GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 352

Exclusive Contracts with Complementary Inputs

Hiroshi Kitamura, Noriaki Matsushima, and Misato Sato

January 2015

GCOE Secretariat
Graduate School of Economics
OSAKA UNIVERSITY
1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Exclusive Contracts with Complementary Inputs∗
Hiroshi Kitamura†

Noriaki Matsushima‡

Misato Sato§

January 6, 2015

Abstract
This study constructs a model of anticompetitive exclusive contracts in the presenceof complementary inputs. A downstream firm transforms multiple complementary inputs
into final products. When complementary input suppliers have market power, upstream
competition within a given input market benefits not only the downstream firm (by lowering the input price) but also complementary input suppliers (by raising complementary
input prices). The downstream firm is thus unable to earn higher profits even whensocially efficient entry is allowed. Hence, the inefficient incumbent supplier can detersocially efficient entry by using exclusive contracts even in the absence of economies ofscale and downstream competition. These results have important implications for antitrust agencies, showing the importance of considering the existence of complementary
inputs when examining cases of potential anticompetitive exclusive dealing.

JEL classifications code: L12, L41, L42.
Keywords: Antitrust policy; Complementary inputs; Exclusive dealing; Multiple inputs.
∗

We thank Eric Avenel, Chiara Fumagalli, Toshihiro Matsumura, Patrick Rey, Dan Sasaki, and Tetsuya
Shinkai for their insightful comments. We also thank Akifumi Ishihara, Akihiko Nakagawa, Ryoko Oki,
Noriyuki Yanagawa, the conference participants at EARIE 2014 (Bocconi University) and the Japanese Economic Association (Seinan Gakuin University), and the seminar participants at Kwansei Gakuin University,
Kyoto University, Osaka University, The University of Rennes 1, and Yokohama National University for helpful discussions and comments. The second author thanks the warm hospitality at MOVE, Universitat Autònoma
de Barcelona where part of this paper was written. We gratefully acknowledge financial support from JSPS
Grant-in-Aid. The usual disclaimer applies.
†
Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto, Kyoto 603-8555,
Japan. Email: hiroshikitamura@cc.kyoto-su.ac.jp‡
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. Email: nmatsush@iser.osaka-u.ac.jp§
Department of Economics, The George Washington University, 2115 G street, NW Monroe Hall 340 Washington DC 20052, USA. Email: smisato@gwmail.gwu.edu

1Introduction

In vertical supply chain relationships, firms often engage in contracts including vertical restraints, such as exclusive contracts, loyalty rebates, slotting fees, resale price maintenance,
quantity fixing, and tie-ins.1 Among vertical restraints, exclusive contracts have long been
controversial.2 Once signed, exclusive contracts deter efficient entrants; they thus may appear to be anticompetitive. However, scholars from the Chicago School oppose this viewby arguing, based on analytic models, that rational economic agents do not sign contracts todeter more efficient entrants (Posner, 1976; Bork, 1978).3 In rebuttals of this argument, postChicago economists indicate specific circumstances under which anticompetitive exclusive
dealing occurs.4 These studies, by extending the single-buyer model upon which the Chicago
School argument depends to a multiple-buyer model, introduce scale economies wherein theentrant requires a certain number of buyers to cover its fixed costs (Rasmusen, Ramseyer,
and Wiley, 1991; Segal and Whinston, 2000a) and competition between buyers (Simpson
and Wickelgren, 2007; Abito and Wright, 2008).
The present study extends this work by noting another economic environment within
which anticompetitive exclusive dealing occurs: when there are complementary inputs. Ina real-world business situation, final-good producers often transform multiple inputs into
final products. More importantly, there exist complementary input suppliers with market
power. In the Intel antitrust case, for example, Microsoft is a supplier with strong market
power.5 Therefore, when applying analysis of anticompetitive exclusive dealing to real-world1

See, for example, Rey and Tirole (1986), Rey and Vergé (2010), and Asker and Bar-Issac (2014). See also
Rey and Tirole (2007) and Rey and Vergé (2008) for surveys of vertical restraints.
2
These exclusive-dealing agreements take various forms, such as restricting territories or rights. See, forinstance, Mathewson and Winter (1984), Rey and Stiglitz (1995), and Matsumura (2003).
3
For analysis of the impact of this argument on antitrust policies, see Motta (2004) and Whinston (2006).
4
In an early contribution, Aghion and Bolton (1987) propose a model in which exclusion does not always
occur, but, when it does, it is anticompetitive. See also Bernheim and Whinston (1998), which explores themarket circumstances under which an exclusive contract can exclude rival incumbents.
5Intel was accused of awarding rebates and various other payments to major original equipment
manufacturers (e.g., Dell and HP). In a single quarter in 2007, conditional rebates and paymentsfrom Intel amounted to 76% of Dell’s operating profit (Gans, 2013). See also Japan Fair Trade
Commission (2005): http://www.jftc.go.jp/eacpf/cases/intel.pdf and the European Commission (2009):

1

situations, the interaction between complementary input suppliers cannot be neglected.
In this study, we develop a model of anticompetitive exclusive dealing in the presenceof complementary inputs. In our model, an upstream incumbent supplier offers an exclusive
contract to a single downstream firm to deter an entrant supplier that is more efficient than
the incumbent supplier—and would thus constitute a socially efficient entry. Because there
are neither scale economies nor downstream competition, the incumbent supplier cannot detersocially efficient entry with exclusive contracts, as in the frameworks of previous studies. The
new dimension in our model is that the downstream firm produces a final product using not
only an input produced by the incumbent supplier but also a complementary input produced
by a supplier with market power.
In a simple setting with linear demand and linear wholesale pricing, we first show thatthe existence of a complementary input supplier with market power allows the incumbent
supplier to deter socially efficient entry via exclusive contracts, even in the case of a single
downstream firm. To intuitively understand this result, consider the effects of socially efficient
entry. Socially efficient entry into an input market generates competition, which reduces the
input’s price and thus allows the downstream firm to earn higher profits. Under the Chicago
School argument, this effect prevents the incumbent supplier from profitably compensating
the downstream firm; Chicago School scholars thus conclude that exclusion is impossible.
However, when there is a complementary input supplier with market power, socially efficient
entry into one input market increases the demand for the complementary input, which benefits
the complementary input supplier by increasing the complementary input price. Hence, compared with the case in which there is no complementary input supplier with market power,socially efficient entry leads to a smaller increase in the downstream firm’s profits. This allows the incumbent supplier to profitably compensate the downstream firm, and therefore
exclusion is possible.
We also check the robustness of the above exclusion outcome and show that it can beobserved in more general settings. First, introducing non-linear demand, we show that the
http://ec.europa.eu/competition/sectors/ICT/intel.html.

2exclusion outcome results as long as the demand curve for final products is not overly convex; that is, exclusion is more likely to be observed under inelastic demand. Furthermore, aunique exclusion equilibrium occurs in a context of non-linear wholesale pricing and a general demand function. 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 (Rasmusen,
Ramseyer, and Wiley, 1991; Segal and Whinston, 2000a; Simpson and Wickelgren, 2007;
Abito and Wright, 2008).6 These studies share a common feature: reaching the exclusion result requires multiple downstream buyers.78 In contrast, this study shows that anticompetitive
exclusive contracts can be signed even under a single-buyer model.
In terms of a single-buyer model of anticompetitive exclusive contracts, this study is closest to Fumagalli, Motta, and Rønde (2012) who explore a model in which exclusive dealing
can both promote relationship-specific investment and foreclose a more efficient supplier.
They show that if relationship-specific investment is possible, inefficient market foreclosure
occurs even in the single-buyer model because the investment promotion effect of exclusive
dealing increases the joint surplus of the incumbent seller and the downstream buyer under
exclusive dealing. In contrast, exclusion in the present study arises because the existence ofcomplementary inputs decreases the downstream firm’s profits when socially efficient entry
6

In addition, certain studies examine pro-competitive exclusive dealings; e.g., non-contractible investments(Marvel, 1982; Besanko and Perry, 1993; Segal and Whinston, 2000b; de Meza and Selvaggi, 2007; de Fontenay, Gans, and Groves, 2010), industry R&D and welfare (Chen and Sappington, 2011), risk sharing (Argenton
and Willems, 2012), and adverse selection (Calzolari and Denicolò, 2013). Exclusive dealing is also considered
as a way to solve the commitment problem posited by Hart and Tirole (1990), which arises when a single upstream firm sells to multiple retailers with two-part tariffs under unobservable contracts. See also O’Brien andShaffer (1992), McAfee and Schwartz (1994), and Rey and Vergé (2004).
7
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’s (2009) study, which corrects the result of
Fumagalli and Motta (2006) in the case of two-part tariffs.
8
For extended models of exclusion with downstream competition, see Wright (2008), Argenton (2010), andKitamura (2010, 2011). Whereas these studies all show that the resulting exclusive contracts are anticompetitive,
Gratz and Reisinger (2013) show potentially pro-competitive effects if downstream firms compete imperfectly
and contract breaches are possible. See also DeGraba (2013) who adapts the model to cover a situation in whicha small rival that is more efficient at serving a portion of the market can make exclusive offers.

3

occurs.
This study is also related to the literature on vertical relationships involving complementary inputs, such as the work of Laussel (2008), Matsushima and Mizuno (2012, 2013), andReisinger and Tarantino (2013).9 These studies demonstrate that vertical integration is notnecessarily profitable because the complementary input supplier extracts the majority of the
profits generated as a result of eliminating double marginalization through vertical integration. The present study extends this work, showing that these ideas can be applied to theliterature on entry deterrence via exclusive contracts.
The remainder of this paper is organized as follows. Section 2 constructs the basic model.
Section 3 presents the main results under linear wholesale pricing and extends the basic modelin several dimensions. Section 4 provides analysis under non-linear wholesale pricing, while
Section 5 offers concluding remarks. Additionally, the appendices support the work with
proofs.

2

ModelThis section develops the basic model environment. We first explain the basic characteristicsof players and the game’s timing in Section 2.1. Section 2.2 then introduces the design of the
anticompetitive exclusive contracts.

2.1

Basic environment

The upstream market is composed of two complementary input markets: A and B (Figure1). Input A is exclusively produced by supplier U A with a constant marginal cost c > 0.
In contrast, input B is produced by an incumbent supplier U IB and an entrant supplier U EB .
Following the Chicago School model, U IB and U EB produce an identical product but withdiffering cost efficiencies; namely, U EB is more efficient than U IB , with a constant marginal
cost cE ∈ [0, c) as opposed to U IB ’s constant marginal cost c > 0.
9

See also Arya and Mittendorf (2007) and Laussel and Long (2012).

4

The downstream market is composed of a single firm. This modeling strategy clarifies therole of U A as having market power; that is, the prevention of socially efficient entry occurs
even in the absence of scale economies and downstream competition, both of which require
more than one downstream firm. A downstream monopolist D transforms inputs A and B intoa final product. D uses Leontief production technology: one unit of the final product is madewith one unit of input A and one unit of input B:10
Q = min{qA , qB },(1)

where qi is the amount of input i ∈ {A, B}. Equation (1) implies that the two inputs are essential to produce a final product in a downstream market—i.e., they are perfect complements.11
The payment for qi units of input i ∈ {A, B} is given as wi qi , where wi is input i’s price. To
simplify the analysis, we assume that D incurs no production costs aside from paying for thetwo inputs. Thus, per unit production cost of D is given by
cD = wA + w B .
We assume that the inverse demand for the final product P(Q) is given by a simple linear
function:P(Q) = a − bQ,(2)where Q is the output of the final product supplied by D, a > 2(2c − cE ), and b > 0. The firstinequality implies that U EB ’s monopoly price is higher than c; namely, the existence of U IB
always acts a constraint on the pricing of U EB when U EB enters the market for input B.12
The model contains four stages (Figure 2). In Stage 1, U IB makes an exclusive offer to
D with fixed compensation x ≥ 0. D decides whether to accept the offer.13 D immediately
Introducing asymmetric production technology where one unit of the final product is made with h > 0 units
of input A and m > 0 units of input B does not qualitatively change the results.
11
Such production technology can be widely observed in real-world manufacturing. For example, producing
a PC requires a CPU and an operating system, and producing a car requires a body and tires.
12
Exclusion will still exist even when U EB is more efficient, but the analysis becomes more complicated. In
Section 4, we consider a case in which the demand function follows a general demand form.
13
Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000a) point out that price commitments
10

5

receives x if it accepts the offer. In Stage 2, U EB decides whether to enter the market for
input B. We assume that the fixed entry cost is sufficiently small that U EB can earn positive
profits. In Stage 3, active suppliers offer linear input prices to D. (In Section 4, we introduce
non-linear pricing.) We assume that if U EB enters the market for input B, then U IB and U EB
become homogeneous Bertrand competitors. The equilibrium price of input i ∈ {A, B} when
D accepts (rejects) the exclusive offer is denoted by wai (wri ) where the superscript ‘a’ (‘r’)
indicates that the exclusive offer is accepted (rejected). In Stage 4, D orders inputs and sells
the final product to consumers. Uk ’s profit in the case where D accepts (rejects) the exclusive
offer is denoted by πak (πrk ), where k ∈ {A, IB, EB}. D’s profit in the case when it accepts(rejects) the exclusive offer is denoted by πaD (πrD ).

2.2

Design of exclusive contracts

For an exclusion equilibrium to exist, the equilibrium transfer x∗ must satisfy the following
two conditions.
First, it must satisfy individual rationality for U IB ; that is, U IB earns higher profits under
exclusive dealing:
πaIB − x∗ ≥ πrIB .

(3)

Second, the exclusive contract must satisfy individual rationality for D; that is, the amountof compensation x induces D to accept the exclusive offer:
πaD + x∗ ≥ πrD .

(4)From the above conditions, it is evident that a unique exclusion equilibrium exists if
and only if inequalities (3) and (4) simultaneously hold. This is equivalent to the following
condition:
πaIB + πaD ≥ πrIB + πrD .

(5)

are unlikely if the product’s nature is not described precisely in advance. In the naked exclusion literature, it
is known that if the incumbent can commit to wholesale prices, then anticompetitive exclusive dealings areenhanced. See Yong (1999) and Appendix B of Fumagalli and Motta (2006).

6

Condition (5) implies that to determine whether anticompetitive exclusive contracts exist, wemust examine whether exclusive contracts increase the joint profits of U IB and D.

3

Linear wholesale pricing

This section analyzes the existence of anticompetitive exclusive contracts under linear wholesale pricing. To intuitively understand the importance of a complementary input supplier
having market power, we first discuss a benchmark case in which complementary input A
is competitively provided (Section 3.1). We then explore the case in which complementary
input A is supplied by a monopolistic supplier and also examine how the curvature of theinverse demand function influences the result (Section 3.2). Finally, Section 3.3 discusses
how market power in input market A influences the result in Section 3.2.

3.1

Benchmark: When a complementary input is competitively supplied

Assume that complementary input A is competitively provided. Then, input A’s price does
not depend on whether the exclusive offer is accepted in Stage 1; i.e., waA = wrA = c. In
this setting, the Chicago School model can be interpreted as a special case in which D canpurchase complementary input A for free; i.e., waA = wrA = 0.14 Therefore, as the Chicago
School approach argues, U IB cannot deter socially efficient entry.
Proposition 1. Suppose that inverse demand is given by a linear function and upstream
suppliers adopt linear wholesale pricing. When complementary input A is competitively provided, U IB cannot deter socially efficient entry by using exclusive contracts.
The result here can be explained using logic similar to that underlying the Chicago School
argument. When D accepts the exclusive offer in Stage 1, it purchases input B from U IB
14
Although a buyer is the final consumer in the Chicago School model, the result does not qualitatively change
if we assume that the buyer is a downstream monopolist. See Lemma 1 of Kitamura, Sato, and Arai (2014).

7

at a higher input price, which allows U IB to earn monopoly profits. However, under linear wholesale pricing, U IB and D cannot maximize their joint profits because of the double
marginalization problem. Thus, the left-hand side of inequality (5) is quite small.
In contrast, when D rejects the exclusive offer in Stage 1, U EB enters the market for input
B in Stage 2. In Stage 3, U IB and U EB compete to manage D. Compared with the case of
exclusive dealing, upstream competition in the B market reduces input B’s price, which solves
the double marginalization problem. Thus, D earns considerably higher rejection profits; thatis, the right-hand side of inequality (5) becomes large. In the absence of scale economies
and downstream competition, U IB cannot profitably compensate D, implying that there is
no x that simultaneously satisfies participation constraints (3) and (4). Therefore, when thedownstream market consists of a monopolist, anticompetitive exclusive dealing cannot occur
if complementary input A is competitively supplied.

3.2

When a complementary input is provided by a monopolist

We now assume that complementary input A is provided exclusively by U A . Input A’s price
now depends on whether entry into the market for input B occurs, unlike in the previous
subsection. As in the case where complementary input A is provided competitively, entry
into the market for input B generates competition within it, reducing B’s price. To understandthe pricing behavior of U A when input B’s price decreases, we first examine the relationshipbetween entry into the B market and input A’s price. The following lemma summarizes therelationship:
Lemma 1. When U A has market power, socially efficient entry into the market for input B
raises the equilibrium price of input A—that is, wrA > waA .
To understand this result, consider the reaction function of the monopolistic supplier of
input i given input j’s price w j ;wi (w j ) =a + c − wj,
2b8

where i, j ∈ {A, B} and i , j. It is easy to see that the strategic interaction between thetwo monopolistic suppliers is strategic substitute; that is, raising input A’s price is the best
response for U A when input B’s price decreases.15 Therefore, entry into the market for input
B induces U A to raise input A’s price and enjoy higher profits. The following propositionshows that this relationship between entry into the market for input B and input A’s priceallows U IB to deter efficient entry by using exclusive contracts:
Proposition 2. Suppose that inverse demand is given by a linear function and upstream
suppliers adopt linear wholesale pricing. If U A is the monopolist for input A, U IB can detersocially efficient entry as a unique exclusionary equilibrium outcome.
Note that the crucial difference as compared to the benchmark case exists in the subgame in which D rejects the exclusive offer. As in the case in which input A is competitively
supplied, entry into the market for input B generates upstream competition in this market,
which benefits D. However, Lemma 1 implies that entry into this market also benefits U A by
increasing input A’s price, which prevents D from earning considerably higher profits upon
rejecting the exclusive offer; that is, the right-hand side of inequality (5) does not become
sufficiently large. This allows U IB to profitably compensate D using its profits under exclusive
dealing. Therefore, the existence of a complementary supplier with market power leads toanticompetitive exclusive dealing even in the absence of scale economies and downstream
competition.
Remark (Non-linear demand)

The previous results rest on the assumption of a linear

inverse demand function. We now relax this assumption to extend Section 3.2’s analysis witha non-linear inverse demand function:
P(Q) = a − bQα .
15(6)

If inputs A and B were substitutes, a reduction in input B’s price would decrease demand for input A, andthus U A would be required to lower A’s price. In contrast, if inputs A and B are complements, a reduction in
input B’s price increases demand for input A.

9

We assume that α > 0 so that firms’ second-order conditions are satisfied. We also assume
that a > ((1 + α)(c − cE ) + 2αc)/α so that U EB ’s monopoly price is higher than c. Comparing
equations (2) and (6) reveals clearly that the linear demand case considered in the previous
subsections is a special case in which α = 1. The price elasticity of demand is given by
−

dQ/Q
p
=
.
d p/p
α(a − p)(7)

Equations (6) and (7) imply that as α increases (decreases), the demand curve becomes concave (convex) or inelastic (elastic). The following proposition shows that the likelihood of
exclusion depends on the curve’s shape.
Proposition 3. Suppose that upstream suppliers adopt linear wholesale pricing. If U A is themonopolistic supplier of input A, U IB can deter socially efficient entry as a unique equilibriumoutcome as long as the demand curve for the final product is not overly convex, α ≥ α '0.40692.
This result has an important implication: exclusion is more likely to be observed when
demand is inelastic. The curvature of the inverse demand curve influences the degree of
demand-expansion following socially efficient entry. When the inverse demand curve is concave (α > 1), the demand-expansion effect of a new upstream entrant is weak; as a result,socially efficient entry does not lead to a large increase in D’s profits, allowing U IB to profitably compensate D. However, as the inverse demand curve becomes convex, the demandexpansion effect becomes significant; in other words, the double marginalization problem ismore serious in the case of convex inverse demand. This leads to a large increase in D’s
profits, preventing U IB from profitably compensating D.

3.3

Discussion

To probe how the monopoly power of a complementary input supplier influences the previous
subsection’s outcome, we briefly discuss the case in which a competitor also supplies complementary input A. We suppose that input A is produced by not only an incumbent supplier,10

U IA , but also an inefficient supplier, U EA . The marginal cost of U EA is dA ∈ (c, a/2). We
assume that U EA enters the market for input A in Stage 2, that U IA and U EA are homogeneous
Bertrand competitors, and that the fixed cost of entry is zero, enabling U EA to enter the market
for input A.16 Note that the analyses in Section 3.1 and 3.2 can be interpreted as the cases inwhich dA = c and dA ≥ a/2, respectively. The following proposition shows that the likelihoodof exclusion depends on the difference in efficiency between the suppliers of complementary
input A.17
Proposition 4. Suppose that input A can be produced by U IA and U EA . U IB can deter socially efficient entry as a unique equilibrium outcome if the efficiency difference between the
suppliers of complementary input A is sufficiently large—that is, dA ≥ d A wherea − 2cdA = a − c − √ .
3Note that c < d A < a/2. The result here fills the gap between Propositions 1 and 2
and confirms the significance of the complementary input supplier being able to control its
input price. When U EA ’s efficiency is high (that is, dA is close to c), the existence of U EA
acts as a constraint on the pricing of U IA and significantly dampens the monopoly power of
U IA . This implies that U IA does not extract sufficient profits from D, regardless of whether
U IB ’s exclusive offer is accepted. In this environment, engaging in exclusive dealing with U IB
significantly reduces D’s profit through double marginalization.
In contrast, when U EA is sufficiently inefficient (that is, dA ∈ ((a+c)/3, a/2)), the existenceof U EA does not necessarily constrain the pricing of U IA . On the one hand, if U IB ’s exclusive
offer is accepted, the existence of U EA never influences U IA ’s pricing decisions. On the other,
if U IB ’s exclusive offer is rejected, the existence of U EA does constrain U IA ’s pricing decisions. In this case, input A’s price depends on U EA ’s efficiency, dA . As noted in Section 3.2,
16

If the entry cost is positive, U EA does not enter input A market because it anticipates zero operating profitsand cannot cover this fixed cost. It may be harder to justify this assumption. However, even if U EA had to incur
the fixed cost, it might enter. For example, this might occur if there were upstream differentiation in input A’s
market. Alternatively, if U EA is already established and operating in other industries, it could supply input A
without additional fixed costs.
17
The result does not change even though we assume that both U IA and U IB independently and simultaneously
make exclusive offers to D with fixed compensation xA (≥ 0) and xB (≥ 0), respectively.

11the stronger the monopoly power of U IA , the lower D’s profitability if U IB ’s exclusive offer isrejected. Therefore, if U EA ’s efficiency is low enough that dA ≥ d A , then U IB and D engage inanticompetitive exclusive dealing even in the presence of competition in the complementary
input market.
Interpretation of these results yields an important policy implication for antitrust agencies:
an increase in the cost efficiency of a dominant input supplier can facilitate anticompetitive
exclusive dealing in complementary input markets. As the dominant input supplier becomes
more efficient, industry output usually increases, which is socially efficient. However, if
we consider the possibility of exclusive contracts in complementary input markets, an improvement in the efficiency of the dominant input supplier may trigger exclusion outcomesin complementary input markets, which is socially inefficient. Therefore, when consideringreal-world exclusive contracts, antitrust agencies should pay attention to not only the structure of the market in which the exclusive contracts are signed but also structural changes inrelated complementary input markets.

4

Non-linear wholesale pricing

In this section, we extend Section 3.2’s model analysis to the case in which upstream suppliers
use two-part tariffs in Stage 3. Two-part tariffs consist of a linear wholesale price and an upfront fixed fee; the two-part tariff offered by input supplier i ∈ {A, IB, EB} when D accepts(rejects) the exclusive offer is denoted by (wai , Fia ) ((wri , Fir )). Under non-linear pricing, thedouble marginalization problem is avoidable, and the joint profit is maximized within thetechnological endowments available to firms. For notational convenience, we define Π∗ (z) as
follows:Π∗ (z) ≡ max(p − z − c)Q(p),pwhere z ≥ 0 and Q(p) is a general demand function. Π∗ (z) can be interpreted as the maximum
joint profit when input B is supplied with marginal cost z. When entry does not occur (occurs),
we have z = c (z = cE ). Hence, the difference between Π∗ (cE ) and Π∗ (c) depends on the12difference between the efficiencies of U EB and U IB . To simplify the analysis, we assume thatΠ∗ (z) is continuous and strictly decreasing in z.
We first consider the case in which the exclusive offer is accepted in Stage 1. Thereaare multiple equilibria in which U A and U IB respectively offer (c, F Aa ) and (c, F IB
) such thata
F Aa + F IB
= Π∗ (c). The firms’ equilibrium profits, excluding the fixed compensation x, are

0 ≤ πaA ≤ Π∗ (c), 0 ≤ πaIB ≤ Π∗ (c), πaD = 0,awhere πaA + πaIB = Π∗ (c). Multiple equilibria consist of any F Aa ≥ 0 and F IB
≥ 0 such thata
F Aa + F IB
= Π∗ (c).

We next consider the case in which the exclusive offer is rejected in Stage 1. In Stage 2,
U EB then enters the market for input B. In Stage 3, U IB offers its best term (c, 0). Anticipatingrr
this term, U A and U EB respectively offer (c, F Ar ) and (cE , F EB), which satisfy Π∗ (cE ) − F EB−r
F Ar ≥ Π∗ (c) − F Ar if F Ar ≤ Π∗ (c) and Π∗ (cE ) − F EB
− F Ar ≥ 0 if F Ar > Π∗ (c). The inequalities thatr
include F EB
indicate that D either prefers U EB ’s offer to that of U IB or treats them equally.
r
By arranging the condition, we conclude that (c, F Ar ) and (cE , F EB
) satisfy the following:rr
0 ≤ F EB≤ Π∗ (cE ) − Π∗ (c) and F Ar + F EB
= Π∗ (cE ). Equilibrium firms’ profits are then given

as follows:Π∗ (c) ≤ πrA ≤ Π∗ (cE ), πrIB = πrD = 0, 0 ≤ πrEB ≤ Π∗ (cE ) − Π∗ (c),where πrA + πrEB = Π∗ (cE ). Multiple equilibria consist of any Π∗ (c) ≤ F Ar ≤ Π∗ (cE ) andrr
0 ≤ F EB≤ Π∗ (cE ) − Π∗ (c) such that F Ar + F EB
= Π∗ (cE ).

Finally, we check for the existence of exclusion. Because we have πaIB +πaD −(πrIB +πrD ) ≥ 0
with equality for πaIB = 0, inequality (5) always holds. Therefore, exclusion is always anequilibrium outcome.
Proposition 5. Suppose that upstream suppliers adopt two-part tariffs. If U A has market
power, then U IB can deter socially efficient entry as a unique equilibrium outcome.
There are a few remarks that must be made regarding this section’s analysis. First, as
pointed out above, there exist multiple equilibria in Stage 3. Second, the profit allocation is13

not consistent with the linear wholesale pricing case. When complementary input suppliers
adopt two-part tariffs, D earns zero operating profits regardless of its decision in Stage 1.
Hence, the main intuition of the Chicago School argument, in which the downstream buyer
enjoys a large surplus, cannot be applied.

5

Concluding remarks

This study explores the existence of anticompetitive exclusive dealing, extending the work ofprevious studies to consider the role of complementary inputs in the upstream market. It isessential that this interaction between complementary input suppliers not be neglected, as it
is common for real-world downstream firms to transform multiple inputs into final products.
Our analysis shows that seemingly small differences in the model’s setting can have crucial ramifications for the results. If the complementary input supplier has market power,
then the inefficient incumbent supplier can deter socially efficient entry by using exclusive
contracts even under the Chicago School argument’s framework in which there is a single
downstream buyer. On checking the robustness of our exclusion outcome, we show that it
does not depend on the assumption of linear demand or linear wholesale pricing; that is, ourresults remain valid when the final product demand curve is not too convex or when nonlinear wholesale pricing is introduced. This study’s analysis can thus be widely applied toreal-world vertical relationships.
Our results also have novel and important implications for antitrust agencies: it is necessary to take into account the existence of complementary inputs when considering the possibility of anticompetitive exclusive dealing. If we discuss the anticompetitiveness of exclusive
contracts while ignoring the existence of complementary input suppliers with market power,
we might over-emphasize the results of the Chicago School argument. This study’s results
suggest that rational economic agents can easily engage in anticompetitive exclusive dealing
in a market requiring multiple inputs to produce a downstream product for which demand isinelastic.

14

Despite these contributions, there remain several outstanding issues requiring further research. The first involves this study’s relationship to other studies of anticompetitive exclusive
dealing. For example, we assume that the downstream firm is a monopolist. We predict that,as Simpson and Wickelgren (2007) and Abito and Wright (2008) discuss, adding downstream
competition to our model would increase the likelihood of reaching an exclusion equilibrium.
In addition, we assume a single entrant in each input market. As Kitamura (2010) discusses,
the likelihood of an exclusion equilibrium decreases if multiple entrants are added to themodel.
The second area for further analysis concerns the generality of our results. For example,
the present study’s analysis assumes Leontief production technology. While this might beappropriate for analyzing certain real-world situations, such as the Intel antitrust case, theresult might also remain valid under more general production technologies. In addition, in
Section 3.3 we assume that the complementary input suppliers are homogeneous Bertrand
competitors. If we were to add differentiation between the complementary input suppliers to
our model, the likelihood of an exclusion equilibrium would increase. A real-world example of this is the case of Eastman Kodak v. Fuji, in which Kodak complained of exclusive
contracts between Fuji and Japanese wholesalers that prevented Kodak from successfully
entering Japan’s photographic film market.18 The Japanese camera market, which complements the film market, is characterized by dominant manufacturers such as Canon and Nikon.
These manufacturers are somewhat differentiated and have market power because consumers
face switching costs when considering changing cameras. Extensions and applications of ourmodel can thus help researchers and policymakers address similar real-world issues.
18

See, for example, Nagaoka and Goto (1997).

15

A

Proofs of All Results

Proof of Proposition 1
Before, we proceed the proof, we explain D’s maximization problem. Given the input prices,wA and wB , the problem is given as
max (a − bQ − wA − wB )Q.
Q(8)The output of the final product supplied by D and the demand for each input are
a − wA − w B
qA = qB = Q(wA , wB ) ≡.
(9)2bWhen input A is competitively provided, we have waA = wrA = c. We first explore the casewhere the exclusive contract is accepted in Stage 1. The outcome in Stage 4 is given in (9).
In Stage 3, U IB sets the input price;
waB = arg max(wB − c)Q(c, wB ).
wB

The equilibrium input prices are waB = a/2 and waA = c. From (9), the equilibrium productionlevels are Qa (c, a/2) = qaA = qaB = (a − 2c)/6b. The firms’ equilibrium profits, excluding the
fixed compensation x, are(a − 2c)2(a − 2c)2, ΠaD =
.
(10)
8b
16b
We next explore the case where the exclusive offer is rejected in Stage 1. Then, U EB
πaA = 0, πaIB =enters the B market in Stage 2. In Stage 3, the competition in the B market reduces input B’s
price to U IB ’s marginal cost; i.e., wrB = c. From (9), the equilibrium production levels are
Qr (c, c) = qrA = qrB = (a − 2c)/2b. The equilibrium profits are(a − 2c)2.
(11)
4bFinally, we check the existence of an exclusion equilibrium. From (10) and (11), we have
πrA = πrIB = 0, πrD =πaIB + πaD − (πrIB + πrD ) = −(a − 2c)2< 0,
16bwhich implies that condition (5) never holds.
Q.E.D.
16

Proof of Lemma 1
Suppose first that the exclusive offer is accepted in Stage 1. The maximization problem of D
in Stage 4 is given as (8). The output of the final product supplied by D and the demand foreach input are given as (9). The maximization problem of supplier i in Stage 3 is given as
max(wi − c)Q(wi , w j ),

(12)

wi ≥c

where i, j ∈ {A, B}, i , j. The first-order conditions lead to the equilibrium input prices:waA = waB =

a+c
.
3

(13)

From (9), the equilibrium production levels are Qa = qaA = qaB = Q((a + c)/3, (a + c)/3) =(a − 2c)/6b. The firms’ equilibrium profits, excluding the fixed compensation x, are
πaA = πaIB =(a − 2c)2(a − 2c)2, πaD =
.
18b
36b

(14)

Suppose next that the exclusive offer is rejected in Stage 1. In this case, U EB enters
the B market in Stage 2. The maximization problem of D in Stage 4 is given as (8). The
output of the final product supplied by D and the demand for each input are given as (9).
The competition in the B market in Stage 3 leads to wrB = c. By contrast, the maximization
problem of U A in Stage 3 is given as (12). The equilibrium price of input A isawrA = .
2

(15)

From (9), the equilibrium production levels are Qr = qrA = qrB = (a − 2c)/4b. The equilibrium
profits of firms are(a − 2c)2(a − 2c)2, πrIB = 0, πrD =.
8b16bFrom (13) and (15), we have the following relation:πrA =waA − wrA = −(16)

a − 2c< 0.
6

Therefore, entry into the B market increases input A’s price.
Q.E.D.
17

Proof of Proposition 2
From (14) and (16), we haveπaIB + πaD − (πrIB + πrD ) =(a − 2c)2> 0,
48bwhich implies that condition (5) always holds.
Q.E.D.

Proof of Proposition 3
We first consider the case where the exclusive offer is accepted in Stage 1. The firms’ equilibrium profits, excluding the fixed compensation x, areπaIB =

α((1 + α)b) α1

a − 2c1 + 2α

! 1+α
α

πaD =

,

α
1

bα

α(a − 2c)
(1 + α)(1 + 2α)

! 1+α
α

.

We next consider the case where the exclusive offer is rejected in Stage 1. The equilibrium
profits of firms are given as
πrIB = 0,

πrD =

α
1

bα

a − 2c(1 + α)2

! 1+α
α

.

Finally, we consider the existence of exclusion. By using above results, we haveα(a − 2c)πaIB + πaD − (πrIB + πrD ) =
1

1+α
α

1+α

2(1+α)(1 + α) α (2 + α) − ((1 + α)(1 + 2α)) α3(1+α)(1 + α) α (1 + 2α) α

bα

1+α

≥ 0,

if and only if α ≥ α. Therefore, condition (5) holds if and only if α < α.
Q.E.D.

Proof of Proposition 4
We first consider the case where U IB ’s exclusive offer is accepted in Stage 1. The existenceof U EA acts as a constraint on the pricing of U IA although U IA is the supplier of input A on
18

the equilibrium path. U IB monopolistically sets input B’s price. The maximization problemof D in Stage 4 is given as (8). The output of the final product supplied by D and the demandfor each input are (9). The maximization problems of U IA and U IB in Stage 3 are given as
a − wA − w B, s.t. wA ≤ dA ,2ba − wA − w B
max (wB − c).
wB
2bmax (wA − c)wA

These maximization problems lead to

a+c



,
dA
if c < dA <



3
a
wA = 

a+c
a+c



if dA ≥,
33

a + c − dA
a+c



if c < dA <
,



2
3
waB = 


a+c
a+c


if dA ≥.

3
3

The equilibrium production levels are a−d −c
a+c

A


, if c < dA <
,



4b
3
qaA = qaB = Qa = 
a+ca − 2c
,
if dA ≥.
6b
3
The equilibrium profits, excluding the fixed compensation x, are given as(a − dA − c)2
a+c



if c < dA <
,



16b
3
a
πD = 
a+c(a − 2c)2
if dA ≥
,

36b
3


(dA − c)(a − dA − c)
a+c



if
c
<
d
<
,
A



4b
3
a
πIA = 
(a − 2c)2
a+c



,
if dA ≥
,

18b
3


a+c
(a − dA − c)2
if c < dA <
,



8b
3
πaIB = (a − 2c)2a+c
if dA ≥.

18b
3
Second, we consider the case where U IB ’s exclusive offer is rejected in Stage 1. As in thefirst case, the existence of U EA acts as a constraint on the pricing of U IA . The equilibrium
19

supplier of input B is U EB and its price is wrB = c. The maximization problem of D in Stage 4
is given as (8). The output of the final product supplied by D and the demand for each input
are given as (9). The maximization problem of U IA in Stage 3 is given as
max (wA − c)wAa − wA − c, s.t. wA ≤ dA .
2bThis maximization problem leads toa
dA if c < dA < ,2
wrA = aa

if dA ≥ .
2
2
The equilibrium production levels are a−d −caA, if c < dA <



2b
2
qrA = qrB = Qr = 


a
−
2c
a



,
if dA ≥ .
4b
2
The equilibrium profits are given asa(a − dA − c)2, if c < dA <



4b
2
πrD = 2
(a − 2c)a
,if dA ≥ ,

16b
2


a
(d
−
c)(a
−
d
−
c)
A
A



, if c < dA <



2b
2
πrIA = 2a(a − 2c)
,
if dA ≥ ,
8b2
πrIB = 0.
From now on, we explore the existence of exclusion. By using above results, we haveπaIB + πaD − (πrIB + πrD )(a − dA − c)2
a+c



−
if c < dA <
,



16b
3



n
o
n
o
√
√
√


 3 dA − ((a − c) − (a − 2c)/ 3) (a − 2c) + 3(a − dA − c)
a+c
a
= 


if
≤ dA < ,



12b
3
2



(a − 2c)2a
if dA ≥ .
48b
2
20

It is easy to see that condition (5) holds if and only if dA ≥ d A .
Q.E.D.

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25

Input A Market

Input B Market

𝑼𝑨

𝑼𝑰𝑩

wA

𝑼𝑬𝑩

wB

𝑫

Figure 1: Market Structures

Stage 1

Stage 2

Stage 3

Stage 4

𝑈𝐼𝐵 makes an
exclusive offer.

𝑈𝐸𝐵 makes
entry decision.

Active suppliers
make input priceoffers.

𝐷 orders inputs 𝐴 and 𝐵
and produces final products.

𝐷 decides.

Figure 2: Timeline26