ISSN (Print) 0473-453X

Discussion Paper No. 1191

ISSN (Online) 2435-0982

THE COUNTERVAILING
POWER HYPOTHESISAND CONTINGENT CONTRACTS
Noriaki Matsushima
Shohei Yoshida

October 2022

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

The countervailing power hypothesis and contingent contracts∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Shohei Yoshida‡
Faculty of Economics, Dokkyo University
October 9, 2022

Abstract
We consider a downstream oligopoly model with one dominant and several fringe
retailers who purchase a manufacturing product from a monopoly supplier. We examine
how contract type inﬂuences the relationship between the dominant retailer’s bargaining
power and the equilibrium retail price. If the contracts between the supplier and fringe
retailers are contingent on the bargaining outcome between the supplier and the dominantretailer, the bargaining power does not aﬀect the retail price. In contrast, if contracts withfringe retailers are not contingent, the relationship between bargaining power and retailprice can be either positive or negative.

JEL codes: L13, D43.
Keywords: Countervailing power, Buyer power, Dominant retailer, Two-part tariﬀ, Contingent contract.

∗

We thank Takeshi Murooka and seminar participants at Osaka University for their valuable discussions andcomments. We gratefully acknowledge the ﬁnancial support from the Japan Society for the Promotion of Science (JSPS), KAKENHI Grant Numbers JP15H03349, JP15H05728, JP19H01483, JP20H05631, JP20K13492,
JP21H00702, JP21K01452, and JP21K18430, Grant-in-Aid for JSPS Fellows Grant Number JP16J02442, andthe program of the Joint Usage/Research Center for “Behavioral Economics” at ISER, Osaka University. Although Matsushima serves as a member of the Competition Policy Research Center (CPRC) at Japan Fair
Trade Commission (JFTC), our paper does not reﬂect the views of JFTC. The usual disclaimer is applicableas well.
†
Noriaki Matsushima, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki,
Osaka, 567-0047, Japan. Phone: (81)-6-6879-8571. Fax: (81)-6-6879-8583. E-mail: nmatsush@iser.osakau.ac.jp‡
Corresponding author: Shohei Yoshida, Faculty of Economics, Dokkyo University, 1-1 Gakuen-Cho, Soka,
Saitama, 340-0042, Japan. E-mail: shoheiyoshida0@gmail.com.

1

I

Introduction

The countervailing power of large retailers over manufacturers can induce lower wholesale
prices, which can cancel out the negative eﬀects of their market power over consumers (e.g.,
Galbraith, 1952), although policymakers have serious concerns about the market power oflarge retailers over consumers. To theoretically investigate the eﬀect of countervailing power
on consumer and social welfare, some researchers of industrial organization provide market
environments in which the countervailing power of retailers contributes to a reduction in retailprices (e.g., Dobson and Waterson, 1997; Iozzi and Valletti, 2014; Gaudin, 2018; Ghosh et al.,2022).
In addition to the above, policymakers also have concerns that the countervailing powerof large retailers may harm small retailers through distortions of those small retailers’ procurement conditions, the so-called ‘waterbed eﬀect’ (e.g., Thomas, 2018). More concretely,
the countervailing power of a large retailer makes it possible to procure a large amount of
input/product owing to its good procurement conditions. Such a great demand of the large
retailer for the supplier’s input/product implies a large disagreement payoﬀ of the supplierin negotiations with small retailers. In other words, a retailer’s countervailing power canforeclose small retailers.1
To investigate the impact of a large retailer’s countervailing power on market outcomes,
we construct an oligopoly model with a dominant retailer as follows: A monopoly suppliersupplies products via two types of retailers. The ﬁrst type of retailer is the dominant retailer
with a positive constant marginal cost. They negotiate the term of the trade (a two-parttariﬀ contract) through Nash bargaining after the determination of a non-discriminatory twopart tariﬀ contract to the fringe retailers. The second type consists of fringe retailers whose
marginal costs increase in quantity. Each fringe retailer takes a retailer price set by the1

Theoretical analyses on the waterbed eﬀect are engaged by Majumdar (2005), Inderst and Wey (2003,

2007), Inderst (2007), and Inderst and Valletti (2011).

2

dominant retailer, as given, sets its own quantity and competes with the dominant retailer formarket demand. Each fringe retailer receives a take-it-or-leave-it contingent contract from themonopoly supplier. The contingent contract includes two types of non-discriminatory twopart tariﬀs for fringe retailers. The ﬁrst type of two-part tariﬀ is applied if the negotiation
between the supplier and dominant retailer reaches an agreement. If the negotiation between
the supplier and the dominant retailer breaks down, the second type of two-part tariﬀ isapplied, and as in Christou and Papadopoulos (2015), the retail price is determined to equalize
the total quantity supplied by fringe retailers.
Contingent contracts are a new element proposed in this study. In the negotiation procedure in the dominant retailer model, as in Chen (2003) (also in Bedre-Defolie and Shaﬀer(2011); Christou and Papadopoulos (2015); Matsushima and Yoshida (2018)), the supplierand the dominant retailer negotiate their trading terms after the non-discriminatory contract
to the fringe retailers is determined. This contract sequence implies that the supplier canuse a contingent contract when it oﬀers a non-discriminatory contract to fringe retailers. As
a non-contingent contract is a special case of a contingent contract, employing a contingent
contract must increase the proﬁtability of the supplier, who can unilaterally set its trading
terms with fringe retailers. Therefore, if it is easy to verify that negotiation with the dominantretailer breaks down, the supplier should oﬀer a contingent contract to the fringe retailers. 2
The modiﬁcation of Chen’s (2003) dominant retailer model contributes to the literatureon the buyer-supplier relationship. Since the seminal work by Binmore et al. (1986), many
related papers in the context of buyer-supplier relations carefully treat disagreement payoﬀsof players (e.g., Horn and Wolinsky, 1988; Dobson and Waterson, 1997; Milliou and Petrakis,
2007; Iozzi and Valletti, 2014; Bacchiega et al., 2018). Following the assumption in Bacchiega
et al. (2018), we propose a possible modiﬁcation to the dominant retailer model originallysuggested by Chen (2003).
2

This is not always true if the contract terms are determined through a negotiation between parties withpositive bargaining power (see, Bacchiega et al., 2018).

3

We show that the countervailing power of the dominant retailer is neutral to the equilibrium retail price if the supplier can oﬀer a contingent contract, which depends on whether
negotiation with the dominant retailer reaches an agreement in the ﬁrst stage. We also showthat the dominant retailer’s countervailing power inﬂuences the equilibrium retail price if the
supplier cannot oﬀer such a contingent contract. Speciﬁcally, increasing the countervailingpower diminishes the equilibrium retail price if the speciﬁed inverse demand function is convex in quantity. Moreover, countervailing power augments the equilibrium retail price if thespeciﬁed inverse demand function is concave in quantity. Our results imply that there is noclear relationship between the countervailing power of dominant retailers and retail price, incontrast to the countervailing power hypothesis.
This study is as an extension of Chen (2003), Bedre-Defolie and Shaﬀer (2011), and Chiristou and Papadopoulos (2015).3 By modifying Chen’s (2003) setting, Bedre-Defolie and Shaﬀer(2011) reconsider the countervailing power hypothesis. Their model diﬀers from that of Chen
(2003) in that (i) the supplier oﬀers a linear contract to fringe retailers, which diﬀers fromthose in the other two studies; and (ii) the supplier’s disagreement payoﬀ is the payment fromthe fringe retailers, in which the retail price is determined by the demand-equal-supply condition, which is also employed in Christou and Papadopoulos (2015).4 Bedre-Defolie and Shaﬀer(2011) show that the convexity/concavity of fringe supply functions inﬂuences the relationshipbetween the retail price and bargaining power of the dominant retailer if fringe retailers areactive. Our study is a direct extension of Christou and Papadopoulos (2015) in that we allow
for more generalized demand and cost functions and consider contract contingency which has
not been considered in the context of countervailing power.
3

In a diﬀerent study, Matsushima and Yoshida (2018) examine a negative relation between the retail priceand the dominant retailer’s bargaining power over the supplier under the assumption that the market demand
shrinks due to a breakdown in bargaining between the supplier and the dominant retailer, who works as asales promoter for the product. Erutku (2005), which is an extension of Chen (2003), and Matsushima (2017)
investigate purchasing power in diﬀerent multichannel models.
4

In addition to the two aspects, Bedre-Defolie and Shaﬀer (2011) further investigate the possibility thatfringe retailers are inactive because of a prohibitively high wholesale price for them.

4The remainder of this paper is organized as follows. The model is described in Section
2. Section 3 presents the equilibrium outcomes and results of comparative statics. Finally,
Section 4 concludes the paper.

II

Model

This section presents the basic model, an extension of Chen (2003), Bedre-Defolie and Shaﬀer(2011), and Christou and Papadopoulos (2015). Consider a monopoly supplier that produces
an intermediate good without any cost and one dominant and n fringe retailers that purchasethe intermediate good from the supplier. Each retailer converts one unit of the intermediate
good to one unit of the ﬁnal good at no cost. However, each retailer incurs an operational
cost to handle ﬁnal goods. The dominant retailer has a constant marginal cost c > 0 to
handle one unit of the ﬁnal good. In contrast, each fringe retailer has an increasing marginal
cost M C(qf ), where qf is its quantity, M C ′ (qf ) > 0 and M C(0) = 0.5 These assumptions
reﬂect that fringe retailers are more eﬃcient at small operational scales, whereas the dominantretailer is more eﬃcient at a large scale.
The demand function for the ﬁnal good is given as D(p), where D′ (p) < 0 and D′ (p) +(p − c)D′′ (p) < 0, as assumed in Chen (2003).6
The dominant retailer negotiates with the supplier over a two-part tariﬀ contract wd qd +Fd ,where wd is the wholesale price, qd is the quantity purchased by the retailer, and Fd is the fee.
In Stage 1, the monopoly supplier unilaterally oﬀers a contingent two-part tariﬀ contract(wfA , FfA ; wfD , FfD ) to each fringe retailer, where wf is the wholesale price, Ff is the fee, andthe superscript A (D) indicates the contract term when the supplier and the dominant retailer
agree (disagree).7 The supplier commits oﬀers. Fringe retailers are price-takers in both the5

The assumption M C(0) = 0 follows those in Chen (2003) and Christou and Papadopoulos (2015). Bedre-

Defolie and Shaﬀer (2011) assume that M C(0) = c, which implies that the dominant retailer is always more
eﬃcient than fringe retailers.
6

Christou and Papadopoulos (2015) employ a linear demand.

7

Bedre-Defolie and Shaﬀer (2011) assume that the supplier oﬀers a common linear contract to each fringe

retailer.

5input and ﬁnal goods markets. Under this assumption, we derive the supply function of eachfringe retailer. It is derived from p = M C(qf ) + wf or qf = M C −1 (p − wf ). Here, we deﬁnes(p − wf ) ≡ M C −1 (p − wf ), where s(0) = 0 and s′ (·) > 0.
In Stage 2, the supplier and the dominant retailer negotiate over the two-part tariﬀ contract, wd qd + Fd . The bargaining power of a dominant retailer is γ ∈ (0, 1).
In Stage 3, considering fringe retailers’ production technology, the dominant retailer sets
the retail price p. Given p, each fringe retailer determines its quantity. If the negotiation
breaks down in Stage 2, p is determined to equalize demand D(p) and the total quantities
supplied by the fringe retailers. This condition is further explained in Section 3.

III

Equilibrium outcome

Our main objective is to discuss how contract type inﬂuences the equilibrium property. First,
we solve the game under a contingent contract using backward induction. Second, we modify
this game by considering a non-contingent contract in that the two-part tariﬀ contract foreach fringe retailer does not depend on whether the dominant retailer signs a contract with the
supplier; that is, we modify the contract space (wfA , FfA ) = (wfD , FfD ) in the second scenario.
Here, we focus only on an interior solution in which the dominant retailer and each of thefringe retailers are active in equilibrium.8

III(i)

Contingent contract

We consider the game under a contingent contract.

8

Bedre-Defolie and Shaﬀer (2011) carefully discuss a possibility that fringe retailers are inactive if a common

wholesale price for them is prohibitively high. We guess that the scenario is less likely to occur in our model
because we assume that the supplier oﬀers a common two-part tariﬀ to fringe retailers, allowing the supplierto fully extract the rents of those fringe retailers.

6

Stage 3

Given the agreement of the negotiation between the supplier and the dominantretailer, we have already derived the supply function of each fringe retailer, s(·):
qf (p − wfA ) = s(p − wfA ).
The residual demand of the dominant retailer is given by D(p) − ns(p − wfA ). Thus, thedominant retailer’s proﬁt becomes
πd = (p − c − wd )[D(p) − ns(p − wfA )] − Fd .
From the ﬁrst-order condition of its proﬁt maximization problem, we obtain the optimal pricep∗ (wd , wfA ) = p∗ such that
[D(p∗ ) − ns(p∗ − wfA )] + (p∗ − c − wd )[D′ (p∗ ) − ns′ (p∗ − wfA )] = 0.

(1)

Note that ∂p∗ (wd , wfA )/∂wd > 0 from the second-order condition.
Stage 2 (bargaining outcome)

Anticipating the third-stage outcome, the supplier andthe dominant retailer negotiate over the two-part tariﬀ contract, wd qd + Fd . Let πs be themonopoly supplier’s proﬁt. The bargaining problem is Bds = {πs , πd }, with the disagreement
payoﬀs for the supplier and the dominant retailer being (Os , Od ), where:πs (wd , wfA , Fd , FfA ) = Fd + wd [D(p∗ (wd , wfA )) − ns(p∗ (wd , wfA ) − wfA )](2)+n[FfA + wfA s(p∗ (wd , wfA ) − wfA )],πd (wd , wfA , Fd ) = (p∗ (wd , wfA ) − c − wd )[D(p∗ (wd , wfA )) − ns(p∗ (wd , wfA ) − wfA )] − Fd ,(3)Os (wfD , FfD ) = n[FfD + wfD s(po − wfD )],where po satisﬁes D(po ) = ns(po − wfD ),(4)

Od = 0.
The equation, D(po ) = ns(po − wfD ) in equation (4), is the demand-equal-supply condition,in which the negotiation between the supplier and the dominant retailer breaks down.
7

By solving the following maximization problem, we derive the outcome of bargaining:max [πs (wd , wfA , Fd , FfA ) − Os (wfD , FfD )]1−γ [πd (wd , wfA , Fd )]γ .

(wd ,Fd )First, Fd must satisfy the following:Fd∗ = (1 − γ)[p∗ − c][D(p∗ ) − ns(p∗ − wfA )]− wd [D(p∗ ) − ns(p∗ − wfA )] − nγ[FfA + wfA s(p∗ − wfA )] + γOs (wfD , FfD ). (5)The problem becomes as follows:
max πs (wd , wfA , Fd , FfA ) + πd (wd , wfA , Fd ) − Os (wfD , FfD ).
wd

That is,max (p∗ − c)[D(p∗ ) − ns(p∗ − wfA )] + n[FfA + wfA s(p∗ − wfA )] − Os (wfD , FfD ).
wd(6)

The ﬁrst-order condition with respect to wd is}∂p∗ {[D(p∗ ) − ns(p∗ − wfA )] + (p∗ − c)[D′ (p∗ ) − ns′ (p∗ − wfA )] + nwfA s′ (p∗ − wfA ) = 0, (7)∂wdOr[D(p∗ ) − ns(p∗ − wfA )] + (p∗ − c)[D′ (p∗ ) − ns′ (p∗ − wfA )] + nwfA s′ (p∗ − wfA ) = 0.

(8)Deﬁne wd∗ = wd (wfA ). The second-order condition is∂ 2 p∗{[D(p∗ ) − ns(p∗ − wfA )] + (p∗ − c)[D′ (p∗ ) − ns′ (p∗ − wfA )] + nwfA s′ (p∗ − wfA )}
2
∂wd
∂p∗
+
{2D′ (p∗ ) + (p∗ − c)D′′ (p∗ ) − 2ns′ (p∗ − wfA ) − n(p∗ − c − wfA )s′′ (p∗ − wfA )} < 0.
∂wd
Because the ﬁrst line becomes zero (by substituting equation (8)) in equilibrium, we have
[2D′ (p∗ ) + (p∗ − c)D′′ (p∗ )] − n[2s′ (p∗ − wfA ) + (p∗ − c − wfA )s′′ (p∗ − wfA )] < 0.

8

(9)

Stage 1

From (2) and (5), and the fact that the supplier extracts the full surplus of each∫ p∗∗ −wfA
fringe retailer through FfA (FfA = 0
s(x)dx), the objective of the supplier is to maximize
the following πS by controlling wfA , wfD , and FfD (we deﬁne p∗∗ ≡ p∗ (wd∗ , wfA ))πs (wd∗ , wfA , wfD , Fd∗ , FfA , FfD )= Fd∗ + wd∗ [D(p∗∗ ) − ns(p∗∗ − wfA )] + n[FfA + wfA s(p∗∗ − wfA )]= (1 − γ)[p∗∗ − c][D(p∗ ) − ns(p∗∗ − wfA )] − wd (wfA )[D(p∗∗ ) − ns(p∗∗ − wfA )][∫ ∗∗ A
]
p

−nγ

−wf

s(x)dx + wfA s(p∗∗ − wfA ) + γOs (wfD , FfD )0+wd (wfA )[D(p∗∗ ) − ns(p∗∗ − wfA )] + n[∫p∗∗ −wfA0{= (1 − γ) [p∗∗ − c][D(p∗∗ ) − ns(p∗∗ − wfA )] + n[∫]s(x)dx + wfA s(p∗∗ − wfA )p∗∗ −wfA]}
s(x)dx + wfA s(p∗∗ − wfA )

(10)

0

+γOs (wfD , FfD ).
From the above maximization problem, γ does not inﬂuence the equilibrium wholesale
prices (wfA , wfD ). Thus, we obtain the following result:
Proposition 1 The countervailing power does not aﬀect the equilibrium wholesale and retailprices.
The key point is the separation of the controls on the total industry proﬁts and the supplier’s
disagreement payoﬀ.

III(ii)

Non-contingent contract

Here, we modify this game in the previous subsection by considering a non-contingent contract,that is, we consider a two-part tariﬀ contract, (wfA , FfA ) = (wfD , FfD ), to each fringe retailer.
We simply replace wfD and FfD with wfA and FfA respectively, and omit superscript A. The
replacement does not change the analytical property in the second and third stages but does9change the supplier’s objective in the ﬁrst stage, as follows:
πs (wd∗ , wf , Fd∗ , Ff ){= (1 − γ) [p∗∗ − c][D(p∗∗ ) − ns(p∗∗ − wf )] + n[∫p∗∗ −wf]}s(x)dx + wf s(p∗∗ − wf )

(11)

0

+γOs (wf , Ff ).
We ﬁnd that only the last terms, γOs (·), in (10) and (11) are diﬀerent. The diﬀerence
substantially changes the relationship between γ and p∗∗ because equation (11) implies thatthe supplier balances the direct proﬁt from the two channels (the ﬁrst term) and the indirect
proﬁt from the disagreement payoﬀ (the second term).
In contrast to the contingent contract case, the neutral result holds no longer. We canverify that the eﬀects of γ on wf , wd , and p are ambiguous. To check the relationshipbetween the prices and bargaining power numerically, we assume that p(Q) = (a − bQ)α ,s(·) = (p−wf βd ) . This functional assumption enables us to examine both concave and convex

demand and supply functions. The parameter settings with α = 1 and β = 1 coincide with
Chiristou and Papadopoulos (2015). Figure 1 shows that the relationship between bargaining
power and retail price can be positive or negative, and the concavity/convexity of demand andsupply functions plays an essential role. Note that when α > 1 and β = 1 (p(Q) is concave),the countervailing power of the dominant retailer can increase the equilibrium retail priceand thus reduce the consumer surplus. Our results imply that there is no clear relationshipbetween the countervailing power of dominant retailers and retail price, in contrast to thecountervailing power hypothesis.

IV

Conclusion

We consider a downstream oligopoly model with one monopoly supplier, one dominant retailer,and fringe retailers by considering a contingent contract for the fringe retailers. This contract
depends on whether the negotiation between the supplier and the dominant retailer reaches
an agreement. We show that the dominant retailer’s countervailing power is neutral to the
10

p

p

p
0.36

0.67

0.45

0.66

0.44

0.65

0.43

0.35

0.42

0.64

0.41

0.34

γ

γ

0.63

(a) [α = 0.5](b) [α = 1.5]γ(c) [α = 2]

Figure 1: Relationship between retail price and bargaining power.
[a = 1, b = 1, c = 1/100, d = 10, n = 4, β = 1]
equilibrium retail price if the supplier can oﬀer a contingent contract, which depends onwhether the negotiation with the dominant retailer reaches an agreement. We also show thatthe countervailing power of the dominant retailer inﬂuences the equilibrium retail price if the
supplier cannot oﬀer such a contingent contract. Speciﬁcally, an increase in the countervailingpower diminishes the equilibrium retail price if the speciﬁed inverse demand function is convex
in quantity. Moreover, countervailing power augments the equilibrium retail price if thespeciﬁed inverse demand function is concave in quantity. Our results imply that there is noclear relationship between the countervailing power of dominant retailers and retail price incontrast to the countervailing power hypothesis.

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