Discussion Paper No.

981

THE COUNTERVAILING
POWER HYPOTHESISWHEN DOMINANT RETAILERSFUNCTION AS SALES PROMOTERS

Noriaki Matsushima
Shohei Yoshida

October 2016
Revised July 2017

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

The countervailing power hypothesis when dominant retailers
function as sales promoters∗
Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Shohei Yoshida‡
Research Fellow of Japan Society for the Promotion of Science
July 4, 2017

Abstract
We consider a downstream oligopoly model with one dominant and several fringe retailers, who purchase a manufacturing product from a monopoly supplier. We then examine
how the supplier’s outside option influences the relation between the dominant retailer’sbargaining power and the equilibrium retail price. If the market demand shrinks due to abreakdown of bargaining between the supplier and the dominant retailer, who works as asales promoter for the product, there is a negative relation between the bargaining power
and the retail price.

JEL codes: L13, D43.
Keywords: Countervailing power, Buyer power, Dominant retailer, Two-part tariﬀ.
∗
We thank the seminar participants at Kyoto Sangyo, Osaka, and Shinshu Universities for their valuable
discussions and comments. We gratefully acknowledge the financial support from the Japan Society for the
Promotion of Science (JSPS), KAKENHI Grant Numbers JP15H03349 and JP15H05728, Grant-in-Aid for
JSPS Fellows Grant Number JP16J02442, and the program of the Joint Usage/Research Center for “Behavioral
Economics” at ISER, Osaka University. The usual disclaimer is applicable.
†
Corresponding author: 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.osaka-u.ac.jp‡
Shohei Yoshida, Institute of Social and Economic Research, Osaka University, Mihogaoka 6-1, Ibaraki, Osaka, 567-0047, Japan. Phone: (81)-6-6879-8552. Fax: (81)-6-6879-8584. E-mail: shoheiyoshida0@gmail.com.

1

1

Introduction

Considering the recent growth of dominant retailers such as Wal-Mart and Toys “R” Us,policy makers are now focusing on the influence of dominant retailers on market outcomessuch as retail price, consumer surplus, and social welfare. Since Galbraith’s (1952) argumenton countervailing power, many researchers have investigated this topic in various models (e.g.,
von Ungern-Sternberg, 1996; Dobson and Waterson, 1997; Chen, 2003; Inderst and Valletti,
2011; Iozzi and Valletti, 2014; Gaudin, 2016, 2017; Chen et al., 2016).1
These previous papers, however, have not considered the dominant retailers as sales promoters. When consumers do not exactly judge the real quality of a product before and/or
after the purchase, famous retailers’ selling that product could enhance its credibility regarding quality. Several empirical studies show that the more favorable the store name perceptions,the higher are the buyers’ product quality perceptions (e.g., Dodds et al., 1991).2 This reputation concern of dominant retailers could influence their negotiations with suppliers. If
dominant retailers have a reputation of possessing high-quality trading products, the breakdown of their negotiations with suppliers could have a negative impact on the reputationof the latter’s products’ quality. This is because such products are not handled by these
dominant retailers. As a result, the market demand for suppliers’ products becomes lesser
than the demand when the dominant retailers handle these products.3 Therefore, this kindof reputation matter provides dominant retailers an additional bargaining power, irrespectiveof that in the context of Nash bargaining.
To investigate the aspect mentioned above, a dominant retailer model in Chen (2003) is
1

Detailed discussions on buyer power are available in Chen (2007), Inderst and Shaﬀer (2007), and Inderst

and Mazzarotto (2008).
2

Marvel and McCaﬀerty (1984), Chu and Chu (1994), and Clerides et al. (2008) also empirically investigate

the eﬀects of famous retailers on quality perception.
3

Another interpretation of this scenario is as follows: since dominant retailers could undertake promotional

activities, if they do not exist, some consumers would be unaware of the existence of the promoted product,
which obviously reduces the market demand.

2

useful because his model distinguishes between two types of retailers—dominant and fringe.
This model assumption allows us to investigate the role of dominant retailers in the context
of the buyer-supplier relationship. Therefore, we incorporate a reputation matter into themodel in Chen (2003), by also considering the recent result in Christou and Papadopoulos(2015), which is a correction of Chen (2003).
The market structure in our model is as follows. A monopoly supplier distributes its
product through two channels. One channel is a dominant retailer that has a positive constantmarginal cost. The dominant retailer and the supplier negotiate their trading terms (a twopart tariﬀ contract) through Nash bargaining. The other channel comprises fringe retailers
that have an increasing marginal cost technology, which is eﬃcient for small quantities. Eachfringe retailer is a price-taker at the retail level and sets its own quantity, which competes withthe demand for the dominant retailer. Each of them is also a price-taker in the trade withthe monopoly supplier, who unilaterally oﬀers nondiscriminatory two-part tariﬀ contracts tothese fringe retailers. Under the framework, if the negotiation between the supplier and thedominant retailer breaks down, the retail price is determined to equalize the total quantities
supplied by the fringe retailers and the market demand that has consequently reduced. The
degree of market shrinkage depends on the importance of the dominant retailer as a sales
promoter of the product.
There are several diﬀerences among Chen (2003), Christou and Papadopoulos (2015), andthis study, regarding the retail market environments during negotiation breakdown between
the dominant retailer and the supplier. In Chen (2003), the retail market is inactive; inChristou and Papadopoulos (2015), it is active and does not shrink; while in our study, it isactive and does shrink.4 Chen (2003) shows that an increase in the buyer power possessedby a dominant retailer could decrease the retail prices for consumers. However, regarding
4

There is another diﬀerence between the three studies. Pertaining to negotiations between the supplier andthe dominant retailer, Christou and Papadopoulos (2015) and our study consider the supplier’s profits fromfringe retailers, unlike Chen (2003).

3

correcting for payoﬀs and outside options in Chen (2003), Christou and Papadopoulos (2015)
show that the countervailing power does not aﬀect the equilibrium prices, that is, they show
the “neutrality result.”5
Our paper shows that contrary to the neutrality result in Christou and Papadopoulos(2015), an increase in the buyer power possessed by a dominant retailer could decrease theretail prices for consumers, as in Chen (2003), even when the demand shrinkage throughnegotiation breakdown is minimal. Furthermore, if the bargaining power of the dominantretailer is strong, an unconventional negative result could arise: Given that the number offringe retailers is large, an increase in their number is more likely to increase the retail price.
The explanation behind the first result is as follows. A change in the bargaining power
generates sales-shifting across retailers. For the supplier, a loss of bargaining power over thedominant retailer leads to a lower share of industry profits. This negative eﬀect of the loss ofbargaining power on the supplier is oﬀset by a higher share of its outside options. In Christou
and Papadopoulos (2015), a possible gain from its outside option completely negates the loss.
In contrast, in our setting, if the dominant retailer functions as a sales promoter, the supplier’s
gain from its outside option is not suﬃcient to negate the loss of bargaining power due todemand shrinkage through negotiation breakdown. This is why an increase in buyer power
decreases retailer prices.
The rest of this paper proceeds as follows. Section 2 describes the model. Section 3
presents the equilibrium outcome and the results of comparative statics. Section 4 concludes
the paper. The Appendix provides the proofs of the lemmas and propositions in Section 3.
5

Erutku (2005) also relaxes several assumptions in Chen (2003). Erutku (2005) changes the following: (i)the bargaining process between a dominant retailer (called “a national retailer” in his paper) and a monopoly
manufacturer; and (ii) strategic interactions between the dominant retailer and its rival. He shows that theprice of the dominant retailer monotonically decreases according to its bargaining power, although that of therival has an inverted U-shaped relation to this parameter. In the context of agricultural industries, Matsushima(2016) considers how purchasing power in one of the two channels influences the profitabilities of upstream
firms.

4

2

Model

This section presents the basic model, which is an extension of Chen (2003) 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 purchase the intermediate goodfrom the supplier. Each retailer converts one unit of the intermediate good to one unit of final
good without any cost. However, each retailer must incur an operational cost to handle thefinal good. The dominant retailer has a constant marginal cost c 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. These assumptions reflect that the fringe
retailers are more eﬃcient at small operational scales, while the dominant retailer 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, those of which are 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 isthe fee. The fringe retailers are price-takers both in the input and final good markets.
In Stage 1, the monopoly supplier unilaterally oﬀers a two-part tariﬀ contract, wf qf + Ff ,to each fringe retailer, where wf is the wholesale price, and Ff is the fee. The supplier commits
its oﬀers. Under the assumption, we can derive the supply function of each fringe retailer. It isderive by p = M C(qf )+wf or qf = M C −1 (p−wf ). Here we define s(p−wf ) ≡ M C −1 (p−wf ),with s(0) = 0 and s′ (·) > 0. We also assume that s′ (·) + (p − c − wf )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 the dominant retailer is γ ∈ (0, 1). If the negotiation
breaks down, the market is still active but shrinks to kD(p) (k ≤ 1). This assumption reflects
the role of the dominant retailer as a sales promoter. The case of k = 1 is equivalent with
6

Christou and Papadopoulos (2015) employ a linear demand.

5

that in Christou and Papadopoulos (2015) except the functional specification in their paper.
In Stage 3, considering the production technology of the fringe retailers, the dominantretailer sets the retail price p. Given p, each fringe retailer determines its quantity. If thenegotiation broke down in Stage 2, p is determined to equalize the shrunken demand kD(p)and the total quantities supplied by the fringe retailers. This condition is further explainedin Section 3.

3

Analysis

We first derive the equilibrium outcome in the game, and then provide the results of thecomparative statics.

3.1

Equilibrium outcome

Our main objective is to discuss how formulating disagreement payoﬀs influences the equilibrium property. Before this, we first solve the game by backward induction.

Stage 3We have already derived the supply function of each fringe retailer, s(·),
qf (p − wf ) = s(p − wf ).

The residual demand for the dominant retailer is given as D(p) − ns(p − wf ). Thus, the profitof the dominant retailer becomes
πd = (p − c − wd )[D(p) − ns(p − wf )] − Fd .
From the first-order condition of its profit maximization problem, we obtain the optimal pricep∗ (wd , wf ) = p∗ such that
[D(p∗ ) − ns(p∗ − wf )] + (p∗ − c − wd )[D′ (p∗ ) − ns′ (p∗ − wf )] = 0.
Note that we have ∂p∗ (wd , wf )/∂wd > 0 from the second-order condition.

6(1)

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 the profitof the monopoly supplier. The bargaining problem is Dds = {πs , πd }, with the disagreement
payoﬀs for the supplier and the dominant retailer being (Os , Od ), respectively, where:πs (wd , wf , Fd , Ff ) = Fd + wd [D(p∗ (wd , wf )) − ns(p∗ (wd , wf ) − wf )](2)+n[Ff + wf s(p∗ (wd , wf ) − wf )],πd (wd , wf , Fd ) = (p∗ (wd , wf ) − c − wd )[D(p∗ (wd , wf )) − ns(p∗ (wd , wf ) − wf )] − Fd ,(3)Os = n[Ff + wf s(po − wf )],where po satisfies kD(po ) = ns(po − wf ) and k ∈ [0, 1],(4)

Od = 0.
The equation, kD(po ) = ns(po − wf ) in Equation (4), is the demand-equal-supply condition,in which the negotiation between the supplier and the dominant retailer breaks down. The
assumption of the supplier’s disagreement payoﬀ reflects the importance of the dominantretailer to promote the supplier’s good.
By solving the following maximization problem, we derive the outcome of bargaining:max [πs (wd , wf , Fd , Ff ) − Os ]1−γ [πd (wd , wf , Fd )]γ .

(wd ,Fd )First, Fd must satisfy the following:
Fd = (1 − γ)[p∗ − c][D(p∗ ) − ns(p∗ − wf )]− wd [D(p∗ ) − ns(p∗ − wf )] − nγwf [s(p∗ − wf ) − s(po − wf )]. (5)The problem becomes as follows:
max = πs + πd − Os = [p∗ − c][D(p∗ ) − ns(p∗ − wf )] + nwf [s(p∗ − wf ) − s(po − wf )].
wd(6)

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

Or
[D(p∗ ) − ns(p∗ − wf )] + (p∗ − c)[D′ (p∗ ) − ns′ (p∗ − wf )] + nwf s′ (p∗ − wf ) = 0.

(8)

Define wd∗ = wd (wf ). The second-order condition is∂ 2 p∗{[D(p∗ ) − ns(p∗ − wf )] + (p∗ − c)[D′ (p∗ ) − ns′ (p∗ − wf )] + nwf s′ (p∗ − wf )}
∂wd2
∂p∗
+
{2D′ (p∗ ) + (p∗ − c)D′′ (p∗ ) − 2ns′ (p∗ − wf ) − n(p∗ − c − wf )s′′ (p∗ − wf )} < 0.
∂wd
Since the first line becomes zero (by substituting Equation (8)) in equilibrium, we have
[2D′ (p∗ ) + (p∗ − c)D′′ (p∗ )] − n[2s′ (p∗ − wf ) + (p∗ − c − wf )s′′ (p∗ − wf )] < 0.

Stage 2 (the properties of the subgame outcome)(9)

Here, we check how wf influencesp∗ (wd (wf ), wf ) and po .
First, we check how wf influences p∗ (wd (wf ), wf ). Using the first-order condition, we
derive how p∗ (wd (wf ), wf ) changes with wf :dp∗ (wd (wf ), wf )n[2s′ + (p − c − wf )s′′ ]
=− ′
.
dwf
2D + (p − c)D′′ − n[2s′ + (p − c − wf )s′′ ](10)

By assumption in Section 2, 2s′ + (p − c − wf )s′′ ≥ 0 (s′ > 0 and s′ + (p − c − wf )s′′ > 0).
From (10), we have the following lemma:
Lemma 1 The second-stage equilibrium price p∗ (wd (wf ), wf ) is increasing with wf . Moreconcretely,0≤dp∗ (wd (wf ), wf )
< 1.
dwf

From Equations (1) and (8), we havewd =ns′ (p∗ − wf )wf .
ns′ (p∗ − wf ) − D′ (p∗ )Because s′ > 0 and D′ < 0, 0 < ns′ < ns′ − D′ . Therefore, wd < wf .

8

(11)

Second, we check the property of po . If k = 1, D(po ) = ns(po − wf ) which implies thatthe demand for the dominant retailer is zero at po = p∗ . Thus, p∗ < po . However, if k is smallenough and if p∗ − wf is positive in equilibrium, then p∗ > po . Also, we havens′ (po − wf )
dpo
= ′
< 1.
dwf
ns (po − wf ) − kD′ (po )(12)This implies the following:
Lemma 2 An decrease in k makes po more sensitive with respect to wf .
The results implies that if k is smaller, an increase in the wholesale price for the fringe
retailers slightly reduces the output when the negotiation breaks down because po − wf does
not decrease so much through the increase in wf . This property deeply relates to Proposition
4, which is explained later.
Due to the analytical complexity, we here specify the functional forms of D(p) and M C(qf )as follows:
Assumption 1 The demand function is linear in price and the marginal cost of each fringe
retailer is also linear in quantity. More concretely,D(p) = a − bp, and M C(qf ) = dqf or equivalently, s(p − wf ) = (p − wf )/d.
Under Assumption 1, the second-stage outcome isnwf,
bd + n(1 − γ)(bdk + n)(ad − (bd + n)c)2 − 4(1 − k)γdn2 wf (a − bwf )Fd (wf ) =.
4d(bd + n)(bdk + n)

wd (wf ) =Stage 1The profit of the monopoly supplier is:

πs (wf , Ff ) = Fd (wf ) + wd [D(p∗ (wd (wf ), wf )) − ns(p∗ (wd (wf ), wf ) − wf )]+n[Ff + wf s(p∗ (wd (wd ), wf ) − wf )].
9

The monopoly supplier sets wf and Ff to maximize its profit, leading to:Ha − b(bdk + n)(bd + n)c,2bH(bd + n)2 (bdk + n − (1 − k)nγ)2 c2
Ff∗ =
,
2dH 2
wf∗ =

where H ≡ (bdk + n)(bd + 2n) − 2(1 − k)(bd + n)nγ.
Using the above outcome, we obtain the profits of the monopoly supplier and the dominantretailer:(1 − γ)(ad − c(bd + n))((a − bc)H − n(bdk + n)bc)
4bdH[
2(bd + n) (bdk + n − (1 − k)nγ)2 c2 n[Ha − b(bdk + n)(bd + n)c]
+
+n
2dH 2
4bdH 2()]
γdkHa
×+ c(bd + n){b(2 − γ)dk + 2(1 − γ)n(γ(k − 1) + 1)} ,bdk + n[]πd∗ = (p∗ − c − wd∗ ) D(p∗ ) − ns(p∗ − wf∗ ) − Fd∗{(ad − (n + bd)c)2(1 − γ)(ad − (n + bd)c)2=−4d(n + bd)
4d(n + bd)()}
2
γ(1 − k)n(b(n + bd)(n + bdk)c)22−a −4b(n + bd)(n + bdk)
H2()γ(1 − k)n2(b(n + bd)(n + bdk)c)2γ(ad − (n + bd)c)2
2
+
a −
.
=
4d(n + bd)
4b(n + bd)(n + bdk)
H2

πs∗ =

Furthermore, we also obtain the important endogenous parameters in equilibrium:
Ha + b(bd + n)(bdk + n − 2(1 − k)nγ)c,2bH
n Ha − b(bdk + n)(bd + n)cwd∗ =,bd + n2bH{()}(1 − γ)(ad − (n + bd)c)2γ(1 − k)n2(b(n + bd)(n + bdk)c)2
∗
2
Fd =
−
a −
,
4d(n + bd)
4b(n + bd)(n + bdk)
H2
c(bd + n)[bdk + n(1 − (1 − k)γ)]ad − (bd + n)c, qf∗ =
.
qd∗ =
2d
dH
p∗ =

3.2

Comparative statics

We provide several results of comparative statics.
First, we examine whether Christou and Papadopoulos’s (2015) neutrality result holds inour setting.
10Lemma 3 Under Assumption 1, an increase in the countervailing power decreases the wholesale prices for the dominant and fringe retailers if k < 1, that is, ∂wd∗ /∂γ < 0 and ∂wf∗ /∂γ < 0
if k < 1.
In contrast to the results of Christou and Papadopoulos (2015) (∂wd /∂γ = 0, ∂wf /∂γ = 0),
the countervailing power aﬀects both wholesale prices when we consider demand shrinkage
under negotiation breakdown. That is, we obtain diﬀerent results in Lemma 3.
The partial derivative of p∗ with respect to γ is:∂p∗(1 − k)(bd + n)(bdk + n)n2 c
= −
.
∂γ
H2
Thus, we obtain the following result:
Proposition 1 Under Assumption 1, an increase in the countervailing power decreases theretail price, and increases consumer surplus if k < 1, that is, ∂p∗ /∂γ < 0 if k < 1.
To explain the mechanism behind Proposition 1, we first explain Christou and Papadopoulos’s (2015) neutrality result. Their result comes from the non-profitability of “sales-shifting”
across retailers. For the supplier, a loss of bargaining power leads to a lower share of theindustry profits, which is nevertheless oﬀset by a higher share of its outside option. If demand
and marginal cost functions are linear, a possible gain from its outside option completely
negates the loss. Thus, they have ∂wf∗ /∂γ = 0.
In contrast, our result shows that their assumption regarding the supplier’s outside option(Os ) is crucial to derive their neutrality result. In other words, if the dominant retailer works
as a sales promoter of the supplier’s product, the supplier’s gain from its outside option is
not suﬃcient to negate the loss of bargaining power. To overcome this loss, the suppliermakes fringe retailers stronger by decreasing wf . Although Lemma 3 and Proposition 1 havecontrasting eﬀects on the quantity of each fringe retailer, the eﬀect in Lemma 3 dominatesthat in Proposition 1, leading to:11

Lemma 4 Under Assumption 1, ∂qf∗ /∂γ > 0 for any k < 1. In addition, ∂qf∗ /∂k < 0.
Since a smaller k reflects a stronger bargaining power of the dominant retailer, the eﬀect of k
on wf∗ , p∗ , and qf∗ has a completely reverse relation to that of γ on wf∗ , p∗ , and qf∗ .
Second, we show how the value of k aﬀects the profitabilities of the monopoly supplierand the dominant retailer. By using the first-order derivative of πs∗ with respect to k, we canshow the following result:
Proposition 2 Under Assumption 1, the profit of the monopoly supplier monotonically increases with an increase in k.
Since a larger k reflects a stronger bargaining position of the monopoly supplier, in contrast
to the eﬀect of γ, the monopoly supplier benefits from an increase in k.
Proposition 3 Under Assumption 1, the profit of the dominant retailer decreases with k if
k is large enough.
This proposition is a flip side of Proposition 2 because a stronger bargaining position of themonopoly supplier is more likely to decrease the profit of the dominant retailer.7
Finally, we examine how the number of fringe retailers, n, aﬀects the retail price. Before
we show the eﬀect, we check the eﬀect of n on the wholesale prices:Lemma 5 Under Assumption 1, an increase in the number of fringe retailers increases thewholesale prices for the fringe retailers if and only if()√(1 − k) 2(1 − k)kγ − k(1 − 2(1 − k)γ)n > bd.
1 − 2(1 − k)kγ
The reason behind the result is as follows. When γ is suﬃciently low, the supplier can exploit
the whole industry profit by two-part tariﬀ. Since the expansion of fringe retailers harms the7

Contrary to the relation between πs and k, there is no clear relation between πd and k because ∂πd∗ /∂kcan be positive for suﬃciently small k.

12monopoly profit by reducing the retail price, the supplier raise the wholesale price for thefringe retailers. On the other hand, when γ is suﬃciently high, the objective of the supplier
is to enhance the profit from the outside option. When n is large, the retail price when thenegotiation breaks down becomes low due to a large production of fringe retailers. To increase
the retail price, the supplier raises the wholesale price.
Lemma 5 leads to the following result:
Proposition 4 Under Assumption 1, an increase in the number of fringe retailers, n, increases the retail price and decreases consumer surplus if and only if:n>√bdk( 2γ(1 − k) + 1)
1
and k < 1 − √ .
22(1 − k) γ − 1
2γ

(13)

The eﬀect of a marginal increase in the number of fringe retailers can be decomposed into
the three terms as follows:
dp∗
=
dn

∂p∗
∂n
|{z}
Direct Eﬀect (-)+∂p∗ ∂wd∂w ∂n| d{z }
Cost Eﬀect (–/+)+∂p∗ ∂wf∂wf ∂n| {z }(14)

Fringe-Shrinkage Eﬀect (–/+)

In fact, the last term is significantly aﬀected by k. The intuition behind the result is as follows.
The condition of the proposition requires that γ is suﬃciently high. When n increases, tosustain the retail price level in which the negotiation breaks down (po ), the supplier raises
the wholesale price for the fringe retailers. Since this reduces the total output of the fringe
retailers, when k is small, an increase in the wholesale price does not reduce the output somuch (see Lemma 2). This implies that when n is suﬃciently large and k is small, the valueof last term becomes positive and large, so that it dominates the direct negative eﬀect. Notethat the right-hand side of the first fraction in (13) decreases with an increase in γ. Thatis, an increase in the number of fringe retailers is less likely to improve the consumer surplus
when the dominant retailer has a strong bargaining power.

13

4Conclusion

We consider a downstream oligopoly model with one monopoly supplier, one dominant retailer,and fringe retailers, by taking into account the role of the dominant retailer as a sales promoterof the supplier’s product. We assume that the existence of the dominant retailer is importantto sustain the downstream demand size at a certain level. More concretely, the demand size
shrinks if the negotiation between the dominant retailer and the supplier breaks down.
We show that an increase in the buyer power possessed by a dominant retailer could
decrease the retail prices for consumers as in Chen (2003), even when the demand shrinkage
through negotiation breakdown is minimal. Furthermore, if the bargaining power of thedominant retailer is strong, an unconventional negative result could arise: An increase in thenumber of fringe retailers is more likely to increase the retail price if their number is large. In
this case, conventional competition policy might backfire on consumer surplus.
We could extend the model to a two-region model in which products in two regions are
geographically diﬀerentiated, and the retail market in each region comprises a dominant andseveral fringe retailers that trade with a local monopoly supplier. That is, each region isconnected through the retail level transactions, which restrict the monopoly power of thedominant retailer in each region. Although the extension complicates the analysis, this could
be a potential avenue for future research.

Appendix: Proofs
Proof of Lemma 3.
Diﬀerentiating wf∗ with respect to γ, we have:
∂wf∗
∂γ

=−

(1 − k)(bdk + n)(bd + n)2 cn[b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ))]2

Note that wd∗ = nwf∗ /(bd + n) also decreases with γ.

14

< 0.

(15)

Proof of Proposition 1.
We have already shown that ∂p∗ /∂γ < 0 if k < 1. We define consumer surplus in equilibrium
as follows:CS =b(qd∗ + nqf∗ )22.

Diﬀerentiating CS with respect to γ, we have∗∗∂(qd + nqf )∂CS∂p∗= b(qd∗ + nqf∗ )= −b2 (qd∗ + nqf∗ )
> 0,
∂γ
∂γ
∂γ

(16)

if k < 1.

Proof of Lemma 4.
Diﬀerentiating qf∗ with respect to γ, we have:
∂qf∗
∂γ

=

bcn(1 − k)(bd + n)(bdk + n)

> 0,

(17)

bcnγ(bd + n)2< 0.
[b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ))]2(18)[b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ))]2

if k < 1. Similarly, diﬀerentiating qf∗ with respect to k, we have:
∂qf∗
∂k

=−

Proof of Proposition 2.
Diﬀerentiating πs∗ with respect to k, we have:
∂πs∗
∂k

=

{
}2
n2 γ b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ)) a2
4b(bdk + n)2 [b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ))]2n2 γb2 (bd + n)2 (bdk + n)2 c2−.
4b(bdk + n)2 [b2 d2 k + 2n2 [1 − (1 − k)γ] + bdn(1 − 2γ + 2k(1 + γ))]2

We thus obtain the following:()
a b2 d2 k + bdn(1 − 2γ + 2(1 + γ)k) + 2n2 (1 − γ(1 − k))
∂πs∗
=0⇔c=
≡ ĉ.
∂k
b(bd + n)(bdk + n)

(19)Note that ∂πs∗ /∂k is decreasing with c. Since we can show that ĉ > c̄, ∂πs∗ /∂k > 0 for allc ∈ (0, c̄).
15

Proof of Proposition 3.
The partial diﬀerential of πd with respect to k is:∂πd∗γn2 a2γn2 b(n + bd)2 ((n + bdk)(2n + bd) + 2γ(1 − k)n(n + bd))c2
= −
+
.
∂k
4b(n + bdk)2
4((n + bdk)(2n + bd) − 2γ(1 − k)n(n + bd))3
Since ∂πd∗ /∂k|k=1 = −γn2 ((2n + bd)2 a2 − b2 (n + bd)2 c2 )/(4b(n + bd)2 (2n + bd)2 ) < 0 for anyc ∈ [0, c̄), the partial diﬀerential is negative around k = 1.

Proof of Lemma 5.
Diﬀerentiating wf∗ with respect to n, we have:
(
)
√
∂wf∗
(1 − k) 2(1 − k)kγ − k(1 − 2(1 − k)γ)
> 0 ⇔ n > bd.
∂n
1 − 2(1 − k)kγ

(20)Note that if dwf∗ /dn > 0, wd∗ also increases with n.

Proof of Proposition 4.
Diﬀerentiating p∗ with respect to n, we have:√∂p∗bdk( 2γ(1 − k) + 1)
>0⇔n>
.
∂n
2(1 − k)2 γ − 1√
The condition that the denominator is positive is k < 1 − 1/ 2γ.

(21)

Diﬀerentiating CS with respect to n, we have:∗∗∂(qd + nqf )∂p∗∂CS= b(qd∗ + nqf∗ )= −b2 (qd∗ + nqf∗ )
.
∂n
∂γ
∂n

(22)

Therefore, the sign of that derivative is converse to that of price.

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