GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 312

How Does Downstream Firms’ Efficiency AffectExclusive Supply Agreements?

Hiroshi Kitamura, Noriaki Matsushima, and Misato Sato

August 2013

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

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

How Does Downstream Firms’ Efficiency AffectExclusive Supply Agreements?∗
Hiroshi Kitamura†

Noriaki Matsushima‡

Misato Sato§

July 21, 2013

Abstract
This study constructs a model for examining anticompetitive exclusive supply contracts that prevent an upstream supplier from selling input to a new downstream firm.
With regard to the technology to transform the input produced by the supplier, as an
entrant becomes increasingly efficient, its input demand can decrease, and thus, the supplier earns smaller profits when socially efficient entry is allowed. Hence, the inefficientincumbent can deter socially efficient entry via exclusive supply contracts, even in theframework of the Chicago School argument where a single seller, a single buyer, and asingle entrant exist.

JEL classifications code: L12, L41, L42.
Keywords: Antitrust policy; Entry deterrence; Exclusive supply contracts; Transformational
technology.
∗

We thank Junichiro Ishida, Shingo Ishiguro, Akifumi Ishihara, Keisuke Kawata, Toshihiro Matsumura,
Masaki Nakabayashi, Takashi Shimizu, Katsuya Takii, the conference participants at the 11th Annual International Industrial Organization Conference, and the seminar participants at Contract Theory Workshop, Hitotsubashi University, Kwansei Gakuin University, Kyoto Sangyo University, and Osaka University for helpful
discussions and comments. We especially thank Ricard Gil, Keizo Mizuno, Patrick Rey, David Salant, andTommaso Valletti for their insightful comments. The second author thanks the warm hospitality at MOVE,
Universitat Autònoma de Barcelona where part of this paper was written. We gratefully acknowledge financial
support from JSPS Grant-in-Aid for Scientific Research (A) No. 22243022, for Scientific Research (C) No.
24530248, and for Young Scientists (B) No. 24730220. The usual disclaimer applies.
†
Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto 603-8555, Japan.
Email: hiroshikitamura@cc.kyoto-su.ac.jp‡
Institute of Social and Economic Research, Osaka University 6-1 Mihogaoka, Ibaraki, Osaka 567-0047,
Japan. Email: nmatsush@iser.osaka-u.ac.jp§
Department of Economics, The George Washington University, 2115 G street, NW Monroe Hall 340 Washington DC 20052, USA. Email: smisato@gwmail.gwu.edu

1Introduction

Among contracts concerned with vertical restraints (e.g., exclusive contracts, loyalty rebates,
slotting fees, resale price maintenance, quantity fixing, and tie-ins),1 exclusive contracts havelong been controversial,2 because, once signed, these can deter efficient entrants. Thus, suchcontracts seem to be anticompetitive—a view opposed by the Chicago School. For instance,by constructing a model of an exclusive contract between an upstream incumbent and a downstream buyer, Posner (1976) and Bork (1978) argue that the rational buyer does not sign sucha contract to deter a more efficient entrant. The Chicago School argument remains highly
influential.3 In rebuttal of the Chicago School argument, post-Chicago economists indicate
specific circumstances under which anticompetitive exclusive dealings occur.4 Their studies,by extending the single-buyer model of the Chicago School argument to a multiple-buyer
model, introduce scale economies wherein the entrant needs a certain number of buyers
to cover its fixed costs (Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston(2000a)) and the competition between buyers (Simpson and Wickelgren (2007) and Abito
and Wright (2008)).
Although these studies investigate situations in which an upstream incumbent makes exclusive offers to downstream firms, in real business situations, downstream firms offer exclusive supply contracts to upstream firms. First, in the relationships between an input supplierand final good producers, the U.S. Federal Trade Commission (FTC) stopped a large drugmaker from enforcing 10-year exclusive supply agreements for an essential ingredient.5 Second, in the relationships between a final good producer and retailers, the FTC stopped a large
1See, for example, Rey and Tirole (1986), Rey and Vergé (2010), and Asker and Bar-Issac (forthcoming).
See also Rey and Tirole (2007) and Rey and Vergé (2008) for surveys of vertical restraints.
2
Exclusive dealing agreements take various forms such as exclusive territories and exclusive rights (see, forinstance, Mathewson and Winter (1984), Rey and Stiglitz (1995), and Matsumura (2003)).
3
For the impact of the Chicago School argument on antitrust policies, see Motta (2004) and Whinston (2006).
4
In an early contribution, Aghion and Bolton (1987) propose a model in which exclusion does not always
occur. However, when it does, it is anticompetitive. See also a study by Bernheim and Whinston (1998), which
explores the market circumstances under which an exclusive contract can exclude rival incumbents.
5
FTC v. Mylan Laboratories, Inc., Cambrex Corporation, Profarmaco S.r.l., and Gyma Laboratories ofAmerica, Inc., No.X990015-1 (http://www.ftc.gov/os/caselist/x990015ddc.shtm).

1toy retailer from preventing toy manufacturers from selling to warehouse clubs.6 More recently, the Japan Fair Trade Commission stopped an online gaming company from preventing
mobile game developers from providing their games through a rival online gaming company.7
Hence, this study aims to ascertain the existence of anticompetitive exclusive supply contracts
that prevent an upstream supplier from selling inputs to a new downstream entrant.
This study presents a model of anticompetitive exclusive supply contracts by inverting thevertical relationship in the Chicago School argument. The presented model comprises one
upstream supplier and one downstream incumbent. A new downstream firm, which needs
an input produced by the upstream supplier, appears as an entrant. The incumbent thenoffers an exclusive supply contract to the upstream supplier, as in the standard models of
anticompetitive exclusive dealing. If the contract is achieved, then the new entrant cannot
enter the market.
Under the standard model setting above, we consider an efficiency measure to evaluate
the efficiency of the incumbent and entrant downstream firms. We introduce the measure thatthe entrant is more efficient than the incumbent in terms of a transformational technology ofan input produced by the upstream supplier; that is, the entrant demands a smaller quantityof inputs from the supplier to produce one unit of final product. Thus, in terms of per unit
production cost, the entrant is more efficient than the incumbent. Note that the presented
model differs not only in relation to the market structure where exclusion occurs, but also
in the efficiency measure of the incumbent and entrant. Previous studies on anticompetitive
exclusive contracts assume that the (exogenous) marginal cost of an upstream entrant is lower
than that of an upstream incumbent. However, these studies do not consider the efficiency
measure employed in our study.
This study shows that when the entrant is efficient in terms of the transformational technology of an input produced by the upstream supplier, the incumbent and the upstream sup6

Toys “R” Us, Inc., v. FTC, No.98-4107 (http://www.ftc.gov/os/adjpro/d9278/toyrus.pdf). Another antitrustcase was The Garment District, Inc., v. Belk Stores Service, Inc., Mathews-Belk Company, Jantzen, Inc., No.852362 (https://bulk.resource.org/courts.gov/c/F2/799/799.F2d.905.85-2362.html). See Comanor and Rey (2000)
for detailed discussions.
7
http://www.jftc.go.jp/en/pressreleases/yearly-2011/jun/individual-000427.html.

2

plier can sign exclusive supply contracts to deter socially efficient entry even in the framework of the Chicago School argument where a single seller, a single buyer, and a single
entrant exist. More precisely, when the entrant and the incumbent have similar efficiency
levels, exclusion never occurs. However, as the entrant becomes increasingly efficient, exclusion can occur. To understand our results, consider the impact of socially efficient entry
from the viewpoint of the upstream supplier. A socially efficient entry generates downstream
competition and increases the final product output. This increases the demand for the input
produced by the upstream supplier and, consequently, its profit. The demand expansion effectof socially efficient entry makes anticompetitive exclusive dealings difficult. However, as theentrant becomes increasingly efficient, it demands a smaller quantity of the input produced by
the upstream supplier. In addition, such entry decreases the market share of the downstream
incumbent, which demands a larger quantity of the input produced by the upstream supplier.8
Therefore, as the entrant becomes efficient, its entry does not lead to a large increase in the
demand for the input produced by the upstream supplier; that is, the upstream supplier does
not welcome the highly efficient entrant. This allows the upstream supplier to engage in anticompetitive exclusive dealings to deter socially efficient entry into the downstream market.
This study also shows that the relation between the likelihood of exclusion and the entrant’s efficiency is non-monotonic; that is, exclusion is more (less) likely to occur if theentrant’s efficiency is at an intermediate (significantly high) level. When the entrant becomes
sufficiently efficient, it can monopolize the downstream market; in other words, the incumbent’s existence does not constrain the entrant’s pricing. Given this significant efficiency difference between the downstream firms, if the entrant’s efficiency improves further, the priceof final products decreases, and this leads to an expansion of the downstream market, which
benefits the upstream supplier. Therefore, exclusion is less likely to occur if the efficiency
level of the entrant is significantly high.
8

If the downstream firms compete in quantity, an improvement in the entrant’s efficiency gradually diminishes the market share of the downstream incumbent. If the downstream firms compete in price and the goods
are perfect substitutes, the market share of the downstream incumbent is zero, that is, a drastic depression of its
market share occurs.

3Moreover, this study shows that exclusion is more likely to occur if the upstream supplier’s efficiency is high, rather than if it is low. Although the existence of an entrant withmore efficient technology than the incumbent decreases the demand for the input and the upstream supplier’s profits, it also reduces the production cost of the upstream supplier, which
improves this supplier’s profits. However, this positive effect does not work well if the upstream supplier is highly efficient, and therefore, exclusion is more likely to occur in thiscase.
This study is related to the literature on anticompetitive exclusive dealings to deter upstream entrants.9 Fumagalli and Motta (2006) propose an extension of the model framed by
Rasmusen, Ramseyer, and Wiley (1991) and Segal and Whinston (2000a) where buyers arecompeting firms.10 They show that intense downstream competition reduces the possibilityof exclusion. However, Simpson and Wickelgren (2007) and Abito and Wright (2008) pointout that this result depends on the assumption that buyers are undifferentiated Bertrand competitors who need to incur epsilon participation fees to stay active. They show that if buyers
are differentiated Bertrand competitors, then intense downstream competition enhances exclusion even in the presence of epsilon participation fees.11
Wright (2008) and Argenton (2010) explore extended models of exclusion with down9

Certain studies examine procompetitive exclusive dealings. Marvel (1982), Besanko and Perry (1993), Segal and Whinston (2000b), de Meza and Selvaggi (2007), and de Fontenay, Gans, and Groves (2010) investigate
the role of exclusive dealing in encouraging non-contractible investments. Chen and Sappington (2011) study
the impact of exclusive contracts on industry R&D and welfare. Fumagalli, Motta, and Rønde (2012) examine
the interaction between procompetitive and anticompetitive effects. They show that the investment promotion
effect of exclusive dealing may facilitate anticompetitive exclusive dealing. In addition, Argenton and Willems(2012) study the trade-off between the positive effect (risk sharing) and the negative effect (exclusion) of exclusive contracts. Another motivation to consider exclusive dealing is to solve the commitment problem of Hart
and Tirole (1990), which arises when a single upstream firm sells to multiple retailers with two-part tariffs underunobservable contracts. See also O’Brien and Shaffer (1992), McAfee and Schwartz (1994), and Rey and Vergé
(2004).
10
Fumagalli and Motta (2008) also show that exclusion with scale economies arises because of coordination
failure among buyers even when the incumbent does not have a first-mover advantage in making exclusive
offers. Doganoglu and Wright (2010) explore exclusion in the presence of network externalities, an example ofscale economies.
11
See also Wright’s (2009) study, which corrects the result of Fumagalli and Motta (2006) in the case of
two-part tariffs.

4stream competition where the incumbent and a potential entrant produce horizontally andvertically, respectively, a differentiated product. Both studies show that the resulting exclusive dealing is anticompetitive.12 Furthermore, economists have recently analyzed anticompetitive exclusive dealings from an experimental perspective (Landeo and Spier, 2009, 2012,
Smith, 2011, and Boone, Müller, and Suetens, forthcoming).
The remainder of this paper is organized as follows. In Section 2, we construct the basic
environment of the model. In Section 3, we analyze the case where downstream firms compete in price. In Section 4, we analyze the case where downstream firms compete in quantity.
In Section 5, we provide discussions and in Section 6, concluding remarks. In Appendix
A and Appendix B, we present the proofs of results under price competition and quantity
competition, respectively.

2

Preliminaries

This section develops the basic environment of the model. We first explain the basic characteristics of players in the model in Section 2.1. Then, the timing of the game is introduced in
Section 2.2. Finally, we introduce the design of exclusive supply contracts in Section 2.3. For
convenience, we consider the relationships between input suppliers and final good producers,
although this model is suitable for a much more general application. For example, the model
can be applied to the relationships between final good producers and retailers.

2.1

Upstream and downstream markets

The downstream market is composed of an incumbent DI and an entrant DE . Each of themproduces a unit of final product using an input exclusively produced by an upstream supplier
12

Kitamura (2010, 2011) also explores the extended model—first, in the presence of multiple entrants, andnext, in the presence of financial constraints. Johnson (2012) extends the models in the presence of adverse
selection. Kitamura, Sato, and Arai (forthcoming) explore the model when the incumbent can establish a direct
retailer. These studies show that the resulting exclusive dealings are anticompetitive. In contrast, Gratz andReisinger (2011) show that exclusive contracts can possibly have procompetitive effects, if downstream firms
compete imperfectly and contract breaches are possible.

5U. For this supplier, the marginal cost is c ≥ 0 and w is the wholesale price of the input
offered.
Downstream firms differ in the production technology. Incumbent DI produces a unit of
final product using one unit of the input. The transformation technology is denoted by
QI = qI ,(1)

where QI (qI ) is the amount of the output (input) for incumbent DI . The per unit production
cost of downstream incumbent DI , wI , is denoted by
wI = w.

(2)

In contrast, entrant DE produces a unit of final product using k units of the input, where k isa positive constant. The transformation technology is denoted by
QE = qE /k,(3)where QE (qE ) is the amount of the output (input) for entrant DE . The per unit production
cost of entrant DE , wE , is denoted by
wE = kw.

(4)Equation (4) implies that entrant DE becomes efficient (that is, the per unit cost of entrant DE
decreases) as k decreases. We assume that 0 < k < 1. On comparing (2) with (4), it is easy to
see that entrant DE is more efficient than incumbent DI in terms of per unit production cost.
There are two interpretations of this assumption. First, in the relationships between an input supplier and final good producers, entrant DE has the efficient technology that allows it toreduce the use of input or to reduce defective products. Second, in the relationships between
a final good producer and retailers, entrant retailer DE is better at supply-chain management
than the incumbent, owing to which it need not hold excess inventories of final products
produced by final good producer U.
The efficiency measure of downstream firms in this study differs from that in previous
studies on anticompetitive exclusive dealing. Previous studies do not focus on the difference6

in the transformational technology of the input, because they explore the existence of entry
deterrence in the upstream market. In measuring the upstream firms’ efficiency, it is naturaland robust to assume that the price of input supplied by competitive sectors differs for the
upstream firms and that the upstream entrant has the smaller per unit production cost because
it has an advantage, namely, the transformational technology of the competitively supplied
input. In contrast, this study focuses on the existence of entry deterrence in the downstream
market. The difference in transformational technology of input produced by the upstream
supplier is an important efficiency measure for downstream firms.

2.2

Timing of the gameThe timing of the game is as follows (see also Figure 1). The model consists of four stages.
In Stage 1, the downstream incumbent DI offers an exclusive supply contract to the upstream
supplier U. This contract involves some fixed compensation x ≥ 0. Supplier U decides
whether to accept this offer. In Stage 2, entrant DE decides whether to enter the downstream
market. We assume that the fixed cost of entry is sufficiently small such that if entrant DE
is active, it could earn positive profits. In Stage 3, supplier U offers a linear wholesale priceof the input, w, to the active downstream firm(s). There are two cases (see Figure 2). If
supplier U accepts the exclusive supply offer in Stage 1, then it offers the input price wa
only to incumbent DI , where the superscript ‘a’ indicates that supplier U accepts the offer. In
contrast, if supplier U rejects the exclusive supply offer in Stage 1, then it offers the input pricewr to all active downstream firms, where the superscript ‘r’ indicates that supplier U rejects
the offer. We assume that supplier U cannot offer different wholesale prices to downstream
firms (in Section 5, we discuss the case where such price discrimination is possible). In Stage4, active downstream firms order the input and compete in the final market. If entry arises inStage 2, then incumbent DI and entrant DE compete. Incumbent DI ’s profit in the case when
supplier U accepts (rejects) the exclusive offer is denoted by ΠaI (ΠrI ), and supplier U’s profit
in the case when it accepts (rejects) the exclusive offer is denoted by πa (πr ).

7

2.3

The design of exclusive supply contracts

Given the equilibrium outcomes in the subgame following Stage 1, we derive the essentialconditions for an exclusive supply contract. For the existence of an exclusion equilibrium,
the equilibrium transfer x∗ needs to satisfy the following two conditions.
First, it has to satisfy individual rationality for the downstream incumbent DI ; that is,incumbent DI must earn higher operating profits under exclusive dealing, such thatΠaI − x ≥ ΠrI .

(5)

Second, it has to satisfy individual rationality for the upstream supplier U; that is, thecompensation amount x must induce supplier U to accept the exclusive supply offer, because
x + πa ≥ πr .

(6)From the above conditions, it is easy to see that an exclusion equilibrium exists if and only
if inequalities (5) and (6) hold simultaneously. This is equivalent to the following condition:
ΠaI + πa ≥ ΠrI + πr .

(7)

Condition (7) implies that for the existence of anticompetitive exclusive supply contracts, weneed to examine whether exclusive supply agreements increase the joint profits of incumbent
DI and supplier U.

3

Price Competition

This section considers the existence of anticompetitive exclusive dealings to deter the sociallyefficient entry of entrant DE when downstream firms are undifferentiated Bertrand competitors. We assume that a general demand function Q(p) is continuous and Q0 (p) < 0. We
assume that demand from the downstream firm Di , where i ∈ {I, E}, depends not only on
its price but also on that of the downstream firm D−i . The quantity that consumers demand
from Di is Q(pi ) when pi < p−i and 0 when pi > p−i . When pi = p−i , the downstream firm8with the lower per unit production cost supplies the entire quantity Q(pi ).13 For notational
convenience, we define p∗ (z) and π∗ (z) as follows:p∗ (z) ≡ arg max(p − z)Q(p),(8)π∗ (z) ≡ (p∗ (z) − z)Q(p∗ (z)),(9)

pwhere z ≥ 0. As often assumed in industrial organization literature, we assume that thesecond-order condition is satisfied, that is,
Assumption 1. The following inequality is satisfied:
2Q0 (p) + (p − z)Q00 (p) < 0.
We first consider the case where supplier U accepts the exclusive offer in Stage 1. In thiscase, it can supply only to incumbent DI . Given the input price wa , incumbent DI optimallychooses paI (wa ) = p∗ (wa ) in Stage 4. By anticipating this pricing, supplier U sets the input
price for incumbent DI to maximize its profit in Stage 3.
wa = arg max(w − c)Q(p∗ (w)).
w(10)

We assume that the second-order condition is satisfied. Because we have wa > c in the equilibrium, the equilibrium price level p∗ (wa ) does not maximize the joint profits of incumbent
DI and supplier U; that is, the double marginalization problem occurs.
ΠaI + πa = (p∗ (wa ) − c)Q(p∗ (wa )) < π∗ (c).

(11)

The entry deterrence allows incumbent DI to earn higher operating profits. However,incumbent DI and supplier U cannot maximize their joint profits, because of the double
marginalization problem.
We next consider the case where supplier U rejects the exclusive supply offer in Stage
1. In this case, entrant DE enters the downstream market in Stage 2. In Stage 4, given the
13This assumption avoids open-set problems in defining equilibria. See, for example, Abito and Wright(2008).

9input price wr , the downstream firms compete in price. Incumbent DI earns zero profits inthis subgame; that is, ΠrI = 0 for all 0 < k < 1. In addition, downstream competition leads
to two types of equilibria in Stage 4. The undifferentiated Bertrand competition leads to thefollowing outcomes:
r(i)
r
r
∗
r
r
Case (i) Incumbent DI offers pr(i)I = w and entrant DE offers pE = w , if p (kw ) ≥ w .

Case (ii) Incumbent DI offers pr(ii)= wr and entrant DE offers pr(ii)= p∗ (kwr ), if p∗ (kwr ) ≤I
E
wr .
In Case (i) (if p∗ (kwr ) ≥ wr ), the marginal cost pricing of incumbent DI binds the pricing ofr∗rr
entrant DE , which leads to pr(i)E = w . In Case (ii) (if p (kw ) ≤ w ), the marginal cost pricing∗rof incumbent DI does not bind the pricing of entrant DE , which leads to pr(ii)E = p (kw ).

By anticipating this pricing in Stage 4, supplier U optimally chooses its input price inStage 3. Note that for each case, we have a unique interior solution (wr(i) , wr(ii) ) ∈ [c, ∞)2 .
Although each interior solution needs to satisfy the constraints (wr(i) ∈ [c, p∗ (kwr(i) )] andwr(ii) ∈ [p∗ (kwr(ii) ), ∞)), we first characterize the properties of each interior solution on thefull domain [c, ∞) in Lemmas 1 and 2. We then consider the constraints of each interior
solution in Lemma 3 and finally characterize the properties of supplier U’s profit in Lemma
4.
From now on, we characterize each interior solution on the full domain [c, ∞) . First, inCase (i), supplier U faces its input demand
r(i)
r
qr(i)
E = kQ(pE ) = kQ(w ).

(12)

Given this input demand, the upstream supplier U optimally chooses input price wr(i) ≡arg maxwr k(wr − c)Q(wr ) in Stage 3. By the maximization problem, we have the profit ofsupplier U as follows:
πr(i) = maxk(wr − c)Q(wr ) = kπ∗ (c).
r
w

(13)

Note that when k = 1, πr(i) = π∗ (c), which implies that entrant DE ’s entry allows supplier
U to earn profits equivalent to the maximized value of the joint profits of supplier U andincumbent DI . From equations (11) and (13), we identify the following properties.
10Lemma 1. Under the interior solution wr(i) ∈ [c, ∞), πr(i) have the following properties:
1. πr(i) is strictly increasing in k but decreasing in c.
2. As k → 1, πr(i) → π∗ (c), which is strictly larger than ΠaI + πa .
3. As k → 0, πr(i) → 0.
Second, in Case (ii), supplier U faces its input demand qr(ii)= kQ(p∗ (kwr )). Given thisE
input demand, supplier U chooses the input price to maximize its profit in Stage 3;πr(ii) = max(wr − c)kQ(p∗ (kwr )) = max {wQ(p∗ (w)) − kcQ(p∗ (w))} .
r
w

w

(14)

From equations (10) and (14), we identify the following properties.
Lemma 2. Under the interior solution wr(ii) ∈ [c, ∞), πr(ii) have the following properties:
1. πr(ii) is strictly decreasing in k and c.
2. As k → 1, πr(ii) → πa .
3. For any c ≥ 0, as k → 0, πr(ii) → πa |c=0 .
4. For c = 0, πr(ii) = πa |c=0 ,where πa |c=0 is supplier U’s profit level under the standard double marginalization problem
when c = 0 (see (10)).
We now characterize these two equilibria on two domains, [c, wr (k)] and [wr (k), ∞), wherewr (k) is the input price satisfyingp∗ (kwr (k)) = wr (k)for each k and is the threshold value at which the mode in Stage 4 changes from Case (i)to Case (ii). The following lemma shows that at least one interior solution exists for all
0 < k < 1.
11

Lemma 3. For Cases (i) and (ii), at least one of the following holds, namely, wr(i) ∈ (c, wr (k))
or wr(ii) ∈ (wr (k), ∞).
Because we have πr(i) = πr(ii) if we set wr(i) and wr(ii) at wr (k), we can conclude that
one of the interior solutions mentioned above becomes the optimal solution of supplier U in
equilibrium. Therefore, exclusion is possible regardless of equilibrium types if we have
ΠaI + πa > max πr(i) , πr(ii) .

(15)

The following lemma characterizes the properties of max πr(i) , πr(ii) .
Lemma 4. max πr(i) , πr(ii) has the following properties.
1. It is strictly decreasing in c.
2. Its functional form is V-shaped with respect to k; that is, there exists a minimized valuek0 ∈ (0, 1). More precisely, we have r(i) r(ii)πr(ii)max π , π=πr(i)

if 0 < k ≤ k0 ,if k0 < k < 1.

(16)

Figure 3 summarizes the property of max πr(i) , πr(ii) . Note that the equilibrium outcomeswhen the exclusive supply offer is accepted do not depend on k. Therefore, exclusion ispossible, if condition (15) holds for k = k0 .
By combining the above arguments, we have the following proposition:
Proposition 1. Suppose that the downstream firms are undifferentiated Bertrand competitors.
Then, there can be an exclusion equilibrium when entrant DE becomes efficient (that is, k <k∗ ), wherek∗ ≡ΠaI + πa.
π∗ (c)

More precisely,
1. For k∗ ≤ k < 1, entry is a unique equilibrium outcome, and
12

(17)

2. For k < k∗ , the possibility of exclusion depends on the efficiency of supplier U;(a) When supplier U is sufficiently efficient, 0 ≤ c < c̃, exclusion is possible for
0 < k < k∗ , where c̃ is a threshold value such that πa |c=0 = ΠaI + πa .
(b) When supplier U is not too efficient, c̃ ≤ c, exclusion is possible for 0 < k00 < k <k∗ , if there exists k00 < k∗ that satisfies πr(ii) = ΠaI + πa .
This proposition implies that the possibility of exclusion depends not only on entrant DE ’s
efficiency but also on that of supplier U. To clarify the property of Proposition 1, we show
the results in Proposition 1 under a linear demand Q(p) = (a − p)/b, where a > c and b > 0.
Remark 1. Under linear demand, exclusion of the highly efficient entrant DE (k < 3/4)
occurs if supplier U is sufficiently efficient (c < 0.18a is sufficient). More precisely,
1. For 3/4 ≤ k < 1, entry is a unique equilibrium outcome, and2. For 0 < k < 3/4, exclusion is a unique equilibrium outcome, if the upstream supplier
U is sufficiently efficient, that is, 0 ≤ c < Ĉ(k) where
√

a 6−2
.
(18)
Ĉ(k) = √6 − 2k√Note that ∂Ĉ(k)/∂k > 0, Ĉ(k) → a(3 − 6)/3 ' 0.1835a as k → 0, and Ĉ(k) → 2a(6 −√6)/15 ' 0.4734a as k → 3/4.
Figure 4 summarizes the result in Proposition 1 under linear demand. Under linear demand,√√
we have k∗ = 3/4, k00 = (2a − (a − c) 6)/2c, and c̃ = (3 − 6)a/3 ' 0.1835a.
The result in Proposition 1 contrasts with those in the previous literature on anticompetitive exclusive dealings. In the previous literature, as the entrant becomes efficient, firms areunlikely to engage in anticompetitive exclusive dealings. In this study, on the contrary, anticompetitive exclusive dealings are likely to be observed as the entrant becomes efficient. In
other words, an exclusive contract operates like the Luddites.14
14

See, for example, Hobsbawm (1952) and Mokyr (1992).

13

The result in Proposition 1 is derived from the negative relationship between entrant DE ’s
efficiency and the demand for the input. Equation (12) implies that the demand for the input
decreases as entrant DE becomes efficient (as k decreases) in Case (i). The socially efficient
entry of entrant DE generates two effects. First, entrant DE ’s entry generates downstream
competition and increases the production level of final goods. This expands the demand forthe input and increases supplier U’s profit. Second, contrarily, entrant DE ’s entry decreases
incumbent DI ’s market share but increases its own market share—note that entrant DE demands a smaller amount of the input, unlike incumbent DI . This reduces the total demand forthe input, and hence, supplier U’s profit. Therefore, the entry of the highly efficient entrant
DE increases the profit of supplier U only slightly. This allows the downstream incumbent
DI to profitably compensate the upstream supplier’s profit when such entry occurs, by using
its monopoly profits under exclusive dealing.
However, Figure 4 also shows that the relation between the likelihood of exclusion andentrant DE ’s efficiency is non-monotonic; that is, exclusion is more (less) likely to be observed for the intermediate (significantly high) level of entrant DE ’s efficiency. When entrant
DE becomes sufficiently efficient, the equilibrium outcome under entry becomes Case (ii) andentrant DE can monopolize the downstream market. In Case (ii), the existence of incumbent
DI does not work as a constraint on entrant DE ’s pricing and a further efficiency improvementof entrant DE decreases the price of final products, which then expands the production levelof entrant DE and the demand for the input. Therefore, for the significantly higher level ofentrant DE ’s efficiency, the upstream supplier U welcomes an improvement in entrant DE ’s
efficiency. This decreases the possibility of anticompetitive exclusion.
Note also that Figure 4 implies that the possibility of anticompetitive exclusive dealings
depends on supplier U’s efficiency: as supplier U becomes inefficient, the possibility of anticompetitive exclusive supply agreements decreases. This is because entrant DE ’s efficienttransformational technology reduces supplier U’s production cost, which improves the latter’s
profit. As supplier U becomes less efficient, the benefit of such cost reduction increases for
supplier U, which decreases the possibility of anticompetitive exclusive supply agreements.

14

4

Quantity Competition

This section considers the existence of anticompetitive exclusive dealings to deter the sociallyefficient entry of entrant DE when downstream firms compete in quantity. In this section, we
use linear demand; that is, the inverse demand for the final product P(Q) is given by a simple
linear function that we use in Remark 1:
P(Q) = a − bQ,
where Q is the output of the final product supplied by downstream firms and where a > cand b > 0. The following proposition shows that exclusion is possible not only under pricecompetition but under Cournot competition as well.
Proposition 2. Suppose that the downstream firms compete in quantity. Then, there can be an
exclusion equilibrium as entrant DE becomes efficient (that is, k < k̂), where k̂ ' 0.921543.
More precisely,
1. For k̂ ≤ k < 1, entry is a unique equilibrium outcome, and2. For 0 < k < k̂, exclusion is a unique equilibrium outcome, if the upstream supplier U
is sufficiently efficient, that is, 0 ≤ c < C(k) and

p
  3

a 2k + 3k2 + 3k − 7 + 27(1 − k)2 (−4k4 − 12k3 + 31k2 − 26k + 10)







(2k − 1)(14k − 13)(k2 − k + 1)
C(k) = if c < Ċ(k),






 Ĉ(k)if c ≥ Ċ(k).

(19)

where Ĉ(k) is in (18) andĊ(k) =

pa k2 − k + 1 − 3(1 − k)2 (k2 − k + 1)
(2 − k)(k2 − k + 1)

.

(20)

Note that ∂Ċ(k)/∂k > 0, Ċ(k) → 0 as k → 1/2, and Ċ(k) → 1 as k → 1 and that ∂C(k)/∂k <√(≥)0 for k > (≤)K̇(cA ), C(k) → a(3 − 6)/3 ' 0.1835a as k → 0, and C(k) → 0 as k → k̂.
15

Figure 5 summarizes the result in Proposition 2. On comparing Figures 4 and 5, we
observe a notable difference that the possibility of anticompetitive exclusion under Cournot
competition is higher; that is, exclusion may arise even when k > 3/4. This result follows from the difference in the degree of demand expansion between the two types of competition. Compared with undifferentiated Bertrand competition, entrant DE ’s entry under
Cournot competition leads to smaller demand expansion, because it partially solves the double marginalization problem. Therefore, entrant DE ’s entry leads to a smaller increase insupplier U’s profit.
In addition, when both downstream firms produce a positive amount of final products (forc < Ċ(k)), the possibility of anticompetitive exclusion increases as entrant DE becomes efficient. Under undifferentiated Bertrand competition, the production level of incumbent DI
is always zero and it does not depend on the degree of entrant DE ’s efficiency. By contrast,
under Cournot competition, the production level of incumbent DI is positive, and more importantly, an improvement of entrant DE ’s efficiency gradually reduces the production levelof incumbent DI that demands a large amount of input. This additional effect leads to further
reduction of demand for input and, thus, increases the benefit of exclusion.

5

Discussion

This section briefly discusses the wholesale pricing of the input and the efficiency of downstream firms. In Section 5.1, we extend the analysis by allowing price discrimination by the
upstream supplier. In Section 5.2, we discuss linear wholesale pricing. In Section 5.3, we
discuss the efficiency measure of downstream firms.

5.1

Price discrimination

Thus far, we assumed that supplier U charges downstream firms a uniform price wr . We
briefly discuss how the results in Section 3 change if supplier U is able to discriminate on
price when entrant DE enters the downstream market. Then, if supplier U chooses input

16

prices wri for Di , where i ∈ {I, E}, the per unit costs of incumbent DI and entrant DE are
denoted by wrI and kwrE , respectively. Although we have assumed that entrant DE must incura sufficiently small fixed cost on entry, we now assume that entrant DE always enters thedownstream market in Stage 2 if the exclusive supply offer is rejected in Stage 1.15
Consider the case where supplier U rejects the exclusive supply offer in Stage 1. In
this case, entrant DE enters the downstream market in Stage 2. In Stage 4, undifferentiatedBertrand competition occurs, which leads to monopolization by the downstream firm withthe lower per unit cost. In equilibrium, supplier U optimally chooses a pair of input prices(wrI , wrE ) such that wrI = kwrE and it earns
πr = max(wrI − kc)Q(wrI ) = π∗ (kc).
r
wI

(21)

In contrast, downstream firms earn zero profits. The result here implies that if supplier U candiscriminate on price, then it can jointly maximize profits with entrant DE and earn all profits
even under linear pricing.16 By comparing such profits with the joint profit with entrant DE
when supplier U accepts the exclusive supply offer (11), it is easy to see that condition (7)never holds, and thus, exclusion becomes impossible.
Proposition 3. Suppose that the downstream firms are undifferentiated Bertrand competitors.
Suppose also that entrant DE always enters the market if the exclusive supply contract isrejected in Stage 1. If supplier U is allowed to discriminate on price, then incumbent DI
cannot deter socially efficient entry by using an exclusive supply contract.
The intuitive logic is as follows. When supplier U can discriminate on price, it can extract
profits of entrant DE by setting wrE = wrI /k. In addition, supplier U can control the final
product price by choosing input price wrI = kwrE = p∗ (kc). Therefore, supplier U can jointly
15

This assumption is significant because the gross profit of entrant DE after entry is zero in this model.
Anticipating the subgame outcome after Stage 2, entrant DE does not enter the market if its sunk entry fee is notcompensated by supplier U. The assumption in this section implies that U pays a side payment to entrant DE tocompensate the entry fee.
16
When downstream firms compete in quantity, joint profit maximization is impossible and these firms earn
positive profits. The result in this section is qualitatively similar to that in which they compete in quantity.

17

maximize profits with entrant DE , when it rejects the exclusive supply offer and does notengage in exclusive supply agreements.
Proposition 3 and the result in Section 3 imply that the imposition of uniform pricing
induces exclusion of an efficient entrant through an exclusive supply contract offered by an
inefficient incumbent. That is, a ban on price discrimination, such as the famous Robinson–
Patman Act, can protect smaller or otherwise weaker competitors. We believe that the result
confirms the main results in Inderst and Valletti (2009), which show that the ban on pricediscrimination in input markets benefits smaller firms but hurts more efficient, larger downstream firms when downstream firms engage in cost-reducing activities. Therefore, we canconclude that this study shows another manner in which a ban on input price discrimination
harms market environments.

5.2

Linear pricing contracts

Note that we believe that linear pricing contracts are common in the real world—for instance,
in gasoline retailing and shipping industries (Lafontaine and Slade, 2013). In the context oflicensing agreements, licenses are often subject to fair, reasonable, and non-discriminatory(FRAND) commitments, which often require that every licensee be able to choose from thesame royalty schedule (Gilbert, 2011). Layne-Farrar and Lerner (2011, p.296) mention thatlicensees pay the patent pool administrator either a percentage of their net sales revenue fromselling the licensed product or a flat fee per unit sold. Gilbert (2011) also mentions the realworld examples of licensing terms that are subject to FRAND commitments and linear pricing
contracts.17 Furthermore, linear pricing contracts are sometimes employed in manufacturingindustries, although non-linear pricing contracts are useful in vertical coordination (Nagle and
Hogan, 2005 and Blair and Lafontaine, 2005). Even in the situation of franchise contracts,
franchisors face several problems in using franchise fees (fixed payments) as their meansof compensation: for instance, wealth constraints of franchisees and franchisor opportunism
17

Following those real-world observations, Tarantino (2012) considers standard setting organizations’ decisions on licensing policy with linear pricing and the standard’s technological specifications.

18with a lump-sum fee (Blair and Lafontaine, 2005).18 Therefore, we believe that this study
provides a useful policy implication in vertical restraints.

5.3

Efficiency measure

We have assumed that entrant DE is more efficient than incumbent DI in terms of a transformational technology of an input produced by the upstream supplier U; that is, entrant DE
demands a smaller quantity of inputs from supplier U to produce one unit of final product.
However, as commonly used in existing literature, we can consider the following efficiency
measure to evaluate the efficiency of downstream firms: entrant DE is more efficient thanincumbent DI in terms of its per unit production cost for several inputs, such as labor, which
are not produced by supplier U. For example, suppose that downstream firms produce final
products using input A that is exclusively supplied by an upstream supplier U A at price wA
and input B supplied by competitive sectors at price cB ≥ 0. Then, we can assume that incumbent DI produces a unit of final product using one unit of input A and one unit of input B
but entrant DE produces a unit of final product using one unit of input A and 0 < m < 1 unitof input B:
wI = wA + cB , wE = wA + mcB .
Under this efficiency measure, we can show that incumbent DI cannot deter socially efficient
entry by using an exclusive supply contract (proof of this result is available upon request).
That is, the difference in measures to evaluate the downstream firms’ efficiency turns out tobe crucial.
18

As also documented in Iyer and Villas-Boas (2003, p.81), in practice, both the magnitude and incidenceof two-part tariffs may be insignificant. Milliou, Petrakis, and Vettas (2009) provide a theoretical reason for
employing linear pricing contracts. Inderst and Valletti (2009) also explain real-world examples in which linear
pricing contracts are employed.

19

6Concluding Remarks

This study examined anticompetitive exclusive supply agreements focusing on the transformational technology of inputs. Previous studies have not differentiated between the incumbent and entrants with regard to the transformational technology of the input produced by
the upstream supplier, because they mainly analyze the entry deterrence in upstream markets. However, our study suggests that when we focus on the entry deterrence in downstream
markets by considering exclusive supply contracts, then the difference in transformational
technology of the input could be an important market element.
We find that when the incumbent and entrant differ with regard to the transformational
technology of the input produced by the upstream supplier, the downstream incumbent andthe upstream supplier sign exclusive supply contracts to deter socially efficient entry, even in
the framework of the Chicago School argument where a single seller, a single buyer, and asingle entrant exist. In addition, the difference in transformational technology of the input
produced by the upstream supplier changes the relationship between the entrant’s efficiency
and the possibility of exclusion: anticompetitive exclusive supply agreements are more likely
to arise if the entrant’s efficiency is at an intermediate level.
This study provides new implications for antitrust agencies: it is necessary to focus on
the efficiency measure when we discuss the anti-competitiveness of exclusive supply agreements. It may be possible to measure downstream firms’ efficiency by checking the defective
rate in relationships between an input supplier and final good producer and the inventory rate
in relationships between a final good producer and retailers. In addition, our exclusion outcomes arise when upstream firms employ simple linear pricing contracts. This result provides
two implications. First, exclusive supply agreements are more likely to be observed in industries where linear pricing contracts are employed. Linear pricing contracts are commonly
employed in manufacturing, gasoline, and shipping industries and are observed for licensing
agreements subject to FRAND commitments. When the presented model is applied to exclusive supply agreements in these industries, antitrust agencies should be careful in measuring

20

downstream firms’ efficiency. Second, a ban on input price discrimination makes exclusion
possible because such a ban works as an enforcement on upstream firms to employ simple
linear pricing contracts.
There are several outstanding concerns requiring further research. The first concern isabout this study’s relationship with other studies on anticompetitive exclusive dealing. For
example, we assume that an upstream supplier firm is a monopolist. By inverting the vertical
relationship analyzed by Simpson and Wickelgren (2007) and Abito and Wright (2008), Oki
and Yanagawa (2011) show that upstream competition allows the downstream incumbent todeter efficient entry with exclusive supply contracts. We predict that if we add upstream
competition to our model, the likelihood of an exclusion equilibrium increases. The second
is about the generality of our results. Although the analysis under quantity competition ispresented in terms of parametric examples, the result might be valid in more general settings.
We hope this study facilitates researchers in addressing these issues.

Appendix A: Proofs of Results in Price Competition
A.1: Proof of Results under General Demand
A.1.1 Proof of Lemma 3
We show that at least one interior solution exists in the profit maximization problems in Case(i) and Case (ii) when the exclusive offer is rejected in Stage 1. For expositional simplicity,
we replace wr (k) with w(k), which satisfies (see the last paragraph before Lemma 3)p∗ (kw(k)) = w(k).
The profit maximization problems of supplier U in the two cases are given as
Case (i)max (w − c)kQ(w) s.t. w ∈ [c, w(k)],Case (ii)max (w − c)kQ(p∗ (kw)) s.t. w ∈ [w(k), ∞).

w

w

21

(22)

The first-order conditions are given as
Case (i)H (i) (w) ≡ Q(w) + (w − c)Q0 (w),Case (ii)H (ii) (w) ≡ Q(p∗ (kw)) + (w − c)kQ0 (p∗ (kw))p∗0 (kw).

Note that each maximization problem has a unique interior solution on the domain [c, ∞).
However, there exists a possibility of a corner solution where the problem in Case (i) has
an interior solution on the domain [w(k), ∞) and the problem in Case (ii) has an interior
solution on the domain [c, w(k)]. In such cases, supplier U’s profit is maximized at the corner,w = w(k). We explore whether the corner solution problem arises. Note that w(k) is the
optimal input price if and only if H (i) (w(k)) > 0 and H (ii) (w(k)) < 0. We show that the two
inequalities do not simultaneously hold. More precisely, we show that H (ii) (w(k)) > 0 if
H (i) (w(k)) > 0.
Suppose that H (i) (w(k)) > 0, that is,
H (i) (w(k)) = Q(w(k)) + (w(k) − c)Q0 (w(k)) > 0.

(23)

By using equation (22), H (ii) (w(k)) can be rewritten as
H (ii) (w(k)) = Q(w(k)) + (w(k) − c)kQ0 (w(k))p∗0 (kw(k)).

(24)

To explore the above equation’s property, we need to derive p∗0 (z). The first-order conditionof the profit maximization problem (8) becomesQ(p∗ (z)) + (p∗ (z) − z)Q0 (p∗ (z)) = 0.
The total differential of this equation leads top∗0 (z) =Q0 (p∗ (z)).
2Q0 (p∗ (z)) + (p∗ (z) − z)Q00 (p∗ (z))(25)

By using equations (22), (23), (24), and (25), we have the following relationship:
H (ii) (w(k)) > H (ii) (w(k)) − H (i) (w(k))(w(k) − c)Q0 (w(k)){2Q0 (p∗ ) + (p∗ − kw(k))Q00 (p∗ ) − kQ0 (p∗ )}=−2Q0 (p∗ ) + (p∗ − kw(k))Q00 (p∗ )(w(k) − c)Q0 (w(k)){kQ0 (p∗ ) + (1 − k)[2Q0 (p∗ ) + p∗ Q00 (p∗ )]}> 0,=−
2Q0 (p∗ ) + (p∗ − kw(k))Q00 (p∗ )
22

(26)

where p∗ ≡ p∗ (kw(k)) = w(k). The last inequality holds because of Q0 (p) < 0 and Assumption 1.
From the above discussion, we have H (ii) (w(k)) > 0 if H (i) (w(k)) > 0. This implies that inCase (ii), an interior solution always exists on the domain (w(k), ∞) if in Case (i), the interior
solution does not exist on the domain [c, w(k)] and the corner solution appears; that is, wealways have wr(ii) ∈ (w(k), ∞) if wr(i) = w(k). This also implies that at least one interior
solution exists and that there are three possibilities concerning the optimal input price forsupplier U:
1. An interior solution exists only on the domain (c, w(k)) in Case (i).
2. An interior solution exists only on the domain (w(k), ∞) in Case (ii).
3. Interior solutions exist on the domains (c, w(k)) in Case (i) and (w(k), ∞) in Case (ii).
In the first and the second cases, we have unique interior solutions. In the third case, we needto check which of the interior solutions is really optimal. The way to check it is provided in
Section 3.
Q.E.D.
A.1.2 Proof of Lemma 4
For a sufficiently small k (as k → 0), we have πr(i) < πr(ii) . However, for k = 1, we have
πr(i) > πr(ii) . Because πr(ii) is strictly decreasing in k but πr(i) is strictly increasing in k, there
exists k0 ∈ (0, 1) such that πr(i) = πr(ii) .
Q.E.D.
A.1.3 Proof of Proposition 3
We first show that input prices that satisfy wrI < kwrE cannot be an equilibrium. Suppose, in
negation, that supplier U optimally chooses wrI = vI and wrE = vE such that wrI < kwrE , which
23

allows DI to supply its product and to choose the price at prI = kwrE . Then, supplier U earns
πr = (wrI − c)Q(kwrE ) = (vI − c)Q(kvE ).

(27)

However, if supplier U instead chooses wrI = kvE and wrE = vI /k, which satisfies wrI > kwrE(that is, kvE > vI ), entrant DE supplies its product and chooses the price at prE = wrI . Under
this input pricing, supplier U earns
πr = (wrE − c)kQ(wrI ) = (vI − kc)Q(kvE ).
This is larger than (27) because k < 1. This is a contradiction.
Second, we also show that we do not have wrI > kwrE in the equilibrium. Suppose, in
negation, that supplier U optimally chooses wrI > kwrE . Then, supplier U earns
πr = (wrE − c)kQ(wrI ) = (kwrE − kc)Q(wrI ) < (wrI − kc)Q(wrI ) = π∗ (kc).
That is, by setting kwrE at wrI , supplier U can increase its profit. This is a contradiction.
Therefore, the optimal input prices of supplier U satisfy wrI = kwrE and its profit is πr = π∗ (kc)in (21). This is higher than the sum of profits in (11). Therefore, Proposition 3 holds.
Q.E.D.

A.2 Proof of Results under Linear Demand
A.2.1 Equilibria in subgames after Stage 1
We consider each of the possible subgames after Stage 1. In this Appendix, we consider
the case of quantity competition. In A.2.1.1, we consider the case where an exclusive offer is
accepted by the upstream supplier. Then, in A.2.1.2, we consider the case where the exclusive
offer is rejected by this supplier.
A.2.1.1 When the exclusive offer is accepted in Stage 1
The equilibrium demand level for the input becomes
qa = QaI =
24

a−c
.
4b

Before compensation through x, the profits of supplier U and incumbent DI are given asπa =(a − c)2(a − c)2, ΠaI =.
8b
16b

A.2.1.2 When the exclusive offer is rejected in Stage 1
As we have seen in Section 3, there are two types of equilibria in the subgame after the
exclusive offer is rejected in Stage 1. We first solve each case on the full domain wr ∈ [c, ∞).
Consider Case (i). The profits of supplier U and incumbent DI are given asπr(i) = k(a − c)2, Πr(i)I = 0.
4b

Next, we consider Case (ii). Given w, entrant DE chooses the following price.
p∗ (w) =a + kw.
2

The equilibrium input price and price of final product are given as
wr(ii) =a − kc3a + kc, pr(ii)
.
E =
2k
4

the profits of supplier U and incumbent DI are given asπr(ii) =(a − kc)2= 0.
, Πr(ii)I
8b

We now determine the optimal input price for supplier U. For 0 < k ≤ 1/2, we alwaysr(i)have πr(i) < πr(ii)< πr(ii) if c > Ć(k), where
A . In contrast, for 1/2 < k < 1, we have π√a(k − 2k(1 − k))Ć(k) =,k(2 − k)and where ∂Ć(k)/∂k > 0, Ć(1/2) = 0, and Ć(1) = 1.
Finally, we examine whether input prices in each equilibrium satisfy the definition of eachcase. We first check Case (i). For 0 < k < 1, we have p∗ (wr(i) ) > wr(i) if c < C̆(k), whereC̆(k) =

ka
,
2−k

25

where ∂C̆(k)/∂k > 0, C̆(0) = 0, and C̆ A (1) = 1. Because we have√a( 2k − k)(1 − k)C̆(k) − Ć(k) => 0,
k(2 − k)for all 0 < k < 1, whenever we have πr(i) > πr(ii) , the equilibrium input price in Case (i)
r(ii)
r(i)
r(ii)
satisfies pr(i).
E = w . We next check Case (ii). For 0 < k < 2/3, we always have pE < w
r(ii)For 2/3 < k < 1, we have pr(ii)if c < C̀(k) whereE < w

C̀(k) =a(3k − 2),k(2 − k)and where ∂C̀(k)/∂k > 0, C̀(2/3) = 0, and C̀(1) = 1. Because we have√a(2 − 2k)(1 − k)> 0,Ć(k) − C̀(k) =k(2 − k)for all 2/3 < k < 1, whenever we have πr(i) < πr(ii) , the equilibrium input price in Case (ii)r(ii)
satisfies pr(ii). Therefore, for all 0 < k ≤ 1/2 we have the equilibrium in Case (ii). On
E < w

the other hand, for 1/2 < k < 1 we have the equilibrium in Case (i) (Case (ii)) if c ≤ Ć(k)(c > Ć(k)).
A.2.2 Proof of Remark 1
If the equilibrium outcomes in Case (i) arise on the equilibrium path when the exclusive
supply offer is rejected, condition (7) holds for k < 3/4. In contrast, if the equilibriumoutcomes in Case (ii) arise on the equilibrium path, condition (7) holds for 0 ≤ c < Ĉ(k).
Q.E.D.

Appendix B: Proofs of Results in Quantity Competition
B.1 Equilibria in Subgames after Stage 1
We consider each of the possible subgames after Stage 1. In this Appendix, we consider
the case of quantity competition. In B.1.1, we consider the case where an exclusive offer is
accepted by the upstream supplier. Then, in B.1.2, we consider the case where the exclusive
offer is rejected by this supplier.
26

B.1.1 When the exclusive offer is accepted in Stage 1
See Appendix A.2.1.1.

B.1.2 When the exclusive offer is rejected in Stage 1
When supplier U rejects the exclusive supply offer, entrant DE enters the downstream market
and supplier U deals with incumbent DI and entrant DE . Given w, the quantities supplied by
the downstream firms are given as a − (2 − k)wa, if w ≤
,


3b
2−k
qrI (w) = 

a


 0,
if w ≥,2−k
aa + (1 − 2k)w, if w ≤
,



3b
2
−
k
r
qE (w) = aa − kw

,
if w ≥,2b2−kAnticipating the outcome, supplier U sets its w to maximize its profit.
max (w − c)(qI (w) + kqE (w)).
wFirst, for w ≤ a/(2 − k), we derive the “local” optimal input price. The maximization
problem is given asa − (2 − k)w
a + (1 − 2k)w
max (w − c)
+k
w
3b
3b

!
s.t. w ≤a.
2−kThis leads to(1 + k)a + 2(1 − k + k2 )c.
4(1 − k + k2 )This is an interior solution if and only if
wr =

c≤

2 − 5k + 5k2.
2(2 − 3k + 3k2 − k3 )The profits of supplier U and incumbent DI are given asπr =(a(1 + k) − 2c(k2 − k + 1))2(a(5k2 − 5k + 2) − 2c(2 − k)(k2 − k + 1))2
r
=
,
Π
.
I
24b(k2 − k + 1)
144b(k2 − k + 1)
27

Second, for w ≥ a/(2 − k), we derive the “local” optimal input price. The maximization
problem is given as
max (w − c)kw

a − kwas.t. w ≥.
2b
2−kThis leads towr =a + kc.
2kThis is an interior solution if and only ifc≥3k − 2.
k(2 − k)The profits of supplier U and incumbent DI are given asπr =(a − kc)2, ΠrI = 0.
8b

For any c ∈ [0, a) such that2 − 5k + 5k2
3k − 2≤c≤,k(2 − k)2(2 − 3k + 3k2 − k3 )the two local optimal input prices exist. We now determine the one that is better for supplier
U. The first input price leads to higher profits for U if and only if c < Ċ(k) (see (20)).
Therefore, if c < Ċ(k), the optimal input price is the first wr , and then, both incumbent DI
and entrant DE are active. Otherwise, it is the second wr , and then, only entrant DE is active.

B.2 Proofs of Results
Proof of Proposition 2
We explore whether an exclusion equilibrium exists by examining whether the inequality in(7) holds. Substituting the result in the previous subsection into the inequality in (7), we havethe condition, 0 < c < C̄(k) (see (19)), in Proposition 2. Therefore, an exclusion equilibrium
exists for 0 ≤ c < C(k).
Q.E.D.
28

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Figure 1: Time line

Figure 2: Wholesale price offers in Stage 334

Figure 3: Properties of max πr(i) , πr(ii)35Figure 4: Results of Proposition 1 under linear demand (a = 1)

Figure 5: Results of Proposition 2 (a = 1)36