GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.20

Exclusionary Vertical Contracts with Multiple Entrants

Hiroshi Kitamura

December 2008

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Exclusionary Vertical Contracts with Multiple Entrants∗
Hiroshi Kitamura†
Graduate School of Economics, Osaka University
December 10, 2008

Abstract
This paper constructs a model of anticompetitive exclusive dealing in the presence of multiple
entrants. Unlike a single-entrant model in the extant literature, an entrant competes not only with
the incumbent to deal with buyers but also with other entrants. The competition among entrants
then plays the role of commitment such that low wholesale prices are oﬀered to buyers when
they deviate from exclusive contracts. We argue that this commitment eﬀect becomes a barrier to
exclusive dealing and that the results diﬀer drastically from the predictions of the single-entrant
framework.

JEL Classifications Code: L12, L41, L42.
Keywords: Vertical Relation; Exclusive Dealing; Multiple Entrants; Antitrust Policy.

∗

This paper has benefited from helpful discussion and constructive comments by many interested and informed persons. The author would especially like to thank Katsuya Takii, Junichiro Ishida, Noriaki Matsushima, David Flath, Shingo
Ishiguro, Toshihiro Matsumura, Keizo Mizuno, Masaki Nakabayashi, Dan Sasaki, Takashi Shimizu, and Tetsuya Shinkai,
conference delegates at the Japanese Economic Association and the Japanese Association for Applied Economics, and seminar participants at Kwansei Gakuin University and Osaka University. The usual disclaimer applies.
†
Address:Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan.
Email address:gge003kh@mail2.econ.osaka-u.ac.jp

1 Introduction
Exclusive contracts have been a controversial issue among economists for some time. Seemingly, theexclusive contract is anticompetitive because it may deter eﬃcient entry and thereby reduce welfare.
However, the Chicago School opposes this view. They show that rational economic agents do not
engage in exclusive dealing for anticompetitive reasons because exclusive dealing does not increase thejoint surplus between contracting parties1 . The Chicago School argument remains highly influential.
The key assumption of the Chicago School argument is that buyers are final consumers. Two recent papers, Simpson and Wickelgren (2007) and Abito and Wright (2008), show that if buyers arecompeting firms and they compete intensively, then an exclusive contract can deter eﬃcient entry because exclusive dealing increases the joint surplus between contracting parties by extracting surplus
from final consumers. More surprisingly, Abito and Wright show that with nonlinear wholesale pricing, exclusion is a unique equilibrium, regardless of the degree of downstream competition and any
cost advantage of the entrant.
The aim of this paper is to reexamine recent studies on exclusive dealing in the framework ofmultiple entrants. Although Simpson and Wickelgren (2007) and Abito and Wright (2008) provide
the mechanism for anticompetitive exclusive dealing, they only analyze the case of a single entrant.
However, industries with high substitutability of production factors, products, or technologies may
be composed of multiple entrants. This paper shows that the existence of multiple entrants serves asa barrier to anticompetitive exclusive dealing and that the results diﬀer drastically from the singleentrant model.
Although our argument is applicable elsewhere, the model presented in this paper follows Abito
and Wright (2008) in amalgamating most existing models of exclusive dealing. Hence, by comparing
our results with Abito and Wright (2008), we clarify the importance of multiple entrants. Abito andWright (2008) construct a model of exclusive dealing where downstream buyers compete imperfectly.
Their approach allows us to analyze the relation between the likelihood of exclusive dealing and thedegree of downstream competition under both linear pricing and nonlinear pricing.
1

See Posner (1976) and Bork (1978). See Whinston (2006) for a survey.

1

To understand the importance of multiple entrants, consider the case of linear wholesale pricing.
Under linear wholesale pricing, exclusive contracts can deter an eﬃcient entrant when buyers compete
intensively. Abito and Wright (2008) show that in the case of a single entrant, this result holds, evenwhen the entrant has a large cost advantage. With intense downstream competition, buyers obtain
almost zero profits when all buyers sign exclusive contracts. Conversely, when one of the buyers
deviates from an exclusive contract, the deviant buyer obtains small profits because the eﬃcient entrant
always oﬀers it a wholesale price slightly lower than the marginal cost of the incumbent. This allows
the incumbent to deter eﬃcient entry by oﬀering an exclusive contract with low transfers, even whenthe entrant is eﬃcient.
The key implicit assumption is that the entrant is unable to commit initially to oﬀer suﬃciently
low wholesale prices when the buyer deviates from an exclusive contract. If the entrant were able
to do so, then the entrant could increase the buyer’s deviation profits and induce them not to sign an
exclusive contract. Therefore, the problem faced by the entrant is that it cannot make the commitment
to oﬀer low wholesale prices, even when it is eﬃcient.
This paper shows that the existence of rival entrants mitigates this commitment problem because
a multiplicity of entrants increases the upstream competition between entrants. This additional competition reduces the wholesale prices oﬀered to deviant buyers to the marginal cost of the second mosteﬃcient entrant, and serves the role of commitment to oﬀer low wholesale prices when buyers deviate
from exclusive contracts.
This finding suggests that while it is useful to express the fundamental mechanism of anticompetitive exclusive dealing in the framework of a single entrant, we need to take into account the possibilityof multiple entrants when we apply the model to any real-world situation. Multiple entrants are likely
to exist in industries with many alternative factor inputs, products, and technologies. For example, inindustries with high technological progress, such as the information industry, a number of alternativetechnologies may arise. If we explore the likelihood of anticompetitive exclusive dealing in theseindustries, the single-entrant framework may lead to misleading predictions.
Although we mainly compare our results with Abito and Wright (2008), where anticompetitive ex-2

clusive dealing arises in the absence of scale economies, we can apply our model to the other exclusive
dealing models where anticompetitive exclusive dealing arises in the presence of scale economies2 .
Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000) show that in the presenceof scale economies, exclusive contracts can deter the eﬃcient entrant. More recently, Fumagalli andMotta (2006) and Wright (forthcoming) examine the model where buyers are competing firms. If weconsider the multiplicity of entrants in these models, the competition between entrants reduces thewholesale price to the deviant buyer to a level where the second most eﬃcient entrant cannot obtain
positive profits. This increases the deviation profits of buyers and becomes a barrier to anticompetitive
exclusive dealing. In addition to exclusive contracts, this paper is related to the literature concernedwith entry deterrence in the presence of multiple entrants (Ashiya (2000) and Ishibashi (2003)). In
those works, the existence of rivals decreases the possibility of entry. In contrast, this paper obtains
the opposite result: the existence of rivals increases the possibility of entry.
The remainder of this paper is organized as follows. Section 2 presents the model. Section 3 introduces the analysis under linear wholesale pricing. Section 4 analyzes the case where the incumbent
and entrants compete with two-part tariﬀs. Section 5 provides an example where the multiplicity ofentrants reduces the likelihood of exclusion with scale economies. Section 6 contains some concluding
remarks. The equilibrium outcomes in the subgame following the buyers’ decisions are in Appendix
A. Appendix B provides the proofs of all results.

2 ModelThis section presents the model. The model we present follows Abito and Wright (2008). The new
dimension here is the multiplicity of entrants. This modeling strategy is designed to clarify the importance of multiple entrants. We characterize an upstream and downstream market in Section 2.1.
The timing of the game is introduced in Section 2.2. Finally, we derive the conditions that exclusive
contracts need to satisfy in Section 2.3.
2

Simpson and Wickelgren (2007) argue that exclusion without scale economies is more likely to arise than exclusion
with scale economies, showing that exclusion is not ineﬃcient if buyers can breach contracts and pay expectation damages.
However, exclusion without scale economies remains ineﬃcient, even if breach is possible. This is one reason why wemainly explore exclusion without scale economies in this paper.

32.1 Upstream and downstream markets
In the upstream market, three firms exist, an incumbent (denoted I), Entrant 1 (E1 ), and Entrant 2(E2 ). These firms produce an identical product but diﬀer in terms of their cost eﬃciency. Both Entrant
i = 1 and 2 are more eﬃcient than the incumbent: they have marginal cost cEi < cI . Without loss ofgenerality, we assume that cE1 < cE2 . Entrants need to incur a fixed cost to start production, F ≥ 0. We
assume that entrants can make wholesale price oﬀers before they incur fixed costs3 . To simplify the
analysis, we assume that F = 0. Note that exclusive dealing here arises in the absence of fixed entry
cost, and we examine the possibility where exclusive dealing arises in the absence of scale economics.
In Section 5, we provide an example where the multiplicity of entrants serves as a barrier to exclusive
dealing, even in the presence of scale economies (F > 0). The upstream firms deal with buyers who
are not final consumers but rather competing firms in the downstream market.
In the downstream market, there are two buyers associated with Buyer 1 (denoted B1 ) and with
Buyer 2 (B2 ). They are diﬀerentiated and compete in prices. The cost of transformation or resale isassumed to be zero for simplicity. Buyers sell to final consumers whose preferences are represented
by the following utility function:
U(q1 , q2 ) = α(q1 + q2 ) −β(q21 + q22 + 2γq1 q2 )
,
2

(1)

where 0 ≤ γ < 1 is a parameter indicating the degree of diﬀerentiation between buyers and q j is theamount of consumption of buyer j. Buyers become homogeneous as the value of γ increases. Whenγ = 0, buyers are independent monopolists. However, when γ = 1, buyers are homogeneous Bertrand
competitors. The inverse demand becomes p j = α − β(q j + γq− j ), where cI < α ≤ 2cI and β > 0.
Buyer j’s demand depends not only on its price but also on buyer − j’s price:
 α−p j−α(1−γ)+p− j
if 0 < p j ≤
,


β
γ


 α(1−γ)−p j +γp− j−α(1−γ)+p− j
qj = 
< p j < α(1 − γ) + γp− j ,ifγ
β(1−γ2 )0
if p j ≥ α(1 − γ) + γp− j .

(2)3This is an important assumption when we consider the likelihood of exclusive dealing in the presence of multiple
entrants. In the single-entrant model, this assumption is not important: the result is unchanged, regardless of the timing of
fixed costs. However, in the presence of multiple entrants, Entrant 2 does not join the upstream competition if entrants needto incur fixed costs before making wholesale price oﬀers. If the buyers are a number of final consumers, then assuming thatthe entrants need to incur fixed costs before they make wholesale price oﬀers may be realistic. In contrast, if the buyers are
firms, then assuming that entrants can make wholesale price oﬀers before they incur fixed costs is more realistic.

4

The interpretation of equation (2) is as follows. When the prices of buyers are suﬃciently close,
both obtain positive demand. However, when the prices of buyers are suﬃciently diﬀerent, the higherpriced buyer loses demand but the lower-priced buyer obtains all demand.
We measure entrant i’s cost advantage by θEi , which satisfies cI = θEi pm (cEi ) + (1 − θEi )cEi , wherepm (cEi ) is the monopoly price for the industry when marginal cost is cEi : pm (cEi ) = (α+cEi )/2. θEi = 0
implies that entrant i has no cost advantage. As θEi increases, entrant i becomes eﬃcient. Following
Abito and Wright (2008), we assume 0 < θEi ≤ 14 . From the definition of θEi , the marginal cost of
entrant i can be expressed as a function of θEi and cI as follows:
cEi = C(θEi , cI ) =2cI − αθEi
for i = 1, 2.
2 − θEi(3)

The advantage in measuring entrant i’s cost advantage with θEi is that it simplifies the analysis. By
using θEi , we obtain results that do not depend on cI , α, and β. We can then analyze the possibility of
exclusive dealing with only three parameters θE1 , θE2 , and γ.
In our framework, the single-entrant case following Abito and Wright (2008) corresponds to thesituation where Entrant 2 has no cost advantage, θE2 = 0. In the following analysis, we show thatthe existence of Entrant 2 and its cost advantage becomes an important factor in determining thepossibility of exclusive dealing.

2.2 Timing of gameThe timing of the game is as follows. The model contains three stages. In stage 1, the incumbent
makes simultaneous and nondiscriminatory exclusive oﬀers to each buyer. This exclusive contract
involves some fixed compensation x ≥ 0. We assume that the incumbent is unable to oﬀer wholesale
prices5 . Buyers simultaneously decide whether to accept this oﬀer. To avoid an open-set problem,
we assume buyers sign an exclusive contract if they are indiﬀerent between signing and not signing
4This assumption implies that the entrants’ monopoly price is higher than cI . Exclusion still exists, even when theentrants are more eﬃcient, but the analysis becomes more complicated.
5
Rasmusen, Ramseyer, and Wiley (1991) point out that price commitment is unlikely if the nature of the product is notprecisely described in advance. In addition, even if the incumbent could set a price in Stage 1, the incumbent may not havean incentive to do so because the optimal pricing is contingent on the buyers’ decision outcome in Stage 1, as we show in thefollowing analysis. The incumbent is then better oﬀ oﬀering wholesale prices after observing the buyers’ decision outcome
in Stage 1.

5the contract. We do not allow all players to breach once the contract is signed. We refer to the buyer
signing the exclusive contract as the signer and the buyer not signing the contract as the free buyer.
The free buyer is able to buy not only from the incumbent but also from entrants in the latter stage.
Let S ∈ {0, 1, 2} be the number of signers.
In Stage 2, upstream firms oﬀer wholesale prices. There are three cases: S = 2, S = 1, and S = 0.
If S = 2, then only the incumbent oﬀers wholesale prices to both buyers. If S = 0, then all upstream
firms oﬀer wholesale prices to both buyers. If S = 1, then the incumbent oﬀers wholesale prices toeach buyer, but entrants oﬀer wholesale prices only to the free buyer. We assume that the incumbent isable to discriminate between buyers that have signed exclusive contracts. To avoid open-set problems,
we assume that free buyers when indiﬀerent deal with eﬃcient upstream firms. Furthermore, we
assume that for the case of S = 1, each buyer is unable to observe the wholesale oﬀer to its rival6 .
This assumption avoids the possibility of multiple equilibria in the subgame for S = 17 .
In Stage 3, the upstream firm(s) start production and buyers compete in prices. Following Abito
and Wright (2008), we assume that buyers do not face a small fixed cost to stay active8 . The incumbent’s profit in the case of S = k, where k ∈ {0, 1, 2}, is denoted by ΠI|S =k , and buyer j’s profit isdenoted by πlj|S =k where l = s( f ) if buyer j is the signer (free buyer).

2.3 Requirement of exclusive contracts
Given equilibrium outcomes in the subgame following Stage 1 (provided in Appendix A), we derive
the conditions that an exclusive contract needs to satisfy. The exclusive contract needs to satisfy thefollowing three conditions. First, it has to satisfy the financial feasibility for the incumbent; that is, theFor S = 0 and S = 2, both buyers deal with the same upstream firms. On the other hand, for S = 1, buyers deal with
diﬀerent upstream firms: the free buyer deals with Entrant 1, but the signer deals with the incumbent in the equilibrium.
Because two groups exist for S = 1, the wholesale price oﬀers that are not publicly observed are more realistic than thosethat are publicly observed.
7
See footnote (15) in Appendix A1.3.
8
The epsilon participation cost for buyers serves the role of a barrier to an exclusive contract in Fumagalli and Motta(2006). However, Simpson and Wickelgren (2007) argue that assuming that buyers are always active is more reasonable forthe following reasons. First, under some diﬀerentiation, both the signer and the free buyer capture positive demand, andtherefore both cover the epsilon participation cost. Second, if we expand the incumbent’s contract space either by allowing
exclusive contracts that are contingent on all buyers signing them or by allowing up-front payments that are contingent on
being active, exclusive contracts again deter entry, even when buyers compete intensively. See the remark in Section III of
Simpson and Wickelgren (2007).
6

6

incumbent cannot oﬀer a x that is larger than half of its profits under exclusive dealing9 ; i.e.:0≤x≤ΠI|S =2.
2(4)

Second, the exclusive contract has to satisfy individual rationality for buyers: put diﬀerently, theamount of compensation x induces each buyer to sign the exclusive contract as a best response when
its rival signs the exclusive contract; i.e.:fx + π sj|S =2 ≥ π j|S =1 for all i = 1, 2.

(5)Finally, the exclusive contract is required to satisfy uniqueness; that is, the amount of compensation x induces each buyer to sign the exclusive contract as a best response when its rival does not sign;i.e.:f

x + π sj|S =1 ≥ π j|S =0 for all i = 1, 2.

(6)

In the following analysis, we explore the existence of a transfer x that simultaneously satisfies theabove conditions. In particular, we examine whether inequalities (4) and (5) are simultaneously satisfied because these conditions are necessary conditions for the existence of anticompetitive exclusive
dealing. Because inequalities (4) and (5) are simultaneously satisfied if and only if:
ΠI|S =2f+ π sj|S =2 ≥ π j|S =1 for all i = 1, 2,
2(7)

we mainly explore whether inequality (7) is satisfied in the following analysis. We then examine
whether inequality (6) also holds.

3 Linear Wholesale Pricing
This section analyzes the possibility of exclusive dealing under linear wholesale pricing. In order tounderstand more easily the commitment eﬀect of the competition between entrants, we first analyze
the case where downstream firms compete intensively (γ → 1) in Section 3.1. We then extend ouranalysis to the case where downstream firms compete less intensively (0 ≤ γ < 1) in Section 3.2.
9
When n buyers exist, the incumbent needs to oﬀer x to all of the n buyers. Therefore, the financial feasibility condition
becomes 0 ≤ nx ≤ ΠI|S =2 .

7

3.1 Intense downstream competition
In this subsection, we examine the possibility of exclusive dealing when buyers compete intensively(γ → 1). We first explore the single-entrant case as the benchmark in Section 3.1.1. We then analyze
the multiple-entrant case in Section 3.1.2.
3.1.1 Benchmark: the single-entrant case
Assume that Entrant 2 does not exist. This corresponds to the case where Entrant 2 has no cost eﬃciency, θE2 = 0. Abito and Wright (2008) show that when buyers compete intensively, the incumbent
can always exclude eﬃcient entry and obtain almost monopoly profits:
Proposition 1 (Abito and Wright (2008) (Proposition 1)). Suppose that Entrant 2 does not exist. If
buyers compete intensively (γ → 1), then under linear wholesale pricing, there is a unique exclusion
equilibrium with the incumbent obtaining almost monopoly profits.
The intuitive logic for this result is as follows. When buyers compete intensively, buyers yield
almost zero profits for the same wholesale prices. Under exclusive dealing (S = 2), the incumbent
oﬀers buyers the same wholesale price that is almost the monopoly level. Buyers then yield almost
zero profits, but the incumbent yields almost monopoly profits (See Figure 1).
When one of the buyers deviates from an exclusive contract (S = 1), the incumbent oﬀers its
marginal cost to the free buyer, but Entrant 1 matches this oﬀer. The free buyer then buys fromEntrant 1 at a slightly lower price than the marginal cost of the incumbent. Conversely, the signer buys
from the incumbent at a wholesale price close to the marginal cost of the incumbent. Because of small
cost diﬀerence, the intense downstream competition induces the free buyer to yield small deviation
profits (See Figure 2). By using profits under exclusive dealing, the incumbent can easily compensate
for the buyers’ deviation profits with a small transfer x > 0. Each buyer is then better oﬀ signing an
exclusive contract. As a result, the incumbent excludes eﬃcient entry and enjoys almost monopoly
profits.
More importantly, this result holds even when Entrant 1 is very eﬃcient, θE1 = 1. This follows
from the determination of the wholesale price to a free buyer for S = 1. For S = 1, Entrant 1 oﬀers
8

the free buyer a wholesale price slightly lower than the marginal cost of the incumbent, regardless of
its eﬃciency. At the beginning of Stage 1, each buyer expects that it will be oﬀered this wholesale
price when it deviates from the exclusive contract and that deviation is then not beneficial.
This result would not hold if Entrant 1 were able to transfer part of its profits to the free buyer for
S = 1. By so doing, dealing with Entrant 1 becomes beneficial to the free buyer for S = 1. When
Entrant 1 is eﬃcient, it obtains high profits and could profitably transfer part of its profits to the freebuyer. If each buyer expected this transfer at the beginning of Stage 1, it would not sign an exclusive
contract. Therefore, Entrant 1 has an ex ante incentive to transfer part of its profits. However, Entrant
1 has an ex post incentive not to do so when the result in Stage 1 is S = 1. Because each buyer knows
Entrant 1’s ex post incentive, it expects a small deviation to profits and signs an exclusive contract.
The problem here is that Entrant 1 cannot commit to transfer part of its profits to the free buyer for
S = 1.
3.1.2 Multiple-entrant caseWe now assume that Entrant 2 exists. In contrast to the single-entrant case, wholesale price competition now exists between entrants when free buyers exist. This additional competition makes it diﬃcult
for the incumbent to deter eﬃcient entry:
Proposition 2. Suppose that multiple entrants exist. If buyers compete intensively (γ → 1), then withlinear wholesale pricing, the incumbent can exclude eﬃcient entry when Entrant 2 is less eﬃcient(θE2 ≤ 2/9). However, the incumbent cannot exclude eﬃcient entry when both Entrant 1 and Entrant
2 are suﬃciently eﬃcient (θE1 > θE2 > 2/9).
This result lies in contrast with the result in the single-entrant case where exclusion is a uniqueequilibrium outcome. The critical diﬀerence arises when one of the buyers deviates from an exclusive
contract (S = 1). Because of the competition between entrants, Entrant 1 oﬀers the free buyer awholesale price that does not match the marginal cost of the incumbent but rather that of Entrant 2.
The reduction of the wholesale price to the free buyer then increases the profits of the free buyer (SeeFigure 3). Therefore, wholesale price competition between entrants serves as the role as a commitment
9to transfer part of the profits of Entrant 1. At the beginning of Stage 1, each buyer expects that it
will deliver large deviation profits as Entrant 2 becomes eﬃcient. Each buyer then does not sign an
exclusive contract if Entrant 2 is suﬃciently eﬃcient.
Note that Proposition 2 also implies that the competition between entrants does not always serve asa barrier to ineﬃcient exclusion; that is, the incumbent can profitably exclude eﬃcient entry if Entrant
2 is ineﬃcient. The incumbent then has an advantage to hold the almost monopoly profits under
exclusive dealing. Therefore, if Entrant 2 is not eﬃcient, the incumbent can profitably compensate
buyers and can exclude eﬃcient entry.
The result here suggests that although the mechanism of anticompetitive exclusive dealing is in
the single-entrant framework, we need to take into account the existence of multiple entrants if wewish to apply the model to real-world situations. In the single-entrant framework model, Abito andWright (2008) provide a very important implication that anticompetitive exclusive dealing is possible
with intense downstream competition. However, once we consider the multiplicity of entrants, thepossibility of exclusive dealing also depends on the eﬃciency of the second most eﬃcient entrant.
As multiple entrants are likely to exist in the industries in the presence of alternative factor inputs,
products, and technologies, when we apply the model to these industries, the analysis in the singleentrant framework may provide misleading predictions.

3.2 Less intense downstream competition
In this subsection, we extend the analysis to all degrees of downstream competition (0 ≤ γ < 1).
The aim of this analysis is to examine the robustness of Proposition 2 under imperfect downstream
competition. Abito and Wright (2008) also analyze the relation between the likelihood of an exclusion
equilibrium and the degree of downstream competition. In a single-entrant framework, they show that
exclusive dealing arises with intense downstream competition. We show that while their prediction iscorrect, it overestimates the likelihood of exclusive dealing as in Proposition 2 when we consider the
possibility of multiple entrants:
Proposition 3. Suppose that upstream firms are restricted to oﬀer linear wholesale prices. The multi-

10plicity of entrants decreases the likelihood of a unique exclusion equilibrium for all degrees of downstream competition where exclusion equilibrium arises in the single-entrant case.
Appendix B summarizes the more precise statement of Proposition 3. Figure 4 depicts the resultsof Proposition 3. In order to understand easily the importance of the existence of Entrant 2, weconstruct Figure 5 where Entrant 2 has no cost eﬃciency (θE2 = 0). Note that this coincides with thesingle-entrant model in Abito and Wright (2008).
By comparing these figures, it is easy to see that exclusion is a unique equilibrium outcome whendownstream competition is not too strong (γ < 0.76). This result coincides with Abito and Wright(2008). One of the main reasons is double marginalization. The double marginalization prevents joint
profit maximization among contracting parties under exclusive dealing (S = 2). In addition, weakercompetition exacerbates the double marginalization problem. Therefore, weaker competition makes
it diﬃcult for the incumbent to compensate the buyers’ deviation profits and makes exclusive dealingdiﬃcult, even in the absence of Entrant 210 .
Conversely, as downstream competition becomes strong (γ ≥ 0.76), exclusion equilibrium arises.
Proposition 3 implies that with strong downstream competition, the multiplicity of entrants becomesa barrier to exclusive dealing. By comparing Figures 4 and 5, it is easy to see that the market environment of a unique exclusion decreases because of the existence of Entrant 2.
One of the significant results here is that the possibility of a unique exclusion equilibrium is most
likely not when buyers compete intensively (γ → 1) but when buyers compete slightly less intensively(γ = 0.94). This result follows from the determination of the free buyer’s profits for S = 1. As
buyers compete less intensively, both the free buyer and the signer obtain positive demands, andduopoly competition arises for S = 1. In the duopoly equilibrium, both buyers yield low (high) profits
with intense (less intense) downstream competition. The incumbent needs to oﬀer a large amountof compensation to buyers as they compete less intensively. Therefore, exclusive dealing becomesdiﬃcult with intense downstream competition when buyers compete less intensively.
On the other hand, as buyers compete intensively, the free buyer monopolizes the downstream
10
In Section 4, we show that the adaptation of nonlinear wholesale pricing solves the double marginalization problem andthat there exists an exclusion equilibrium, even when downstream competition is suﬃciently relaxed.

11market. In order to monopolize the downstream market, free buyers are required to choose a high (low)price with intense (less intense) downstream competition11 . This implies that under monopolization by
the free buyer, the free buyer’s profits increase as buyers compete intensively. From the incumbent’sview, the amount of compensation increases with the intensity of downstream competition. Hence,
exclusion equilibrium is more likely to arise when downstream competition is slightly relaxed.

4 Two-part Tariﬀs
In this section, we extend the analysis to allow upstream firms to adapt two-part tariﬀs. Two-part tariﬀs
consist of a linear wholesale price and an upfront fixed fee, denoted by (w, ψ) ∈ R2+ . In the singleentrant model, Abito and Wright (2008) provide the strong result that the adaptation of nonlinear
wholesale pricing allows the incumbent always to exclude eﬃcient entry. However, we show that thisstrongly depends on the assumption of a single entrant.
In order to understand our logic, we first review the result in Abito and Wright (2008) and explain
why exclusion is a unique equilibrium outcome in the framework of a single entrant. We then showthat a multiplicity of entrants does not always lead to the exclusion equilibrium. The result in Abito
and Wright (2008) is summarized as follows.
Proposition 4 (Abito and Wright (2008) (Proposition 5)). Suppose that Entrant 2 does not exist. If
upstream firms can oﬀer two-part tariﬀs (w, ψ), then exclusion is a unique equilibrium outcome for all
degrees of downstream competition and cost eﬃciencies of the entrant.
Figure 6 summarizes the above proposition. There are two main reasons why nonlinear wholesale pricing allows the incumbent always to exclude eﬃcient entry in the single-entrant case. First,
nonlinear wholesale pricing solves the double marginalization problem and achieves joint profit maximization among contracting parties under exclusive dealing (S = 2). Therefore, compared with linear
wholesale pricing, nonlinear wholesale pricing increases the joint profits between the incumbent andbuyers for S = 2; see the left-hand side of inequality (7).

11

See Appendix A.1.3 Case (C) and (D).

12

Second, another eﬀect exists in the single-entrant case such that nonlinear wholesale pricing reduces the free buyer’s profits when one of the buyers deviates from an exclusive contract (S = 1); see
the right-hand side of inequality (7). With linear wholesale pricing, the free buyer yields high profits
as Entrant 1 becomes eﬃcient. However, with nonlinear wholesale pricing, Entrant 1 can withdraw
the free buyer’s profits with the fixed free ψ > 0. Entrant 1 then chooses a two-part tariﬀ that induces
the free buyer to yield exactly the same profits as if it dealt with the incumbent. Therefore, the freebuyer’s profits in the single entrant do not depend on the cost advantage of Entrant 1; rather, they are
the normal duopoly profits when both buyers compete by buying from the incumbent at the marginal
cost of the incumbent.
Note that the maximized joint profits of the incumbent and each buyer are always higher than
the normal duopoly profits of each buyer. Therefore, Abito and Wright (2008) conclude that theincumbent can always exclude eﬃcient entry in the case of a single entrant. However, as with linear
wholesale pricing, this result would not hold if Entrant 1 oﬀered a suﬃciently low fixed fee for S = 1.
By lowering the fixed fee, Entrant 1 could increase the deviation profits of each buyer and thereby
induce each buyer not to sign an exclusive contract. Therefore, the problem here is that Entrant 1
excessively withdraws the free buyer’s profits with a fixed fee and it cannot commit to oﬀer a low
fixed fee for S = 1. This paper shows that as with linear wholesale pricing, the multiplicity of entrants
solves this commitment problem under nonlinear wholesale pricing:
Proposition 5. Suppose that upstream firms adapt two-part tariﬀs (w, ψ). The multiplicity of entrants
decreases the likelihood of a unique exclusion equilibrium for all degrees of downstream competition.
Appendix B provides a precise statement of Proposition 5. Figure 7 depicts the results of Proposition 5. Proposition 5 implies that as with linear wholesale pricing, the existence of Entrant 2 serves therole of a barrier to ineﬃcient exclusive dealing. Because of the wholesale price competition between
entrants, Entrant 1 chooses a fixed fee ψ so that the free buyer yields profits so as to deal not with theincumbent but rather with Entrant 2 for S = 1. Therefore, the existence of Entrant 2 serves the role ofa commitment to reduce the fixed fee. This makes the free buyer’s profits for S = 1 depend on the costadvantage of Entrant 2. If Entrant 2 is suﬃciently eﬃcient, then each buyer expects that it is better13

oﬀ deviating from an exclusive contract at the beginning of Stage 1 and so does not sign an exclusive
contract.
In addition, the comparison between linear wholesale pricing and nonlinear wholesale pricing
leads to two main findings. First, compared with linear wholesale pricing, the adaptation of nonlinear
wholesale pricing increases the possibility of exclusive dealing: ΘNL (γ) > ΘL (γ) for all 0 ≤ γ <1. This follows from the achievement of the joint profit maximization for S = 2. The joint profit
maximization allows the incumbent to compensate buyers for a larger amount of profits than linear
wholesale pricing. This increases the possibility of exclusion equilibrium.
Second, compared with linear wholesale pricing, the multiplicity of entrants serves the more important role as a barrier to anticompetitive exclusive dealing under nonlinear wholesale pricing. Underlinear wholesale price, the incumbent cannot exclude eﬃcient entry, regardless of the existence ofmultiple entrants if downstream competition is not too strong (γ < 0.76). In contrast, under nonlinear
wholesale pricing, the incumbent can always exclude eﬃcient entry in the absence of the multiplicity
of entrants. Therefore, under nonlinear wholesale pricing, exploring the possibility of anticompetitive exclusive dealing in the single-entrant case may lead to more misleading predictions than linear
wholesale pricing.

5 Scale Economies
Although we can examine exclusive dealing in the absence of scale economies in Simpson and Wickelgren (2007) and Abito and Wright (2008), our logic is applicable to exclusive dealing in the presenceof scale economies. In order to show the applicability, this section provides an example of how themultiplicity of entrants reduces the likelihood of exclusive dealing with scale economies in Fumagalli
and Motta (2006). To simplify the analysis, we assume that α = β = 1 and cI = 1/2. In order tocoincide with Fumagalli and Motta (2006), we analyze the case where buyers are independent monopolists12 γ = 0. To simplify the analysis, Entrant 1 is eﬃcient enough that θE1 = 1 (cE1 = 0). We first12

When buyers are independent monopolists in Fumagalli and Motta (2006), buyers’ profits depend only on the wholesale
prices oﬀered to themselves. This model structure coincides with Rasmusen, Ramseyer, and Wiley (1991), where buyers are
final consumers whose surplus depends only on their own prices.

14explore the single-entrant case in Section 5.1. We then examine multiple entrants in Section 5.2. We
assume that upstream firms are restricted to linear wholesale prices.

5.1 Single-entrant case
Suppose now that Entrant 2 does not exist. In the absence of scale economies (F = 0), we havef

πSs =2 = 1/64, ΠI|S =2 = 1/16 and ΠE|S =2 = 0 for S = 2, πS =0 = 1/16, ΠI|S =0 = 0, and ΠE|S =0 = 1/4
f

for S = 0, and πS =1 = 1/16, πSs =1 = 1/64, ΠI|S =1 = 1/32, and ΠE|S =1 = 1/8 for S = 1.
Suppose now that the fixed cost is suﬃciently high that Entrant 1 requires both buyers to be
profitable: 1/8 < F < 1/4. If the incumbent can only make simultaneous and nondiscriminatory
exclusive oﬀers to each buyer, then there exist both an exclusion equilibrium and an entry equilibrium.
However, if buyers can coordinate, they prefer the entry equilibrium to the exclusion equilibrium
because upstream competition raises their profits. Therefore, the entry equilibrium is more likely.
The exclusion equilibrium becomes a unique equilibrium outcome if we extend the contract space.
f

For example, if the incumbent can make discriminatory oﬀers where x1 = πS =0 − πSs =2 + ϵ to Buyer 1
but x2 = ϵ to Buyer 2 for any small ϵ > 0, then there is a unique exclusion equilibrium13 . This resultffollows from πS =0 < πSs =2 + ΠI|S =2 .
Note that this result would not hold if Entrant 1 were able to oﬀer low wholesale prices for S = 0.
By so doing, each buyer’s profits for S = 0 would increase. This might make it diﬃcult for theincumbent to exclude Entrant 1, even if it could use discriminatory oﬀers. For example, if Entrant 1√foﬀered wE1 |S =0 < (4 − 5)/4  0.441, then buyers would be better oﬀ not signing exclusive contractsfwith discriminatory or sequential oﬀers: πS =0 > πSs =2 + ΠI|S =2 . In addition, if Entrant 1 oﬀered√fwE1 |S =0 = 2/5 < (4 − 5)/4, then Entrant 1 would yield ΠE1 |S =0 = 6/25. In this case, Entrant 1
could profitably deal with each buyer for S = 0 if 1/8 < F < 6/25. Therefore, if entrants were able
to commit to make such wholesale prices for S = 0, the incumbent could not exclude Entrant 1 as aunique equilibrium outcome, even with discriminatory oﬀers.

13
Abito and Wright (2008) show that this result holds for all degrees of downstream competition (0 ≤ γ < 1) in Proposition 3.

15

5.2 Multiple-entrant case
Suppose now that Entrant 2 does exist. We assume that Entrant 2 is also eﬃcient, θE2 = 1 − ϵ for
any small ϵ > 0 and F = 1/6. The existence of Entrant 2 does not aﬀect the equilibrium outcomes
for S = 2 and S = 1. However, the existence of Entrant 2 changes the equilibrium outcomes for
S = 0. For S = 0, Entrant 2 oﬀers wholesale prices at the level where its profit is equal to fixed cost:√fΠE2 |S =0 = 1/6 and Entrant 1 profitably matches this oﬀer wE1 |S =0 = (3 − 3)/6  0.211. With this√ffwholesale price, each buyer yields πS =1 = (2 + 3)/24 > 1/16. Because πS =0 − (πSs =2 + ΠI|S =2 ) =√(1 + 8 3)/192 > 0, the incumbent cannot profitably compensate one of the buyers, even when it uses
discriminatory oﬀers.
Note that the only diﬀerence is the existence of Entrant 2. This adds competition between entrants
in the upstream competition for S = 0 and reduces wholesale prices to buyers, and increases buyers’
profits for S = 0. The incumbent cannot compensate for these profits, even with discriminatoryoﬀers, if Entrant 2 is suﬃciently eﬃcient. Therefore, exploring the likelihood of exclusion with scaleeconomies in the framework of a single entrant may also lead to misleading prediction.

6 Concluding Remarks
This paper has explored recent studies on exclusive dealing in the framework of multiple entrants.
Unlike a single-entrant model, an entrant needs to compete not only with the incumbent but also with
its rivals. We find that the competition between entrants serves as a barrier to exclusive dealing, andthe results diﬀer drastically from the single-entrant model.
This paper provides new implications for antitrust agencies: put simply, we need to take into
account the existence of multiple entrants when we apply the model to real-world situations. The
findings here imply that earlier results obtained in the single-entrant framework may depend on theassumption of a single entrant. However, multiple entrants may exist in industries in the presence ofalternative factor inputs, products, or technologies. Although the fundamental mechanism of exclusive
dealing is in the single-entrant framework, it may overestimate the possibility of exclusion and maylead to misleading predictions.

16Our result is diﬀerently interpreted in that anticompetitive exclusive dealing is more likely to beobserved in the industries with few alternative factor inputs, products, and technologies14 . In suchindustries, there is less opportunity for multiple entrants to appear, and exploring the possibility of
exclusion in a single-entrant framework may be more suitable. Therefore, the possibility of anticompetitive exclusive dealing increases.
Several outstanding issues require future work. First, there is concern about the generality of ourresults. While the analysis is in terms of a parametric example, the results may extend to settings
that are more general. Second, there is a concern about the incumbent’s behavior needed to achievea market environment where exclusive dealing is possible. Previous studies have mainly explored theexistence of exclusive dealing for anticompetitive reasons. However, the incumbent may be able toaﬀect the market environment to exclude the more eﬃcient entry. We trust this study will assist future
researchers in addressing these important issues.

A Equilibria in the subgame following Stage 1
This Appendix provides the characterization of equilibria in the subgame following Stage 1. We
consider each of the possible subgames for S = 2, S = 0, and S = 1, respectively. We first explore
the case of linear wholesale pricing in Section A.1. We then examine the case of nonlinear wholesalepricing in Section A.2.

A.1 Linear wholesale pricing
A.1.1 S = 2
When both buyers sign exclusive contracts in Stage 1, they both deal with the incumbent. The incumbent sets wholesale prices for each buyer that maximize its profit by taking into account the buyers’
pricing in Stage 4 given its wholesale price: i.e.:
w sj|S =2 = arg max
w j ≥cI∑(w j − cI )q j (p1 (w1 , w2 ), p2 (w1 , w2 )),(8)

j∈{1,2}14

For example, MDS Nordion, which produces Molybdenum-99, a radioisotope, has a 10-year exclusive contract with two
Japanese companies, Nihon Medi-Physics and Daiichi Radioisotope Laboratory. These companies produce Technetium-99mfrom Molybdenum-99. The key point is that Molybdenum-99 is the only factor input available to produce Technetium-99m.
This may make exclusive dealing easier.

17

subject to:p j (w j , w− j ) = arg max (p j − w j )q j (p j , p− j ),p j ≥w j(9)given w− j for j = 1, 2. In the equilibrium, the incumbent yields:
ΠI|S =2 =(α − cI )2,2β(1 + γ)(2 − γ)(10)

π sj|S =2 =(α − cI )2 (1 − γ).
4β(1 + γ)(2 − γ)2(11)and each buyer j = 1, 2 yields:

A.1.2 S = 0
When neither buyer signs an exclusive contract in Stage 1, each becomes a free buyer and deals with
Entrant 1. Because of the competition between entrants, the equilibrium wholesale price decreases tofthe marginal cost of Entrant 2: w j|S =0 = cE2 for all j = 1, 2. Given this wholesale price, each buyer
chooses the price to maximize its profits: i.e.:fp j|S =0 = arg max (p j − cE2 )q j (p j , p− j ),p j ≥cE2(12)

for all j = 1, 2. In the equilibrium, the incumbent obtains zero profits: ΠI|S =0 = 0. On the other hand,each buyer j = 1, 2 yields:fπ j|S =0 =(α − cE2 )2 (1 − γ)j
≥ 4πS =2 .
β(1 + γ)(2 − γ)2(13)

A.1.3 S = 1
When one of buyers signs the exclusive contract but the other does not, only the signer deals with theincumbent. Without loss of generality, assume that buyer − j signs the exclusive contract. Then, buyer
j becomes the free buyer and deals with Entrant 1 in equilibrium. Entrant 2 oﬀers its marginal cost to
the free buyer. On the other hand, the incumbent and Entrant 1 choose the profit maximization priceby taking into account the buyers’ pricing in the final stage given their wholesale prices: i.e.
(*)
fw j|S =1 = arg max (w j − cE1 )q j (p j (w j , p− j ), p− j ),w j ≥cE1

18

(14)

and:
w−s j|S =1 = arg max (w− j − cI )q j (p j , p− j (p j , w− j )),(15)

w j ≤ cE2 ,(16)

p j (w j , p− j ) = arg max (p j − w j )q j (p j , p− j ),(17)

p− j (p j , w− j ) = arg max (p− j − w− j )q− j (p j , p− j ).

(18)

w− j ≥cI

subject to:

p j ≥w j

p− j ≥w− j

Let γL (θE1 ), γ M (θE1 ), and γH (θE1 ) be defined such that:
1. for γ ≤ γL (θE1 ), Entrant 1 sets its wholesale price to the free buyer at cE2 regardless of theeﬃciency of Entrant 2;
2. for γ > γ M (θE1 ), the incumbent sets its wholesale price to the signer at cI if entrants are almost
identical: θE1 − θE2 ≤ ϵ for any ϵ > 0; and3. for γ ≥ γH (θE1 ), the incumbent sets its wholesale price to the signer at cI regardless of theeﬃciency of Entrant 2.
In order to derive γL and γH , we assume that no constraints are active in the problem (*). Then,f

we obtain the pair (w j|S =1 , w−s j|S =1 ):w j|S =1 =8(α + cI ) − 3γ2 (2α + cI ) − 2γ(α − cE1 ),16 − 9γ2(19)

w−s j|S =1 =8(α + cE1 ) − 3γ2 (2α + cE1 ) − 2γ(α − cI ).
16 − 9γ2

(20)

f

and:

f

f

Because cE2 < cI , we have w j|S =1 ≥ cE2 if w j|S =1 ≥ cI . Therefore, the critical value γL is derived byfsolving w j|S =1 = cI with respect to γ, which implies:
γL =

θE1 +

√

196 − 340θE1 + 145(θE1 )2 − 2.
3(4 − θE1 )(21)

For 0 ≤ γ ≤ γL , Entrant 1 sets its wholesale price to the free buyer at cE2 even when Entrant 2 has low
eﬃciency.
19

The critical value γH is derived by solving w−s j|S =1 = cI with respect to γ, which implies:√
49 − 48θE1 + 12(θE1 )2 − 1.
γH =3(2 − 3θE1 )(22)For γH ≤ γ < 1, the incumbent sets its wholesale price to the signer at cI regardless of the eﬃciency
of Entrant 2. The free buyer then monopolizes the downstream market in equilibrium.
f

In order to derive γ M , we assume that the entrants are almost identical so that w j|S =1 = cE2 inproblem (*). Then, we obtain:f

w−s j|S =1 (w j|S =1 = cE2 ) =4(α + cI ) − γ2 (2α + cI ) − 2γ(α − cE2 ).
8 − 3γ2

The critical value γ M is derived by solving w−s j|S =1 = cI with respect to γ, which implies:√
9 − 8θE1 + 2(θE1 )2 − 1.
γM =2 − θE1

(23)

(24)

For γ ≤ γ M , the incumbent sets its wholesale price to the signer at w−s j|S =1 > cI even if the entrants arealmost identical. On the other hand, for γ > γ M , the incumbent sets its wholesale price to the signerat w−s j|S =1 = cI if Entrant 2 is suﬃciently eﬃcient: γ > γ M (θE2 ).
Note that γL < γ M < γH and all critical values are strictly decreasing in θE1 ∈ [0, 1]. We define
four cases as follows: (A) 0 ≤ γ ≤ γL , (B) γL < γ ≤ γ M , (C) γ M < γ < γH , and (D) γH ≤ γ < 1. We
explore each case as follows.
Case (A)f (A)In case (A), w j|S =1 = cE2 but the other constraints are not active in problem (*). In the equilibrium,
the incumbent yields:
Π(A)=I|S =12((α − cI )(2 − γ2 ) − γ(α − cE2 ))2,β(1 − γ2 )(8 − 3γ2 )2(25)

and the signer yields π−s(A)= Π(A)/2. On the other hand, Entrant 1 yields:
j|S =1
I|S =1Π(A)=E1 |S =1

(cE2 − cE1 )((α − cE2 )(4 − 3γ2 ) − γ(α − cI )),β(1 − γ2 )(8 − 3γ2 )(26)((α − cE2 )(4 − 3γ2 ) − γ(α − cI ))2.
β(1 − γ2 )(8 − 3γ2 )2(27)

and the free buyer yields:f (A)π j|S =1 =20

Case (B)
f (B)

f (B)

In case (B), w−s(B)> cI but w j|S =1 is not always restricted by the marginal cost of Entrant 2. w j|S =1 is
j|S =1f (B)not restricted by the marginal cost of Entrant 2 if the entrant is not eﬃcient. More precisely, w j|S =1 <cE2 if and only if θE2 < Θiw where:θE1 (16 − γ(2 + 9γ)) − 4(1 − γ)(4 + 3γ).
(2 − γ)(4 + 3γ) − γθE1

Θiw (γ, θE1 ) =In the equilibrium, the incumbent yields:2((4−3γ2 )(α−cI )−γ(α−cE1 ))2




(B)
β(1−γ2 )(16−9γ2 )2
ΠI|S =1 = 


Π(A)
I|S =1

if 0 ≤ θE2 < Θiw
if Θiw ≤ θE2 < θE1 ,

(28)

(29)

and the signer yields π−s(B)= Π(B)/2. On the other hand, Entrant 1 yields:
j|S =1
I|S =1
2((4−3γ2 )(α−cE1 )−γ(α−cI ))2




(B)
β(1−γ2 )(16−9γ2 )2
ΠE1 |S =1 = 


Π(A)
E |S =1
1

if 0 ≤ θE2 < Θiw
if Θiw ≤ θE2 < θE1 ,(30)

and the free buyer yields:


(B)
iw


ΠE1 |S =1 (0 ≤ θE2 < Θ )/2f (B)
π j|S =1 = π f (A)j|S =1

if 0 ≤ θE2 < Θiw
if Θiw ≤ θE2 < θE1 .

(31)

Case (C)In case (C), the incumbent and signer do not always obtain positive demands. The incumbent and thesigner yield positive demands if Entrant 2 is not eﬃcient. More precisely, w−s(C)> cI if 0 ≤ θE2 <j|S =1Θ−1
M (γ) where:
Θ−1
M (γ) =

2(2 + γ)(1 − γ),
2 − γ2(32)

and where Θ−1
M (γ) is an inverse function of γ M (θ). In contrast, if Entrant 2 is suﬃciently eﬃcient,
s(C)
15
Θ−1
M (γ) ≤ θE2 < θE1 , then the incumbent and the signer yield zero demands w− j|S =1 = cI . Note that
15
Fumagalli and Motta (2006) point out in their Section 3 that multiple equilibria exist when the free buyer monopolizes
the downstream market S = 1. When the free buyer monopolizes the downstream market, the signer does not capture any
demand and profits in the subgame for S = 1. From the viewpoint of the incumbent, regardless of the price the incumbent
sets for the signer, the signer obtains no demand and the incumbent yields zero profits. Therefore, there exists a multiplicity
of equilibria in the subgame for S = 1. Fumagalli and Motta (2006) then show that both entry and exclusion equilibriums
exist. However, this depends on the assumption that the wholesale price to the signer is publicly observable and that the

21

iw
−1
iw
Θ−1
M > Θ if and only if γ < γH . Therefore, in case (C), we have Θ M > Θ . When the free buyerf (C)monopolizes the downstream market, it sets its price at p− j|S =1 = (cI − α(1 − γ))/γ, which is suﬃcientf (C)for monopolization. Entrant 1 sets its wholesale price to the free buyer at w j|S =1 < cE2 if and only if
θE2 < Θiw as in case (B). In the equilibrium, the incumbent yields:
 (B)


ΠI|S =1 (0 ≤ θE2 < Θiw ) if 0 ≤ θE2 < Θiw



 (A)
Π(C)
=
ΠI|S =1
if Θiw ≤ θE2 < Θ−1

I|S =1
M



0
<
θ
if Θ−1
≤
θ
E1 ,
E2
M
and the signer yields π−s(C)= Π(C)/2. On the other hand, Entrant 1 yields:
j|S =1
I|S =1 (B)ΠE1 |S =1 (0 ≤ θE2 < Θiw ) if 0 ≤ θE2 < Θiw





Π(A)
if Θiw ≤ θE2 < Θ−1
Π(C)
=
M

E1 |S =1
E1 |S =1
−1 ≤ θ
 (cE2 −cE1 )(α−cI )
<
θ
if
Θ
E1 ,
E2
M
βγ

(33)

(34)

and the free buyer yields:

 (B)


ΠE1 |S =1 (0 ≤ θE2 < Θiw )/2π f (A)f (C)π j|S =1 = 
j|S =12 (α−cI ) (γ(α−c2E2 )−(α−cI ))βγ

if 0 ≤ θE2 < Θiw
if Θiw ≤ θE2 < Θ−1M(35)

if Θ−1M ≤ θE2 < θE1 .

Case (D)In case (D), the incumbent and the signer never obtain positive demand. They set w−s(D)= p−s(D)=j|S =1
j|S =1
cI . In order to induce the free buyer to monopolize the downstream market, Entrant 1 needs to set its
f (D)

f (D)

wholesale price at w j|S =1 = min{cE2 , ((2−γ2 )cI −α(1−γ)(2+γ))/γ}. w j|S =1 is restricted by the marginal
cost of Entrant 2 if and only if θE2 ≥ Θ−1
M . The free buyer sets its price at p j|S =1 = (cI − α(1 − γ))/γ.
f (D)In the equilibrium, both the incumbent and the buyers yield zero profits. However, Entrant 1 yields:
(α−cI )(γ(α−cE1 )−(2−γ2 )(α−cI ))
if 0 ≤ θE2 < Θ−1

(D)
M
βγ2
(36)
ΠE1 |S =1 = −1 ≤ θ
−1 ≤ θ

Π(C)
<
θE1 ,
)
if
Θ
<
θ
(Θ
E
E
E
2
1
2
M
M
E1 |S =1
and the free buyer yields:

(α−cI )2 (1−γ2 )



f (D)
βγ2
π j|S =1 = 

f (C)

π j|S =1 (Θ−1M ≤ θE2 < θE1 )if 0 ≤ θE2 < Θ−1M
if Θ−1M ≤ θE2 < θE1 .

(37)

incumbent can commit not to change the wholesale price to the signer. Note that when the incumbent oﬀers a high wholesale
price to the signer, the free buyer chooses a price slightly lower than the wholesale price or its monopoly price. Therefore,
the incumbent can yield positive profits by charging a wholesale price lower than the free buyer’s price. Our assumption ofunobservable wholesale prices implies that the incumbent cannot solve this commitment problem. Under the unobservable
wholesale price case, the unique equilibrium outcome in the subgame for S = 1 is that the incumbent oﬀers cI to the signer.

22

A.2 Two-part tariﬀs
A.2.1 S=2
When both buyers sign exclusive contracts in Stage 1, each deals with the incumbent. The incumbent
maximizes its profit by setting its wholesale price so that each of buyers chooses the joint profit
maximizing price pSs =2 = (α + cI )/2. This wholesale price is wSs =2 = cI + γ(α − cI )/2. The incumbent
extracts all of the buyers’ profits by using a fixed fee and yields all profits:
ΠI|S =2 =(α − cI )2.
2β(1 + γ)(38)

On the other hand, each of the buyers yields zero profits: π sj|S =2 = 0 for j = 1, 2.
A.2.2 S=0
When neither buyer signs an exclusive contract in Stage 2, all upstream firms compete to deal witheach buyer. Because Entrant 1 is the most eﬃcient firm, it attracts both buyers in the equilibrium. The
incumbent oﬀers its best terms (cI , 0) to both buyers. Entrant 2 oﬀers its best terms (cE2 , 0) to bothbuyers. Entrant 1 only has to match Entrant 2’s oﬀer to attract both buyers. In the equilibrium, bothbuyers yield the duopoly profit:f

π j|S =0 =(α − cE2 )2 (1 − γ)j
≥ 4πS =2 (γ).
2β(1 + γ)(2 − γ)(39)

for j = 1, 2. On the other hand, the incumbent yields zero profits: ΠI|S =0 = 0. These equilibriumoutcomes are identical to linear wholesale pricing.
A.2.3 S=1
When one of the buyers signs the exclusive contract but the other does not, only the signer deals with
the incumbent. Without loss of generality, assume that buyer − j signs the exclusive contract. Then,buyer j becomes the free buyer. In the equilibrium, the free buyer j deals with Entrant 1. Entrant 2
then oﬀers its best terms (cE2 , 0) to the free buyer, and Entrant 1 matches this to attract the free buyer
j. Therefore, the free buyer j’s profit is determined by the profit oﬀ the equilibrium path where it

23

accepts oﬀers from Entrant 2. The incumbent oﬀers the wholesale price w−s j|S =1 = cI and extracts allfof signer − j’s profits. Downstream firms compete in prices given w j|S =1 = cE2 and w−s j|S =1 = cI :max (p j − cE2 )q j (p j , p− j ),(40)

max (p− j − cI )q− j (p j , p− j ).

(41)

p j ≥cE2

p− j ≥cI

Note that p−s j|S =1 > cI if and only if γ < γ M . We define two cases as follows: (a) 0 ≤ γ < γ M and (b)γ M ≤ γ < 1. We explore each case as follows.
Case (a)For γ < γ M , the incumbent yields:
Π(a)=I|S =1((2 − γ2 )(α − cI ) − γ(α − cE2 ))2.
β(1 − γ2 )(4 − γ2 )2(42)((2 − γ2 )(α − cE2 ) − γ(α − cI ))2.
β(1 − γ2 )(4 − γ2 )2(43)

In contrast, free buyer j yields:f (a)π j|S =1 =Case (b)For γ ≥ γ M , free buyer j monopolizes the downstream market and chooses the monopoly price. In
the monopoly equilibrium, the incumbent and signer − j yield zero profits: Π(b)= π−s(b)= 0. InI|S =1
j|S =1
contrast, the free buyer yields:f (b)π j|S =1 =(α − cI )2 (γ(α − cE2 ) − γ(α − cE2 ))
.
βγ2

(44)

B Proofs of all results
Proof of Proposition 2
See Proof of Proposition 3.
Q.E.D.

24

Precise statement of Proposition 3
Suppose that upstream firms are restricted to oﬀer linear wholesale prices. The possibility of exclusive
dealing is determined by the degree of downstream competition and the cost eﬃciency of entrants(γ, θE1 , θE2 ) as follows.
1. If downstream competition is not too strong (γ < 0.76), then the incumbent cannot exclude
eﬃcient entry regardless of the cost eﬃciency of entrants.
2. For suﬃciently strong downstream competition (γ > 0.94):(a) the incumbent cannot exclude eﬃcient entry if both Entrant 1 and Entrant 2 are suﬃciently
eﬃcient (θE1 > θE2 > ΘL (γ)); or(b) the incumbent can exclude eﬃcient entry if Entrant 2 is less eﬃcient (θE2 ≤ ΘL (γ)).
3. For an intermediate level of downstream competition:(a) the incumbent cannot exclude eﬃcient entry if either Entrant 1 is suﬃciently eﬃcient(θE1 > Θ s (γ) and θE1 > 0.65 is suﬃcient) or if Entrant 2 is suﬃciently eﬃcient (θE2 >ΘL (γ) and θE2 > 0.32 is suﬃcient); or(b) the incumbent can exclude eﬃcient entry either if Entrant 1 is not eﬃcient (θE1 ≤ ΘL (γ)) or
if Entrant 1 is not too eﬃcient and Entrant 2 is not eﬃcient (Θ s (γ) ≥ θE1 > ΘL (γ) ≥ θE2 ),where√
2(4 + 3γ)((4 − 3γ) (1 − γ)(3 − 2γ) + 2(1 − γ)(2 − γ))Θ (γ) =,
(45)√
(16 − 9γ2 ) (1 − γ)(3 − 2γ) − 2γ(2 − γ)√2((1−γ)(192+γ(64+160γ−64γ2 −39γ3 +18γ4 ))−2(2−γ)(8−3γ2 )(4−3γ2 )) (1−γ)(3−2γ))
for 0 ≤ γ < 0.94,192−γ(320+32γ−256γ2 +73γ3 +45γ4 −18γ5 )ΘL (γ) = 
2 (4γ2 −14γ+9))2(16(1−γ)−γ
for 0.94 ≤ γ < 1.
2s(4−γ)(4+γ−2γ )

(46)

25

Proof of Proposition 3
We first explore the existence of an exclusion equilibrium when the wholesale price to the free buyer
is not restricted by the marginal cost of Entrant 2 (θE2 < Θiw ). For γ < 0.94, inequality (7) holds forall θE1 ∈ [0, 1]. For γ < 0.76, inequality (7) never holds. For an intermediate level of downstream
competition, inequality (7) holds if and only if θE1 ≤ Θ s (γ).
Second, we explore the existence of exclusion equilibrium when the wholesale price to the freebuyer is restricted by the marginal cost of Entrant 2 (θE2 ≥ Θiw ). For γ < 0.76, inequality (7) neverholds. For γ ≥ 0.76, inequality (7) holds if and only if θE2 ≤ ΘL (γ). When exclusion is possible, thefincumbent oﬀers x ≥ πS =1 − πSs =2 .
Finally, we prove that the incumbent excludes Entrant 1 and Entrant 2 as a unique equilibriumfoutcome by oﬀering x = πS =1 − πSs =2 . To do this, we show that for all for all (θE1 , θE2 , γ) ∈ [0, 1] ×
f

f

f

f

[0, θE1 ] × [0, 1], πS =1 + πSs =1 ≥ πS =0 + πSs =2 . Let H = πS =1 + πSs =1 − (πS =0 + πSs =2 ). Note that:fπS =0 (γ, θE2 ) + πSs =2 (γ) =(α − cI )2 (1 − γ)(20 − 4θE2 + (θE2 )2 ), for all (θE2 , γ) ∈ [0, 1] × [0, 1]. (47)4β(1 + γ)(2 − γ)2 (2 − θE2 )2f

On the other hand, πS =1 + πSs =1 depends on γ. There are four cases, as shown in Appendix A.1.3.
Case (A)f (A)(A)
When 0 ≤ γ ≤ γL , πS =1 + πSs(A)=1 depends on γ and θE2 . By diﬀerentiating H (θE2 , γ) with respect toθE2 , we have ∂H (A) (θE2 , γ)/∂θE2 ≥ 0 for all (θE2 , γ) ∈ [0, 1] × [0, 1]. Because:
H (A) (0, γ) =(α − cI )2 γ(1 − γ)(128 + γ(32 − 48γ − 5γ2 ))≥ 0 for all 0 ≤ γ ≤ 1,β(1 + γ)(2 − γ)2 (8 − 3γ2 )2(48)

we always have H (A) (θE2 , γ) ≥ 0 in case (A).
Case (B)For γL < γ ≤ γ M , there are two possibilities: θE2 ≥ Θiw and θE2 < Θiw . Because inequality (48) holdsf (B)for all 0 ≤ γ ≤ 1, we have H (B) (θE2 , γ) > 0 for θE2 ≥ Θiw . On the other hand, for θE2 < Θiw , πS =1 +πSs(B)=1fdepends on γ and θE1 . Therefore, H (B) is a function of γ, θE1 , and θE2 . Because πS =0 (θE2 , γ) is strictlyincreasing in θE2 , we examine the case θE2 = Θiw (θE1 , γ) where obtaining H (B) (θE1 , θE2 , γ) ≥ 0 is26most diﬃcult in θE2 ≤ Θiw . By diﬀerentiating H (B) (θE1 , Θiw (θE1 , γ), γ) with respect to θE1 , we have
∂H (B) (θE1 , Θiw (θE1 , γ), γ)/∂θE1 ≥ 0 for all (θE2 , γ) ∈ [0, 1] × [0, γH ]. Because:iw−1
H (B) (Θ−1
L , Θ (ΘL (γ), γ), γ) =(α − cI )2 γ(1 − γ)(128 + γ(32 − 48γ + 5γ2 ))≥ 0 for all 0 ≤ γ ≤ 1,
4β(1 + γ)(2 − γ)2 (8 − 3γ2 )2(49)

we have H (B) (θE1 , θE2 , γ) ≥ 0 for θE2 < Θiw . Therefore, we always have H (B) (θE1 , θE2 , γ) ≥ 0 in case
(B).
Case (C)−1For γ M < γ ≤ γH , there are three possibilities: 0 ≤ θE2 < Θiw , Θiw ≤ θE2 < Θ−1
M , and Θ M ≤ θE2 .

Because inequalities (48) and (49) hold for all 0 ≤ γ ≤ 1, we have H (C) (θE1 , θE2 , γ) > 0 for 0 ≤
s(C)
(C)
−1
θE2 < Θ−1
M . On the other hand, for Θ M ≤ θE2 , πS =1 = 0 and πS =1 depends on θE2 and γ. Hence, H
f (C)is a function of γ, θE1 , and θE2 . Because πS =0 (θE2 , γ) is strictly increasing in Θ−1
M ≤ θE2 < θE1 , wef (C)examine the case θE2 = θE1 where obtaining H (C) (θE1 , θE2 , γ) ≥ 0 is most diﬃcult in Θ−1
M ≤ θE2 ≤ θE1 .
By diﬀerentiating H (C) (θE1 , θE1 , γ) with respect to θE1 , we have ∂H (C) (θE1 , θE1 , γ)/∂θE1 ≥ 0 for all(θE1 , γ) ∈ [Θ−1M , 1] × [γ M , 1]. Because:−1
H (C) (Θ−1M (γ), Θ M (γ), γ) =(α − cI )2 γ(1 − γ)(16 + γ(3 − 8γ))
≥ 0 for all γ M ≤ γ ≤ 1,
4β(1 + γ)(2 − γ)2

(50)

(C)
we have H (C) (θE1 , θE2 , γ) ≥ 0 for Θ−1
M ≤ θE2 ≤ θE1 . Therefore, we always have H (θE1 , θE2 , γ) ≥ 0 incase (C).
Case (D)−1For γH ≤ γ < 1, there are two possibilities: θE2 ≥ Θ−1
M and θE2 < Θ M . Because inequality (50) holds−1for γH ≤ γ ≤ 1, we have H (D) (θE1 , θE2 , γ) ≥ 0 for θE2 ≥ Θ−1
M . On the other hand, for θE2 < Θ M ,
f (D)

f (D)

(D) is a function of θ
πSs(D)
E2 and γ. Because πS =0 (γ) is=1 = 0 and πS =1 depends only on γ. Hence, H(D) (Θ−1 , γ) ≥ 0 is
−1strictly increasing in θE2 < Θ−1M
M , we examine the case θE2 = Θ M where obtaining H(D) (Θ−1 , γ) = H (C) (Θ−1 , Θ−1 , γ), inequality (50) implies thatmost diﬃcult in θE2 ≤ Θ−1
MM
M
M . Because H(D) (θ , θ , γ) ≥ 0 in case
we have H (D) (θE1 , θE2 , γ) ≥ 0 for θE2 < Θ−1
E1 E2
M . Therefore, we always have H

(D).
27

f

From cases (A), (B), (C), and (D), for all (θE1 , θE2 , γ) ∈ [0, 1] × [0, θE1 ] × [0, 1], we have πS =1 +f

πSs =1 ≥ πS =0 + πSs =2 . Therefore, the incumbent excludes Entrant 1 and 2 as a unique equilibriumfoutcome by oﬀering x = πS =1 − πSs =2 .
Q.E.D.

Precise statement of Proposition 5
Suppose that upstream firms adapt two-part tariﬀs (w, ψ). The incumbent can exclude eﬃcient entry
either if Entrant 1 is not eﬃcient (θE1 ≤ ΘNL (γ)) or if Entrant 2 is not eﬃcient (θE2 ≤ ΘNL (γ) < θE1 ).
The incumbent cannot exclude eﬃcient entry if Entrant 2 is suﬃciently eﬃcient (θE2 > ΘNL (γ) andθE2 > 0.32 is suﬃcient), where:(1−γ)(γ3 +2(4−γ2 )−2(γ3 +2(1+γ)(2−γ))




NL
16(1−γ)−γ2 (2−γ)2 (3+γ)Θ (γ) = 2(4−3γ2 ) (2+γ)2

for 0 ≤ γ < 0.94,
for 0.94 ≤ γ < 1.

(51)

Proof of Proposition 5
We first explore the existence of exclusion equilibrium. Inequality (7) hold if and only if θE2 ≤ ΘNL (γ).
Because θE1 > θE2 , exclusion exists if θE1 ≤ ΘNL (γ). On the other hand, when θE1 > ΘNL (γ), exclusionfexists if θE2 ≤ ΘNL (γ). When exclusion is possible, the incumbent oﬀers x ≥ πS =1 .
Next, we prove that the incumbent can exclude Entrants 1 and 2 as a unique equilibrium outcomefff

by oﬀering x = πS =1 . We show that for all (θE2 , γ) ∈ [0, 1] × [0, 1], πS =1 ≥ πS =0 . We examine case (a)and case (b) respectively.
Case (a)For S = 0, the free buyer yields duopoly profits where its and its rival’s wholesale price is the marginal
cost of Entrant 2. However, for S = 1, the free buyer yields duopoly profits where its wholesale priceis the marginal cost of Entrant 2 but its rival’s wholesale price is the marginal cost of the incumbent.
Because the buyer’s profit is strictly increasing in its rival’s wholesale price, the free buyer yields
f (a)

f

higher profits for S = 1. Hence, πS =1 ≥ πS =0 for all (θE2 , γ) ∈ [0, 1] × [0, γ M ].
28

Case (b)
f

f

f (b)

f (b)

f (b)
Note that πS =1 (γ) ≥ πS =1 (γ M ) and πS =0 (γ) ≤ πS =0 (γ M ) for all γ ≥ γ M . Because πS =1 (γ M ) =
f (a)

f (b)

f

f

πS =1 (γ M ) ≥ πS =0 (γ M ), we have πS =1 ≥ πS =0 for all (θE2 , γ) ∈ [0, 1] × [γ M , 1]. Therefore, for allff(θE2 , γ) ∈ [0, 1] × [0, 1], πS =1 ≥ πS =0 .
Q.E.D.

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Ashiya, M., 2000. Weak Entrants Are Welcome. International Journal of Industrial Organization
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Bork, R.H., 1978. The antitrust paradox: A policy at war with itself. New York: Basic Books.
Fumagalli, C., and Motta, M., 2006. Exclusive Dealing and Entry, when Buyers Compete. American
Economic Review 96(3), 785–795.
Ishibashi, I., 2003. A Note on Credible Spatial Entry Deterrence. International Journal of Industrial
Organization 21(2), 283–289.
Posner, R.A., 1976. Antitrust law: An economic perspective. Chicago: University of Chicago Press.
Rasmusen, E.B., Ramseyer, J.M., and Wiley, John, S. Jr., 1991. Naked Exclusion. American Economic
Review 81(5), 1137–1145.
Segal, I.R., and Whinston, M.D., 2000. Naked Exclusion: Comment. American Economic Review
90(1), 296–309.
Simpson, J., and Wickelgren, A.L., 2007. Naked Exclusion, Eﬃcient Breach, and Downstream Competition. American Economic Review 97(4), 1305–1320.

29

Whinston, M.D., 2006. Lectures on Antitrust Economics. Cambridge, MA: MIT Press.
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30

pΠ I |S = 2

π Ss = 2

Π E |S = 2

α + cI2

P = α − βq
cI

cE1
α −cI
2β

α −cI
β

q

Figure 1: S=2pΠ I |S =1

π Sf =1

Π E|S =1P = α − βq
cI

cE1
α −cI
β

q

Figure 2: S=1 (Single entrant case)pΠ I |S =1

π Sf =1

Π E|S =1P = α − βq

cI
c E2
cE1
α −cI
β

Figure 3: S=1 (Multiple entrant case)

q

θE 1
1

Θs (γ )
Exclusion if
EntryθE ≤ ΘL (γ )2Entry if

θE > ΘL (γ )2ΘL (γ )
Exclusion

10γ
Figure 4: Linear wholesale pricing (Multiple entrant case)θE 11Θs (γ )

Entry

Exclusion

10γ
Figure 5: Linear wholesale pricing (Single entrant case)

θE 1
1

Exclusion

10γ
Figure 6: Non-linear wholesale price (Single entrant case)θE 1
1

Exclusion if θ E2 ≤ ΘNL (γ )Entry if θ E2 > ΘNL (γ )ΘNL (γ )Exclusion

0
Figure 7: Non-linear wholesale price (Multiple entrant case)1γ