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

Noriaki Matsushima‡

Misato Sato§

July 10, 2023
Abstract
We develop a bilateral monopoly model with a downstream entrant to examine anticompetitive exclusive supply contracts that prevent the upstream supplier from selling inputs to thedownstream entrant. When the entrant is efficient and needs a lesser amount of input producedby the supplier than the incumbent, the input demand does not increase significantly following
the entry. Therefore, the socially efficient entry does not increase the supplier’s profits significantly, allowing the downstream incumbent to deter socially efficient entry through exclusive
supply contracts, even in the simplest framework, which is composed of a single seller, buyer,
and entrant.

JEL classifications code: L12, L41, L42, C72.
Keywords: Antitrust policy; Entry deterrence; Exclusive supply contracts; Transformational technology; Input price discrimination.
* We have benefited from many constructive comments from the General Editor and two anonymous referees.
We especially thank Yuki Amemiya, Ricard Gil, Akifumi Ishihara, Lars Persson, Keizo Mizuno, Patrick Rey, David
Salant, Mark Schankerman, and Tommaso Valletti for their insightful comments. We also thank Zhijun Chen, Chiaki Hara, Kazumi Hori, Junichiro Ishida, Shingo Ishiguro, Atsushi Kajii, Keisuke Kawata, Toshihiro Matsumura,
Masaki Nakabayashi, Tatsuhiko Nariu, Ryoko Oki, Tadashi Sekiguchi, Takashi Shimizu, Katsuya Takii, conference
participants at Asia-Pacific Industrial Organisation Conference, the European Association for Research in Industrial
Economics, the International Industrial Organization Conference, the Japan Association for Applied Economics, theJapanese Economic Association, and seminar participants at Contract Theory Workshop, Hitotsubashi University,
Kwansei Gakuin University, Kyoto University, and Osaka University for helpful discussions and comments. The second author appreciates the warm hospitality at MOVE, Universitat Autònoma de Barcelona, where part of this paper
was written, and acknowledges financial support from the “Strategic Young Researcher Overseas Visits Program for
Accelerating Brain Circulation” of JSPS. We gratefully acknowledge financial support from JSPS KAKENHI Grant
Nos. JP22243022, JP24530248, JP24730220, JP15H03349, JP15H05728, JP15K17060, JP17H00984, 18H00847,
18K01593, 19H01483, JP20K13485, JP20H05631, JP21K01452, and JP22H00846, and the program of the Joint Usage/Research Center for ‘Behavioral Economics’ at ISER, Osaka University. Although Matsushima serves as the
director of the Competition Policy Research Center (CPRC) at the Japan Fair Trade Commission (JFTC), our paper
does not necessarily reflect the views of the JFTC. The usual disclaimer applies.
†
Professor, Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku, Kyoto 603-8555,
Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp‡(Corresponding author) Professor, Institute of Social and Economic Research, Osaka University, 6-1 Mihogaoka,
Ibaraki, Osaka 567-0047, Japan. E-mail: nmatsush@iser.osaka-u.ac.jp§
Associate Professor, Faculty 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 often observe exclusive supply contracts between an input supplier and a final product producer in antitrust cases. For example, a large-scale pharmaceutical company enforces 10-year
exclusive supply agreements for an essential ingredient.1 In the relationship between a final good
producer and retailer, an established toy retailer also prevents toy manufacturers from selling towarehouse clubs.2 More recently, an online gaming company prohibits mobile game developersfrom providing their games through a rival online gaming company.3 Despite these observations,in the literature of anticompetitive exclusive dealings, many papers focus on exclusion in upstream
markets. Thus, this study aims to focus on exclusion in downstream markets.
We construct a model of anticompetitive exclusive supply agreements in which a downstream
incumbent prevents an upstream supplier from selling inputs to a potential downstream entrant
whose efficiency is higher than the incumbent in terms of the amount of necessary inputs producedby the upstream supplier. We can interpret the efficiency difference as differences in the defective
rate in relationships between an input supplier and final product producers. The technology difference can also be explained by lower input use or lower wasted materials. The source of such
efficiency differences can become a crucial issue for the upstream supplier because the efficiencyof downstream firms can affect the demand for input produced by the upstream supplier.
Under linear wholesale pricing (Iozzi and Valletti, 2014), we show that the downstream incumbent can deter socially efficient entry through exclusive supply contracts even in the simplest
setting, where a single seller, buyer, and entrant exist. More precisely, when the entrant and incumbent have similar efficiency levels, we never have exclusion results; however, as the entrant’s
efficiency increases, exclusion can occur. The results imply that anticompetitive exclusive supply
1

FTC v. Mylan Laboratories, Inc., Cambrex Corporation, Profarmaco S.r.l., and Gyma Laboratories of America,
Inc., No.X990015-1 (http://www.ftc.gov/os/caselist/x990015ddc.shtm).
2
Toys“R”Us, Inc., v.
FTC, 0910082 (https://www.ftc.gov/enforcement/cases-proceedings/
091-0082/toys-r-us-inc).
3
See Cease and Desist Order against DeNA Co., Ltd (http://www.jftc.go.jp/en/pressreleases/
yearly-2011/jun/individual-000427.html).

1

agreements are attainable when the downstream entrant has a lower defective rate.
To understand our results, consider the impact of socially efficient entry. Socially efficient
entry generates downstream competition and increases the final product output, which increases
the demand for inputs produced by the upstream supplier and, consequently, its profit. However,as the entrant becomes efficient, it demands a smaller quantity of input produced by the upstream
supplier. Therefore, the entry does not significantly increase the input demand. As a result, theupstream supplier cannot earn considerably high profits, allowing the downstream incumbent todeter socially efficient entry by profitably compensating the upstream supplier.
The findings here provide the following important implication for anticompetitive exclusive
dealing: When we consider entry deterrence in downstream markets, the source of differencesin efficiency among downstream firms can be an important issue for upstream suppliers. Generally, upstream suppliers welcome the entry of efficient downstream firms because the efficiencyof downstream firms usually increases input demand through lower downstream prices. However, if downstream entrants are efficient in production technology to use inputs produced by upstream suppliers, those upstream suppliers are less enthusiastic about the entrants’ technologicalefficiency, which may lead to anticompetitive exclusive dealing.
We then extend our analysis by allowing input price discrimination and two-part tariffs to contribute to the literature of input price discrimination (Katz, 1987; DeGraba, 1990; Yoshida, 2000;
Inderst and Shaffer 2009). We show that input price discrimination and two-part tariffs reduce thepossibility of exclusion because the upstream supplier can extract all of the industry profits. However, if input price discrimination is imperfect because of input arbitrage between the downstream
firms, the exclusion result can be sustainable. Thus, anticompetitive exclusive supply agreements
are more likely to arise when arbitrage is easy because of higher product storability or when a dominant downstream firm offers a price parity clause, inducing the supplier to use uniform pricing.
The structure of the paper is as follows. Section 2 presents a literature review on anticompetitive exclusive dealings. Section 3 contains the preliminaries. Section 4 provides the main results.
2Section 5 provides some discussions. Section 6 offers concluding remarks. Appendix A introduces
the property of general demand. Appendix B presents the proofs of results.

2

Literature Review

This study is related to the literature on anticompetitive exclusive dealings.4
4

Certain studies examine procompetitive exclusive dealings. See Marvel (1982), Besanko and Perry (1993), Segal
and Whinston (2000a), de Meza and Selvaggi (2007), de Fontenay, Gans, and Groves (2010), and Chen and Sappington(2011). Below, we explain the literature on exclusive dealings and our contribution to it.
In the literature on anticompetitive exclusive contracts, using a simple setting with an upstream incumbent, an
upstream entrant, and a downstream buyer, the Chicago School argument in the 1970s (Posner, 1976; Bork, 1978)
points out that rational economic agents never sign exclusive contracts for anticompetitive reasons, if we consider all
members’ participation constraints in the contracting party.5 The argument casts doubt about an intuitive view that
exclusive contracts can deter efficient entrants.6
In rebuttal to the Chicago School argument, many papers find that rational economic agents agree with exclusive
contracts for anticompetitive reasons in certain market environments. We categorize those papers into the following
three: (i) increasing the number of downstream buyers from one; (ii) changing the nature of upstream competition; (iii)converting the vertical relations and focusing on downstream entry.7 Our paper is closely related to the third category.
The papers in the first category add other downstream buyers to the Chicago School argument. Some of those papers
focus on the presence of scale economies, wherein the upstream entrant needs a certain number of buyers to cover fixed
costs (Rasmusen, Ramseyer, and Wiley, 1991; Segal and Whinston, 2000b). Also, other papers consider downstream
competition, wherein the upstream entry does not generate considerably high profits for downstream firms because
downstream competition transfers most of the gain from entry to final consumers, who are third parties (Simpson andWickelgren, 2007; Abito and Wright, 2008).8
The literature in the first category progresses even nowadays. In the original setting in the Chicago School argument(an ex-ante bilateral monopoly) with an inefficient outside retailer that can be an outside option for the efficient entrant,
Kitamura, Matsushima, and Sato (forthcoming(a)) show that excluding an upstream entry emerges if the efficiency of
the upstream entrant is high. In addition, by extending the Chicago School setting to the case of durable goods,
Kitamura, Matsushima, and Sato (forthcoming(b)) show that exclusive contracts can be signed to deter upstream entry
because such entry exacerbates the intertemporal competition between a downstream durable goods monopolist now
and itself in the future.
The papers in the second category, which focus on the nature of upstream competition, point out that exclusion is
attainable when the incumbent supplier can set liquidated damages for the case of entry (Aghion and Bolton, 1987),
the entrant is capacity constrained (Yong, 1996), suppliers compete à la Cournot (Farrell, 2005), suppliers can merge(Fumagalli, Motta, and Persson, 2009), the incumbent supplier makes relationship-specific investments (Fumagalli,
Motta, and Rønde, 2012), there exists a complementary input supplier with market power (Kitamura, Matsushima,and Sato, 2018a).9The papers in the third category discuss downstream entrants.10 Comanor and Rey (2000) consider a market withan incumbent supplier, a downstream incumbent having external suppliers, and a downstream entrant. The existenceof external suppliers limits the downstream incumbent’s purchase price offered to the incumbent supplier, inducing theefficient downstream entrant to offer a low purchase price in response to the incumbent’s one. Therefore, the upstream
supplier cannot earn higher profits even when efficient entry occurs, inducing the upstream supplier to engage inanticompetitive exclusive dealings. Oki and Yanagawa (2011) also show an exclusion outcome in a market with two
upstream suppliers.
By contrast, the present study considers neither the outside option of the downstream incumbent nor upstream
competition but explores how the difference between downstream firms’ production technology affects anticompetitive
exclusive supply agreements.

3

3Preliminaries

This section develops the basic environment of the model. For convenience, we consider the
relationship between input suppliers and final product producers, although we can also apply thismodel to the relationship between final good producers and retailers.

3.1

Upstream and downstream markets

The downstream market is composed of an incumbent DI and entrant DE . Each downstream firm
produces a homogeneous final product, using only an input monopolistically produced by an upstream supplier U. For this supplier, the marginal cost is c ≥ 0, and w is the wholesale price offeredby it.
The downstream firms differ in production technology. DI produces one unit of the final product using one unit of input. The per-unit production cost of DI , cI , becomescI = w.

(1)

By contrast, DE produces one unit of the final product using k ∈ (0, 1) units of input. The per-unit
production cost of DE , cE , becomescE = kw.

(2)

Equation (2) implies that DE becomes more efficient than DI as k decreases.
We can interpret the production-technology assumption in two ways. First, between an input
supplier and final product producers, entrant producer DE has the efficient technology that allows it
to reduce input use or the number of defective products. Second, between a final product producer
U and retailers, entrant retailer DE is better at supply-chain management than the incumbent inthat it needs lower wasted products produced by the final product producer U.

4

3.2

Timing of the gameThe model consists of four stages (Figure 1). In Stage 1, DI offers an exclusive supply contract
to U.11 This contract involves some fixed compensation of x ≥ 0. U decides whether to accept
this offer. We use the scripts a and r to indicate the cases in which U accepts and rejects the offer,respectively. In Stage 2, after observing U’s decision, DE decides whether to enter the downstream
market. We assume that the fixed cost of entry f (> 0) is sufficiently small, such that if DE is active,
it could earn positive profits.
In Stage 3, U offers a common linear wholesale price of the input, w, to the active downstream
firm(s). In case a, U offers input price wa only to DI . In case r, U offers input price wr to all active
downstream firms. In Section 5, we discuss the case where input price discrimination is possible.
In Stage 4, active downstream firms order inputs, produce final products, and sell them to final
consumers. In case r, DI and DE compete. DI ’s profit when U accepts (rejects) the exclusive offer
is denoted by πaI (πrI ) and U’s profit when it accepts (rejects) the exclusive offer is denoted by πaU(πrU ).
Stage 1

Stage 2

Stage 3

Stage 4DI makes an
exclusive offer.

DE makes
entry decision.

U makes input
price offers.

Active downstream firms
order the input andcompete.

U decides.

Figure 1: Time line

3.3

Design of exclusive supply contracts

Given the equilibrium outcomes in the subgame following Stage 1, we derive the essential conditions for an exclusive supply contract. For an exclusion equilibrium, the equilibrium transfer x∗
11

The assumption on the first-mover advantage of DI follows the Chicago School argument. Some studies showthat exclusion results arise even in the absence of a first-mover advantage. See Choi and Stefanadis (2018), who
extend the model of exclusion with scale economies. Recently, Kitamura, Matsushima, and Sato (2018b) show that
exclusive-offer competition leads to anticompetitive exclusive dealing even in the framework of the Chicago School
argument. According to their results, if DE is able to make exclusive offers in the model here, DE excludes DI in theequilibrium, and such exclusion is still socially inefficient.

5

must satisfy the following two conditions simultaneously.
First, it must satisfy the individual rationality condition for DI ; that is, DI must earn higher
operating profits under exclusive dealing:
πaI − x∗ ≥ πrI or x∗ ≤ πaI − πrI .

(3)

Second, it must satisfy the individual rationality for U; that is, the compensation amount x∗
must induce U to accept the exclusive supply offer:x∗ + πaU ≥ πrU or x∗ ≥ πrU − πaU .

(4)

The inequalities (3) and (4) imply
πaI + πaU ≥ πrI + πrU .

(5)

Condition (5) implies that for the existence of anticompetitive exclusive supply contracts, we must
examine whether exclusive supply agreements increase the joint profits of DI and U. Therefore,
the existence of exclusion equilibria does not depend on who makes the offer; in other words, theresults do not change even if we allow U to make the exclusive supply offer.

4

Main Results

We consider the existence of anticompetitive exclusive dealings to deter the socially efficient entryof DE when the downstream firms engage in homogeneous good price competition with a wellbehaved general demand Q(p), where p is the retail price. Q(p) is continuous and Q′ (p) < 0.
The quantity that consumers demand for Di is Q(pi ) when pi < p j and 0 when pi > p j , wherei, j ∈ {I, E} and j , i. When pi = p j , the downstream firm with the lower per unit production cost
supplies the entire quantity Q(pi ). We define p∗ (z) and Π∗ (z) for z ≥ 0 as follows:p∗ (z) ≡ argmax (p − z)Q(p), Π∗ (z) ≡ (p∗ (z) − z)Q(p∗ (z)).
p

We assume that the demand function satisfies the following conditions.
6

Assumption 1. The demand function has the following properties. (i) For all p > z, (p − z)Q(p)is strictly and globally concave in p: 2Q′ (p) + (p − z)Q′′ (p) < 0; (ii) Q(p) is log-concave:−d(Q(p∗ (z))/Q′ (p∗ (z)))/dz ≤ 0.
Assumption 1(i) is the standard second-order condition and (ii) secures that the pass-throughrate, p∗′ (z), is less than 1. The upper bound of the pass-through rate is a key to obtaining the main
result (Proposition 1), and the curvature of the demand function (the elasticity of demand) in itself
is not a key factor in obtaining this result.
We assume that p∗ (z) has the following properties.
Assumption 2. The level of c is not too high:p∗ (0) > c.

(6)

In the following, we solve the game using backward induction, starting from Stage 4.

4.1

U accepts the exclusive supply offer in Stage 1 (case a)

We first consider the case where U accepts the exclusive offer in Stage 1. In this case, it can supply
only to DI . Given input price wa , DI optimally chooses paI (wa ) = p∗ (wa ) in Stage 4. By anticipating
this pricing, U sets the input price for DI to maximize its profit in Stage 3.
wa = argmax(w − c)Q(p∗ (w)).

(7)

w

To have a unique solution, we assume that (w − c)Q(p∗ (w)) is strictly and globally concave in w.
Because we have wa > c in the equilibrium, the equilibrium price level p∗ (wa ) does not maximize the joint profits of DI and U; that is, the double marginalization problem occurs.
πaI + πaU = (p∗ (wa ) − c)Q(p∗ (wa )) < Π∗ (c).

(8)

Although entry deterrence allows DI to earn higher operating profits, DI and U cannot maximize
their joint profits because of the double marginalization problem.
7

4.2

U rejects the exclusive supply offer in Stage 1 (case r)

We next consider the case where U rejects the exclusive supply offer in Stage 1. In this case, DE
enters the downstream market in Stage 2.
In Stage 4, given the input price wr , the downstream firms compete in price. DI earns zeroprofits in this subgame; that is, πrI = 0 for all 0 < k < 1. In addition, downstream competition leads
to two types of equilibria in Stage 4.
r(i)
r
r
∗
r
r
Case (i) DI offers pr(i)I = w and DE offers pE = w if p (kw ) ≥ w .
∗
r
∗
r
r
Case (ii) DI offers pr(ii)= wr and DE offers pr(ii)I
E = p (kw ) if p (kw ) ≤ w .

In Case (i) (if p∗ (kwr ) ≥ wr ), the marginal cost pricing of DI binds the pricing of DE , which leadsr∗rr
to pr(i)E = w . In Case (ii) (if p (kw ) ≤ w ), the marginal cost pricing of DI does not bind the∗r 12
pricing of DE , which leads to pr(ii)E = p (kw ).

By anticipating this pricing in Stage 4, U optimally chooses its input price in Stage 3. Notethat for each case, we have a unique interior solution; that is, we have wr(i) ∈ [c, ∞) and wr(ii) ∈[c, ∞). Each interior solution must satisfy the constraints (wr(i) ∈ [c, p∗ (kwr (k))] and wr(ii) ∈[p∗ (kwr (k)), ∞)), where wr (k) is the input price such thatp∗ (kwr (k)) ≡ wr (k),for each k. wr (k) is the threshold value at which the mode in Stage 4 changes from Case (i) to Case(ii). Under Assumption 1, we can show that wr (k) is unique for each k (see Appendix A).
In the rest of this subsection, we first characterize the properties of each interior solution in thefull domain [c, ∞) in Lemmas 1 and 2. We then consider the constraints of each interior solution in
Lemma 3 and characterize the properties of U’s profit in Lemma 4. Finally, in the next subsection,
we explore the existence of an exclusion equilibrium.
From here on, we characterize each interior solution in the full domain [c, ∞).
12

The properties of cases (i) and (ii) are similar to the Arrow’s terminology of nondrastic and drastic innovation.

8

First, in Case (i), U faces its input demand
r(i)
r
qr(i)
E = kQ(pE ) = kQ(w ).

(9)Given this input demand, U optimally chooses input price wr(i) ≡ argmaxwr k(wr − c)Q(wr ) in Stage
3. With the maximization problem, the profit of U is as follows:πr(i)k(wr − c)Q(wr ) = kΠ∗ (c).
U = maxrw(10)

From equations (8) and (10), we identify the following properties.
Lemma 1. Under the interior solution wr(i) ∈ [c, ∞), πr(i)U has the following properties:
1. πr(i)U is strictly increasing in k but decreasing in c.
∗aa2. As k → 1, πr(i)U → Π (c), which is strictly larger than πI + πU .

3. As k → 0, πr(i)U → 0.
The notable feature in Case (i) is that U earns lower profits as DE becomes efficient. Such a
counterintuitive result can be explained by the relationship between DE ’s efficiency and its input
demand. In Case (i), the equilibrium retail price becomes pr(i) = wr ; DE ’s efficiency does not affectthe retail price. Because of this property, the production level of final products is unchanged evenwhen DE becomes efficient. Thus, as in equation (9), DE ’s input demand becomes smaller as DE
becomes efficient, which induces U to earn a lower profit.
Second, in Case (ii), U faces its input demand qr(ii)= kQ(p∗ (kwr )). Given this input demand,E
U sets an input price to maximize its profit in Stage 3:
πUr(ii) = max(wr − c)kQ(p∗ (kwr )) = max (w − kc)Q(p∗ (w)).
rw

w

From equations (7) and (11), we identify the following properties.
Lemma 2. Under the interior solution wr(ii) ∈ [c, ∞), πUr(ii) has the following properties:
9(11)

1. πr(ii)U is strictly decreasing in k and c.
a
2. As k → 1, πr(ii)U → πU .
a
3. For any c ≥ 0, as k → 0, πr(ii)U → πU |c=0 .
a
4. For c = 0, πr(ii)U = πU |c=0 ,where πaU |c=0 is U’s profit level under the standard double marginalization problem when c = 0(see equation (7)).
The notable feature in Case (ii) differs from Case (i) in that U earns higher profits as DE becomes efficient. In Case (ii), U and DE face the double marginalization problem. The equilibrium
retail price becomes p(ii) = p∗ (kwr ); in contrast to Case (i), as DE becomes efficient, it sets thelower retail price, increasing the output level of final products and the input demand. Thus, U andDE earn higher profits as DE ’s efficiency increases.
We now characterize these two equilibria on two domains, [c, wr (k)] and [wr (k), ∞).
Lemma 3. For Cases (i) and (ii), at least one of the following holds, wr(i) ∈ [c, wr (k)] or wr(ii) ∈[wr (k), ∞).
□Proof. See Appendix B.1.

r(ii)
r(i)
Because πr(i)= wr(ii) = wr (k), one of the above-mentioned interior solutions
U = πU for w

becomes U’s optimal solution in equilibrium. Therefore, exclusion is possible regardless of equilibrium type if we haver(ii)πaI + πaU ≥ max πr(i)U , πU .
r(ii)The following lemma characterizes the properties of max πr(i)(see Figure 2).
U , πUr(ii)Lemma 4. max πr(i)has the following properties.
U , πU
1. It is strictly decreasing in c.
10

(12)

Profits

P * (c )p Ua |c =0

p Ur (i )p Ur (ii )

p Ua

0

k'

1

k


r(ii)
Figure 2: Properties of max πr(i)U , πU

2. Its functional form is V-shaped with respect to k. More precisely, we have
 r(ii)
if 0 < k ≤ k′ ,
 r(i) r(ii)πU
max πU , πU = πr(i)
if k′ < k < 1.
U□

Proof. See Appendix B.2.

r(ii)The characteristic of max πr(i)in Lemma 4 comes from the first items in Lemmas 1 andU , πU2, meaning that this value depends on k. However, the joint profit of DI and U under exclusion,
πaI + πaU , does not depend on k (see equation (8)) because DE is inactive. From the facts, exclusionis possible if condition (12) holds for k = k′ , at which max πUr(i) , πUr(ii) takes the lowest value.

4.3

The existence of an exclusion equilibrium

By combining the arguments in Sections 4.1 and 4.2, we finally explore the existence of an exclusion equilibrium by focusing on c.
r(ii)and πaI +πaU for c = 0
We start from the simplest case of c = 0. The properties of max πr(i)U , πUis summarized in the left side of Figure 4.3. By checking condition (12), we have πaI + πaU > πr(ii)U

11Profits

Profits

P * (c )p Ia + p Ua
p Ur (ii )P * (c )p Ur (i )p Ua |c =0(= p Ua )p Ua |c =0(= p Ia + p Ua )

0

k'

k*

1

0

k

p Ur (i )p Ur (ii )

p Ua

k'

k*

1

k


r(ii)
Figure 3: Properties of max πr(i)and πaI + πaU for c ≤ c̃U , πU
Note: c = 0 (left) and c = c̃ (right)′∗
for 0 < k ≤ k′ and πaI + πaU > πr(i)U for k ≤ k ≤ k , where

πaI + πaUk ≡.
Π∗ (c)∗

Thus, when c = 0, exclusion is an equilibrium for 0 < k ≤ k∗ .
r(ii)As c increases from c = 0, max πr(i)and πaI + πaU decrease, while πaU c=0 is unchanged.
U , πU
There exists a unique threshold value c̃ such that πaU c=0 = πaI + πaU (see the right-hand side of
r(i)
′
a
a
Figure 4.3). When 0 < c ≤ c̃, we have πaI + πaU ≥ πaU c=0 > πr(ii)U for 0 < k ≤ k and πI + πU ≥ πU

for k′ ≤ k ≤ k∗ , which implies that condition (12) always holds in Case (ii). Thus, by combining
the result for c = 0, we conclude that when 0 ≤ c ≤ c̃, exclusion is an equilibrium outcome for
0 < k ≤ k∗ .
We next consider the case of c > c̃. For a slight increase in c from c̃, there exists a thresholdvalue k′′ ∈ (0, k∗ ] such that πUr(ii) = πaI + πaU in Case (ii) (see the left-hand side of Figure 4.3) because
DE with sufficiently small k significantly contributes to the reduction of U’s real marginal cost,kc, (see (11)), in particular when c is high. As k becomes small, U can set a sufficiently highwholesale price and earn a large profit from the trade with DE . Thus, exclusion is impossible when
k is sufficiently small (0 < k < k′′ ), whereas it is possible when k is not too small (k′′ ≤ k ≤ k∗ );12Profits

Profitsp Ur (i )*P (c )

p Ua |c =0
p Ia + p Ua

p Ua |c =0

p Ur (ii )p Ur (ii )pp Ur (i )aU

P * (c )p Ia + p Ua
p Ua
0

k''

k'

k*

1

0

k

k*

k'

k''

1

k


r(ii)
Figure 4: Properties of max πr(i)and πaI + πaU for c > c̃U , πU
Note: not sufficiently large c (left) and sufficiently large c (right)that is, condition (12) does not always hold in Case (ii).
By contrast, for sufficiently large c, k∗ < k′ < k′′ may hold as in the right-hand side of Figure
4.3.13 In this case, condition (12) does not hold even for k = k′ ; that is, entry is an equilibriumoutcome for all 0 < k < 1. Moreover, from the above discussion, we can conclude that if k∗ < k <1, exclusion never occurs for any c. Proposition 1 summarizes the discussion:
Proposition 1. Suppose that the downstream firms are undifferentiated Bertrand competitors. The
possibility of exclusion depends on the marginal cost of U, c, and the efficiency of DE , k.
1. When the marginal cost of U is sufficiently low, 0 ≤ c ≤ c̃, exclusion is possible if 0 < k ≤ k∗ .
r(ii)
′
′
2. When it is not too low, c̃ < c < c̃′ where c̃′ satisfies πaI + πaU = πr(i)U |k=k = πU |k=k , exclusion

is possible if k′′ ≤ k ≤ k∗ .
3. For any level of c, entry is a unique equilibrium outcome if k∗ < k < 1.
Note that condition (12) is a sufficient condition. Therefore, there may exist an exclusion equilibrium even when condition (12) does not hold. We provide necessary and sufficient conditions
for the exclusion equilibrium, by introducing linear demand:
13

This is observed in the case of linear demand.

13• 1.0
0.9
0.8

" (!)

0.7

Entry

0.6
0.5
0.4

• (!)

0.3
0.2

Exclusion

0.1
0.0
0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Figure 5: Results of Remark 1 (α = 1)

Remark 1. Suppose that Q(p) = (α − p)/β, where α > c and β > 0. Exclusion is a unique√√equilibrium outcome if and only if 0 < k ≤ 3/4 and 0 ≤ c ≤ Ĉ(k), where Ĉ(k) = 6 − 2 α/( 6 −2k); otherwise, entry is a unique equilibrium outcome. Note that ∂Ĉ(k)/∂k > 0, Ĉ(k) → (3 −√√6)α/3 ≃ 0.1835α as k → 0, and Ĉ(k) → 2(6 − 6)α/15 ≃ 0.4734α as k → 3/4.
√√
Under linear demand, we have k∗ = 3/4, k′′ = (2α−(α−c) 6)/2c, c̃ = (3− 6)α/3 ≃ 0.1835α,√
and c̃′ = 2(6 − 6)α/15 ≃ 0.4734α. For 0 < k ≤ 1/2, we have equilibrium in Case (ii) when entry
occurs. Conversely, for 1/2 < k < 1, we have the equilibrium in Case (i) (Case (ii)) if c ≤ Ć(k)√(c > Ć(k)), where Ć(k) = α(k − 2k(1 − k))/k(2 − k).14 Figure 5 summarizes the result in Remark 1
for α = 1. Note that we can obtain a similar result under a demand system with constant elasticity
of demand, Q = αp−ε , where ε > 1.15
We explain the features of Proposition 1 (Remark 1), by classifying the situation into the threecases: (i) c > c̃′ , (ii) c ≤ c̃′ and k > k∗ , and (iii) c ≤ c̃′ and k ≤ k∗ . Suppose that the marginal cost of
U is high enough (c > c̃′ ). The double marginalization problem is significant under exclusion (seeπaI +πaU in the righthand side of Figure 4.3). In Case (i) (k > k′ (> k∗ )), rejecting an exclusive contract
accelerates downstream competition, which mitigates the double marginalization problem as inUnder linear demand, πaU < πrU holds for all (c, k) ∈ [0, α) × (0, 1), which implies that U always welcomes DE ’s
entry if it rejects the exclusive offer. Thus, the fixed transfer, x, is crucial to attaining exclusive dealing.
15
If we consider perfectly inelastic consumer demand (Q(p) = q(> 0) if p ≤ p̄(> 0), otherwise, Q(p) = 0), theexclusion does not occur because the shrinkage of input demand through entry is completely offset by the higher input
price, which exceeds the consumers’ willingness to pay, p̄. The result is available upon request.
14

14

the Chicago School argument. In Case (ii) (k < k′ ), downstream entry improves the downstream
efficiency; namely, wa − kwr is large. Thus, downstream entry induces U to earn a high profit,
which makes exclusion impossible.
When the marginal cost of U is not high enough (c ≤ c̃′ ), the gain from the downstream
competition still leads to a large amount of input demand expansion as long as DE demands a large
amount of input for k ((k′ <)k∗ < k < 1) in Case (i); exclusion never occurs if k > k∗ , otherwise(k < k∗ ), it can occur depending on U’s cost, c. The equilibrium property contrasts the results inthe literature on anticompetitive exclusive dealings indicating that firms are unlikely to engage inanticompetitive exclusive dealings as the entrant becomes efficient. Hence, an exclusive contract
in this study works like the Luddites (e.g., Hobsbawm, 1952; Mokyr, 1992) and resistance totechnological change (e.g., Parente and Prescott, 1994, 1999; Desmet, Greig, and Parente, 2020).
Finally, we consider the interaction between the values of c and k for c ≤ c̃′ and k ≤ k′ inCase (ii). DE ’s efficiency improvement has two effects on the input demand. One is the demand
shrinkage effect in which, DE ’s efficiency improvement directly reduces the input demand kQ(pr ).
The other is the demand expansion effect in which DE ’s efficiency improvement indirectly expands
the input demand by reducing DE ’s marginal cost kw and the retail price.
As the marginal cost of U becomes higher (higher c within c ≤ c̃′ ), the demand expansion
effect becomes stronger because the reduction of DE ’s marginal cost is larger. By contrast, thedemand shrinkage effect becomes weaker because higher c leads to smaller input demand, and theamount of input demand shrinkage is smaller. That is, as the marginal cost of U becomes higher,
the gain from the higher efficiency of DE becomes larger, and the loss from the higher efficiencyof DE becomes smaller. Therefore, the threshold value of k, k′′ , becomes larger as c increases.
We discuss DE ’s efficiency management because we show that exclusion is more likely tooccur if k is small. One may consider how DE increases k to avoid exclusion by DI . We discuss
two scenarios. First, DE could offer U a per-unit “kickback” of k∗ − k + ε, benefitting U andincreasing the effective marginal cost of DE to k∗ + ε. To implement the kickback, DE needs to15

commit to it before DI offers U an exclusive supply contract. The scenario does not seem to matchthe timeline in Figure 1. Second, DE could simply buy larger quantities of the input than it actuallyneeds to use—and destroys enough, so that the effective k(> k∗ ). DE needs to credibly commit toprocuring excessive quantities before DI ’s exclusive offer.
We think that our exclusion mechanism implies that DE needs to choose not a radical but amoderately efficient technology when it decides the entry and announces its entry decision before
Stage 1. In other words, too advanced technology diminishes the importance of U, inducing it tosign an inefficient exclusive supply contract with DI .

4.4

Robustness of the results

Finally, we check the robustness of these results.16 We first examine the existence of anticompetitive exclusive dealing under quantity competition with P(Q) = α − βQ. There can be an exclusion
equilibrium for k ≤ k̂ ≃ 0.92 > 3/4 under quantity competition, which implies that the possibilityof anticompetitive exclusion under Cournot competition is higher. This result follows from thedifference in the degree of demand expansion between the two types of competition. Comparedwith undifferentiated Bertrand competition, DE ’s entry under Cournot competition leads to smaller
demand expansion. Therefore, DE ’s entry leads to a smaller increase in U’s profit; exclusion outcomes are more likely to be observed.
We also examine the case of price competition with differentiated products. The demand for
Di ’s product is qi = (α′ − pi + γp j )/(1 − γ2 ) (i, j ∈ {I, E}, j , i, α′ > c, γ ∈ [0, 1)) (Singhand Vives, 1984). We find that an exclusion equilibrium exists for lower k whose threshold value
increases with the degree of product substitution γ and converges to 3/4 when γ → 1; namely,
product differentiation decreases the possibility of exclusion. As downstream firms’ products aredifferentiated, input demand expansion becomes larger, making DI ’s compensation to U difficult.
As another extension, we introduce an inefficient upstream supplier UO with marginal cost c+co
16

The detailed analyses in those extended settings are available upon request.

16

(co > 0) to the main model. We assume that c/k < c + co < α/2 under which upstream competition
binds U’s pricing and the entry of DE occurs only when it can trade with U. Using linear demand,
we find that DI can achieve exclusive dealing even for k > 3/4; namely, the existence of an
inefficient supplier facilitates downstream exclusion. When upstream competition exists, the entryof DE triggers both upstream and downstream competition, which gives most of the gain fromentry to final consumers (third parties), reducing DI and U’s joint profits, as in upstream exclusion
with downstream competition (Simpson and Wickelgren, 2007; Abito and Wright, 2008). Thus,
exclusion is more likely to be observed.

5

Discussion

This section briefly introduces the discussion on different pricing policies and exclusive dealing in
toy retailing markets.

5.1

Wholesale pricing

This section briefly discusses the wholesale pricing of the input. Thus far, we have assumed that U
charges downstream firms a uniform price wr when DE enters the downstream market. We consider
how the results in Section 4 change if U is able to (i) charge different input prices and to (ii) adopt
two-part tariffs. The two discussions clarify the impact of input price discrimination on the market
structure, contributing to the literature on input price discrimination.
We first explore the case of input price discrimination. When U chooses input prices wri for
Di , where i ∈ {I, E}, the per unit costs of DI and DE are denoted by wrI and kwrE . To clarify thedifference from uniform pricing, we focus on the case where U rejects the exclusive supply offer
in Stage 1 and DE enters the downstream market in Stage 2. In Stage 4, given the input prices
set in Stage 3, undifferentiated Bertrand competition occurs, which leads to monopolization by
the downstream firm with a lower per unit cost. In equilibrium, U optimally chooses a pair of
input prices (wrI , wrE ), such that wrI = kwrE = p∗ (kc) in Stage 3, and earns πrU = (wrE − c)kQ(wrI ) =17

(p∗ (kc) − kc)Q(p∗ (kc)) = Π∗ (kc) < πaI + πaU ; thus, exclusion is not achievable.
The impossibility of exclusion under price discrimination follows U’s control ability of downstream competition. Under uniform price, U cannot control the downstream competition for thecase of entry, which prevents U from achieving the joint profit maximization with DE , which plays
a key role in an exclusion outcome. By contrast, when price discrimination is possible, U can control the downstream competition. When undifferentiated Bertrand competition DI and DE occurs,
U can jointly maximize profit with DE , and more importantly, it can earn all profits.17 Thus there
is no room for an exclusion equilibrium.
Next, we consider the case in which U adopts two-part tariffs, which are publicly observable,
and makes take-it-or-leave-it offers.18 Two-part tariffs consist of a linear wholesale price, w, and an
upfront fixed fee, F. The two-part tariff offered by U to Di when U accepts (rejects) the exclusive
supply offer is denoted by (wai , Fia ) and ((wri , Fir )), where i ∈ {I, E}. When U accepts the exclusive
supply offer, it sets (waI , F Ia ) = (c, Π∗ (c)), which allows U and DI to jointly earn πaI +πaU = Π∗ (c). By
contrast, when U rejects the exclusive supply offer, it sets (wri , Fir ) = (c, Π∗ (kc)). In the equilibrium,
we have πrU = Π∗ (kc), which implies that as in discrimination under linear pricing, condition (5)never holds. Thus, exclusive supply contracts are not attainable.
When U can adopt two-part tariffs, it can avoid the double marginalization problem, which
benefits U. As in price discrimination under linear pricing at the beginning of this subsection,
two-part tariffs allow U to jointly maximize profits with DE and to earn all the profits. Because DE
is more efficient than DI , the joint profit of U and DE is higher than that of U and DI . Thus, DIcannot achieve exclusion.
We introduce two remarks on the above results. First, we should emphasize that in each case,downstream firms earn zero operating profits when U rejects the exclusive supply offer. Anticipating this outcome, DE does not enter the downstream market in Stage 2 even when U rejects
17

When downstream firms compete in quantity, joint profit maximization is impossible and DE earns a positive
profit, which is lower than that when U employs a uniform price.
18
Under private offers, U cannot maximize joint profits because of opportunistic behavior as posited by Hart andTirole (1990). See also O’Brien and Shaffer (1992), McAfee and Schwartz (1994), and Rey and Vergé (2004).

18

the exclusive supply offer in Stage 1. Therefore, when U charges different input prices and adopts
two-part tariffs, there is a price commitment problem; that is, U is unable to commit initially to an
input price offer that allows DE to cover the fixed cost f (> 0).19
Second, introducing input arbitrage between DI and DE , we can combine the two polar cases
discussed above: (i) U cannot price discriminate; (ii) U can price discriminate. Each downstream
firm can sell its purchased input to its rival by incurring per-unit arbitrage cost t. U sets its prices
for DI and DE while anticipating the possibility of input arbitrage. Using linear demand, we findthat DI successfully excludes DE through an exclusive supply contract if t is not large enough andk is small. That is, if there is a chance of input arbitrage, an exclusive supply contract between
DI and U is attainable even when U can price discriminate.20 t would be low under low transport
costs relative to the values of those products (or high values of products relative to transport costs).
Pharmaceuticals are such a prominent example of potential arbitrage.21
From the above discussions, we can conclude that this study is most suitable for a discussion on
the anticompetitiveness of exclusive supply agreements in industries where input price discrimination is less implementable. Such cases are more likely to be observed when dominant downstream
firms impose a price parity clause or when arbitrage is easy because of higher product storability.
Therefore, anticompetitive exclusive supply agreements are more likely to occur when DI offers
price parity clauses or when product storability is high.22

5.2

Exclusive dealing in toy retailing markets

We briefly consider the link between exclusive supply agreements in this study and the case ofToys “R” Us (TRU), which was the largest toy retailer in the US in the late twentieth century.23 In
19

The commitment problem is mitigated if we consider imperfect downstream market competition such as differentiated Bertrand competition and quantity competition or general bargaining with two-part tariffs.
20
The results are available upon request.
21
We thank the General Editor for pointing out the scenario.
22
Recently, several papers discuss price parity clauses (e.g., Johnson, 2017 and references therein).
23
See Federal Trade Commission (1997), available at https://www.ftc.gov/sites/default/files/
documents/cases/1997/09/toysrus_0.pdf. See also Comanor and Rey (2000) and Scherer (2009).

19

the early 1990s, TRU faced a competitive threat from warehouse clubs, such as Costco and Sam’s
Club. To avoid competition from warehouse clubs, TRU approached toy suppliers, and “Suppliers
agreed not to sell to the clubs the same toys that TRU carried.”24This case is related to our study in the following aspects. The first aspect is the efficiency
difference between TRU and warehouse clubs. TRU sold a large variety of toy products; it stored
16000 stock-keeping units in the early 1990s.25 Such large stock-keeping units were usually costly
in terms of supply chain management. Conversely, warehouse clubs had lower operating costs
by increasing inventory turnover ratios, contributing to a low risk of obsolescence, similar to low
defective rates.26 Therefore, warehouse clubs were more efficient.
The second aspect is the possibility of price discrimination.27 Toy products are usually storable
goods, which implies that arbitrage among retailers is not very difficult. From the discussion in
Section 5.1, suppliers seemed to have difficulty in adopting price discrimination with no arbitrage
when they sell toy products to not only TRU, but also warehouse clubs.
Note that both aspects play an essential role in achieving exclusive dealing in this study. Moreover, a similar case is observed in the exclusive supply agreements by Belk Stores, a department
store chain in the US. Belk Stores induced Jantzen sportswear to stop selling to Garment District,a discount retailer in the US.28 Therefore, our exclusion mechanism can apply to the situationin which a specialty retailer makes an exclusive supply offer to manufacturers to exclude large
discount retailers.
24

See page 1 of Federal Trade Commission (1997).
See page 3 of Federal Trade Commission (1997).
26
See page 6 of Federal Trade Commission (1997). Moreover, Costco is famous for efficient supply chain management because of higher inventory turnover ratios (lower risk of obsolescence) than other retailers. See, for example,“What Costco’s Inventory Turnover Says About Its Moat” The Motley Fool August 22, 2016 (https://www.fool.
com/investing/2016/08/22/what-costcos-inventory-turnover-says-about-its-moa.aspx).
27
Scherer (2009, pp.445-453) summarizes the key issues in the case of TRU.
28
See The Garment District, Inc., v. Belk Stores Service, Inc., Mathews-Belk Company, Jantzen, Inc., 799 F.2d 905.
(https://law.justia.com/cases/federal/appellate-courts/F2/799/905/117747/). See also Comanorand Rey (2000) for detailed discussions.
25

20

6

Concluding Remarks

This study examined anticompetitive exclusive supply agreements focusing on the necessary amountof inputs to produce one unit of final product. Previous studies have not differentiated between theincumbent and entrants with regard to the necessary amount of inputs, because they mainly analyzed entry deterrence in upstream markets. However, our study suggests that when we focus on
entry deterrence in downstream markets by considering exclusive supply contracts, the differencein the necessary amount of inputs can be an important market element.
We find that when the incumbent and entrant differ in the necessary amount of inputs, the inefficient downstream incumbent and supplier sign exclusive supply contracts to deter socially efficient
entry, even in the three-player model with a single seller, buyer, and entrant. In addition, the difference in the necessary amount of inputs changes the relationship between the entrant’s efficiency
and possibility of exclusion: anticompetitive exclusive supply agreements are more likely to arise
if the entrant’s efficiency is at an intermediate level.
These results provide new implications for antitrust agencies: it is necessary to focus on the efficiency measure when discussing the anticompetitiveness 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 producers.
We also find that exclusive supply agreements based on the difference in the necessary amountof inputs are more likely to arise when upstream firms have difficulty adopting input price discrimination. Although perfect price discrimination, where no input arbitrage exists, reduces the possibility of anticompetitive exclusive supply agreements, exclusion results are achievable if there is achance of input arbitrage. These results provide the following implications. First, exclusive supply
agreements are more likely to be observed when the product is storable, making input arbitrage
easy. Second, the analysis here can be applied when the dominant downstream firm offers priceparity clauses, which induces the upstream supplier to use uniform pricing.

21

As further research, we could incorporate a more sophisticated bargaining procedure into themain model because this study is related to the literature on buyer power in the sense that buyers
can unilaterally offer exclusive contracts to suppliers as in Miklós-Thal, Rey, and Vergé (2011).
We hope this study facilitates researchers in addressing the issue.

22

A

Property of Demand Function

By definition, p∗ (z) satisfies the first-order condition Q(p∗ (z)) + (p∗ (z) − z)Q′ (p∗ (z)) = 0, following
Fabinger and Weyl (2012), which can be rewritten asp∗ (z) − z = −Q(p∗ (z))≡ µ(p∗ (z)).
Q′ (p∗ (z))

When Q(p) is log-concave, we haveµ′ (p∗ (z)) ≤ 0.

(13)

By totally differentiating the first-order condition, we obtain the pass-through ratep∗′ (z) =Q′ (p∗ (z))
1
=
.
′
∗
∗
′′
∗
′
2Q (p (z)) + (p (z) − z)Q (p (z))) 1 − µ (p∗ (z))(14)

Under condition (13), 0 < p∗′ (z) ≤ 1 always holds.
We now check the existence and uniqueness of wr (k). Because d p∗ (kw)/dw = kp∗′ (w) ≤ kholds under condition (13), p∗ (kw) is strictly increasing in w and its pass-through rate is smaller
than or equal to k(< 1), which is smaller than the increase in w (see also Figure 6). In addition, we
have p∗ (kc) > c for all 0 < k < 1 when condition (6) holds. Therefore, there exists unique wr (k)for each k.
p

p=w

p = p * (kw)p * (kwr (k ))p * (kc)p * (0)
c

45o

0

c

wr (k )

Figure 6: Properties of p∗ (kw)

23

w

B

Proofs of Results

B.1

Proof of Lemma 3

We show that at least one interior solution exists in the profit maximization problems in Cases (i)and (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).

(15)

The profit maximization problems of 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

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

Note that each maximization problem has a unique interior solution on domain [c, ∞). However,there exists a possibility of a corner solution, where the problem in Case (i) has an interior solution
on domain [w(k), ∞) and the problem in Case (ii) has an interior solution on domain [c, w(k)].
In such cases, 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, weshow 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)Q′ (w(k)) > 0.

(16)

Using equation (15), H (ii) (w(k)) can be rewritten as
H (ii) (w(k)) = Q(w(k)) + (w(k) − c)kQ′ (w(k))p∗′ (kw(k)).
24

(17)

Using equations (16) and (17), we have the following relationship:
H (ii) (w(k)) > H (ii) (w(k)) − H (i) (w(k))= (w(k) − c)Q′ (w(k))[kp∗′ (kw(k)) − 1] > 0,
The last inequality holds because Q′ (p) < 0, 0 < k < 1, and 0 < p∗′ (kw(k)) ≤ 1 always hold undercondition (13).
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 domain (w(k), ∞) if in Case (i), the interior solution
does not exist on domain [c, w(k)] and the corner solution appears; that is, we always have wr(ii) ∈(w(k), ∞) if wr(i) = w(k). This also implies that at least one interior solution exists and there arethree possibilities concerning the optimal input price for U:
1. An interior solution exists only on domain (c, w(k)) in Case (i).
2. An interior solution exists only on domain (w(k), ∞) in Case (ii).
3. Interior solutions exist on domain (c, w(k)) in Case (i) and (w(k), ∞) in Case (ii).
In the first and second cases, we have unique interior solutions. Therefore, the statements inLemma 3 hold.
Q.E.D.

B.2

Proof of Lemma 4

By Lemmas 1 and 2, the first statement in Lemma 4 holds. We now prove the second statement.
r(ii)
r(ii)
r(i)
For a sufficiently small k (as k → 0), we have πr(i)U < πU . However, for k = 1, we have πU > πU .
r(i)′Because πr(ii)U is strictly decreasing in k but πU is strictly increasing in k, there exists k ∈ (0, 1)r(ii)such that πr(i)U = πU .

Q.E.D.

25

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