ISSN (Print) 0473-453X

Discussion Paper No. 1115

ISSN (Online) 2435-0982

WHICH IS BETTERFOR DURABLE GOODS PRODUCERS,EXCLUSIVE
OR OPEN SUPPLY CHAIN?

Hiroshi Kitamura
Noriaki Matsushima
Misato Sato

January 2021

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

Which is Better for Durable Goods Producers, Exclusive
or Open Supply Chain?∗
Hiroshi Kitamura†

Noriaki Matsushima‡

Misato Sato§

January 19, 2021

Abstract
We explore the supply chain problem of a downstream durable goods monopolist,
who chooses one of the following trading modes: an exclusive supply chain with anincumbent supplier or an open supply chain, allowing the monopolist to trade with a
new efficient entrant in the future. The predicted retail price reduction in the future
dampens the profitability of the original firms. An efficient entrant’s entry magnifies
such a price reduction, causing a further reduction of original firms’ joint profits. In
equilibrium, the downstream monopolist chooses the exclusive supply chain to escapefurther price reductions, although it anticipates efficient entry.

JEL classification codes: L12, L41, L42.
Keywords: Antitrust policy; Durable goods; Exclusive supply chain; Vertical relation.
∗
An earlier version of this paper is entitled “Exclusive Contracts in Durable Goods Markets.” We thankKeisuke Hattori, Akifumi Ishihara, Hirokazu Ishise, Hiroshi Kinokuni, Gordon Klein, Inés Macho-Stadler,
Chrysovalantou Milliou, Tomomichi Mizuno, Takeshi Murooka, as well as the conference participants at
the 3rd Asia-Pacific Industrial Organisation Conference (The University of Melbourne), EARIE 2019 (Universitat Pompeu Fabra), XXXIV Jornadas de Economı́a Industrial (Universidad Complutense de Madrid),
Japan Association for Applied Economics, Spring Meeting (Nanzan University), and Japanese Economic
Association, Autumn Meeting (Kobe University), and seminar participants at Osaka University (ISER and
OSIPP) and Summer Workshop on Economic Theory (Otaru Chamber of Commerce & Industry). We gratefully acknowledge the financial support from JSPS KAKENHI grant numbers JP15H05728, JP17H00984,
JP17K13729, JP18H00847, JP18K01593, JP19H01483, and JP20H05631, The Japan Center for Economic
Research, and the program of the Joint Usage/Research Center for “Behavioral Economics” at the ISER,
Osaka University. The usual disclaimer applies.
†
Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto-City 603-8555,
Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp‡
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. E-mail: nmatsush@iser.osaka-u.ac.jp§
Graduate School of Humanities and Social Sciences, Okayama University, Tsushima-naka 3-1-1, Kita-Ku,
Okayama 700-8530, Japan. E-mail: msato@okayama-u.ac.jp

1

Introduction

We commonly observe an exclusive supply chain in many markets for durable goods, including industrial machinery/equipment and electronic and electric equipment (Heide, Dutta,
and Bergen, 1998),1 although the recent progress of information communication technology(ICT) helps firms find new trading partners worldwide, which affects the organizational formation of firms in various ways (Granot and Sošić, 2005; World Bank, 2009, Chapter 6;
Bloom et al., 2014). The exclusive supply chain in the smartphone processor market is atypical example.2 Apple, for instance, has selected an exclusive supply chain with Taiwan
Semiconductor Manufacturing Company (TSMC) for Apple’s processors.3 The argument of
supply chain openness is also related to the discussion on exclusive contracts in the context
of competition policy (see the earlier discussions of Posner, 1976 and Bork, 1978). In reality, we have observed anticompetitive exclusive contracts in the market for durable goods
such as aluminum (the United States of America v. Aluminum Co. of America in the US,
1945.), furniture (Paramount Bed Case in Japan, 1998), artificial teeth (the United Statesof America. v. Dentsply International, INC., in the US, 2005), and CPUs (Intel Case in
the US, 2005).4 Despite these observations, previous studies on exclusive vertical chains and1

See also Mollgaard and Lorentzen (2004), who explore exclusive dealings in the Eastern European carcomponent industry. Moreover, in the aviation industry, Boeing Company and Airbus sometimes award
exclusivity to one or two jet engine makers over other makers. See “GE Unit Lands Exclusive Boeing Pact
For Developing Commercial Jet Engine” The Wall Street Journal, July 8, 1999 (https://www.wsj.com/
articles/SB931391538252682453), and “Airbus selects Rolls-Royce Trent 7000 as exclusive engine for the
A330neo” Rolls-Royce, July 14, 2014 (https://www.rolls-royce.com/media/press-releases/yr-2014/
140714-a330neo.aspx).
2

For the case of Qualcomm’s exclusive supply chain with Samsung, see “Samsung beats TSMC to win
new Qualcomm order to make mobile chips” The Korea Economic Daily, October 6, 2020 (https://www.
kedglobal.com/newsView/ked202010100034).
3

See “Taiwan’s TSMC to continue as Apple’s exclusive A-series chip supplier: reports” Taiwan News,
October 13, 2018 (https://www.taiwannews.com.tw/en/news/3551431).
4

For each case, see the United States of America v. Aluminum Co. of America, 148 F.2d 416(1945, https://law.justia.com/cases/federal/appellate-courts/F2/148/416/1503668/); Blair and
Sokol (2015); United States of America. v. Dentsply International, INC., 399 F.3d (2005, https://www.

1exclusive contracts investigate only perishable goods markets. In this study, to contribute tosupply chain management and competition policy, we consider the problem of an exclusive
supply chain in durable goods markets.
We consider a two-period durable goods market, as in Bulow (1986), Denicolò and Garella(1999), and Desai, Koenigsberg, and Purohit (2004), by introducing the entrant supplier inthe second period. We explore the situation in which a downstream durable goods monopolist
chooses one of two trading modes in the first period: (i) a two-period exclusive supply chain
with an existing incumbent supplier, or (ii) an open supply chain, which causes competition
between the incumbent and a potential supplier. We investigate whether the incumbent
supplier and the downstream firm sign a two-period exclusive contract in the first period. To
do it, we combine the model framework of durable goods markets (Bulow, 1986; Denicolò andGarella, 1999; Desai, Koenigsberg, and Purohit, 2004) with the scenario of entry deterrence
through exclusive contracts in the Chicago School argument (Posner, 1976; Bork, 1978).
By introducing two-part tariffs as in Desai, Koenigsberg, and Purohit (2004), we showthat exclusion is a unique equilibrium outcome in a general demand setting; in other words,
the durability of products induces a downstream firm to choose an exclusive supply chain.
As in the Chicago School framework, second-period entry generates upstream competition,
which allows the downstream firm to procure inputs at a lower wholesale price in the second
period. Thus, exclusion seems difficult. However, in durable goods markets, a retail pricereduction through upstream competition in the second period makes some of final consumers
refrain from purchasing in the first period, leading to a low retail price and low joint profits forthe contracting party in the first period. That is, second-period upstream entry exacerbates
the intertemporal pricing problem in the downstream market. Because exclusive contracts
leagle.com/decision/2005580399f3d1811565); Advanced Micro Devices, INC., a Delaware corporation,
and AMD International Sales & Services, LTD., a Delaware corporation, v. Intel Corporation, a Delaware corporation, and Intel Kabushiki Kaisha, a Japanese corporation, Civil Action No. 05-441-JJF (2005, https://www.amd.com/system/files/amd-intel-full-complaint.pdf, respectively.

2

can mitigate such a pricing problem, the contracting party can enjoy higher joint profits
in the first period. Therefore, the incumbent supplier can profitably make a two-period
exclusive offer to the downstream firm in durable goods markets in the first period.
The exclusion mechanism in our study provides an important implication for competition
policy. The Chicago School argument, which introduces the impossibility of exclusive contracts for anticompetitive reasons, does not necessarily apply to durable goods markets. Our
study’s exclusion mechanism arises from the nature of a durable goods monopolist, initiallyargued by Coase (1972).5 Moreover, the exclusion outcomes in our study are derived undera general demand setting. The supplementary appendix also shows that exclusion outcomesare attainable in various extended settings under a linear demand system. Hence, we canapply the findings to diverse real-world exclusive supply chains in durable goods markets.
Because the exclusive contract in this study is a tool to deter the entry of efficient entrants
in the future, this study is suitable for a situation where a local firm faces the threat of entryof highly efficient multinational firms.6 For example, Vist, a Russian personal computermaker, develops exclusive distribution agreements with several key retailers as a survival
strategy toward the entry of multinational firms such as Compaq, IBM, and Hewlett-Packard(Dawar and Frost, 1999). More importantly, multinational firms usually spend some time toactually enter the markets after the news of their entry (Bao and Chen, 2018). Such news
in the media allows every economic agent to predict future entry and the incumbent firm to
respond to the threat of future entry, both of which are necessary for this study’s exclusion
mechanism. Therefore, the exclusion mechanism can apply to such situations. In Section 5,
we introduce the Intel case (2005) as an example of exclusive dealing in detail and consider
the linkage with the results in this study.
5

For a model analysis of Coase conjecture, see Stokey (1981), Bulow (1982), Gul, Sonnenschein, andWilson (1986), and Hart and Tirole (1988).
6

Note that multinational firms usually have high productivity, and they are more efficient than domestic
firms (Helpman, Melitz, and Yeaple, 2004; Kimura and Kiyota, 2006).

3

The findings in this study also provide important predictions for information societies andthe openness of supply chains. Recent developments in ICT seemingly facilitate the opennessof supply chains because such progress allows downstream firms to find alternative trading
partners more easily. However, our findings imply that such a view does not necessarily
remain valid in durable goods markets; downstream durable goods producers may choose todevelop an exclusive supply chain even when an efficient supplier appears in the future.7
The remainder of this paper is organized as follows. Section 2 provides the literature review. Section 3 constructs the model. Section 4 analyzes the existence of exclusion outcomesunder two-part tariffs. Section 5 provides the extension analysis and discusses the linkage
between this study and the Intel case (2005). Section 6 offers concluding remarks. Appendix
A provides the proofs of the results.

2

Literature Review

This study is related to the literature on entry deterrence in durable goods markets.8 Bycomparing selling with renting, Bucovetsky and Chilton (1986) show that a durable goods
monopolist may choose selling to deter future entry. Bulow (1986) also shows that a durable
goods monopolist has an incentive to increase durability to prevent future entry from theviewpoint of planned obsolescence. These studies focus on how vertically integrated durable
goods monopolists influence the demand for future entrants. In contrast, this study discusses entry deterrence by an upstream firm that trades with a downstream durable goods
monopolist by focusing on exclusive contracts.
The market environment in our model is also related to those in which consumers have7

Our focus is also related to the exclusiveness of keiretsu in the Japanese automobile industry (e.g., Aoki
and Lennerfors, 2013) and channel coordination (e.g., Jeuland and Shugan, 1983; Coughlan, 1985; Gupta
and Loulou, 1998; Gupta, 2008).
8

Several studies analyze firms’ strategies to deter future entry in the perishable goods market because ofcost uncertainty (Milgrom and Roberts, 1982), quality uncertainty (Schmalensee, 1982), and switching costs(Klemperer, 1987).

4

multi-period opportunities to purchase final products. In those models, Besanko and Winston (1990) and Dudine, Hendel, and Lizzeri (2006) discuss storable goods markets, andCoase (1972), Bulow (1982, 1986), Denicolò and Garella (1999), Bruce, Desai, and Staelin(2006), Agrawal et al. (2012), and Gilbert, Randhawa, and Sun (2014) investigate durable
goods markets. Our paper contributes to the research stream on those product markets withvertical supply chains (storable goods markets (Desai, Koenigsberg, and Purohit, 2010; Lin,
Parlaktürk, Swaminathan, 2018; Kabul and Parlaktürk, 2019), leasing versus selling (Purohit, 1995; Desai and Purohit, 1999; Bhaskaran and Gilbert, 2009, 2015), and secondary
markets (Shulman and Coughlan, 2007; Oraiopoulos, Ferguson, and Toktay, 2012)).
In particular, this study is related to the literature on anticompetitive exclusive contracts
that deter the socially efficient entry of a potential entrant.9 The literature on anticompetitive exclusive contracts starts from the Chicago School argument in the 1970s (Posner, 1976;
Bork, 1978). Using a simple setting, they point out that rational economic agents neversign exclusive contracts for anticompetitive reasons if we consider all members’ participation constraints in the contracting party.10 In rebuttal to the Chicago School, post-Chicago
economists find that rational economic agents agree with exclusive contracts for anticompetitive reasons in certain market environments. Some papers extend a single-buyer model in the
Chicago School argument to a multiple-buyer model. For instance, the entrant needs a certain number of buyers to cover its fixed costs (Rasmusen, Ramseyer, and Wiley, 1991; Segal
and Whinston, 2000), and buyers compete in downstream markets (Simpson and Wickelgren, 2007; Abito and Wright, 2008).11 In these studies, negative externalities exist; signing
9

Several studies focus on the fact that active firms may compete for exclusivity and explore the welfare
effect (Mathewson and Winter, 1987; O’Brien and Shaffer, 1997; Bernheim and Whinston, 1998). Recently,
Calzolari and Denicolò (2013, 2015) introduced asymmetric information in this literature.
10

For the analysis of the impact of this argument on antitrust policies, see Motta (2004), Whinston (2006),
and Fumagalli, Motta, and Calcagno (2018).
11

In the literature on exclusion with downstream competition, Fumagalli and Motta (2006) show that theexistence of participation fees to remain active in the downstream market plays a crucial role in exclusion
if buyers are undifferentiated Bertrand competitors. See also Wright (2009), who reconsiders the result of5exclusive contracts reduces the possibility of entry under scale economies, and upstream
entry reduces industry profits in the presence of downstream competition.12 Furthermore,in the framework of a single downstream firm, several studies point out that the intensity of upstream competition plays a crucial role in the Chicago School argument.13 Theyshow that the exclusion result is attainable in the cases where the incumbent sets liquidated
damages for the case of entry (Aghion and Bolton, 1987), where the entrant is capacity
constrained (Yong, 1996), where upstream firms compete à la Cournot (Farrell, 2005), andwhere upstream firms can merge (Fumagalli, Motta, and Persson, 2009).14 To the best of ourknowledge, existing papers in this literature consider only perishable goods markets. Thus,
we construct the model here to clarify that the exclusion mechanism in this study depends
on the nature of durable goods markets; exclusion occurs because of the negative externalitythat future entry reduces current industry profits.
Furthermore, the model formulation of our paper has links to the models of durable goods
markets concerning vertical channel coordination (Desai, Koenigsberg, and Purohit, 2004;
Arya and Mittendorf, 2006; Su and Zhang, 2008; Bhaskaran and Gilbert, 2009, 2015; Yang,2012; Gümüş, Ray, and Yin, 2013; Ramanan and Bhargava, 2014). Desai, Koenigsberg, andPurohit (2004) employ two-period durable goods monopoly models with a separate retailchannel to discuss the effect of commitment on the vertical trading term. They show thatthe commitment to the vertical contract overcomes the Coase problem, leading to a higher
Fumagalli and Motta (2006) in the case of two-part tariffs.
12

For the extended model of exclusion with scale economies, see Choi and Stefanadis (2018). By contrast, for extended models of exclusion with downstream competition, see Wright (2008), Argenton (2010),
Kitamura (2010), and DeGraba (2013) who show the anticompetitiveness of the realized exclusive contracts.
Gratz and Reisinger (2013) show procompetitive effects if downstream firms compete imperfectly and contract breaches are possible.
13

For another mechanism of anticompetitive exclusive dealing, see Fumagalli, Motta, and Rønde (2012),
who focus on the incumbent’s relationship-specific investments. See also Kitamura, Matsushima, and Sato(2018), who focus on a complementary input supplier with market power.
14

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

6

profit for the monopoly manufacturer. Arya and Mittendorf (2006) consider a multi-period
durable goods monopoly market. The monopolist determines if it separates its input sector
to overcome the Coase problem. The trading term between the input and retail sectors is alinear wholesale price. They show the possibility that vertical separation by the monopolist
occurs in equilibrium.15 In line with the discussions by Desai, Koenigsberg, and Purohit(2004) and Arya and Mittendorf (2006), Su and Zhang (2008) investigate a newsvendor
problem with demand uncertainty to discuss channel coordination. Bhaskaran and Gilbert(2009, 2015) follow Bulow (1982) to investigate the durability choice by a monopoly manufacturer under four scenarios ((i) centralized or decentralized channel and (ii) selling or leasing).
Gümüş, Ray, and Yin (2013) discuss return policy under demand uncertainty in decentralized durable goods markets. The authors of those papers do not address the openness of the
supply chain in durable goods markets, which is the primary topic of our article.

3

Model

We explain the market environment. There is a common discount factor δ ∈ (0, 1) for all
players who are active in two periods t = 1, 2. The final good is perfectly durable; that is,
the final good produced and used in period 1 can be used in period 2, without depreciation.
For simplicity, there is no resale market. We assume that all firms cannot commit to future
prices and that there is no possibility of renting products.
The rest of this section is organized as follows. We first explain players’ characteristics
and the game’s timing in Section 3.1. Section 3.2 then introduces the design of exclusive
contracts.
15

Yang (2012) extends Arya and Mittendorf (2006) to downstream oligopoly. Ramanan and Bhargava(2014) also employ the framework of Arya and Mittendorf (2006) to discuss consumers’ uncertainty on theirvaluations of the product.

7

3.1

Basic environment

Consumers There are a number of consumers, each of whom buys at most one unit of
the final good. For simplicity, we assume that the mass of consumers is normalized to one.
We index consumers by y ∈ [0, ȳ], whose distribution depends on F (y), where F ′ (y) > 0
for all y ∈ [0, ȳ), F (0) = 0, and F (ȳ) = 1. The consumer type y’s willingness to pay,
which is stationary for all periods, is denoted by v(y), where v ′ (y) > 0 for all y ∈ [0, ȳ). If
a consumer y purchases a unit of the final good in period t(= 1, 2), the consumer’s gross
lifetime discounted utility evaluated in period 1 becomesut (y) = δ t−1 v(y),which implies that each consumer can use the final good for eternity.16 In the supplementary
appendix, we introduce the analysis under linear demand in which F (y) = y and v(y) = y;that is, y is uniformly distributed and the consumer’s willingness to pay is a linear function ofy. This setting allows us to explore the existence of exclusion outcomes in various situations
such as linear wholesale pricing and vertical product differentiation.
In period 2, a consumer type y, who does not purchase the final good in period 1,purchases the final good if and only if the consumer surplus is nonnegative, i.e., v(y)−p2 ≥ 0.
By rationally predicting p2 , the consumer purchases the final good in period 1 if and only
if v(y) − p1 ≥ δ(v(y) − p2 ). The indifferent consumer type y1 is defined by v(y1) − p1 =δ(v(y1 ) − p2 ). Let q1 be the mass of consumers purchasing in period 1. By definition, we
have q1 = 1 − F (y1). Then, by using these two equations, we have the static inverse demandin period 1, which is a function of q1 and p2 , as follows:
P1 (q1 , p2 ) ≡ (1 − δ)v(y(q1 )) + δp2 ,16

Denicolò and Garella (1999) also use a similar formulation to discuss a durable goods market. Alternatively, we can consider the setting in which v̄(y) represents a value of per period use; the surplus of consumer
type y is (1 + δ)v̄(y) − p1 for the purchase in period 1, while it is v̄(y) − p2 for the purchase in period 2. In
the supplementary appendix, we explore such a setting under linear demand and derive exclusion results.

8where y1 (q1 ) ≡ F −1 (1 − q1 ). As y1′ (q1 ) = −1/F ′ (y1 ) < 0 always holds, P1 (q1 , p2 ) has thefollowing properties:∂P1 (q1 , p2 )∂P1 (q1 , p2 )= (1 − δ)v ′ (y1 )y1′ (q1 ) < 0,
= δ > 0.
∂q1
∂p2
The above properties imply that the static inverse demand in period 1 is downward slopingwith respect to the current production level and becomes larger when consumers anticipate
a higher future price.
Firms The upstream market consists of an incumbent supplier UI and an entrant supplier
UE . UI and UE produce an identical input but differ in terms of cost efficiency. The constantmarginal costs of UI and UE are cI (< v(ȳ)) and cE ∈ [0, cI ), respectively. To guarantee thatsome consumers prefer to purchase the final good in period 1 (q1 > 0), we assume that thedifference in suppliers’ marginal costs is not too large. In Section 4.3, we extend the model tothe case in which the difference in the efficiency is too large and no consumer purchases the
final good in period 1 when UE ’s entry is anticipated. Each supplier offers a two-part tariff,which consists of a linear wholesale price w and an upfront fixed fee ψ to the downstream
market if it is active.
The downstream market is composed of a downstream monopolist D. This modelingstrategy clarifies the role of durable goods; namely, the prevention of efficient entry occurs
even in the absence of scale economies and downstream competition, both of which require
more than one downstream firm. D transforms one unit of the input into one unit of the final
good, which is durable. To simplify the analysis, we assume that the cost of transformationis zero. Thus, if D faces a wholesale price wt in period t, its per-unit production cost inperiod t becomes wt . The assumption on the downstream monopolist follows that in Desai,
Koenigsberg, and Purohit (2004).

9

Timing Following Bucovetsky and Chilton (1986) and Bulow (1986), in period 1, only UI
exists in the upstream market. This may be because of a patent right, superior technology,
efficient marketing, or an industry protection policy. Period 1 consists of three stages. In
period 1.1, UI makes a two-period exclusive offer to D, with fixed compensation x ≥ 0. This
assumption simply follows the standard literature on naked exclusion. In Section 3.2, weshow that for the existence of exclusion outcomes, it does not matter whether UI or D makes
exclusive offers. Following the standard literature on naked exclusion, we assume that theexclusive offer does not contain the term of wholesale prices.17 After observing the exclusive
offer, D decides whether to accept the offer. D immediately receives x if it accepts the offer.
Let ω ∈ {a, r} be D’s decision in period 1.1, where the superscript “a” (“r”) indicates thatthe exclusive offer is accepted (rejected). In period 1.2, UI offers a two-part tariff contract
to D. In period 1.3, D orders the input and sells the final good to consumers.
At the beginning of period 2, UE appears in the upstream market. Period 2 consists ofthree stages. In period 2.1, UE decides whether to enter the market.18 We assume that the
fixed entry cost is sufficiently small such that UE can earn positive profits. In period 2.2,
active suppliers offer two-part tariff contracts to D. For the case of UE ’s entry, UI and UE
become homogeneous Bertrand competitors. We assume that if they charge the same input
price, the efficient supplier, UE , supplies its input to D. In period 2.3, D orders the input
and sells the final good to consumers.
ω
Ui ’s profit in period t when D’s decision in period 1.1 is ω ∈ {a, r} is denoted by πi|t,ωwhere i ∈ {I, E}. Likewise, D’s profit in period t is denoted by πD|t
. We assume that UI
17

Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000) point out that price commitments are unlikely if the nature of the final good is not precisely described in advance. In the naked
exclusion literature, it is known that if the incumbent can commit to wholesale prices, then the possibilityof anticompetitive exclusive dealing is enhanced. See Yong (1999) and Appendix B of Fumagalli and Motta(2006).
18

The result does not change if we consider the possibility that UI makes exclusive offers in period 2 if
its exclusive offer in period 1 is rejected. In such a case, the Chicago School argument can be applied; UI
cannot make exclusive offers to profitably compensate D in period 2.

10

ω
ω
ω
ω
and D maximize the present discounted value of their profits πI|1
+ δπI|2and πD|1+ δπD|2ω
respectively, while UE maximizes the second-period profits πE|2
for the case of entry.

3.2

Design of exclusive contracts

For an exclusion equilibrium to exist, the equilibrium transfer x∗ must simultaneously satisfy
the following two conditions.
First, the exclusive contract must satisfy individual rationality for D; that is, the amountof compensation x∗ induces D to accept the exclusive offer:
a
a
r
r
r
r
a
a
πD|1
+ x∗ + δπD|2≥ πD|1+ δπD|2or x∗ ≥ πD|1
+ δπD|2− (πD|1+ δπD|2).

(1)

Second, the exclusive contract must satisfy individual rationality for UI ; that is, UI earns
higher profits under exclusive dealing:
a
a
r
r
a
a
r
r
πI|1
+ δπI|2− x∗ ≥ πI|1
+ δπI|2or x∗ ≤ πI|1
+ δπI|2− (πI|1+ δπI|2).

(2)

From the above conditions, it is evident that an exclusion equilibrium exists if and only
if inequalities (1) and (2) simultaneously hold. This is equivalent to the following condition:
a
a
a
a
r
r
r
r
πI|1
+ δπI|2+ πD|1+ δπD|2≥ πI|1+ δπI|2+ πD|1+ δπD|2.

(3)

Condition (3) implies that anticompetitive exclusive contracts are attained if exclusive contracts increase the overall joint profits of UI and D. The condition also implies that whether
UI or D makes an exclusive offer does not affect the existence of exclusion outcomes.

4

Analysis

We characterize the properties of subgame perfect Nash equilibria by solving the game using
backward induction. Starting from the game in period 2, we explore the equilibrium outcomesin the subgame after D’s decision in period 1.1. We then explore the contractual decision inperiod 1.1 to examine the existence of exclusion outcomes.
11

4.1

Period 2

Let y2 denote the indifferent consumer in period 2, which satisfies v(y2)−p2 = 0. By defining
q2 as the mass of consumers purchasing in period 2, the sum of the production level in both
periods becomes q1 + q2 = 1 − F (y2). From these two equations, we obtain the residual
inverse demand function in period 2 given q1 :P2 (q2 |q1 ) ≡ v(y2 (q2 |q1 )),where y2 (q2 |q1 ) ≡ F −1 (1 − q1 − q2 ). Note that P2 (q2 |q1 ) has the following property:∂P2 (q2 |q1 )∂P2 (q2 |q1 )v ′ (y2 (q2 |q1 ))
=
=− ′
< 0.
∂q1
∂q2
F (y2 (q2 |q1 ))(4)

For notational convenience, we define q2∗ (q1 , z), P2∗ (q1 , z), and Π∗2 (q1 , z) as follows:q2∗ (q1 , z) ≡ argmax (P2 (q2 |q1 ) − z)q2 ,

(5)

q2 ≥0

P2∗ (q1 , z) ≡ P2 (q2∗ (q1 , z)|q1 ),Π∗2 (q1 , z) ≡ (P2∗ (q1 , z) − z)q2∗ (q1 , z),where 0 ≤ z < P2 (0|q1 ).
In period 2.3, given the two-part tariff contract (w2 , ψ2 ), D chooses q2∗ (q1 , w2) to maximize
its second-period profit, which leads to the equilibrium price P2∗ (q1 , w2 ). D earns
πD|2 (q1 , w2) = Π∗2 (q1 , w2 ) − ψ2 .
The following lemma summarizes the properties of q2∗ (q1 , w2 ), P2∗ (q1 , w2 ), and Π∗2 (q1 , w2 ).
Lemma 1. q2∗ (q1 , w2 ), P2∗ (q1 , w2 ), and Π∗2 (q1 , w2 ) have the following properties:∂P2∗ (q1 , w2)∂Π∗2 (q1 , w2 )∂q2∗ (q1 , w2 )
< 0,
> 0,
< 0.
∂w2
∂w2
∂q1
Proof. See Appendix A.1.
In period 2.2, the supplier(s) consider the profit maximization problem(s), given q1 . First,
we discuss the case in which the exclusive offer is accepted in period 1.1. Second, we explore
the case in which the exclusive offer is rejected in period 1.1. Finally, we compare both cases.
12

Exclusive supply chain When the exclusive supply chain is chosen, only UI exists in theaaupstream market in period 2. In period 2.2, UI offers the trading term, (w2|I, ψ2|I) = (cI , ψ2a ),such that Π∗2 (q1 , w2 ) − ψ2a = 0. D accepts this trading term, and the resulting profits of UI
and D are given by
a
a
= 0.
πI|2
= Π∗2 (q1 , cI ), πD|2

The realized price in period 2 is pa2 = P2∗ (q1 , cI ).
Open supply chain When the open supply chain is chosen, UE enters the upstream
market in period 2.1. In period 2.2, the upstream competition between UI and UE inducesrr
UI to offer the best trading term, (w2|I, ψ2|I) = (cI , 0), which allows D to earn Π∗2 (q1 , cI ).
rrr
Anticipating this trading term by UI , UE sets (w2|E, ψ2|E) = (cE , ψ2|E
) such that Π∗2 (q1 , cE ) −r= Π∗2 (q1 , cI ). D accepts this trading term, and the resulting profits of D, UI , and UE
ψ2|E

are
r
r
r
πD|2
= Π∗2 (q1 , cI ), πI|2
= 0, πE|2= Π∗2 (q1 , cE ) − Π∗2 (q1 , cI ).

The realized price in period 2 is pr2 = P2∗ (q1 , cE ).
Comparison Finally, we compare the above outcomes in period 2 given q1 . For the pairof UI and D, their joint profit under the exclusive supply chain is the same as that under
the open supply chain. The difference between these two cases appears in the realized prices
in period 2. Using the results in Lemma 1, we have the following property:
Lemma 2. Given q1 , the realized price under the exclusive supply chain is higher than that
under the open supply chain: P2∗ (q1 , cI ) > P2∗ (q1 , cE ) always holds.
The lower realized price in period 2 under the open supply chain follows from the efficiencyof UE ; namely, entry of the efficient upstream entrant leads to the lower market price. In
the following analysis, we show that this property plays an important role in choosing an
exclusive or open supply chain.
13

4.2

Period 1

Anticipating the outcome in period 2, D determines q1 given the trading term (w1 , ψ1 ) inperiod 1.3. For expositional simplicity, we use P1 (q1 |cI ) ≡ P1 (q1 , P2∗ (q1 , cI )) and P1 (q1 |cE ) ≡P1 (q1 , P2∗ (q1 , cE )) to express the static inverse demand functions under the exclusive supply
chain and the open supply chain in period 1, respectively. The difference in these two
inverse demand functions arises from the difference in the realized prices in period 2. Using
the properties of ∂P1 (q1 , p2 )/∂p2 > 0 and P2∗ (q1 , cI ) > P2∗ (q1 , cE ), the two inverse demands
in period 1 have the following property:
Lemma 3. The static inverse demand in period 1 under the exclusive supply chain is always
strictly larger than that under the open supply chain; i.e., P1 (q1 |cI ) > P1 (q1 |cE ) holds forany q1 .
The result in Lemma 3 implies that the exclusive supply chain allows D to face larger
consumer demand than the open supply chain, which can be explained by the intertemporalexternal effect of entry in period 2. Under the open supply chain, the entry of UE occurs inperiod 2, allowing consumers to purchase the final good at a low price in period 2. Such alow price in period 2 induces some consumers to refrain from purchasing in period 1. Hence,the consumer demand in period 1 under the open supply chain becomes smaller than that
under the exclusive supply chain. In other words, the contracting party suffers from thesmall consumer demand in period 1 under the open supply chain.
In the rest of this subsection, we derive the equilibrium outcomes. Because firms’ profit
maximization problems depend greatly on the contractual decision in period 1.1, we explore
the profit maximization problem in periods 1.2 and 1.3 for the cases of the exclusive supply
chain and the open supply chain, separately.
Exclusive supply chain We derive the equilibrium outcome starting from D’s profit
maximization problem in period 1.3. When the exclusive supply chain is chosen in period14a1.1, D anticipates that it earns zero profit in period 2; namely, πD|2
= 0. Then, given

the two-part tariff contract (w2 , ψ2 ), which is offered by UI in period 1.2, D chooses the
production level q1a (w1 ) to maximize its overall profits in period 1.3:
q1a (w1 ) ≡ argmax (P1 (q1 |cI ) − w1 )q1 − ψ1 .

(6)

q1 ≥0

The first-order condition of the maximization problem in (6) isP1 (q1a (w1 )|cI ) − w1 +∂P1 (q1a (w1 )|cI ) a
q1 (w1 ) = 0.
∂q1

(7)

a
We now consider UI ’s profit maximization problem in period 1.2. By anticipating πI|2=Π∗2 (q1a (w1 ), cI ), UI ’s overall profits become(w1a , ψ1a ) ≡ argmax (w1 − cI )q1a (w1 ) + ψ1 + δΠ∗2 (q1a (w1 ), cI ).
w1 ,ψ1

UI determines the highest ψ1 such that D prefers accepting (w1 , ψ1 ) to declining it; that is,ψ1 = (P1 (q1a (w1 )|cI ) − w1 )q1a (w1 ). Under this two-part tariff contract, D’s overall profits areaazero; πD|1
+ δπD|2= 0. Then, UI ’s profit maximization problem is rewritten as

w1a ≡ argmax (P1 (q1a (w1 )|cI ) − cI )q1a (w1 ) + δΠ∗2 (q1a (w1 ), cI ).

(8)

w1

The first-order condition of the maximization problem in (8) is∂Π∗2 (q1a (w1a ), cI ) a ′ a∂P1 (q1a (w1a )|cI ) a aaaq1 (w1 ) = 0.
q1 (w1 ) + δ(P1 (q1 (w1 )|cI ) − cI ) +
∂q1
∂q1

(9)

Using the first-order conditions in (7) and (9), we obtain the following:
w1a = cI − δ∂Π∗2 (q1a (w1a ), cI ).
∂q1

From Lemma 1, we have ∂Π∗2 (q1a (w1a ), cI )/∂q1 < 0, which implies that w1a > cI holds. The
following proposition summarizes the discussion.
Proposition 1. When an exclusive supply chain is chosen for both periods, the equilibrium
wholesale price in period 1 is strictly higher than the marginal cost of UI ; that is, w1a > cI .
15

Proof. The above discussion is based on q1a ′ (w1a ) < 0. The proof of q1a ′ (w1a ) < 0 is provided
in Appendix A.2.
This proposition is a generalization of Desai, Koenigsberg, and Purohit (2004) in thatthe demand system in our paper is a general form. Under the exclusive supply chain, Dchooses the first period production level without considering the profit in period 2 because
it anticipates that it earns nothing in period 2. To control D’s pricing behavior in period 1,
UI sets w1a > cI to fulfill the overall joint profit maximization of UI and D.
Note that the realized production level in period 1, q1a (w1a ), is the same as q1∗ , which isthe solution of the following maximization problem:q1∗ ≡ argmax (P1 (q1 |cI ) − cI )q1 + δΠ∗2 (q1 , cI ).
q1 ≥0This means that UI indirectly controls q1 through its wholesale price, w1 , to maximize theoverall joint profits of UI and D. As a result, the overall joint profits of UI and D become
a
a
a
a
πI|1
+ δπI|2+ πD|1+ δπD|2= (P1 (q1∗ |cI ) − cI )q1∗ + δΠ∗2 (q1∗ , cI ).

(10)

Open supply chain When the open supply chain is chosen in period 1.1, D anticipates itrearns πD|2
= Π∗2 (q1 , cI ) in period 2. Then, given the two-part tariff contract (w2 , ψ2 ), which

is offered by UI in period 1.2, D chooses the production level q1r (w1 ) to maximize its overallprofits in period 1.3:
q1r (w1 ) ≡ argmax (P1 (q1 |cE ) − w1 )q1 − ψ1 + δΠ∗2 (q1 , cI ).

(11)

q1 ≥0

The first-order condition of the maximization problem in (11) is
P1 (q1r (w1 )|cE ) − w1 +∂P1 (q1r (w1 )|cE ) r∂Π∗ (q r (w1 ), cI )q1 (w1 ) + δ 2 1
= 0.
∂q1
∂q1

By anticipating this reaction, UI ’s profit maximization problem in period 1.2 becomes(w1r , ψ1r ) ≡ argmax (w1 − cI )q1r (w1 ) + ψ1
w1 ,ψ116(12)

Note that under the open supply chain, D can earn δΠ∗2 (0, cI ) in period 2 through the competition between UI and UE if it rejects the two-part tariff contract (w1 , ψ1 ). UI determines
the highest ψ1 such that D prefers accepting (w1 , ψ1 ) to declining it; the optimal level of
upfront fixed payment satisfies ψ1 = (P1 (q1r (w1 )|cI ) − w1 )q1r (w1 ) + δΠ∗2 (q1 , cI ) − δΠ∗2 (0, cI ).
Under this two-part tariff contract, UI cannot extract all the overall joint profits of UI and D;rr
unlike the exclusive supply chain, D earns positive overall profits, πD|1
+ δπD|2= δΠ∗2 (0, cI ).

Then, UI ’s profit maximization problem is rewritten as
w1r ≡ argmax (P1 (q1r (w1 )|cE ) − cI )q1r (w1 ) + δΠ∗2 (q1r (w1 ), cI ) − δΠ∗2 (0, cI ).

(13)

w1

The first-order condition of the maximization problem in (13) is∂P1 (q1r (w1r )|cE ) r r∂Π∗2 (q1r (w1r ), cI ) r ′ r
rr(P1 (q1 (w1 )|cE ) − cI ) +q1 (w1 ) + δ
q1 (w1 ) = 0.
∂q1
∂q1

(14)

Using the first-order conditions in (12) and (14), we obtain the following:
w1r = cI .
The following proposition summarizes the discussion.
Proposition 2. When UE enters the upstream market in period 2, the equilibrium linear
wholesale price in period 1 equals the marginal cost of UI ; that is, w1r = cI .
Proof. The above discussion is based on q1r ′ (w1r ) < 0. The precise proof of q1r ′ (w1r ) < 0 isprovided in Appendix A.3.
Under the open supply chain, from (12) and (14), we find that D’s maximization problemaligns with UI ’s if and only if UI sets w1r = cI as in the standard two-part tariff pricing.
Note that the realized production level in period 1, q1r (w1r ), is the same as q1∗∗ , which isthe solution of the following maximization problem:
q1∗∗ ≡ argmax (P1 (q1 |cE ) − cI )q1 + δΠ∗2 (q1 , cI ).
q1 ≥0

17

Thus, the overall joint profits of UI and D become
r
r
r
r
πI|1
+ δπI|2+ πD|1+ δπD|2= (P1 (q1∗∗ |cE ) − cI )q1∗∗ + δΠ∗2 (q1∗∗ , cI ).

(15)

Comparison We finally examine the contractual decision in period 1.1. We compare
equations (10) and (15): the overall joint profits in the two trading modes. For q1 , theoverall joint profits in period 1 under the exclusive supply chain areΠa (q1∗ ) ≡ (P1 (q1 |cI ) − cI )q1 + δΠ∗2 (q1 , cI ),(16)

while those under the open supply chain areΠr (q1 ) ≡ (P1 (q1 |cE ) − cI )q1 + δΠ∗2 (q1 , cI ).

(17)

The only difference between Πa (q1 ) and Πr (q1 ) is in the static inverse demand in period
1, P1 (q1 |·). From Lemma 3, we have P1 (q1 |cI ) > P1 (q1 |cE ) for any q1 , which implies thatΠa (q1 ) > Πr (q1 ) holds for all q1 . Recall that, under the two modes, UI indirectly controls
q1 through its wholesale price, w1 . Therefore, under the exclusive supply chain, by setting
w1 = w1a , UI achieves the overall joint profits, Πa (q1∗ ), which are strictly higher than thoseunder the open supply chain, Πr (q1∗∗ ); that is, condition (3) always holds.
Theorem 1. In durable goods markets, UI and D choose an exclusive supply chain for any
cI , cE , and δ if D produces the positive production level in period 1 when consumers predict
UE ’s entry in period 2.
Theorem 1 implies that if at least some consumers purchase final goods in period 1 under
the open supply chain, the two-period exclusive supply chain, which deters the entry of an
efficient entrant in period 2, is always established in durable goods markets. This result canbe explained by the intertemporal negative externality in durable goods markets. In durable
goods markets, future entry intertemporally affects the current market outcome. Efficient18

entry in period 2 discourages final consumers from purchasing final goods in period 1 because
they predict that entry leads to a future price reduction. Such a property prevents thecontracting party from choosing the optimal pair of prices (p1 , p2 ) to maximize overall joint
profits under the open supply chain. In other words, using the two-part tariff contract, UI
can indirectly control the pair of prices to achieve overall joint profit maximization under the
exclusive supply chain, whereas it cannot control the prices in period 2 because of the entryof UE under the open supply chain. Therefore, although consumers benefit from future entry,
such entry is harmful to the contracting party; namely, in durable goods markets, rational
economic agents choose the exclusive supply chain, which deters efficient future entry.
In addition, from the viewpoint of competition policy, the findings here provide a new
insight for anticompetitive exclusive contracts. If we modify our model to the case of perishable goods markets, the modified model coincides with the framework of the Chicago School
argument; exclusive contracts cannot deter the entry of an efficient entrant. Theorem 1
implies that the Chicago School argument cannot be applied to durable goods markets; thenature of durable goods markets allows the inefficient incumbent supplier to deter efficient
future entry thorough anticompetitive exclusive contracts.19

4.3

Highly efficient entrant

Thus far, we explore the case in which at least some consumers purchase final goods in period
1. However, if UE is sufficiently efficient, the inverse demand in period 1, P1 (q1 |cE ), becomesso small that no consumer purchases final goods in period 1 under the open supply chain;that is, q1r = 0. In this case, the overall joint profits in period 1 under the open supply chainare Πr (0) = δΠ∗2 (0, cI ). From (16) and (17), we haveΠa (q1∗ ) > Πa (0) = Πr (0),19

In the supplementary appendix, we compare the social surpluses under the exclusive supply chain andthe open supply chain, by introducing linear demand. We show that the exclusive supply chain to deter theefficient entrant is always undesirable from the viewpoint of social surplus.

19

which implies that the exclusive supply chain leads to higher overall joint profits of UI andD than the open supply chain even when UE is highly efficient.
Theorem 2. In durable goods markets, UI and D choose an exclusive supply chain for any
cI , cE , and δ even if D does not produce the positive production level in period 1 when
consumers expect UE ’s entry in period 2.
The results in Theorems 1 and 2 imply that in durable goods markets, the exclusive
supply chain is always established regardless of the difference in cost efficiency. Note thatthe results in this study depend on the assumption that upstream suppliers can adopt twopart tariffs. In the supplementary appendix, we extend the model of linear demand to thecase of linear wholesale pricing. We show that in contrast to the case of two-part tariffs,
linear wholesale pricing does not lead to exclusion outcomes for a high discount factor. The
major difference between the two types of wholesale pricing is the existence of a double
marginalization problem, which is often observed for the case of linear wholesale pricing.
When the exclusive supply chain is chosen, the double marginalization problem occurs forboth periods, which reduces the overall joint profits of UI and D under the exclusive supply
chain. By contrast, when the open supply chain is chosen, efficient entry in period 2 canmitigate such a problem; D can earn considerably high profits in period 2. If the discountfactor is high, this effect becomes dominant and thus, the open supply chain is chosen in theequilibrium.

5Intel CaseIn this section, we briefly consider the linkage between exclusive dealing in this study andthe Intel case (2005). In the CPU market, Intel’s rival firms develop competitive products
such as “Crusoe” developed by Transmeta in the early 2000s and “Ryzen” developed by
AMD and “M1” developed by Apple based on ARM architecture around 2020. Hence,20

although Intel maintains a dominant position in the CPU market, it commonly faces thethreat of competitive products developed by rival firms. More importantly, Ryzen and ARMbased processors have been adopted recently by several personal computer makers becauseof their high performance.20 Because of these adoptions, for example, AMD’s market sharefor notebook CPUs grows from 11.4% in 2019 to 20.1% in 2020.21 Against the market
environment change, Intel would currently have an incentive to exclude these rivals throughexclusive contracts if such contracts were lawful. Thus, considering the linkage between
the exclusion mechanism in this study and the Intel case (2005) allows us to predict thefeasibility of the exclusive dealing in the current and future CPU markets.
In the Intel case (2005), a competitive product also existed.22 Intel had a dominantposition in the market for the 32-bit version of x86 microprocessors to run on the Microsoft
Windows and Linux families of operating systems. However, in 2003, AMD achieved technological leadership for the 64-bit version of the x86 microprocessors, which are expected to
become the next-generation standard.23 The US District Court for the District of Delaware
pointed out that “Bested in a technology duel over which it long claimed leadership, Intel
increased exploitation of its market power to pressure customers to refrain from migrating
to AMD’s superior, lower-cost microprocessors.”24 As one of the countermeasures against
20

See “Apple unleashes M1” Apple, November 10, 2020 (https://www.apple.com/newsroom/2020/11/
apple-unleashes-m1/), “Lenovo brings AMD to its gaming laptops” CNET, July 16, 2020 (https://www.
cnet.com/news/lenovo-brings-amd-to-its-gaming-laptops/), and “HP Envy x360 13 (2020) review:
This small 2-in-1 is more premium than its price” CNET, September 23, 2020 (https://www.cnet.com/
reviews/hp-envy-x360-13-2020-review/).
21

See “Global Notebook Computer Shipment Expected to Reach 217 Million Units in 2021, with Chromebooks Accounting for 18.5% of Total Shipment, Says TrendFoce” TrendForce, January 6, 2021 (https://www.trendforce.com/presscenter/news/20210106-10633.html).
22

See also “Intel and AMD: A long history in court” CNET, July 8, 2005 (https://www.cnet.com/news/
intel-and-amd-a-long-history-in-court/).
23

In fact, 64-bit microprocessors started to become mainstream in CPU markets after Intel launched Core
2 processors, its first 64-bit microprocessors for general consumers, on July 27, 2006 (https://www.intel.
com/pressroom/archive/releases/2006/20060727comp.htm).
24

See page 3 of the report on Advanced Micro Devices, Inc., a Delaware corporation, and AMD Inter-21

AMD, “Intel had forced major customers into exclusive-or near exclusive deals.”25 From
the above evidence, Intel’s exclusive dealings can be interpreted as its strategy to protect
its dominant position in the CPU market in the future, which corresponds to this study’s
assumption in which the incumbent supplier, a currently dominant firm, is inferior to theentrant supplier in the future. In addition, compared with other industries, firms and consumers can easily access information on the perspective on the change of upstream markets
in computing industries, which implies that all economic agents are more likely to anticipate
the market structure in the future, which is also required in the exclusion mechanism in thisstudy. Therefore, the exclusive mechanism in this study is more likely to work effectively in
the CPU market; namely, using exclusive contracts, a dominant CPU supplier can exclude arival CPU supplier, which has efficient technology regarded as the next-generation standard.
In addition, Intel’s conduct as described above benefits not only Intel but also personal computer manufacturers because these manufacturers obtain a large amount from theawarding of rebates and various other payments. For example, in a single quarter in 2007,
conditional rebates and payments from Intel amounted to 76% of Dell’s operating profit(Gans, 2013).26 From the viewpoint of downstream personal computer manufacturers, thiskind of offer from Intel is very attractive. It is an adequate incentive for downstream firms
to choose exclusive trading.
Finally, we note that in the analysis of the previous section, input suppliers differ in termsof cost efficiency. Considering the Intel case (2005), it is more suitable to construct a modelin which inputs produced by suppliers are vertically differentiated. In the supplementary
appendix, we extend the model of linear demand to the case in which input suppliers produce
national Sales & Services, Ltd., a Delaware corporation, v. Intel corporation, a Delaware corporation, andIntel Kabushiki Kaisha, a Japanese corporation, Civil Action No. 05-441-JJF, 2005 (https://www.amd.
com/system/files/amd-intel-full-complaint.pdf).
25

See page 2 of the report.

26

See also Japan Fair Trade Commission (2005): http://www.jftc.go.jp/eacpf/cases/intel.pdf,
and the European Commission (2009): http://ec.europa.eu/competition/sectors/ICT/intel.html.

22

vertically differentiated inputs, following Argenton (2010). We show that the exclusion
mechanism in this study remains valid; namely, inferior input suppliers can deter superior
input suppliers through exclusive contracts in durable goods markets.

6

Conclusion

This study has explored a supply chain problem in durable goods markets. We consider thesituation in which a downstream durable goods monopolist chooses one of the two trading
modes: (i) an exclusive supply chain with an existing incumbent supplier, or (ii) an open supply chain, which causes competition between the existing incumbent supplier and a potential
supplier in the future. The problem is also related to the discussion of anticompetitive exclusive contracts. Therefore, this study contributes to the literature in terms of both supply
chain management and competition policy. Extending the static framework of the Chicago
School argument to a two-period durable goods model without price commitment, we showthat the downstream durable goods monopolist chooses the exclusive supply chain by signinglong-term exclusive contracts; the potential entrant cannot enter the upstream market evenwhen it is efficient. This study’s exclusion mechanism is explained by the time-inconsistency
problem (Coase, 1972).
Our result has new policy implications for antitrust agencies; the Chicago School argument, which is based on static perishable goods markets, is not necessarily applicable todurable goods markets. Because of the nature of durable goods, rational economic agentscan engage in anticompetitive exclusive contracts to exclude the efficient entrant in the future even under the simplest setting in which exclusion never occurs in static perishable
goods markets. When we discuss the anticompetitiveness of exclusive contracts, we need toconsider the durability of final goods in the market in which exclusion occurs. Otherwise, we
may overestimate the Chicago School argument, which may lead to misleading predictions.
The results here could explain why some vertical relations are too stable, which results23

in inefficiency. In the literature on business and management, long-standing organizationties cause lower incentives for opportunistic behavior (Williamson, 1975), information search(Uzzi, 1997), and necessary restructuring (Ernst and Bamford, 2005). Such organizations areless likely to search for new partners and capabilities and pursue existing business practices,which result in a decrease in benefits within interorganizational ties or a negative impact
on organization performance (Uzzi, 1997; Goerzen, 2007; Poppo, Zhou, and Zenger, 2008).27
In our model, the downstream firm has an incentive to sign a two-period contract with theincumbent supplier, even by anticipating the appearance of a new efficient supplier in thenear future. The resulting outcome leads to inefficiency of the vertical chain. Thus, although
the advancement of informatization and globalization seemingly expands the open supply
chain, these business environment changes may have a smaller impact on the openness ofvertical relations in durable goods markets.
Despite these contributions, there remain some issues requiring future research. First,
we predict that the exclusion results are more likely to be observed if we introduce productdurability into the other models of anticompetitive exclusive dealing based on perishable
goods in the literature. Second, for the analysis in this study, we restrict our attention toa particular industry structure—a single potential entrant supplier—for clarity, but we cancertainly assume multiple entrant suppliers. As in Kitamura (2010), if we assume multiple
entrants, the competition between entrants induces the downstream firm to earn higher
profits under the open supply chain, which enables us to predict that the exclusive supply
chain may not be chosen if multiple entrants are sufficiently efficient. Thus, extensions andapplications of our model can help researchers and policy makers address similar real-world
issues.
27

In addition, as those organizations accumulate relationship-specific experiences, such organizational
inertia becomes stronger, and the partners share relationship-specific routines (Kim, Oh, and Swaminathan,2006). Moreover, the specificity and efficiency of a relationship-specific investment may discourage thedownstream firm from switching to a new trading partner (Kitamura, Miyaoka, and Sato, 2016).

24

A

Proofs of the results

A.1

Proof of Lemma 1

In this proof, we often use q2∗ ≡ q2∗ (q1 , w2 ) for notational simplicity. We first explore theproperty of ∂q2∗ (q1 , w2 )/∂w2 . For z = w2 , the first-order condition of D’s profit maximization
problem (5) becomes
P2 (q2∗ |q1 ) − w2 +∂P2 (q2∗ |q1 ) ∗q2 = 0.
∂q2(18)

The second-order condition of the maximization problem in (5) leads to
2∂P2 (q2∗ |q1 ) ∂ 2 P2 (q2∗ |q1 ) ∗q2 < 0.
+∂q2∂q22

Then, using (18), the implicit function theorem shows that1dq2∗
= ∂P2 (q∗ |q1) ∂ 2 P2 (q∗ |q1 ) < 0.
2
2
dw2
2
+
q2∗
2
∂q2

(19)

∂q2

Hence, we have ∂q2∗ (q1 , w2 )/∂w2 < 0. We next consider the property of ∂P2∗ (q1 , w2 )/∂w2 .
Using (4) and (19), partial differentiation leads to∂P2∗ (q1 , w2 )∂P2 (q2∗ (q1 , w2 )|q1 )∂P (q2∗ |q1 ) ∂q2∗ (q1 , w2 )
=
=
> 0.
∂w2
∂w2
∂q2
∂w2
Finally, we examine the property of Π∗2 (q1 , w2 ). Using (4) and (18), we have
∂P2 (q2∗ |q1 ) ∗∂Π∗2 (q1 , w2)=
q2 < 0.
∂q1∂q1
Q.E.D.

A.2

Proof of Proposition 1

We only show that q1a ′ (w1 ) < 0 holds. The second-order condition of D’s profit maximization
problem (6) leads to
2∂P1 (q1a (w1 )|cI ) ∂ 2 P1 (q1a (w1 )|cI ) a
q1 (w1 ) < 0.
+
∂q1
∂q12
25

Using (7), the implicit function theorem shows that1
dq1a
= ∂P1 (qa (w1 )|c ) ∂ 2 P1 (qa (w1 )|c )
< 0.
I
I
a
1
1
dw1
+
2
q
(w
)
2
1
1
∂q1

∂q1

Q.E.D.

A.3

Proof of Proposition 2

We only show that q1r ′ (w1 ) < 0 holds. The second-order condition of the maximization
problem in (11) leads to∂P1 (q1r (w1 )|cE ) ∂ 2 P1 (q1r (w1 )|cE ) r∂ 2 Π∗2 (q1r (w1 ), cI )2
q1 (w1 ) + δ
< 0.
+
∂q1
∂q12
∂q12
Using (12), the implicit function theorem shows that
1
dq1r
< 0.
= ∂P1 (qr (w1 )|c ) ∂ 2 P1 (qr (w1 )|c )∂ 2 Π∗2 (q1r (w1 ),cI )
I
I
r
1
1
dw1
2
+
q
(w
)
+
δ
2
2
1
1
∂q1

∂q1

∂q1

Q.E.D.

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Online Appendix for Which is Better for Durable Goods
Producers: Exclusive or Open Supply Chain?
Hiroshi Kitamura∗

Noriaki Matsushima†

Misato Sato‡

January 19, 2021

Abstract
In this Appendix, we introduce a linear demand system to show a wide rangeof extensions to the analyses of Kitamura, Matsushima, and Sato (2021). Section 1
constructs a model. Section 2 explores the case of two-part tariffs as a benchmark
analysis. Section 3 examines the case of a highly efficient entrant. Section 4 provides
the analysis under linear wholesale pricing. Section 5 considers the case in which
suppliers produce vertically differentiated inputs. Section 6 investigates the case of aforeign supplier’s entry. Finally, Section 7 extends the analysis by introducing anotherformulation of consumers’ utility.

JEL classification codes: L12, L41, L42.
Keywords: Antitrust policy; Durable goods; Exclusive supply chain; Vertical relation.

∗

Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto-City 603-8555,
Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp†
Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. E-mail: nmatsush@iser.osaka-u.ac.jp‡
Graduate School of Humanities and Social Sciences, Okayama University, Tsushima-naka 3-1-1, Kita-Ku,
Okayama 700-8530, Japan. E-mail: msato@okayama-u.ac.jp

1

Model

The basic model structure in this appendix coincides with that in Kitamura, Matsushima,and Sato (2021). Thus, we briefly explain the model under linear demand.

1.1

Basic environment

Consumers There is a mass of consumers for every period. Each consumer has a different
preference for a final good. Let v be the consumer type in terms of willingness to pay, whichis stationary for all periods and uniformly distributed on the interval [0, 1].1 For simplicity,
we assume that the mass of consumers is normalized to one. In durable goods markets, eachconsumer purchases a final good once by considering the final good price in both periods, p1
and p2 . To clarify the exclusion mechanism in a simple way, we assume that each consumer
can use the final good for eternity; a consumer type v always enjoys v − pt by purchasing thefinal good in period t.2 In period 2, a consumer type v, who does not purchase the final goodin period 1, purchases the final good if and only if the consumer surplus is nonnegative, i.e.,v − p2 ≥ 0. By rationally predicting p2 , the consumer purchases the final good in period 1
if and only if v − p1 ≥ δ(v − p2 ). By defining the indifferent consumer’s willingness to pay
v1 as v1 − p1 = δ(v1 − p2 ), we havev1 =p1 − δp2
.
1−δ

(1.1)

Then, the willingness to pay of consumers who purchase the final good in period 1 is distributed in v ∈ [v1 , 1]; thus, the demand for the final good at each period Qt becomes
Q1 = 1 − v1 , Q2 = v1 − p2 .
1

In the model, we assume that F (y) = y and v(y) = y as in Kitamura, Matsushima, and Sato (2021).

2

Alternatively, we can consider the setting in which v represents a value of per period use; the surplusof consumer type v is (1 + δ)v − p1 for the purchase in period 1, while it is v − p2 for the purchase in period
2. In Section 7, we explore such a case and derive exclusion results.

1

Firms The upstream market consists of the incumbent supplier UI and the entrant supplier
UE . UI and UE produce an identical input but differ in terms of cost efficiency; that is, UE
is more efficient than UI , with constant marginal cost cE ∈ [0, cI ), as opposed to UI ’s
constant marginal cost cI < 1. We explore the case in which suppliers’ inputs are verticallydifferentiated in Section 5. Following Abito and Wright (2008) and Kitamura (2010), we
measure UE ’s cost advantage by η, where cI = ηpm (cE ) + (1 − η)cE and pm (cE ) is themonopoly price for the industry when the marginal cost is cE and the market demand is
Q = 1 − p: pm (cE ) = (1 + cE )/2. η = 0 implies that UE has no cost advantage. As η
increases, UE becomes efficient. We assume that η ∈ (0, 2).3 From the definition of η, themarginal cost of UE can be expressed as a function of η and cI as follows:cE =2cI − η.
2−ηThe downstream market is composed of a downstream monopolist D. D transforms one
unit of the input into one unit of the final good, which is durable. To simplify the analysis,
we assume that the cost of transformation is zero.
Timing The timing of the game coincides with that of Kitamura, Matsushima, and Sato
(2021).

1.2

Design of exclusive contracts

From Kitamura, Matsushima, and Sato (2021), an exclusion equilibrium exists if and only
if the following condition holds:
a
a
a
a
r
r
r
r
πI|1
+ δπI|2+ πD|1+ δπD|2≥ πI|1+ δπI|2+ πD|1+ δπD|2.
3(1.2)

When cE = 0, we have cI = η/2. In this case, as η → 0, cI → 0 and as η → 2, cI → 1. We impose therestriction η ∈ (0, 1) for the case of linear wholesale pricing in Section 4.

2

2

Two-part tariffs

This section considers the case of two-part tariffs, which consist of a linear wholesale pricew and an upfront fixed fee ψ. A two-part tariff offered by Ui in period t when D’s decisionωω
is ω ∈ {a, r} is denoted by (wi|t, ψi|t).

To guarantee that at least some consumers purchase the final goods in period 1, we
assume the following condition:
Assumption 1. v1 < 1 holds for the case in which the open supply chain is chosen in period
1.1; i.e.,
0 < δ < δ̂(η) ≡2(2 − η)
,
4−η

(2.1)

where δ̂ ′ (η) < 0 for all η ∈ (0, 2) and where as η → 0, δ̂(η) → 1 and as η → 2, δ̂(η) → 0.
If condition (2.1) does not hold, all consumers are attracted to the purchase in period 2,
when the entry of UE is predicted. In Section 3, we explore the case in which condition (2.1)
does not hold.
In the rest of this section, we solve the game using backward induction, starting fromperiod 2.3. In Section 2.1, we first explore the equilibrium outcomes in the subgame following
D’s decision in period 1.1. Section 2.2 then explores the contractual decision in period 1.1.

2.1

Equilibrium outcomes after period 1.1

We characterize the properties of the subgame perfect Nash equilibria after the subgamein period 1.1 from the perspective of joint profit maximization between each supplier and
D. This approach simplifies the analysis. In the last part of this subsection, we showthat the overall joint profit maximization is achieved under two-part tariff contracts in theequilibrium, derived in Appendix A.1.
We first consider the game in period 2. For notational convenience, we define p∗2 (z, v1 )3and Π∗2 (z, v1 ) as follows:p∗2 (z, v1 ) ≡ argmax(p2 − z)(v1 − p2 ),

(2.2)

p2 ≥z

Π∗2 (z, v1 ) ≡ (p∗2 (z, v1 ) − z)(v1 − p∗2 (z, v1 )).
By solving the joint profit maximization problem (2.2), we havep∗2 (z, v1 ) =(v1 − z)2
v1 + z, Π∗2 (z, v1 ) =.
2
4Note that Π∗2 (z, v1 ) is the jointly maximized profit of UI and D in period 2 when the supplier
has a marginal cost equal to z ≥ 0. When the exclusive supply chain is chosen in period
1.1, UI becomes the equilibrium supplier in period 2; that is, the joint profit of UI and D
becomes Π∗2 (cI , v1 ) and the equilibrium price of the final good is p∗2 (cI , v1 ) under exclusive
dealing. By contrast, when the open supply chain is chosen, UI and UE compete in period
2. UI makes its best offer (cI , 0) to D. UE offers (cE , Π∗2 (cE , v1 ) − Π∗2 (cI , v1 )) and it becomesthe equilibrium supplier in period 2. In the equilibrium, D chooses p∗2 (cE , v1 ). As a result,
D earns Π∗2 (cI , v1 ) and UI earns nothing, which implies that the joint profit of UI and D
becomes Π∗2 (cI , v1 ) for the case of entry. Therefore, the joint profit of UI and D in period 2
can be expressed by Π∗2 (cI , v1 ) regardless of D’s decision in period 1.1. However, the value ofthe indifferent consumer’s willingness to pay v1 depends on D’s decision in period 1.1 because
the equilibrium final good price depends on the efficiency of the equilibrium supplier. By
substituting the equilibrium price in period 2 into equation (1.1), we havev1 (p1 , z) =2p1 − δz
,
2−δ

(2.3)

where z = cI (cE ) when the exclusive (open) supply chain is chosen in period 1.1. Equation(2.3) shows that in durable goods markets, the equilibrium outcome in period 2 intertemporally affects the indifferent consumer’s willingness to pay, which cannot be observed in theperishable goods market. Equation (2.3) implies that the price reduction due to entry in4period 2 increases v1 (p1 , z); therefore, the contracting party needs to reduce the final goodprice to maintain the same output level in period 1 when entry in period 2 is predicted;
namely, in durable goods markets, the possibility of future entry reduces current consumer
demand.
We next consider the game in period 1. Using equation (2.3), we define p∗1 (z), Π∗1 (z), andΠ∗J (z) as follows:p∗1 (z) ≡ argmax(p1 − cI )(1 − v1 (p1 , z)) + δΠ∗2 (cI , v1 (p1 , z)),

(2.4)

p1

Π∗1 (z) ≡ (p∗1 (z) − cI )(1 − v1 (p∗1 (z), z)), Π∗J (z) ≡ Π∗1 (z) + δΠ∗2 (cI , v1 (p∗1 (z), z)).
Note that Π∗J (z) is the jointly maximized overall profits of D and UI when the marginal costof an equilibrium supplier in period 2 is z. By solving the joint profit maximization problem(2.4), we have the following joint profit maximizing prices in the equilibrium:p∗1 (z) =(2 − δ)(1 + cI ) + 4(1 − δ)z(2 − δ)2 (1 + cI ) + 2δ(1 − δ)z, p∗2 (z, v1 (p∗1 (z), z)) =.
2(4 − 3δ)2(4 − 3δ)
(2.5)

Under these prices, the equilibrium overall joint profit of UI and D becomesno{(2 − δ)2 − (4 − 2δ − δ 2 )cI + 2δ(1 − δ)z} (2 − δ)(1 − cI ) − δ(1 − z),Π∗1 (z) =2(4 − 3δ)22(2 − δ)(1 − cI ) + δ(cI − z)∗∗
Π2 (cI , v1 (p1 (z), z)) =,4(4 − 3δ)2
(2 − δ)2 (1 − cI )2 − δ(cI − z) 4(1 − δ)(1 − cI ) − δ(cI − z)
∗
.
(2.6)
ΠJ (z) =4(4 − 3δ)The following lemma shows that the joint profit maximization prices (2.5) are obtained as
the subgame perfect Nash equilibrium outcomes.
Lemma 2.1. Suppose that suppliers adopt two-part tariffs and condition (2.1) holds. In thesubgames after period 1.1, the pair of final goods prices becomes (p∗1 (cI ), p∗2 (cI , v1 (p∗1 (cI ), cI )))when the exclusive supply chain is chosen and (p∗1 (cE ), p∗2 (cE , v1 (p∗1 (cE ), cE ))) when the open
supply chain is chosen.
5Proof. See Appendix B.1.

2.2

Choice of supply chain

By using the above results, we now consider the game in period 1.1. By definition, we have
a
a
a
a
r
r
r
r
Π∗J (cI ) = πI|1
+ δπI|2+ πD|1+ δπD|2and Π∗J (cE ) = πI|1
+ δπI|2+ πD|1+ δπD|2. Thus, we

examine whether Π∗J (cI ) ≥ Π∗J (cE ) holds. From equation (2.6), this condition is equivalentto 4(1 − δ)(1 − cI ) ≥ δ(cI − cE ). By solving this condition with respect to δ, we have0 < δ ≤ δ(cE ) ≡4(1 − cI ),4 − 3cI − cE′

where δ(cE ) ∈ (0, 1), δ (cE ) > 0, and δ(cE ) → 1 as cE → cI . Therefore, the exclusion supply
chain is chosen for a sufficiently small discount factor. The following proposition shows thatthe exclusive supply chain is always chosen as long as condition (2.1) holds:
Proposition 2.1. Suppose that condition (2.1) holds. If suppliers adopt two-part tariffs indurable goods markets, UI and D choose an exclusive supply chain for all (η, δ) ∈ (0, 2) ×(0, δ̂(η)].
Proof. See Appendix B.2.

3

Highly efficient entrant

In the previous section, we considered the case in which at least some consumers purchase
final goods by assuming condition (2.1) holds. However, if all players are sufficiently patientand UE is sufficiently efficient, condition (2.1) does not hold; UI cannot earn positive profits
when the open supply chain is chosen in period 1.1. Although such a situation seemingly
induces UI and D to choose the open supply chain, the following proposition shows that UI
and D choose the exclusive supply chain, which deters socially efficient entry:

6

Proposition 3.1. Suppose that condition (2.1) does not hold. If suppliers adopt two-parttariffs in durable goods markets, UI and D choose an exclusive supply chain for all (η, δ) ∈(0, 2) × (δ̂(η), 1].
Proof. See Appendix B.3.

4

Linear wholesale pricing

In this section, we extend the case of a two-part tariff to that of linear wholesale pricing;ωUi offers (wi|t, 0) in period t when D’s decision is ω ∈ {a, r}. In this subsection, we assume

that η ∈ (0, 1) under which the equilibrium wholesale price in period 2 always becomesrr
the marginal cost of UI for the case of entry; that is, wI|2
= wE|2= cI .4 The equilibriumoutcomes in the subgame after D’s decision in period 1.1 are provided in Appendix A.2. The
following proposition shows that in the case of linear wholesale pricing, the exclusive supply
chain is chosen under certain conditions.
Proposition 4.1. Suppose that suppliers offer linear wholesale pricing and η ∈ (0, 1). In
durable goods markets, UI and D choose the exclusive supply chain for δ ≤ δ̃ ≃ 0.79537.
Proof. See Appendix B.4.
The intuitive explanation for the above result is provided in Section 4 of Kitamura,
Matsushima, and Sato (2021).

5

Vertically differentiated inputs

In this subsection, we extend Section 2’s analysis to the case in which input suppliers produce
vertically differentiated inputs. The basic model structure here follows Argenton (2010),
who examines the existence of anticompetitive exclusive contracts when upstream suppliers’
4This assumption implies that UE ’s monopoly price is higher than cI . The exclusive supply chain canstill be chosen even when UE is more efficient, while the analysis becomes more complicated.

7

products are vertically differentiated. Although Argenton (2010) explores the case in which
two downstream firms exist in the perishable goods market, we explore the case in which asingle downstream firm exists in durable goods markets. The quality of the final product with
the input supplied by Ui is denoted by µi , where i ∈ {I, E}. Following Argenton (2010), we
assume that UE produces a higher quality input, µE > µI , and that for simplicity, all firms’
marginal costs are zero; that is, cI = cE = cD = 0. Consumer preferences follow the standard
in the literature on vertical product differentiation. There is a unit mass of consumers,
indexed by θ, which is uniformly distributed on the interval [0, 1]. Consumers decide whether
to purchase one unit of final goods. To purchase one unit of the final good with quality µi
in period t, a consumer of type θ pays pi|t and enjoys consumer surplus θµi − pi|t .5 As in theprevious section, each consumer purchases the final good once by considering the prices ofavailable final goods in both periods pI|1 and pi|2 . In period 2, a consumer type θ, who does
not purchase the final good in period 1, purchases the final good if and only if θµI − paI|2 ≥ 0
when the exclusive supply chain is chosen (max{θµI − prI|2 , θµE − prE|2 } ≥ 0 when the open
supply chain is chosen). By rationally predicting pi|2 , the consumer purchases the final goodwith the input supplied by UI in period 1 if and only if θµI − paI|1 ≥ δ(θµI − paI|2 ) whenthe exclusive supply chain is chosen (θµI − prI|1 ≥ max{δ(θµI − prI|2 ), δ(θµE − prE|2 )} when
the open supply chain is chosen). We define the indifferent consumer’s type θ1a and θ1r as
θ1a µI − paI|1 = δ(θ1a µI − paI|2 ) and θ1r µI − prI|1 = δ(θ1r µE − prE|2 ), respectively. To obtain interior
solutions, we assume the following condition:
Assumption 5.1.
µE2−δ
< ψ(δ) ≡
,
µI
δ

(5.1)

where ψ ′ (δ) < 0 for all δ ∈ (0, 1) and where δ → 0, ψ(δ) → ∞ and as δ → 1, ψ(δ) → 1.
If condition (5.1) holds, the second-order condition and 0 < θ1r < 1 simultaneously hold.656Consumer preferences here coincide with those in the previous section for µI = µE = 1.
This condition is derived from θ1r < 1. The other conditions always hold if this condition holds. More

8

The equilibrium outcomes in the subgame after D’s decision in period 1.1 are provided
in Appendix A.3. The following proposition shows that the exclusive supply chain is always
chosen.
Proposition 5.1. Suppose that suppliers adopt two-part tariffs and condition (5.1) holds.
In durable goods markets, the incumbent supplier with the low quality input can achieve an
exclusive supply chain, which excludes the future entrant supplier with the high quality input.
Proof. See Appendix B.5.

6

Entry of a foreign firm

Thus far, we have assumed that UE is a domestic firm. In this subsection, we extend Section
2’s analysis to the case in which UE is a foreign firm. In reality, a domestic incumbent supplier
may use exclusive contracts to deter the entry of a foreign firm with high efficiency. For
example, Vist, a Russian personal computer manufacturer, develops exclusive distributionagreements with several key retailers as a survival strategy toward the entry of foreign firms
such as Compaq, IBM, and Hewlett-Packard (Dawar and Frost, 1999), which is described in
the introduction of Kitamura, Matsushima, and Sato (2021).
Note that condition (1.2) implies that the possibility of exclusion depends only on theoverall joint profit of UI and D; the nationality of UE does not affect the possibility of
exclusion. Hence, we only focus on the social inefficiency of the exclusive supply chain, which
deters entry of a foreign firm, by comparing the social surpluses. The following propositionshows that the exclusive supply chain to deter the foreign firm, which is more efficient than
UI , is always undesirable from a viewpoint of social surplus in the domestic market.
Proposition 6.1. Suppose that UE is a foreign supplier and suppliers adopt two-part tariffs.
Then, the exclusive supply chain to deter the entry of UE is always socially inefficient.
importantly, as in Proposition 3.1, the exclusive supply chain is chosen even when condition (5.1) does not
hold.

9

Proof. See Appendix B.6.
Note that if UE is a domestic supplier, UE ’s profit is also contained in the social surplus
for the case of entry. Therefore, we have the following proposition:
Proposition 6.2. Suppose that UE is a domestic supplier and suppliers adopt two-parttariffs. Then, the exclusive supply chain to deter the entry of UE is always socially inefficient.

7

When v represents the value of per period use

In this section, we extend Section 2’s analysis to the case in which v is the per period use
value. In this setting, the consumer purchases the final good in period 1 if and only if(1 + δ)v − p1 ≥ δ(v − p2 ), which is the only difference from the model introduced in Section
1. The indifferent consumer’s willingness to pay becomes
v1 = p1 − δp2 .
Compared with equation (1.1), the model setting here leads to a lower value of v1 ; namely,
consumers are more attracted to the purchase in period 1 when they enjoy v for per perioduse. Thus, the demand for the final good in period 2 may be too small to obtain positive
sales for the higher value of cI . To avoid this problem, we make the following assumption:
Assumption 7.1. v1 > cI holds; i.e.,
0 < cI < c(δ) ≡2+δ
,2 + 3δ

(7.1)

where c′ (δ) < 0 for all δ ∈ (0, 1) and where δ → 0, c(δ) → 1 and δ → 1, c(δ) → 3/5.
Under condition (7.1), at least some consumers purchase the final good in period 2 regardlessof the acceptance decision of exclusive contracts in period 1.1.
When suppliers adopt two-part tariffs, the joint profit maximization problems in period
2 and period 1 in this appendix coincides with joint profit maximization problems (2.2)
10and (2.4).7 By solving these joint profit maximization problems, we obtain the joint profit
maximizing prices in the equilibrium:(2 + δ) 2 + δ + (2 − δ)cI + 2δz
2 + δ + (2 − δ)cI + 4z∗
p1 (z) =, p∗2 (z, v1 (p∗1 (z), z)) =.
2(4 + δ)2(4 + δ)Under these prices, the equilibrium production levels of final goods are given byQ∗1 (z) =2(1 + cI ) + δ(1 − cI ) − 2(2 + δ)z2(1 − cI ) + δ(cI + z)
, Q∗2 (z) =,4+δ
2(4 + δ)which implies that even when the open supply chain is chosen, at least some consumers
purchase the final good in period 1 for all 0 ≤ cE < cI < c(δ). Then, the equilibrium overall
joint profit of UI and D becomesΠ∗1 (z) =
2(1 − cI ) + δ(cI + z) 4(1 − cI ) − (2 + δ)cI + δ + 2z,
2(4 + δ)2Π∗2 (cI , v1 (p∗1 (z), z)) =22(1 − (1 + δ)cI ) + δ(1 − z),
4(4 + δ)2


2
2
2 2
4
+
4δ
+
δ
−
2(4
+
2δ
+
δ
)c
+
2(2
+
δ
)c
+
4(1
−
c
)
+
δ(2c
+
z)
δz
I
I
I
I
. (7.2)
Π∗J (z) =4(4 + δ)We now examine the contractual decision in period 1.1. From equation (7.2), we have
δ 4(1 − cI ) + δ(3cI + cE ) (cI − cE )∗∗ΠJ (cI ) − ΠJ (cE ) => 0,
4(4 + δ)for all (δ, cE ) ∈ (0, 1) × [0, cI ), which implies that condition (1.2) always holds. Thus, wehave the following proposition.
Proposition 7.1. Suppose that v represents the value of per period use and condition (7.1)
holds. If suppliers adopt two-part tariffs in durable goods markets, UI and D choose an
exclusive supply chain for all (δ, cE ) ∈ (0, 1) × [0, cI ).

7

Like in Section 2, we can show that overall joint profit maximization is achieved under two-part tariff
contracts. The calculation results are available upon request.

11A

Equilibrium outcomes in subgames after period 1.1

We derive the equilibrium outcomes after the game in period 1.1 using backward induction.
Appendix A.1 introduces the equilibrium outcomes under two-part tariffs. Appendix A.2
provides the equilibrium outcomes under linear wholesale pricing. Finally, Appendix A.3
derives the equilibrium outcomes when inputs are vertically differentiated.

A.1

Two-part tariffs

We first consider the case where the exclusive supply chain is chosen (equivalently, theexclusive offer is accepted) in period 1.1. In this case, UI becomes an upstream monopolist
in both periods. As discussed in Section 2, in period 2.2, UI offers (cI , Π∗2 (cI , v1 )) to D. In
period 2.3, D chooses p∗2 (cI , v1 ). UI earns Π∗2 (cI , v1 ) and D earns nothing. By substitutingz = cI into (2.3), we have v1 (p1 , cI ). By anticipating these reaction functions and given(wI|1, ψI|1 ), in period 1.3, D chooses the final good price to maximize its overall profits:
pa1 (wI|1) = argmax(p1 − wI|1 )(1 − v1 (p1 , cI )) − ψI|1 .
p1 ≥wI|1a
Because UI can extract all D’s profit by setting ψI|1
= (pa1 (wI|1) − wI|1)(1 − v1 (pa1 (wI|1), cI )),UI ’s profit maximization problem in period 1.2 becomesawI|1= argmax(pa1 (wI|1 ) − cI )(1 − v1 (pa1 (wI|1), cI )) + δΠ∗2 (cI , v1 (pa1 (wI|1 ), cI )).
wI|1

In the equilibrium, we have
a
wI|1
=

δ(2 − δ) + (8(1 − δ) + δ 2 )cI> cI ,
2(4 − 3δ)which leads to (2.5) with z = cI . The firms’ equilibrium profits, excluding the fixed compensation x, are(2 − δ)2 (1 − cI )2aa
πI|1 + δπI|2 =,4(4 − 3δ)

12

a
a
a
πE|2
= 0, πD|1+ δπD|2
= 0.

(A.1)

Second, we consider the case in which the open supply chain is chosen in period 1.1. In
this case, UI is an upstream monopolist in period 1 but it needs to compete with UE in period
2 because socially efficient entry occurs. Given v1 , in period 2.2, UI makes its best offer (cI , 0)and earns nothing. In contrast, UE offers (cE , Π∗2 (cE , v1 ) − Π∗2 (cI , v1 )), which induces D toearn Π∗2 (cI , v1 ) in period 2. Because UE is the equilibrium supplier, the indifferent consumer’s
willingness to pay becomes v1 (p1 , cE ). By anticipating these reaction functions and given(wI|1, ψI|1 ), in period 1.3, D chooses the final good price to maximize its overall profits:pr1 (wI|1 ) = argmax(p1 − wI|1)(1 − v1 (p1 , cE )) + δΠ∗2 (cI , v1 (p1 , cE )) − ψI|1 .
p1

If D rejects UI ’s two-part tariffs in period 1.2, D earns nothing in period 1 but (1 − cI )2 /4
in period 2 because of the competition between UI and UE . By anticipating this outside
profit, in period 1.2, UI sets two-part tariffs that induce D to earn δ(1 − cI )2 /4 by settingr= (pr1 (wI|1) − wI|1)(1 − v1 (pr1 (wI|1), cI )) + δΠ∗2 (cI , v1 (pr1 (wI|1 ), cI )) − δ(1 − cI )2 /4. UI ’s
ψI|1

profit maximization problem in period 1.2 becomesrwI|1= argmax(pr1 (wI|1 ) − cI )(1 − v1 (pr1 (wI|1), cE )) + δΠ∗2 (cI , v1 (pr1 (wI|1), cE )) −wI|1

δ(1 − cI )2.
4r
In the equilibrium, we have wI|1
= cI , which leads to (2.5) with z = cE . The resulting firms’

profits are(4(1 − δ) − η(2 − δ))(1 − cI )2
η(4(2 − δ) + ηδ)(1 − cI )2
r
,
π
=
,
E|2
4(2 − η)2 (4 − 3δ)4(2 − η)2 (4 − 3δ)δ(1 − cI )2rr
πD|1 + δπD|2 =
.
4
r
r
πI|1
+ δπI|2
=

A.2

(A.2)

Linear wholesale pricing

We first consider the case where the exclusive supply chain is chosen in period 1.1. In
this case, UI becomes an upstream monopolist in both periods. In period 2.3, D’s profit
maximization problem given wI|2 and v1 is given bypa2 (wI|2, v1 ) = argmax(p2 − wI|2)(v1 − p2 ).
p2 ≥wI|2

13

By solving this problem, we have pa2 (wI|2 , v1 ) = (wI|2 + v1 )/2. By anticipating D’s reaction
function, UI ’s profit maximization problem in period 2.2 becomesawI|2(v1 ) = argmax(wI|2 − cI )(v1 − pa2 (wI|2 , v1 )).
wI|2 ≥cIa
By solving this problem, we have wI|2
(v1 ) = (v1 + cI )/2. The final good price of given v1 inperiod 2 becomespa2 (v1 ) =3v1 + cI
.
4

(A.3)

a
a
Period 2 profits given v1 are given by πI|2
(v1 ) = (v1 − cI )2 /8 and πD|2
(v1 ) = (v1 − cI )2 /16.

By substituting (A.3) into (1.1), we havev1a (p1 ) =4p1 − δcI.
4−δ

By anticipating these reaction functions and given wI|1 , in period 1.3, D chooses the final
good price to maximize its overall profits:
pa1 (wI|1 ) = argmax(p1 − wI|1 )(1 − v1a (p1 )) + δ
p1 ≥wI|1(v1a (p1 ) − cI )2.
16

By anticipating this reaction, in period 1.2, UI chooses the input priceawI|1= argmax(wI|1 − cI )(1 − v1a (pa1 (wI|1))) + δ
w1 ≥cI(v1a (pa1 (wI|1)) − cI )2.
8

The equilibrium prices and indifferent consumer’s willingness to pay are+ (128 − 24δ − 11δ 2 )cI
24 − 7δ − (40 − 17δ)cIa, wI|2=,32(8 − 3δ)
8(8 − 3δ)3(24 − 7δ) + (56 − 27δ)cI(4 − δ)(24 − 7δ) + (32 + 4δ − 7δ 2 )cI, pa2 =,pa1 =
16(8 − 3δ)
16(8 − 3δ)
24 − 7δ + (8 − 5δ)cI
v1a =.
4(8 − 3δ)128 − 72δ + 11δ
a
wI|1
=

2

The firms’ equilibrium profits, excluding the fixed compensation x, are(8 − δ)(1 − cI )2a, πE|2= 0,64(8 − 3δ)(1024 − 576δ + 32δ 2 + 19δ 3 )(1 − cI )2
a
a
.
πD|1
+ δπD|2=
256(8 − 3δ)2aaπI|1+ δπI|2
=

14

(A.4)

Second, we consider the case where the open supply chain is chosen in period 1.1. In thiscase, UI is an upstream monopolist in period 1 but it needs to compete with UE in period 2.
rGiven v1 , the competition between UI and UE in period 2.2 leads to wE|2
= cI . Given this

input pricing, D’s maximization problem in period 2.3 becomespr2 (v1 ) = argmax(p2 − cI )(v1 − p2 ).
p2 ≥cIThe solution of this problem ispr2 (v1 ) =v1 + cI
.
2

(A.5)

r
r
Period 2 profits given v1 are given by πI|2
(v1 ) = 0 and πD|2
(v1 ) = (v1 −cI )2 /4. By substituting(A.5) into (1.1), we havev1r (p1 ) =2p1 − δcI.
2−δ

By anticipating these reaction functions and given wI|1 , in period 1.3, D chooses the final
good price to maximize its overall profits:pr1 (wI|1) = argmax(p1 − wI|1)(1 − v1r (p1 )) + δ
p1 ≥wI|1(v1r (p1 ) − cI )2.
4

By anticipating pr1 (wI|1), in period 1.2, UI optimally chooses the input price:rwI|1= argmax(wI|1 − cI )(1 − v1r (pr1 (wI|1))).
wI|1 ≥cI

By solving these problems, the equilibrium prices and indifferent consumer’s willingness to
pay are1 − δ + (1 + δ)cI
(2 − δ)(3 − 2δ) + (2(1 − δ 2 ) + δ)cIr
, wE|2= cI , pr1 =,2
2(4 − 3δ)3 − 2δ + (1 − δ)cI
3 − 2δ + (5 − 4δ)cI, v1r =.
pr2 =2(4 − 3δ)4 − 3δ

r
wI|1=

The firms’ equilibrium profits are(1 − δ)2 (1 − cI )2(1 − δ)(cI − cE )(1 − cI )r
, πE|2=,2(4 − 3δ)2(4 − 3δ)
(1 + 2δ(1 − δ))(1 − cI )2
r
r
.
πD|1
+ δπD|2=
4(4 − 3δ)rr
πI|1
+ δπI|2
=

15

(A.6)

A.3

Vertical product differentiation

For the sake of convenience, we derive the outcomes when Ui monopolizes the upstream
market in period 2 and D’s decision in period 1.1 is ω ∈ {a, r}. The type of consumer who
is indifferent between purchasing the final good with Ui ’s input and not in period 2 satisfies
θ2 µi − pi|2 = 0. From this condition, the demand for the final good with Ui ’s input in period
2 given θ1ω becomesQωi|2 (pi|2 , θ1ω ) = θ1ω −

pi|2
.
µi

In period 2.3, D’s profit maximization problem given (wi|2, ψi|2 ) is given bypωi|2 (wi|2 , θ1ω ) = argmax(pi|2 − wi|2 )QωI|2 (pi|2 , θ1ω ) − ψi|2 .

(A.7)

pi|2 ≥wi|2

By solving this problem, we have pωi|2 (wi|2, θ1ω ) = (wi|2 + θ1ω µi )/2. In period 2.2, Ui extractsωωω
D’s profits by setting ψi|2
= (pi|2 (wi|2 , θ1ω ) − wi|2 )Qi|2 (pi|2 (wi|2 , θ1ω ), θ1ω ) − ki|2
, where ki|2
is D’soutside profits. Then, Ui ’s profit maximization problem in period 2.2 becomes
ω
ω
.
wi|2
(θ1ω ) = argmax pωi|2 (wi|2, θ1ω )Qωi|2 (pi|2 (wi|2 , θ1ω ), θ1ω ) − ki|2

(A.8)

wi|2

ω
In the equilibrium, wI|2
(θ1ω ) = 0, which leads to the following final good price with Ui ’s input

given θ1ω in period 2:
pωi|2 (θ1ω ) =θ1ω µi.
2(A.9)

The rest of this appendix is organized as follows. Appendix A.3.1 explores the case where
the exclusive supply chain is chosen in period 1.1. Appendix A.3.2 introduces the case where
the open supply chain is chosen in period 1.1.
A.3.1

Exclusive supply chain

When the exclusive supply chain is chosen in period 1.1, UI becomes an upstream monopolist
in both periods. We first consider period 2. D and UI ’s profit maximization problems are16

ωgiven by (A.7) and (A.8) with i = I, ω = a, and ki|2
= 0. Period 2 profits given θ1a areaπI|2(θ1a ) =(θ1a )2 µIa, πD|2(θ1a ) = 0.
4

We next consider period 1. By substituting (A.9) into θ1a µI − pI|1 = δ(θ1a µI − paI|2 ), wehaveθ1a (pI|1 ) =2pI|1.
(2 − δ)µI

Then, the demand for the final goods in period 1 becomes
QaI|1 (pI|1 ) = 1 −2pI|1.
(2 − δ)µI

Given (wI|1 , ψI|1 ), in period 1.3, D chooses the final good price to maximize its profits:
paI|1 (wI|1 ) = argmax(pI|1 − wI|1 )QaI|1 (pI|1 ) − ψI|1 .
pI|1 ≥wI|1

By solving this problem, we have paI|1 (wI|1) = (2wI|1 + (2 − δ)µI )/4. In period 1.2, UI canaextract all of D’s overall profits by setting ψI|1
= (paI|1 (wI|1) − wI|1)QaI|1 (paI|1 (wI|1)). Then,

UI ’s profit maximization problem in period 1.2 becomesaa(θ1a (paI|1 (wI|1))).
wI|1= argmax paI|1 (wI|1 )QaI|1 (paI|1(wI|1 )) + δπI|2
wI|1

The equilibrium prices are
a
wI|1
=

δ(2 − δ)µI
(2 − δ)2 µI(2 − δ)µIa, wI|2= 0, paI|1 =, paI|2 =.
2(4 − 3δ)2(4 − 3δ)2(4 − 3δ)The firms’ equilibrium profits, excluding the fixed compensation x, areaaπI|1+ δπI|2=

A.3.2(2 − δ)2 µIaaa, πE|2= 0, πD|1+ δπD|2= 0.
4(4 − 3δ)(A.10)

Open supply chain

When the open supply chain is chosen in period 1.1, UI is an upstream monopolist in period
1, while it needs to compete with UE in period 2.
17

We first consider period 2. By considering the possibility of upstream monopoly andduopoly, we derive the equilibrium outcomes in period 2. Assuming the monopoly outcome
in period 2, we consider the case where D purchases only UE ’s input. D and UE ’s profit
maximization problems are given by (A.7) and (A.8) with i = E and ω = r. In this case,r(m)UE offers wE|2 = 0. The joint profits between UE and D under upstream monopoly become
r(m)

πE

r(m)

(θ1r ) + πD

(θ1r ) =(θ1r )2 µE.
4(A.11)

By contrast, assuming the duopoly outcome in period 2, we can derive the following demand
for the final good with each supplier. The type of consumer who is indifferent between
purchasing the final good with UI ’s input and purchasing nothing in period 2, denoted byθ̂r , satisfies θ̂r µI − pI|2 = 0. In addition, the type of consumer who is indifferent between
purchasing the final good with UE ’s input and purchasing the final good with UI ’s input,
denoted by θ̃r , satisfies θ̃r µE − pE|2 = θ̃r µI − pI|2 . By solving these equations, we haveθ̂r = pI|2 /µI and θ̃r = (pE|2 − pI|2 )/(µE − µI ). Assuming θ̂r ≤ θ̃r ≤ θ1r , the demand for the
final good with Ui ’s input is given byr(d)QE|2 (pI|2, pE|2 , θ1r ) = θ1r −pE|2 − pI|2 pI|2
pE|2 − pI|2r(d), QI|2 (pI|2 , pE|2, θ1r ) =−.
µE − µIµE − µIµI

In period 2.3, given (wi|2 , ψi|2 ), D optimally chooses the prices of the final goods:
r(d)

r(d)

(pI|2 (wI|2 , wE|2, θ1r ), pE|2 (wI|2 , wE|2, θ1r )) = argmax
pri|2

X n

i∈{I,E}or(d)(pi|2 −wi|2 )Qi|2 (pI|2 , pE|2, θ1r )−ψi|2 .

By solving this problem, we havewI|2 + θ1r µIr(d)r,pI|2 (wI|2, wE|2, θ1 ) =2

wE|2 + θ1r µEr(d)rpE|2 (wI|2 , wE|2, θ1 ) =,2(A.12)

which leads to the following demand for each input:
r(d)

qI

(wI|2 , wE|2, θ1r ) =wI|2 − wE|2 + θ1r (µE − µI )µI wE|2 − µE wI|2r(d), qE (wI|2, wE|2, θ1r ) =.
2(µE − µI )2(µE − µI )(A.13)18

By comparing upstream monopoly and duopoly, we now show that D never purchasesr
UI ’s input. When UE offers wE|2
= 0, the first equation in (A.13) implies that UI ’s input isrnever purchased for wI > 0. By contrast, when UI offers wI|2
≤ 0, the joint profits between

UI and D when D purchases both UE ’s and UI ’s inputs by substituting (A.12) and (A.13)
become(θr )2 µE
r(d)
r(d)
−
πI (wI|2 , wE|2, ψI|2 , θ1r ) + πD (wI|2, wE|2, ψI|2 , ψE|2 , θ1r ) = 1
42µE wI|2

4µI (µE − µI )− ψE|2 ,for wE|2 = 0. The joint profits between UI and D are maximized at wI|2 = 0, under which
D’s demand for UI ’s input is zero. The comparison with (A.11) implies that UI cannot
increase the industry profits; namely, D never purchases UI ’s input on the equilibrium path.
The equilibrium outcomes in period 2 are summarized as follows. In period 2.2, UI offers
r
r
r
(wI|2
, ψI|2) = (0, 0) and earns nothing. In contrast, UE offers (0, ψE|2), which induces D torearn the outside profit when it sells the final good with UI ’s input; that is, kE|2
= (θ1r )2 µI /4.8
rThis leads to ψE|2
= (θ1r )2 (µE − µI )/4. Under these offers, the equilibrium price of the final

good with UE ’s input becomes
prE|2(θ1r ) =θ1r µE.
2(A.14)As a result, period 2 profits given θ1r arer
r
πI|2(θ1r ) = 0, πE|2
(θ1r ) =(θ1r )2 (µE − µI )(θr )2 µIr, πD|2(θ1r ) = 1.
4
4

We next consider period 1. By substituting (A.14) into θ1r µI − pI|1 = δ(θ1r µE − prE|2), wehaveθ1r (pI|1 ) =2pI|1.
2µI − δµE

Then, the demand for the final good with UI ’s input in period 1 becomes
QrI|1 (pI|1) = 1 −
8

2pI|1
.
2µI − δµE

The profit maximization problem of D in such a case corresponds with the maximization problem (A.7)
by substituting (wi|2 , ψi|2 ) = (0, 0) and θ1ω = 1.

19

r
r
Given (wI|1, ψI|1), in period 1.3, D optimally chooses the final good price with UI ’s input tomaximize its overall profits:rr(θ1r (pI|1 )) − ψI|1.
prI|1 (wI|1) = argmax(pI|1 − wI|1)QrI|1 (pI|1 ) + δπD|2
pI|1

By solving this problem, we haveprI|1 (wI|1) =(2µI − δµE )(2(wI|1 + µI ) − δµE ).
2((4 − δ)µI − 2δµE )If D rejects UI ’s two-part tariffs in period 1.2, D earns nothing in period 1 but µI /4 in period
2 because of the competition between UI and UE . By anticipating this outside profit, inrperiod 1.2, UI sets two-part tariffs that induce D to earn δµI /4 by setting ψI|1
= (pI|1 (wI|1)−

wI|1 )QrI|1 (pI|1(wI|1 ))+δ(θ1r (pI|1(wI|1 )))2 µI /4−δµI /4. Then, UI ’s profit maximization problem
in period 1.2 becomesrwI|1= argmax prI|1(wI|1 )QrI|1 (prI|1 (wI|1)) +no2rδ θI|1(prI|1 (wI|1 )) µI

wI|1

4

−

δµI
.
4

r
In the equilibrium, we have wI|1
= 0. The firms’ overall profits in the equilibrium are((2 − δ)µI − δµE )2
δ(µE − µI )(2µE − δµI )2r, πE|2 =,4((4 − δ)µI − 2δµE )4((4 − δ)µI − 2δµE )2
δµI
r
r
.
πD|1
+ δπD|2
=
4
r
r
πI|1
+ δπI|2=

B

Proofs of the results

B.1

Proof of Lemma 2.1(A.15)

See the results in Appendix A.1.
Q.E.D.

B.2

Proof of Proposition 2.1

Substituting z(η) = (2cI − η)/(2 − η) into Π∗J (z), we haveΠ∗J (z(η)) =(8 − η 2 δ 2 + 2η 2 + 2ηδ 2 + 4ηδ − 8η + 2δ 2 − 8δ)(1 − cI )2.
2(2 − η)2 (4 − 3δ)20By differentiating Π∗J (η) with respect to η, we have∂ΠJ (z(η))δ(2(2 − η) − (4 − η)δ)(1 − cI )2
=−
< 0,
∂η
(2 − η)3 (4 − 3δ)for all (η, δ) ∈ (0, 2) × (0, δ̂(η)). Hence, Π∗J (cI ) > Π∗J (cE ) holds for cE < cI , which implies
that condition (1.2) holds.
Q.E.D.

B.3

Proof of Proposition 3.1

If condition (2.1) does not hold, UI and D cannot earn positive profits in period 1 for therrcase in which the open supply chain is chosen in period 1.1; that is, πI|1
= πD|1= 0. Inrr
period 2, UI and D earn πI|2
= 0, πD|2= (1 − cI )2 /4. By substituting these equations and(A.1), we find that
a
a
a
a
r
r
r
r
πI|1
+ δπI|2+ πD|1+ δπD|2− (πI|1+ δπI|2+ πD|1+ δπD|2)=(1 − δ)2 (1 − cI )2> 0,
4 − 3δ

which implies that condition (1.2) always holds.
Q.E.D.

B.4

Proof of Proposition 4.1

By substituting (A.4) and (A.6), we find that
a
a
a
a
r
r
r
r
πI|1
+δπI|2+πD|1+δπD|2−(πI|1+δπI|2+πD|1+δπD|2)=

δ(768 − 1280δ + 412δ 2 − 21δ 3 )(1 − cI )2.
256(4 − 3δ)(8 − 3δ)2

Condition (1.2) holds for δ ≤ δ̃.
Q.E.D.

21

B.5

Proof of Proposition 5.1

By substituting (A.10) and (A.15), we find that
a
a
a
a
r
r
r
r
πI|1
+δπI|2+πD|1+δπD|2−(πI|1+δπI|2+πD|1+δπD|2)=

δ(µE − µI )((8(1 − δ) + δ 2 )µE + δ(4 − 3δ)µI ).
4(4 − 3δ)((4 − δ)µI − 2δµE )Condition (1.2) holds if and only if µE /µI ≤ ψ̂(δ) ≡ (8(1−δ)+δ 2 )/δ(4−3δ). As ψ̂(δ)−ψ(δ) =2(1−δ)/(4−3δ) > 0 for all δ ∈ (0, 1), exclusion always becomes a unique equilibrium outcome
as long as condition (5.1) holds.

B.6

Proof of Proposition 6.2

We first derive the social surplus when the exclusive supply chain is chosen. The consumer
surplus in this case becomesa

CS (δ) =Z Qa1
0(1 − x − pa1 )dx + δ
2

=

Z Qa2
0

(v1a − x − pa2 )dx

2(4 − 3δ )(1 − cI ).
8(4 − 3δ)The producer surplus in the domestic market becomes ΠJ (cI ). By summing these surpluses,
the social surplus under the exclusive supply chain becomes
W a (δ) =(12 − 8δ − δ 2 )(1 − cI )2.
8(4 − 3δ)Second, we derive the social surplus when the open supply chain is chosen for δ < δ̂(η).
The consumer surplus becomes
Z Qr1
Z Qr2
r
r
CS (η, δ|δ < δ̂(η)) =(1 − x − p1 )dx + δ(v1r − x − pr2 )dx
0

0

2

2

(16 − 16η + 4η − 12δ + η 2 δ 2 + 16ηδ − 4ηδ 2 − 4η 2 δ)(1 − cI )2=.
8(2 − η)2 (4 − 3δ)Domestic firms earn ΠJ (cE ) in this case. By summing these surpluses, the social surplus
under the open supply chain for δ < δ̂(η) becomesW r (η, δ|δ < δ̂(η)) =(48 − 48η − 32δ + 12η 2 − 4δ 2 + 32ηδ − 4η 2 δ + 4ηδ 2 − 3η 2 δ 2 )(1 − cI )2.
8(2 − η)2 (4 − 3δ)22Third, we derive the social surplus when the open supply chain is chosen for δ ≥ δ̂(η).
The consumer surplus becomesCS1r (η, δ|δ ≥ δ̂(η)) + δCS2r (η, δ|δ ≥ δ̂(η)) = 0 + δ

Z Qr2
0

(v1r − x − pr2 )dx

δ(1 − cE )2.
=8
Domestic firms’ two-period joint profit becomes δ(1 − cI )2 /4. By summing these surpluses,
the social surplus under the open supply chain for δ ≥ δ̂(η) becomesW r (η, δ|δ ≥ δ̂(η)) =δ(6 − 4η + η 2 )(1 − cI )2.
4(2 − η)We now consider the social inefficiency of the exclusive supply chain. First, we explore
the case of δ < δ̂(η). In this case, we haveW r (η, δ|δ < δ̂(η)) − W a (δ) =η 2 δ(2 − δ)(1 − cI )2> 0,
4(2 − η)2 (4 − 3δ)for all (η, δ) ∈ (0, 2) × (0, δ̂(η)). Second, we explore the case of δ ≥ δ̂(η). We define Γ(η, δ)asΓ(η, δ) ≡ W r (η, δ|δ ≥ δ̂(η)) − W a (δ)=(−48 + 48η − +80δ − 12η 2 − 32δ 2 − 64ηδ + 16η 2 δ + 20ηδ 2 − 5η 2 δ 2 )(1 − cI )2.
8(2 − η)2 (4 − 3δ)Note that Γ(η, δ) has the following two properties∂Γ(η, δ)δ(1 − cI )2
=
> 0,
∂η
(2 − η)3
Γ(η, δ̂(η)) =η 2 (1 − cI )2> 0,(4 − η)(4 − η 2 )for all η ∈ (0, 2), which implies that we have Γ(η, δ) > 0 for all (η, δ) ∈ (0, 2) × [δ̂(η), 1).
Thus, exclusion of UE is always anticompetitive even when UE is a foreign firm.
Q.E.D.
23

References
Abito, J.M., and Wright, J., 2008. Exclusive Dealing with Imperfect Downstream Competition. International Journal of Industrial Organization 26(1), 227–246.
Argenton, C., 2010. Exclusive Quality. Journal of Industrial Economics 58(3), 690–716.
Dawar, N., and Frost, T., 1999. Competing with Giants: Survival Strategies for Local Companies in Emerging Markets. Harvard Business Review 77(2), 119–129.
Kitamura, H., 2010. Exclusionary Vertical Contracts with Multiple Entrants. International
Journal of Industrial Organization 28(3), 213–219.
Kitamura, H., Matsushima, N., and Sato, M., 2021. Which is Better for Durable Goods Producers, Exclusive or Open Supply Chain? mimeo.

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