a single downstream firm to clarify the role of exclusive-offer competition. Following previous
studies, an exclusive offer involves a fixed compensation. After the downstream firm’s decision on whether to accept exclusive offers, the industry profit allocation is determined bynegotiations between the downstream firm and each existing upstream firm through generalized Nash bargaining. In this setting, we compare the case where one of the upstream firms
is a potential entrant that cannot make exclusive offers (benchmark analysis) with the casewhere both upstream firms are existing firms (main analysis).
By introducing non-linear wholesale pricing and a general demand function, we first showthat exclusion never occurs and that the upstream market always becomes a duopoly in thebenchmark analysis; that is, the Chicago School argument can be applied. When the exclusive offer is rejected, upstream competition induces the downstream firm to earn higher
profits. By considering the industry profit allocation under upstream duopoly, exclusive dealing is not profitable for the upstream incumbent because the exclusive offer acceptable for thedownstream firm is costly for any bargaining power allocation.
We then show that in the main analysis, exclusion can be an equilibrium outcome. Under
Royce Trent 7000 as exclusive engine for the A330neo” Rolls-Royce, July 14, 2014 (link).
5

See, for example, “‘Cola Wars’ Foaming On College Campuses” Chicago Tribune, November 6, 1994
(link). The cola wars at restaurants, cinemas, and universities are discussed in Section 4.2.

2

exclusive-offer competition, an upstream firm’s profit depends on the rival’s offer. Whenthe rival upstream firm makes an exclusive offer acceptable for the downstream firm, theupstream firm is excluded unless it succeeds in exclusion. That is, exclusive-offer competition
prevents the upstream firm from enjoying positive profits under upstream duopoly. As aresult, compared with the benchmark analysis, the upstream firm has a strong incentive for
exclusive dealing; the upper bound of the exclusive offer increases. Therefore, there exists a
possibility that each upstream firm can profitably make a higher exclusive offer acceptable forthe downstream firm. We find that this exclusion mechanism works if the downstream firm
has relatively weak bargaining power because it earns lower profits for which the upstream
firm can compensate easily.
We also check the robustness of the above exclusion outcome by extending the model.
First, our exclusion logic can be applied in the case of linear wholesale pricing. The exclusion equilibrium exists when upstream firms are sufficiently differentiated. Second, inAppendix D, we show that the exclusion equilibrium exists for the case where downstream
firms competing in quantity make exclusive supply offers to a single upstream firm. Note thatin both cases, exclusion cannot be an equilibrium outcome in the absence of exclusive-offer
competition. Therefore, the exclusion outcome identified in this study can be widely applied
to diverse real-world vertical relationships.
This study is related to the literature on anticompetitive exclusive contracts that deter
the socially efficient entry of a potential entrant. By extending the Chicago School argument’s single-buyer model to a multiple-buyer model, mainstream studies introduce scaleeconomies, wherein the entrant needs a certain number of buyers to cover its fixed costs(Rasmusen, Ramseyer, and Wiley, 1991; Segal and Whinston, 2000), and competition between buyers (Simpson and Wickelgren, 2007; Abito and Wright, 2008).6 In these studies,
negative externalities exist; signing exclusive contracts reduces the possibility of entry under6

In the literature on exclusion with downstream competition, Fumagalli and Motta (2006) show that theexistence of participation fees to remain active in the downstream market plays a crucial role in exclusion if
buyers are undifferentiated Bertrand competitors. See also Wright (2009), who corrects the result of Fumagalli
and Motta (2006) in the case of two-part tariffs.

3scale economies and upstream entry reduces industry profits in the presence of downstream
competition.7 By contrast, this study shows that anticompetitive exclusive contracts can besigned even under a single-buyer model because of a negative externality that the high exclusive offer by an upstream firm reduces the rival upstream firm’s profits for the case of failingexclusive dealing.
In the framework of a single downstream firm, this study is related to the literature on
anticompetitive exclusive contracts that deter the potential entrant focusing on the nature of
upstream competition.8 The studies in this strand of the literature point out that the intensityof upstream competition plays a crucial role in the Chicago School critique. They show thatthe exclusion result is obtained in the cases where the incumbent sets liquidated damages forthe case of entry (Aghion and Bolton, 1987), where the entrant is capacity constrained (Yong,1996), where upstream firms compete à la Cournot (Farrell, 2005), and where upstream firms
can merge (Fumagalli, Motta, and Persson, 2009).9 Our study complements these works inthe sense that we show an alternative route through which the lower intensity of upstream
competition due to product differentiation leads to anticompetitive exclusive dealing in thepresence of exclusive-offer competition.
Few studies address exclusive dealing that aims to exclude existing firms.10 By extending the model of exclusion with downstream competition, DeGraba (2013) and Shen (2014)7

For extended models of exclusion with downstream competition, see Wright (2008), Argenton (2010),
and Kitamura (2010). Whereas these studies all show that the resulting exclusive contracts are anticompetitive,
Gratz and Reisinger (2013) show potentially procompetitive effects if downstream firms compete imperfectly
and contract breaches are possible.
8

For another mechanism of anticompetitive exclusive dealing, see Fumagalli, Motta, and Rønde (2012),
who focus on the incumbent’s relationship-specific investments. See also Kitamura, Matsushima, and Sato(2018), who focus on the existence of a complementary input supplier with market power.
9

See also Kitamura, Matsushima, and Sato (2017a), who show that anticompetitive exclusive dealing canoccur if the downstream buyer bargains with suppliers sequentially.
10

Choi and Stefanadis (2017) explore the exclusive-offer competition between upstream firms before they
enter the market. By extending the model of exclusion with scale economies, they point out that exclusion
becomes a unique coalition-proof subgame-perfect equilibrium outcome when a derivative innovator can enter
the market only if the incumbent innovator enters the market.

4explore exclusive-offer competition. In their studies, exclusion arises because of downstream
competition. By contrast, this study explores anticompetitive exclusive dealing in the absenceof downstream competition and shows that exclusive-offer competition leads to anticompetitive exclusive dealing.
In terms of exclusive-offer competition, this study is also close to the benchmark modelof Bernheim and Whinston (1998, Sections II and III).11 In their study, the exclusive offer
involves wholesale prices for the case where the downstream firm rejects the exclusive offer, which is suitable for exploring short-term exclusive contracts.12 In such offers, upstream
firms can commit not to sell their products to the downstream firm by setting considerablyhigh wholesale prices for the case of rejection, which removes the downstream firm’s opportunity to deal with both upstream firms. By contrast, following the standard naked exclusion
literature, we assume that each upstream firm cannot commit to wholesale prices when its
exclusive offer is rejected, which is suitable for long-term exclusive contracts.13 In reality, as
shown by one of the cola wars’ examples, PepsiCo outbid Coca-Cola for a $14 million, 12year monopoly contract at Pennsylvania State University in 1992, which implies that it is along-term exclusive contract. In this setting, when the downstream firm rejects both exclusive
offers, it can deal with both upstream firms and earn considerably higher profits, under which
the Chicago School model does not lead to anticompetitive exclusive dealing. Therefore, thisstudy is suitable for long-term exclusive dealing and it clarifies the role of exclusive-offer
competition in the literature on naked exclusion.
The remainder of this paper is organized as follows. Section 2 constructs the model.
Section 3 analyzes the existence of exclusion outcomes under two-part tariffs. Section 4
provides a discussion and Section 5 offers concluding remarks. Appendix A provides theproofs of the results. Appendix B provides the parametric results when manufacturers operate
11

See also Calzolari and Denicolò (2013, 2015), who explore upstream firms making exclusive offers in thepresence of adverse selection, while we assume complete information.
12

See the discussion by Whinston (2006, p. 166).

13

In addition to the commitment problem, we also consider cases in which each upstream firm does notalways have full bargaining power over the downstream firm when it determines its wholesale price.

5

at the same marginal costs under linear demand. Appendix C introduces the parametric results
when manufacturers operate at different marginal costs under linear demand. Appendix D
explores the case of exclusive supply agreements.

2 ModelThis section develops the basic setting of the model. The upstream market consists of two
manufacturers U1 and U2 . Each manufacturer operates at the same marginal cost c ≥ 0
and produces a final product, which is differentiated. We explore the case of the asymmetric cost function in Section 3.4. The downstream market is composed of a downstream
retailer D, which sells the manufacturers’ products. This modeling strategy clarifies the roleof exclusive-offer competition because we can easily compare the result of this study withthat of the Chicago School argument; exclusion never occurs in the benchmark analysis.14
To simplify the analysis, we assume that D incurs no operating cost aside from paying forthe product of Ui . Therefore, given wholesale price wi , the resale cost of D when it sells qi
Pamount of Ui ’s final product to final consumers is given by C D (q1 , q2) = i wi qi .

The demand system has the following properties. Given the pair of manufacturers’ prod-

uct prices (p1 , p2 ), demand for U1 ’s product is denoted by Q(p1 , p2 ). By assuming symmetricdemand, demand for U2 ’s product is denoted by Q(p2 , p1 ). When the prices of these manufacturers’ products are sufficiently close, both obtain positive demand. However, when theseprices differ sufficiently, the higher priced manufacturer loses demand, while the lower priced
manufacturer obtains all demand. The degree of product substitution between manufacturers’
products is represented by γ ∈ (0, 1). Manufacturers’ products become homogeneous as thevalue of γ increases. For γ = 0, manufacturers produce independent goods. Alternatively,for γ = 1, manufacturers produce perfectly substitutes. In addition, when U j is excluded,
demand for Ui ’s product does not depend on γ, where i, j ∈ {1, 2} and i , j. We denotedemand for Ui ’s product in the monopoly case by Q(pi ) ≡ Q(pi , ∞).
14

Although a buyer is the final consumer in the Chicago School model, the results in all the propositions inthis paper do not change if we assume that the buyer is a downstream monopolist.

6For the sake of the analysis under generalized Nash bargaining, we assume that industry profits under exclusive dealing (pi − c)Q(pi ) and those under non-exclusion cases(pi − c)Q(pi , p j ) + (p j − c)Q(p j , pi ) are globally and strictly concave and satisfy the secondorder conditions. We define pm and pd as follows:pm ≡ arg max(pi − c)Q(pi ),pi(pd , pd ) ≡ arg max(pi − c)Q(pi , p j ) + (p j − c)Q(p j , pi ).
pi ,p jWe define Πm and Πd as the net profit of each vertical chain under upstream monopoly andunder upstream duopoly:Πm ≡ (pm − c)Q(pm ), Πd ≡ (pd − c)Q(pd , pd ).
We assume the following relationship:
Assumption 1. For all 0 < γ < 1,
2Πd > Πm > Πd ,(1)

where ∂Πm /∂γ = 0, ∂Πd /∂γ < 0, Πd → Πm as γ → 0, and 2Πd → Πm as γ → 1.
The first inequality of Condition (1) is the key property in this study, which implies thatan increase in the number of product varieties generates an additional industry value except
when γ = 1. In addition, the second inequality in Condition (1) implies that an increase in
the number of product varieties reduces the net profit per vertical chain except when γ = 0.
Note that the properties introduced above hold under standard linear demand with a representative consumer, which is introduced when we explore the case of linear wholesale pricing in
Section 4.
The model contains three stages. In Stage 1, U1 and U2 make an exclusive offer to D
with fixed compensation xi ≥ 0. Following the standard literature on naked exclusion, we

7assume that each exclusive offer does not contain the term of wholesale prices.15 D can rejectboth offers or it can accept one of the offers. Let ω ∈ {R, E1, E2} be D’s decision in Stage
1. D immediately receives xi if it accepts Ui ’s exclusive offer. If D is indifferent between
two exclusive offers and acceptance leads to higher profits, it accepts one of the offers withprobability 1/2. In Stage 2, active manufacturers offer a two-part tariff contract. We extend
the model to the case of linear wholesale pricing in Section 4. In Stage 3, D orders the final
product and sells it to consumers at pωi . Ui ’s profit is denoted by πωUi . Likewise, D’s profit isdenoted by πωD .

3 Two-part tariffs
This section analyzes the existence of anticompetitive exclusive contracts under two-parttariffs, which consist of a linear wholesale price and an upfront fixed fee; the two-part tariff
offered by Ui when D’s decision is ω ∈ {R, E1, E2} is denoted by (wωi , F iω ), where i ∈ {1, 2}.
We assume that the industry profit allocation after Stage 1 is given by the Nash bargaining
solution and that the net joint surplus is divided between D and each manufacturer in the
proportion β to 1 − β, where β ∈ (0, 1) represents D’s bargaining power.
The rest of this section is organized as follows. Section 3.1 derives the equilibrium outcomes after the game in Stage 1 by using backward induction. Section 3.2 examines thegame in Stage 1 by introducing the benchmark analysis in which one of the manufacturers
is a potential entrant as in the Chicago School model. Section 3.3 then explores the casewhere both manufacturers make exclusive offers. Section 3.4 finally examines the effect of
cost asymmetry on the existence of an exclusion equilibrium.
15

Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000) point out that price commitments
are unlikely if the product’s nature is not precisely described in advance. In the naked exclusion literature, it is
known that if the incumbent can commit to wholesale prices, then the possibility of anticompetitive exclusive
dealing is enhanced. See Yong (1999) and Appendix B of Fumagalli and Motta (2006).

8

3.1 Equilibrium outcomes after Stage 1
We first consider the case in which Ui ’s exclusive offer is accepted in Stage 1. Note that fornotational simplicity, we do not discuss explicitly how the wholesale price is determined ineach instance of bargaining because we can easily show that marginal cost pricing is achievedin all cases by using the envelope theorem. In Stage 2, D negotiates with Ui and makes a twopart tariff contract, (c, F iEi ). The bargaining problem between D and Ui is described by the
payoff pairs (Πm − F iEi , F iEi ) and the disagreement point (0, 0). The solution is given by
F iEi = arg max β log[Πm − F i ] + (1 − β) log F i .
Fi

The maximization problem leads to
F iEi = (1 − β)Πm .
The firms’ equilibrium profits, excluding the fixed compensation xi , areEi
πUi= (1 − β)Πm , πUEij = 0, πEi
D = βΠm .

(2)

Depending on the bargaining power β, Ui and D split the monopoly profit, Πm .
We next consider the case in which D rejects both exclusive offers in Stage 1. In this case,D sells both manufacturers’ products. We assume that the bargaining in Stage 2 takes theform of simultaneous bilateral negotiation; that is, when negotiating with two manufacturers,
D simultaneously and separately negotiates with each of them. D and Ui then make a twopart tariff contract, (c, F iR ). The outcome of each negotiation is given by the Nash bargaining
solution based on the belief that the outcome of the bargaining with the other party is determined in the same way. The bargaining problem between D and Ui is described by the payoff
pairs (2Πd − F Rj − F iR , F iR ) and the disagreement point (z j , 0), where z j ≡ Πm − F Rj is D’s profit
when it sells only U j ’s product under two-part tariff contract (c, F Rj ). The solution is given by
F iR = arg max β log[2Πd − F j − F i − z j ] + (1 − β) log F i .
Fi

The maximization problem leads to
F iR = (1 − β)(2Πd − Πm ),9for each i ∈ {1, 2}. The resulting profits of the firms are given as
πRUi = (1 − β)(2Πd − Πm ), πRD = 2((1 − β)(Πm − Πd ) + βΠd ).

(3)

Ui obtains its additional contribution weighted by its bargaining power 1 − β, and D earns theremaining industry profit under upstream duopoly after subtracting the payments for U1 andU2 (that is, 2Πd − πRU1 − πRU2 ).

3.2 Benchmark analysis
Assume that U j is a potential entrant and only Ui can make an exclusive offer as in the
Chicago School model. In this subsection, we modify the timing of Stage 1 as follows. In
Stage 1.1, Ui makes an exclusive offer xi and D decides whether to accept the offer. Afterobserving D’s decision, U j decides whether to enter the upstream market in Stage 1.2. The
fixed cost of entry is sufficiently small so that U j earns positive profits.
To start the analysis, we derive the essential conditions for an exclusive contract when
only one manufacturer makes exclusive offers. For an exclusion equilibrium to exist, theequilibrium transfer x∗i must satisfy the following two conditions.
First, the exclusive contract must satisfy individual rationality for D; that is, the amountof compensation x∗i induces D to accept the exclusive offer:
∗
R
∗
R
Ei
πEi
D + xi ≥ πD or xi ≥ ∆πD ≡ πD − πD ,(4)

where ∆πD is the absolute value of D’s profit loss under exclusive dealing.
Second, it must satisfy individual rationality for Ui ; that is, Ui earns higher profits under
exclusive dealing:
Ei
Ei
πUi
− x∗i ≥ πRUi or x∗i ≤ ∆πU ≡ πUi− πRUi ,(5)

E1
where ∆πU is Ui ’s profit increase under exclusive dealing. Note that ∆πU = πU1− πRU1 =E2
πU2− πRU2 .

From the above conditions, it is evident that an exclusion equilibrium exists if and only if
inequalities (4) and (5) simultaneously hold. This is equivalent to the following condition:
Ei
RR∆πU ≥ ∆πD or πUi+ πEi
D ≥ πUi + πD .

10

(6)

Condition (6) implies that anticompetitive exclusive contracts are attained if exclusive contracts increase the joint profits of Ui and D or equivalently if Ui ’s profit increase is higher
than D’s profit loss under exclusive dealing.
By using the subgame outcomes derived in the previous subsection, we now consider thegame in Stage 1. By substituting Equations (2) and (3), we find that under Condition (1)∆πU − ∆πD = −β(2Πd − Πm ) < 0,which implies that exclusion never occurs.
Proposition 1. Suppose both manufacturers adopt two-part tariffs. If U j is a potential entrant
and only Ui can make an exclusive offer, Ui cannot exclude U j through exclusive contracts.
Proposition 1 confirms the robustness of the Chicago School argument when we extend
its model to the case where manufacturers produce differentiated products and adopt two-parttariffs. Under the non-linear pricing scheme, following the bargaining procedure, the firms
split the total industry profit. Except for the cases of β = 0 and γ = 1, entry by U j generates
some additional profits for D, those of which are sufficient to eliminate the incentives of D
and Ui to reach exclusion.16 Therefore, exclusion does not occur when only one manufacturer
can make the exclusive offer.

3.3 When exclusive-offer competition exists
In contrast to the previous subsection, we now assume that both manufacturers are existing
firms and can make exclusive offers. Compared with the case where exclusive-offer competition does not exist, the difference arises in the upper bound of Ui ’s exclusive offer xmax
, whichidepends on U j ’s offer, where
xmax
≡
i

(

Ei
πUi
∆πU

if x j ≥ ∆πD ,if x j < ∆πD .

16

(7)

When β = 0, U j obtains all its additional contribution, implying that entry leaves nothing to D. Whenγ = 1, U j does not add any contribution to the industry because of the perfect substitutability of products.

11

Ei
Note that πUi> ∆πU and that xmax
= ∆πU in the benchmark case. The feature of xmax
is exiiplained by D’s decision on whether to accept the exclusive offer by U j . Figure 1 summarizes
D’s decision in response to both manufacturers’ offers in Stage 1. When both exclusive offers
are lower than ∆πD , D rejects both. By contrast, when at least one of the exclusive offers ishigher than or equal to ∆πD , D accepts the better offer; more concretely, at least one of xi andx j satisfies Condition (4) in the shadowed area of Figure 1. Hence, D’s behaviors affect bothmanufacturers’ exclusive offers as follows. When U j offers x j < ∆πD , Ui can be active andearn πRUi (> 0) even when it fails to exclude U j . By comparing this profit with its net profitEiunder exclusion πUi− xi , Ui does not offer xi (> ∆πU ) as in the benchmark case. On the conEj
trary, when U j ’s exclusive offer satisfies x j ≥ ∆πD , Ui is out of the market and earns πUi=0

if it fails to exclude U j . In this case, the exclusion of U j is profitable for Ui if πUEii − xi ≥ 0.
Therefore, Ui makes a higher exclusive offer if x j ≥ ∆πD .
[Figure 1 about here]
Figures 2 and 3 summarize the set of each manufacturer’s feasible offer (x1 , x2 ) that satisfies xi ∈ [0, xmax ] for each i ∈ {1, 2}. Each manufacturer’s offer is feasible in the shadowed
area of Figures 2 and 3, which can be a candidate for the set of exclusion offers in the exclu∗∗
sion equilibrium (x∗∗1 , x2 ); in other words, other areas cannot be the exclusion equilibrium.
Eiand ∆πD , we have two cases. First,
Depending on the magnitude relationship between πUi
Ei< ∆πD , summarized in Figure 2, each manufacturer’s exclusive offer is feasible in only
if πUione region because D’s rejection profit is considerably high and each manufacturer cannotEi≥ ∆πD , sumcompensate D profitably when its rival makes the higher offer. Second, if πUi

marized in Figure 3, each manufacturer’s exclusive offer is feasible in two regions. Because
D’s rejection profit is not too high in this case, Ui can profitably offer xi (≥ ∆πD ) when U jmakes the high offer x j ≥ (∆πD ).
[Figures 2 and 3 about here]
To explore the existence of an exclusion equilibrium, we now combine the results in
Figures 1, 2, and 3. Figures 4 and 5 combine these figures and D’s decision in the shadowed
12

Ei
areas in Figures 2 and 3. Figure 4 implies that exclusion never occurs if πUi< ∆πD . In thiscase, there exist only non-exclusion equilibria in which each manufacturer offers xi ∈ [0, ∆πD )and D rejects both offers. By contrast, Figure 5 shows that an exclusion equilibrium exists if
Ei
πUi≥ ∆πD . The candidate for the equilibrium offer is the area in which (x1 , x2 ) ∈ [∆πD , πUEii ]2holds. Obviously, xi > x j ≥ ∆πD and xi = x j < πUEii cannot be an equilibrium because at least
one of the manufacturers has an incentive to deviate. There exists the exclusion equilibrium
Ei
in which each manufacturer offers x∗∗i = πUi and D accepts one of the offers. Note that evenEiwhen πUi≥ ∆πD , there also exists the non-exclusion equilibria in which each manufacturer

offers xi ∈ [0, ∆πD ) and D rejects both offers.
[Figures 4 and 5 about here]
We finally consider the existence of an exclusion equilibrium. From the above discussion,Ei
we need to check whether πUi≥ ∆πD holds. By substituting Equations (2) and (3), we obtainEi
πUi− ∆πD = (1 − 2β)(2Πd − Πm ) ≥ 0,

if and only if β ∈ (0, 1/2], which implies that an exclusion equilibrium exists for the weakbargaining power of D.
Proposition 2. Suppose that both manufacturers make exclusive offers in Stage 1 and adopt
two-part tariffs in Stage 2. If D has strong bargaining power (β > 1/2), exclusion cannot be
an equilibrium outcome. By contrast, if D has weak bargaining power (β ≤ 1/2), there exist
both an exclusion equilibrium and non-exclusion equilibria.
Proposition 2 shows that under exclusive-offer competition, an exclusion equilibrium exists depending on the bargaining power of D over manufacturers. For the weak bargaining
power of D, D earns a lower profit when it rejects both exclusive offers in Stage 1. Therefore,
each manufacturer can compensate D profitably. Moreover, the existence of an exclusion
equilibrium does not depend on the degree of product substitution γ under non-linear wholesale pricing. Note that the result here highly depends on the assumption that manufacturers’
13costs are symmetric. In the following subsection, we explore the case of an asymmetric cost
structure and show that the exclusion equilibrium is more likely to be observed for lower γ.
Note that Proposition 2 shows that exclusion is not a unique equilibrium outcome. By
comparing the two types of equilibria, the manufacturers strictly prefer the non-exclusion
equilibria to the exclusion equilibrium. Seemingly, the cola wars capture the exclusion equilibrium because both Coca-Cola and PepsiCo pay a large monetary transfer. The likelihoodof exclusion here may depend on the market history. If a president of one upstream firm hasa managerial incentive to maximize market share rather than profit, exclusion is more likely
to occur. Once exclusion occurs, it is more likely to be observed continuously—even whenthe managerial incentive changes. In addition, D has a strong incentive to yield the exclusion
outcome. Because Condition (4) holds with strict inequality under the exclusion equilibrium,
D prefers the exclusion equilibrium to the non-exclusion equilibrium. Hence, D may try to
do something to yield the exclusion outcome.

3.4 Cost asymmetry
This subsection briefly discusses the effect of cost asymmetry on the existence of an exclusion
equilibrium. Thus far, we have assumed that each manufacturer operates at the same marginal
cost c ≥ 0. We now extend the model to the case in which manufacturers operate at different
marginal costs. Without loss of generality, we assume that the marginal cost of U1 is lower
than that of U2 , namely 0 ≤ c1 < c2 . We define pmi and pdi as follows:pmi ≡ arg max (pi − ci )Q(pi ),pi(pdi , pd j ) ≡ arg max (pi − ci )Q(pi , p j ) + (p j − c j )Q(p j , pi ).
pi ,p jWe define Πmi and Πdi as the net profit of Ui ’s vertical chain under upstream monopoly andupstream duopoly:Πmi ≡ (pmi − ci )Q(pmi ), Πdi ≡ (pdi − ci )Q(pdi , pd j ).

14

Since our focus is the existence of anticompetitive exclusive dealing, we only consider thecase where the upstream market becomes a duopoly in the absence of exclusive dealing,
namely Πdi > 0 for each i ∈ {1, 2}. For the sake of notational convenience, we define∆Πi ≡ Πdi + Πd j − Πm j ,which can be interpreted as the level of increment in the industry profit when Ui ’s product isalso launched in the upstream market monopolized by U j . As in Assumption 1, we assume
the following relationships:
Assumption 2. Πmi and Πdi have the following properties:
1. Trading with U1 leads to higher profits than that with U2 ;
Πm1 > Πm2 , Πd1 > Πd2 .

(8)

2. For each i ∈ {1, 2} and γ ∈ (0, 1),Πd1 + Πd2 > Πmi > Πdi ,(9)where ∂Πmi /∂γ = 0, ∂Πdi /∂γ < 0, Πdi → Πmi as γ → 0, and Πd2 → 0 and Πd1 → Πm1
for sufficiently high γ.
3. ∆Π1 is decreasing in c1 but ∆Π2 is increasing in c1 :
∂∆Π2
∂∆Π1
< 0,
> 0.
∂c1
∂c1

(10)

Note that Conditions (8) and (9) imply that∆Π1 > ∆Π2 > 0.

(11)

By using the above definitions, we can derive the equilibrium profits under asymmetriccosts. As in Section 3.1, the negotiation between Di and Ui leads to marginal cost pricing in
15

all cases. Under generalized Nash bargaining, the firms’ equilibrium profits under exclusive
dealing, excluding the fixed compensation xi , areEi
πUi= (1 − β)Πmi , πUEij = 0, πEi
D = βΠmi .

(12)

By contrast, the firms’ equilibrium profits under non-exclusive dealing are
πRUi = (1 − β)∆Πi , πRD = (1 − β)(Πmi − Πdi + Πm j − Πd j ) + β(Πdi + Πd j ).

(13)

From Condition (10), we have ∂πRU1 /∂c1 < 0 but ∂πRU2 /∂c1 > 0, which is observed in thelinear demand model.17
We now consider the existence of an exclusion equilibrium. We first explore the case inwhich only Ui can make an exclusive offer. By substituting Equations (12) and (13), we findthat under Condition (9)
Ei
R
R
πUi
+ πEi
D − (πUi + πD ) = −β∆Π j < 0,which implies that exclusion never occurs.
Proposition 3. Suppose both manufacturers adopt two-part tariffs. If U j is a potential entrant
and only Ui can make an exclusive offer, Ui cannot exclude U j through exclusive contracts
even under asymmetric costs.
Proposition 3 implies that U1 cannot deter the entry of U2 as long as entry increases theindustry profit. Therefore, the result confirms the robustness of the Chicago School argumentin the case where the incumbent manufacturer cannot deter the entry of a potential entrant
manufacturer, which is even less efficient.
We next consider the case in which both manufacturers make exclusive offers. Note thatEiR
the exclusion equilibrium exists if and only if πUi+ πEi
D ≥ πD holds for each i ∈ {1, 2}. By
EiR
substituting Equations (12) and (13), we have πUi+ πEi
D − πD ≥ 0 if and only ifβ ≤ βi ≡17∆Πi∆Πi + ∆Π j

See Appendix C, which introduces the results under the linear demand model.

16

(14)

for each i ∈ {1, 2}. From Conditions (11) and (14), βi have the following relationships:
0 < β2 <1< β1 < 1,2(15)

where β1 → 1 and β2 → 0 as ∆Π2 → 0. Condition (15) shows that β1 > β2 always holds;
thus, the exclusion equilibrium exists if and only if β ≤ β2 . Because β2 < 1/2 always holds,
cost asymmetry reduces the possibility of the exclusion equilibrium. More precisely, bydifferentiating βi with respect to c1 , we have
!
∂∆Π j
∂βi
1
∂∆Πi
=
∆Π j −∆Πi .
∂c1 (∆Πi + ∆Π j )2 ∂c1∂c1
Under Condition (10), we have ∂β1 /∂c1 < 0 and ∂β2 /∂c1 > 0. Therefore, as U1 becomes
more efficient, β2 decreases; in other words, the exclusion equilibrium is less likely to exist.
The following proposition summarizes the results provided above.
Proposition 4. Suppose that both manufacturers make exclusive offers in Stage 1 and adopt
two-part tariffs in Stage 2. As the degree of cost asymmetry increases, exclusion is less likely
to be an equilibrium outcome.
The result in Proposition 4 implies that the exclusion mechanism in this study is more
likely to work well when each manufacturer has a similar cost structure. When U1 ’s efficiency
increases, the industry profit under duopoly Πd1 + Πd2 increases, which allows D to earnhigher profits under upstream duopoly because ∂πRD /∂c1 = −(1 − β)∂∆Π2 /∂c1 + β(∂Πd1 /∂c1 +∂Πd2 /∂c2 ) < 0. By contrast, the increase in U1 ’s efficiency does not affect U2 ’s monopoly
profit under exclusive dealing; hence, U2 has difficulty in compensating D. Therefore, thepossibility of exclusion becomes lower under cost asymmetry.
Finally, we explore the relationship between the existence of an exclusion equilibrium
and the degree of product substitution γ. By differentiating βi with respect to γ, we have!
Πm j − Πmi ∂Πdi ∂Πd j∂βi> 0 if and only if Πmi > Πm j .
=+∂γ(∆Πi + ∆Π j )2 ∂γ∂γ
From Condition (8), we have ∂β1 /∂γ > 0 and ∂β2 /∂γ < 0, which lead to the following
proposition.
17

Proposition 5. Suppose that both manufacturers make exclusive offers in Stage 1 and adopt
two-part tariffs in Stage 2. Under cost asymmetry, the exclusion equilibrium is more likely tobe observed for the cases in which the manufacturers produce highly differentiated products.
The result in Proposition 5 implies that under cost asymmetry, the existence of an exclusion equilibrium is determined by the degree of product substitution γ; in other words, theresult in Proposition 2 highly depends on the symmetric cost structure. The result here is explained by the property of bargaining when D rejects both exclusive offers. By differentiating
πRD with respect to γ, we have!
∂πRD∂Πd1 ∂Πd2> 0 for β < 1/2,
= (2β − 1)
+
∂γ
∂γ
∂γ
which implies that as the manufacturers produce less differentiated products, D earns highprofits under upstream duopoly for the weak bargaining power of D. The degree of productsubstitution affects D’s profit under upstream duopoly in two ways. First, as γ increases, theindustry profit Πdi + Πd j directly decreases, which has the negative effect of decreasing πRD .
Second, because Ui ’s additional contribution decreases, it earns lower profits πRUi = F iR , whichindirectly increases D’s outside option profit under the bargaining with U j , z j = Πmi −F iR . This
indirect effect increases πRD and becomes dominant for lower β because the strong bargaining
power of Ui decreases F iR largely. Under this relationship, as the manufacturers produce
more differentiated products when the downstream firm has weak bargaining power, U2 cancompensate D more easily; thus, the exclusion equilibrium is more likely to be observed.18

4 Discussion
This section briefly discusses the wholesale pricing and real-world examples of exclusiveoffer competition. Section 4.1 extends the analysis to the case of linear wholesale pricing.
18

When the downstream firm and each manufacturer have the same bargaining power (β = 1/2), we have

∂πRD /∂γ = 0; thus, the degree of product substitution does not affect D’s profit under upstream duopoly. Since
the threshold value of D’s bargaining power under symmetric costs is 1/2, the likelihood of exclusion undersymmetric costs does not depend on product substitution.

18

Section 4.2 introduces some examples of exclusive-offer competition.

4.1 Linear wholesale pricing
This subsection explores the existence of anticompetitive exclusive dealing under linear wholesale pricing by assuming standard linear demand with a representative consumer, in which
demand for Ui ’s product is provided by−a(1 − γ) + p j
a − pi



if
0
<
p
,

i ≤


b
γ




 a(1 − γ) − p + γp−a(1 − γ) + p jijQ(pi , p j ) = if< pi < a(1 − γ) + γp j ,


2)

b(1
−
γ
γ





 0
if pi ≥ a(1 − γ) + γp j ,(16)

where i, j ∈ {1, 2} and i , j. As in the previous section, we assume that the industry profit
allocation after Stage 1 is given by the Nash bargaining solution.
We first explore the existence of an exclusion equilibrium when only Ui can make an
exclusive offer. Like the case of two-part tariffs, exclusion never occurs.
Proposition 6. Suppose both manufacturers offer linear wholesale prices. If U j is a potential
entrant and only Ui can make an exclusive offer, Ui cannot exclude U j through exclusive
contracts for any pair of bargaining power allocation and the degree of product substitution.
Proof. See Appendix A.1.

Proposition 6 implies that in the absence of exclusive-offer competition, anticompetitive
exclusive dealing cannot occur, which can be explained by the logic underlying the Chicago
School argument.
We next investigate the existence of an exclusion equilibrium when both manufacturers
can make exclusive offers. In this case, an exclusion equilibrium exists under some conditions.
Proposition 7. Suppose that both manufacturers make exclusive offers in Stage 1 and linear
wholesale prices are determined through Nash bargaining in Stage 2. When the products
19are less differentiated (γ > γ̃ ≃ 0.77393), exclusion cannot be an equilibrium outcome. By
contrast, when those are sufficiently differentiated (γ ≤ γ̃), there exist both an exclusion
equilibrium and non-exclusion equilibria for a sufficiently weak bargaining power of D (β ≤β̂(γ)), whereβ̂(γ) ≡

4φ2 + 2γ(1 + γ)(5γ − 4)φ + 4γ2 (1 + γ)(γ3 + 3γ2 + 3γ − 5),
6γ2 (1 + γ)φandhφ ≡ γ3 (1 + γ)2 (γ4 + 4γ3 + 6γ2 − 32γ + 19)
1
q
 3
6
2
2
5
4
3
3
2
+3 6γ (1 + γ) (1 − γ ) γ + 5γ + 10γ + 5γ − 12γ − 11γ + 9 .
Proof. See Appendix A.2.

Note that β̂(γ) has a single peaked property with β̂(γ) → 1/3 as γ → 0, β̂(γ) → 0 as γ → γ̃,and the maximized value β̂(γ∗ ) ≃ 0.413049 at γ∗ ≃ 0.469146.
[Figure 6 about here]
Figure 6 summarizes Proposition 7. The notable result in Proposition 7 is that linear
wholesale pricing leads to the low possibility of the anticompetitive exclusion equilibrium;
exclusion never occurs for the intermediate level of D’s bargaining power or for less differentiated manufacturers’ products.
The major difference between two types of pricing is the existence of double marginalization problems, which arise not only under exclusive dealing but also under upstream duopoly.
These two double marginalization problems reduce the possibility of exclusive outcomes in
the following ways. Under exclusive dealing, the contracting party cannot achieve the joint
profit maximization due to the double marginalization problem. Because upstream duopoly
can mitigate such a problem, manufacturers have difficulty in compensating D for the intermediate level of D’s bargaining power. Moreover, the mitigation effect of upstream duopoly

20depends on the degree of product substitution γ. When the manufacturers produce almost homogeneous products, the double marginalization problem under upstream duopoly is not too
serious, which allows D to earn large rejection profits πRD . Hence, the exclusion equilibrium
does not exist when the products are less differentiated.

4.2 Cola wars in the real world
In this subsection, we provide examples of exclusive-offer competition in the soft drinks industry. For instance, cola wars have continued for decades between Coca-Cola and PepsiCo,with each aiming to be the exclusive beverage provider to fast food restaurants.19 Throughexclusive-offer competition, some customers shift from one manufacturer to the other. For
example, Arby’s Restaurant Group Inc. decided to switch from PepsiCo to Coca-Cola starting from early 2018 after more than a decade-long contract with PepsiCo.20 Subway had hada partnership with PepsiCo as its primary beverage provider since 1988; however, in 2003, it
decided to make a transition to Coca-Cola in its worldwide restaurants.21 From 2015, PepsiCo Canada started to serve Subway Canada as its exclusive beverage and snack provider.22
Another example of the exclusive-offer competition between these two giant suppliers canbe observed on university campuses, as noted in Introduction.23 The cola wars forming on
university campuses are widespread in the United States and contain some of the key featuresin our analysis. First, exclusive-offer competition involves a large monetary transfer in returnfor a long-term monopoly position; for example, in 1998, The University of Maryland at Col19

Cola wars have also been observed in the relationship between the cola providers and cinemas. See “CocaCola Lures Regal Cinemas From Rival Pepsi in Latest Steal” The Wall Street Journal, April 9, 2002 (link).
20

See “Coca-Cola Wins Arby’s Away From PepsiCo in Latest Showdown” Bloomberg, August 18, 2017
(link).
21

See “Coke Wins a 10-Year Contract From Subway, Ousting PepsiCo” The Wall Street Journal, November
28, 2003 (link).
22

See the second paragraph from the bottom in “Coca-Cola Wins Arby’s Away From PepsiCo in Latest
Showdown”
23

Regarding universities’ switch between Coca-Cola and PepsiCo, see “Coke vs. Pepsi: University Chooses
Side In Cola Wars” Montclair Patch, July 12, 2016 (link).

21

lege Park signed a 15-year exclusive contract with PepsiCo worth $57.5 million.24 Second,
universities receive not only an annual royalty fee but also a commission fee from the retailsale of some products. The commission rates, which could be correlated to β in our analysis,vary across universities. In the case of Ohio State University and Rutgers University, theyrange from about 20 percent to over 50 percent of the retail sale of drinks and snacks.25 Finally, universities are usually local monopolists; hence, exclusive-offer competition is more
likely to play an essential role in exclusion. Although the objective of universities may not
be to maximize their profits, our results remain valid even when universities are consumersurplus maximizers.26 Therefore, we think that our analysis fits the cola wars on university
campuses well.
Although the above examples are not antitrust cases, the cola wars sometimes lead to
antitrust cases. For example, in 1998, PepsiCo filed an antitrust lawsuit against Coca-Cola,
alleging that Coca-Cola did not allow food-service distributers who already distribute Coketo distribute Pepsi.27 The notable point here is that Coca-Cola was stronger in the foodservice distribution sector of the soft drinks market than in the overall market; the foodservice distributors’ share of Coca-Cola was 65 percent and that of PepsiCo was 22 percent,while in the overall soft drinks market Coca-Cola had a 43.9 percent share and PepsiCo had
a 30.9 percent share. Therefore, if one side has an extremely higher market share in one area
because of exclusive dealing, it will be more likely to take the cola wars to court.
24

See “Thirsting For Cash, Colleges Take Sides In Corporate Cola Wars” The Washington Post, December
23, 1997 (link).
25

For the case of Ohio State University, see “Refreshing or restricting? Ohio States $32M deal with CocaCola brings up questions of transparency” The Lantern, December 19, 2013 (link). In addition, for the case of
Rutgers University, see “As of May 2005, Rutgers University no longer has a contract with Coca-Cola” Rutgers
University, May, 2005 (link).
26

The results are available upon request.

27

See “Taking The ’Cola Wars’ Into Court” The Washington Post, May 31, 1998 (link).

22

5 Conclusion
This study has explored the existence of anticompetitive exclusive dealing when all upstream
firms can make exclusive offers. Most previous studies consider anticompetitive exclusive
dealing to deter a potential entrant, which cannot make an exclusive offer. However, in thereal-world situation, existing firms are often excluded. Therefore, we need to consider how
the existence of exclusive-offer competition affects the possibility of exclusion to apply themodel to these cases.
We show that a seemingly small difference in the setting turns out to be crucial. In contrast
to the case where one of the upstream firms is a potential entrant, the existence of exclusiveoffer competition eliminates upstream firms’ opportunity to earn positive profits when they
fail to exclude the rival upstream firm. We point out that this induces upstream firms to
make higher exclusive offers and show that when the downstream firm has weak bargaining
power, anticompetitive exclusive dealing can be an equilibrium outcome in the two-part tariff
setting of the general demand function and Nash bargaining. Moreover, this result holds invarious settings and thus the exclusion outcome identified in this study can be widely applied
to diverse real-world vertical relationships.
The finding here provides new implications for antitrust agencies; anticompetitive exclusive dealing is more likely to be observed when upstream firms are existing firms. In addition,
because the downstream firm has a strong incentive to engage in anticompetitive exclusive
dealing, it is more likely to lead the negotiation of anticompetitive exclusive dealing when
upstream firms are existing firms.
Despite these contributions, there remain several outstanding issues requiring future research. First, there is a concern about upstream firms’ behavior to achieve a market environment where an exclusion equilibrium does not exist. Although we assume that the level ofproduct substitution or bargaining power is exogenously given, upstream firms could control
these parameters. Second, there is a concern about this study’s relationship with other studiesof anticompetitive exclusive dealing. We predict that if we add exclusive-offer competition23

into previous studies, exclusion becomes less costly. We hope that this study will assist future
researchers in addressing these issues.

24

A Proofs of the results
A.1

Proof of Proposition 6

Before proceeding to the proof, we derive firms’ equilibrium profits in the subgame after D’s
decision in Stage 1.
We first explore the case in which Ui ’s exclusive offer is accepted in Stage 1. Under
exclusive dealing, the final consumer’s demand for Ui ’s product becomes Q(pi ) = (a − pi )/b.
We solve the game by using backward induction. In Stage 3, given wi determined in Stage
2, D optimally chooses the price of Ui ’s product, namely p∗ (wi ) ≡ arg max pi (pi − wi )Q(pi ) =(a + wi )/2. The optimal production level of Ui ’s product supplied by D given wi becomesQ∗ (wi ) ≡ Q(p∗ (wi )) = (a − wi )/2b. In Stage 2, Ui and D negotiate and make a contract for the
linear wholesale price wiEi . By defining D’s profit given wi as Π∗ (wi ) ≡ (p∗ (wi )−wi )Q∗ (wi ), thebargaining problem between D and Ui is described by the payoff pairs (Π∗ (wi ), (wi −c)Q∗ (wi ))and the disagreement point (0, 0). The solution is given bywiEi = arg max β log Π∗ (wi ) + (1 − β) log[(wi − c)Q∗ (wi )].
wi

The maximization problem leads towiEi =

a + c − β(a − c).
2

The firms’ equilibrium profits, excluding the fixed compensation xi , are
Ei
πUi
=

(1 + β)2 (a − c)2(1 − β2 )(a − c)2, πUEij = 0, πEi
=
.
D
8b
16b

(17)

We next explore the case in which D rejects both exclusive offers in Stage 1. In Stage 3,
given the wholesale prices wi and w j determined in Stage 2, D optimally chooses the pricesof each manufacturer’s product (p∗ (wi , w j ), p∗ (w j , wi )), where(p∗ (wi , w j ), p∗ (w j , wi )) ≡ arg max(pi − wi )Q(pi , p j ) + (p j − w j )Q(p j , pi ),
pi ,p j25

where i, j ∈ {1, 2} and i , j. The production level of each final product supplied by D given
wi and w j is given byQ∗ (wi , w j ) ≡ Q(p∗ (wi , w j ), p∗ (w j , wi )) =a − wi − γ(a − w j ).
2(1 − γ2 )b

In Stage 2, U1 , U2 , and D make contract(s) for the linear wholesale prices wR1 and wR2 . By
defining D’s profit from selling Ui ’s product given (wi , w j ) as Π∗ (wi , w j ) ≡ (p∗ (wi , w j ) −wi )Q∗ (wi , w j ), the bargaining problem between D and Ui is described by the payoff pairs(Π∗ (wRi , wRj ) + Π∗ (wRj , wRi ), (wRi − c)Q∗ (wRi , wRj )) and the disagreement point (Π∗ (wRj ), 0), whereΠ∗ (wRj ) is D’s profit when it sells only U j ’s product given the linear wholesale price wRj . The
solution is given by
wRi = arg max β log[Π∗ (wi , w j ) + Π∗ (w j , wi ) − Π∗ (w j )] + (1 − β) log[(wi − c)Q∗ (wi , w j )].
wi

The maximization problem leads towRi =a(1 − γ) + c − β(a(1 − γ) − c),2 − γ(1 − β)for each i ∈ {1, 2}. The resulting profits of the firms are given as
πRUi =(1 − β2 )(1 − γ)(a − c)2(1 + β)2 (a − c)2
R
,
π
=
.
D
2b(1 + γ)(2 − γ(1 − β))22b(1 + γ)(2 − γ(1 − β))2(18)

We now consider the existence of an exclusion equilibrium. We show that Condition (6)never holds; in other words, by substituting Equations (17) and (18), we have
∆πU − ∆πD = −(a − c)2 (1 + β)(8 − (1 + γ)(2 − γ(1 − β))(3 − β))< 0,16b(1 + γ)(2 − γ(1 − β))(19)for all (β, γ) ∈ (0, 1)2 . Let η(β, γ) ≡ −8 + (1 + γ)(2 − γ(1 − β))(3 − β). Note that η(β, γ) < 0 if
and only if Condition (19) holds. By differentiating η(β, γ) with respect to β and γ, we haveηβ (β, γ) R 0 ⇔ β ⋚ K(γ) ≡−1 + 2γ,γ

ηγ (β, γ) R 0 ⇔ β R L(γ) ≡−1 + 2γ.
1 + 2γ26

Note that for γ ∈ (1/2, 1], K ′ (γ) > L′ (γ) > 0 and K(γ) > L(γ) > 0 and that K(1/2) =L(1/2) = 0 and K(1) = 1 and L(1) = 1/3. Figure 7 summarizes the properties of ηβ (β, γ)and ηγ (β, γ). There are six regions in (β, γ) ∈ [0, 1]2 such that (i) ηβ (β, γ) = ηγ (β, γ) = 0, (ii)ηβ (β, γ) < 0, ηγ (β, γ) > 0, (iii) ηβ (β, γ) = 0, ηγ (β, γ) > 0, (iv) ηβ (β, γ) > 0, ηγ (β, γ) > 0, (v)ηβ (β, γ) > 0, ηγ (β, γ) = 0, and (vi) ηβ (β, γ) > 0, ηγ (β, γ) < 0. The arrows in Figure 7 indicate
the direction of the increase in η(β, γ) for each region. From Figure 7, for (β, γ) = (0, 1/2),
η(β, γ) takes the locally maximized value in region (i), where we have η(β, γ) = −5/4 < 0.
More importantly, Figure 7 shows that η(β, γ) is globally maximized in the domain (β, γ) ∈[0, 1]2 when (β, γ) = (1, 1), where we have η(1, 1) = 0. Therefore, η(β, γ) < 0 for all(β, γ) ∈ (0, 1)2 .
Q.E.D.

A.2

Proof of Proposition 7

We check whether πUEii ≥ ∆πD holds. By substituting Equations (17) and (18), we obtainEi− ∆πD ≥ 0 if and only if γ ≤ γ̃ and β ≤ β̂(γ) < 1/2.
πUi

Q.E.D.

B Results under linear demand and symmetric costs
This appendix introduces the analysis of the model in Section 3.1–3.3 under the linear demand function (16). Under the linear demand function, we have(a − c)2(a − c)2Πm =, Πd =.
4b(1 + γ)b(20)

Then, the firms’ equilibrium profits under exclusive dealing, excluding the fixed compensation xi , are
Ei
πUi
=

β(a − c)2
(1 − β)(a − c)2, πUEij = 0, πEi
=
.
D
4b
4b

27

(21)

The profits of firms under no exclusive dealing are given as
πRUi =(β(1 − γ) + γ)(a − c)2(1 − β)(1 − γ)(a − c)2, πRD =.
4b(1 + γ)
2b(1 + γ)(22)

We now explore the existence of an exclusion equilibrium. For the case in which only
Ui can make an exclusive offer, we check whether Condition (6) holds. By substituting
Equations (21) and (22), we have
∆πU − ∆πD = −β(1 − γ)(a − c)2< 0,
4b(1 + γ)(23)

for all γ ∈ [0, 1) and β ∈ (0, 1); as with linear wholesale pricing, exclusion never occurs. This
result is consistent with Proposition 1.
By contrast, for the existence of an exclusion equilibrium when both manufacturers canmake exclusive offers, we check whether πUEii ≥ ∆πD holds. By substituting Equations (21)and (22), we have

(1 − 2β)(1 − γ)(a − c)2
≥ 0,
(24)
4b(1 + γ)
for β ∈ (0, 1/2]. Therefore, an exclusion equilibrium exists if β ≤ 1/2, which is consistentEi
πUi− ∆πD =

with Proposition 2.

C Results under linear demand and asymmetric costs
This appendix introduces the analysis of the model in Section 3.4 under the linear demand
function (16). We measure U1 ’s cost advantage by θ, where c2 = θpm1 + (1 − θ)c1 andpm1 = (a + c1 )/2. θ = 0 implies that U1 has no cost advantage. As θ increases, U1 becomesefficient. We assume the following relationship:
0 < θ < min{2(1 − γ), 1}.

(25)

If Condition (25) holds, the upstream market becomes a duopoly if the exclusive offer isrejected. When D accepts U1 ’s exclusive offer, the firms’ equilibrium profits, excluding the
fixed compensation x1 , are
E1
πU1
=

(1 − β)(a − c2 )2β(a − c2 )2
E1
E1
,
π
=
0,
π
=
.
D
U2
(2 − θ)2 b(2 − θ)2 b
28

(26)

Likewise, when D accepts U2 ’s exclusive offer, the firms’ equilibrium profits, excluding the
fixed compensation x2 , are
E2
πU2
=

β(a − c2 )2(1 − β)(a − c2 )2E2, πU1= 0, πE2
=
.
D
4b
4b

(27)

By contrast, when D rejects both exclusive offers, the firms’ equilibrium profits are(θ2 + 4(1 − γ)(2 − θ) − (1 − β)((2(1 − γ) + θγ)2 + (2(1 − γ) − θ)2 ))(a − c2 )2,4b(1 − γ2 )(2 − θ)2(1 − β)(2(1 − γ) + θγ)2 (a − c2 )2(1 − β)(2(1 − γ) + θ)2 (a − c2 )2R
πRU1 =,π=.
U2
4b(1 − γ2 )(2 − θ)24b(1 − γ2 )(2 − θ)2
πRD =

(28)

We now consider the existence of an exclusion equilibrium when both manufacturers
EiR
make exclusive offers. By substituting (26), (27), and (28), πUi+ πEi
D − πD ≥ 0 if and only ifβ ≤ βi (γ, θ), where
β1 (γ, θ) ≡(2(1 − γ) − θ)2(2(1 − γ) + θγ)2
,
β
(γ,
θ)
≡
.
2
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )The following lemma summarizes the properties of βi (γ, θ).
Proposition C.1. βi (γ, θ) has the following properties:
1. 0 < β2 < 1/2 < β1 < 1.
2. ∂β1 /∂γ > 0 and ∂β2 /∂γ < 0.
3. ∂β1 /∂θ > 0 and ∂β2 /∂θ < 0.
4. As γ → (2 − θ)/2, β1 → 1 and β2 → 0.
5. As θ → 0, β1 → 1/2 and β2 → 1/2.
Proof. We examine the first property. Note that β2 > 0 is obvious. Then, we haveβ1 −(θ(4 − θ)(1 − γ))21 1
= − β2 => 0,
2 2
2(4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 ))29

1 − β1 =(2(1 − γ) − θ)2> 0.
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )Therefore, the first property holds. The second and third properties can be derived by thefollowing results; under Condition (25),∂β1 2θ(4 − θ)(2(1 − γ) + θγ)(2(1 − γ) − θ)
=
> 0,
∂γ
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )∂β22θ(4 − θ)(2(1 − γ) + θγ)(2(1 − γ) − θ)
=−
< 0,
∂γ
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )∂β1 4(1 − γ2 )(2(1 − γ) + θγ)(2(1 − γ) − θ)
=
> 0,
∂θ
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )4(1 − γ2 )(2(1 − γ) + θγ)(2(1 − γ) − θ)
∂β2
=−
< 0.
∂γ
4(1 − γ)2 (2 − θ) + θ2 (1 + γ2 )The fourth and fifth properties are obtained by substituting γ = (2 − θ)/2 and θ = 0 intoβi (γ, θ), which is continuous in θ and γ.

D Exclusive supply contracts when downstream firms compete in quantity
This appendix introduces another case where exclusive-offer competition plays an essential
role in exclusive dealing. The upstream market is composed of an upstream monopolist U,
whose marginal cost is c ≥ 0. The downstream market is composed of two downstream firmsthat produce homogeneous products. Each downstream firm produces one unit of the final
product by using one unit of input produced by U. For simplicity, we assume that the costof transformation is zero for each Di ; given the input price w, the per unit production costof Di is given by cDi = wi , where i ∈ {1, 2}. D1 and D2 compete in quantity. Let Qi be theproduction level of Di . We assume that inverse demand for the final product P(Q) is given bya simple linear function:
P(Q) = a − bQ,
where Q ≡ Q1 + Q2 is the output of the final product, a > c, and b > 0.
30

The model in this appendix contains three stages. In Stage 1, D1 and D2 make exclusive
supply offers to U with fixed compensation yi ≥ 0, where i ∈ {1, 2}. U can reject both offers or
accept one of the offers. As defined in Section 2, let ω ∈ {R, E1, E2} be U’s decision in Stage
1. If U is indifferent between these two exclusive offers and acceptance is more profitable, it
accepts one of the offers with probability 1/2. In Stage 2, U offers linear wholesale price w
to active downstream firms. The equilibrium wholesale price offered by U is denoted by wω .
In Stage 3, active downstream firms order inputs and determine the production level of the
final product Qi . Di ’s profit is denoted by πωDi . Likewise, U’s profit is denoted by πωU .

D.1 Equilibrium outcomes after Stage 1
We first explore the case in which Di ’s exclusive supply offer is accepted by U in Stage 1.
In Stage 3, given w, Di optimally chooses the production level QiEi (w) ≡ arg maxQi (P(Qi ) −w)Qi = (a − w)/2b. Then, input demand for U becomes QEi (w) = QiEi (w) = (a − w)/2b. In
Stage 2, by anticipating these results, U optimally chooses input price wEi ≡ arg maxw (w −c)Q(w) = (a + c)/2. The equilibrium production levels become QEi = QiEi = (a − c)/4b andQEi
j = 0, where i, j ∈ {1, 2} and i , j. The firms equilibrium profits, excluding the fixedcompensation yi , are
πEi
Di =

(a − c)2(a − c)2Ei, πEi.
D j = 0, πD =
16b
8

(29)

We next explore the case in which U rejects the exclusive supply offers in Stage 1. In
Stage 3, given w, Di competes in quantity. Standard Cournot competition leads to QRi (w) =(a − w)/3b. Then, input demand for U becomes QR (w) = 2(a − c)/3b. In Stage 2, by
anticipating these results, U optimally chooses input price wR ≡ arg maxw (w − c)QR (w) =(a + c)/2. The equilibrium production levels become QR1 = QR2 = (a − c)/6b. The firms’
equilibrium profits are
πRDi =(a − c)2(a − c)2R, πU =
.
36b
6b

31

(30)

D.2 Benchmark analysis
As in Section 3.2, we assume that D2 is a potential entrant and only Di can make an exclusive
offer in Stage 1. For an exclusion equilibrium to exist, the equilibrium transfer y∗i must satisfy
the following two conditions.
First, the exclusive contract must satisfy individual rationality for U:y∗i ≥ ∆πcU ,(31)

where ∆πcU ≡ πRU − πUEi .
Second, it must satisfy individual rationality for Di :y∗i ≤ ∆πcD ,(32)

R
where ∆πcD ≡ πEi
Di − πDi .

From the above conditions, it is evident that an exclusion equilibrium exists if and only if
inequalities (31) and (32) simultaneously hold. This is equivalent to the following condition:∆πcD ≥ ∆πcU .

(33)

We now consider the game in Stage 1. By substituting Equations (29) and (30), we obtain∆πcD − ∆πcU =(a − c)2
< 0,
144b

(34)

which implies that Condition (33) never holds. Therefore, the exclusion outcomes cannot be
observed.
Proposition D.1. Suppose that downstream firms D1 and D2 compete in quantity by purchasing inputs from upstream monopolist U. If D2 is a potential entrant and only D1 can make an
exclusive offer, D1 cannot exclude D2 via exclusive contracts.
The result here coincides with that of Appendix B in Kitamura, Matsushima, and Sato(2017b).28
28

More precisely, both models coincide for k = 1 in their model.

32

D.3 When exclusive-offer competition exists
Assume that both downstream firms make exclusive offers. As in Section 3.3, the upperbound of Di ’s exclusive offer ymax
depends on D j ’s offer, where
i
( Ei
πDi
if y j ≥ ∆πcU
maxyi ≡∆πD if y j < ∆πcU
and where πEi
Di > ∆πD .
max
For y j < ∆πcU , we have ymax = ∆πcD . With this offer, Ui earns πEi< πRU because
Di + yinequality (34) holds; hence, the individual rationality constraint for U does not hold. Therefore, as in Section 3.3, the non-exclusion equilibrium always exists. For the existence of ancexclusion equilibrium, we check whether πEi
Di ≥ ∆πU holds. By substituting Equations (29)and (30), we obtain(a − c)2Eic
πDi − ∆πU =
> 0,
48b

which implies that the exclusion outcomes can be observed.
Proposition D.2. Suppose that downstream firms D1 and D2 compete in quantity by purchasing inputs from upstream monopolist U. When both downstream firms can make exclusive
offers, there exist both an exclusion equilibrium and a non-exclusion equilibrium. In the exclusion equilibrium, both D1 and D2 offer y∗i = πEi
Di > ∆πU and U earns all the industry
profits.

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36

‫ݔ‬ଶ

‫ʹܧ‬

ሺ‫ݔ‬ଶ ൐ ‫ݔ‬ଵ ൐ οߨ஽ ሻ‫ʹܧ‬ሺ‫ݔ‬ଶ ൐ οߨ஽ ൐ ‫ݔ‬ଵ ሻ

οߨ஽

‫ͳܧ‬

ሺ‫ݔ‬ଵ ൐ ‫ݔ‬ଶ ൐ οߨ஽ ሻܴሺοߨ஽ ൐ ‫ݔ‬ଶ ൐ ‫ݔ‬ଵ ሻ‫ͳܧ‬ሺ‫ݔ‬ଵ ൐ οߨ஽ ൐ ‫ݔ‬ଶ ሻܴͲ‫ݔ‬ଵ ൌ ‫ݔ‬ଶ ሺ‫ ݅ܧ‬withprobability 1/2)ሺοߨ஽ ൐ ‫ݔ‬ଵ ൐ ‫ݔ‬ଶ ሻοߨ஽Figure 1: Individual Rationality for D

37

‫ݔ‬ଵ

‫ݔ‬ଶ

‫ݔ‬ଵ୫ୟ୶

οߨ஽‫ݔ‬ଵ୫ୟ୶

ா௜
ߨ௎௜

οߨ௎

Ͳ

οߨ௎

‫ݔ‬ଶ୫ୟ୶

‫ݔ‬ଶ୫ୟ୶

ா௜ߨ௎௜‫ݔ‬௜ ൑ ‫ ݔ‬୫ୟ୶ holds for each ݅ ‫ א‬ሼͳǡʹሽ.

‫ݔ‬ଵ

οߨ஽

Ei
Figure 2: Area of Feasible Offers for Ui (πUi
< ∆πD )

‫ݔ‬ଶ‫ݔ‬ଵ୫ୟ୶

ா௜
ߨ௎௜

οߨ஽
οߨ௎

Ͳ

‫ݔ‬ଵ୫ୟ୶

‫ݔ‬௜ ൑ ‫ ݔ‬୫ୟ୶ holds for each ݅ ‫ א‬ሼͳǡʹሽ.

‫ݔ‬ଶ୫ୟ୶

οߨ௎‫ݔ‬ଶ୫ୟ୶

οߨ஽

ா௜
ߨ௎௜

‫ݔ‬ଵ

Ei
Figure 3: Area of Feasible Offers for Ui (πUi
≥ ∆πD )38‫ݔ‬ଶ‫ݔ‬ଵ୫ୟ୶

οߨ஽‫ݔ‬ଵ୫ୟ୶

ா௜
ߨ௎௜

οߨ௎

Ͳ

ܴ

οߨ௎

‫ݔ‬ଶ୫ୟ୶
‫ݔ‬ଶ୫ୟ୶

ா௜
ߨ௎௜

‫ݔ‬ଵ

οߨ஽

Ei
Figure 4: Existence of an Exclusion Equilibrium for πUi
< ∆πD

‫ݔ‬ଶ

‫ݔ‬ଵ୫ୟ୶

ா௜
ߨ௎௜

‫ʹܧ‬

οߨ஽
οߨ௎

Ͳ

‫ݔ‬ଵ୫ୟ୶

ܴ‫ͳܧ‬‫ݔ‬ଵ ൌ ‫ݔ‬ଶ ሺ‫ ݅ܧ‬withprobability 1/2)‫ݔ‬ଶ୫ୟ୶

Exclusion Equilibrium‫ݔ‬ଶ୫ୟ୶

οߨ௎

οߨ஽

ா௜
ߨ௎௜

‫ݔ‬ଵ

Ei
Figure 5: Existence of an Exclusion Equilibrium for πUi
≥ ∆πD

39

ߚ

0.5

݊‫ܦܧ݋‬

0.4

0.3‫݅ܦܧ‬Ȁ݊‫ܦܧ݋‬

0.2

0.1

መ
ߚሺߛሻ

0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

ߛ

Figure 6: Existence of an Exclusion Equilibrium under Linear wholesale pricingߚ1.0‫ܭ‬ሺߛሻ

0.9
0.8

ߟఉ ൌ Ͳߟఊ ൐ Ͳ

0.7

ߟఉ ൏ Ͳߟఊ ൐ Ͳ

0.6
0.5
0.4
0.3
0.2
0.1
0.0
0.0

0.1

0.2

ߟఉ ൌ ߟఊ ൌ Ͳ

0.3

0.4

0.5

‫ܮ‬ሺߛሻ
0.6

ߟఉ ൐ Ͳߟఊ ൌ Ͳ
0.7

0.8

ߟఉ ൐ Ͳߟఊ ൐ Ͳߟఉ ൐ Ͳߟఊ ൏ Ͳ
0.9

Figure 7: Properties of ηβ (β, γ) and ηγ (β, γ)40

1.0ߛ