ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E19-10

Galbraith meets Weyl and Fabinger:
Countervailing power and imperfect competitionin the retail market
byTakanori Adachi

September 2019

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Galbraith meets Weyl and Fabinger: Countervailingpower and imperfect competition in the retail market∗
Takanori Adachi†
September 5, 2019

Abstract
I study how a change in countervailing power on the retailer side (Galbraith 1952)
affects the retail and wholesale prices by employing Weyl and Fabinger’s (2013) conductparameter approach in a model of vertical relationships with Nash cooperative bargaining.
I argue that the effects of countervailing power are affected by its relationship with theindustry’s conduct.

Keywords: Distribution Channels; Bargaining; Countervailing Power.
JEL Classification: D43; L13; L49.

∗I am grateful to Germain Gaudin for helpful discussions on earlier version of the paper. I also thank a
Grant-in-Aid for Scientific Research (C) (15K03425,18K01567) from the Japan Society for the Promotion ofScience. All remaining errors are mine.
†School of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. E-mail:adachi.t@soec.nagoya-u.ac.jp

1Introduction

I study a tractable model of vertical relationships between N symmetric manufacturers and M
symmetric retailers à la Gaudin (2016) to analyze two sources of Galbraith’s (1952) countervailing power by retailers: (i) a decrease in the number of retailers as a result of consolidation,and (ii) an increase in the retailer’s bargaining power. I particularly consider the effects on theretail sector’s conduct by using the conduct parameter approach à la Weyl and Fabinger (2013)
to model imperfect competition in the retail sector. By focusing on symmetric equilibrium retail and wholesale prices, I show that a decrease in the number of retailers as a way of creating
countervailing power does not necessarily lower the retail price if the effects on the industry’s
conduct are taken into account. In contrast, the effects of a change in the retailer’s bargaining
power on the prices depend on whether it improves or worsens the industry’s conduct.

2ModelSuppose that there are N ≥ 1 symmetric upstream firms (manufacturers) whose marginal costof production is constant, cU ≥ 0. Here, N upstream firms are horizontally differentiated. There
are also M ≥ 1 symmetric common retailers: each downstream retailer i ∈ M ≡ {1, 2, .., M }transacts with all N upstream firms. Each manufacturer produces one type of product, andfor each manufacturer’s product, the retailer incurs a constant marginal cost of sales, cD ≥ 0.
Each common retailer sells the manufacturers’ products to the final market, and the demandfor product ij (brand j ∈ N ≡ {1, 2, .., N } sold by retailer i) is sij (P), where p11 p12 . . . p1N ..  p21 . . .
. 
P= .
.
.. ...
 ..
. 

pM 1 . . . . . . p M N
Each common retailer i pays the unit price wij to manufacturer j: its total profit is writtenPDUas ΠDi ≡j∈N (pij − wij − c )sij (P). Similarly, each retailer j’s profit is given by π j ≡1Pi∈M (wij − cU)sij (P).

In the following analysis, we focus on symmetric equilibrium prices p and w, and thus
denote by s(p) the per-product market demand corresponding to p: s(p) ≡ sij (p, ..., p). where
p is the symmetric price for M × N products. Note also that the wholesale and final prices are
determined simultaneously as a pair {w, p}: this assumption is standard in empirical studies ofbargaining in vertical relationships such as Draganska, Klapper, and Villas-Boas (2010), Mezaand Sudhir (2010), Crawford and Yurukoglu (2012), Grennan (2013, 2014), Gowrisankaran,Nevo, and Town (2015), Ho and Lee (2017), Crawford, Lee, Whinston, and Yurukoglu (2018),and De los Santos, O’Brien, and Wildenbeest (2018).1 I also employ the simplifying assumptionthat each bargaining is played one of the M delegates from a manufacturer and one of theN delegates from a retailer, and each bargaining is unobservable from the other delegates.
Additionally, it is also assumed that players hold “passive beliefs” in the sense that even if aplayer in one bargaining process observes out-of-equilibrium price offer, the player still holdsthe belief that the equilibrium is played (in the bargaining and the pricing decisions) by theplayers outside of this bargaining process (McAfee and Schwartz 1994).
Under these assumptions, we focus on the bargaining process over wij . Given the players’
belief that the symmetric equilibrium {w, p} is played, it is determined to maximize the Nash
λU 1−λproduct, [∆ΠDwith respect to wij , where λ ∈ [0, 1] is the common retailer’s Nashij ] [∆π ij ]bargaining weight,2DD∆ΠDij ≡ (p − wij − c )s(p) − (N − 1)(p − w − c )∆s(p)∆π Uij ≡ (wij − cU )s(p) − (M − 1)(w − cU )∆s(p),and ∆s(p) ≡ se(p) − s(p) > 0 is the market share difference, with se(p) being the market share of1

Collard-Wexler, Gowrisankaran, and Lee (2019) provide a non-cooperative foundation for Nash’s (1950)
modeling. This timing assumption would be innocuous when the frequencies of price revisions are similar forwholesale and retail prices. In other cases, retail prices may be revised more frequently: it seems more natural
to assume that retail prices are chosen after wholesale prices are determined. Nonetheless, this assumption isutilized in empirical studies to ease computational burden. See Iozzi and Valletti (2014) for a study of richer
timing and information structure.
2
The bargaining weight λ can be understood as the factor that summarizes “the tactics employed by thebargainers, the procedure through which negotiations are conducted, the information structure, and the player’s
discount rates.” (Muthoo 1999, p.35).

2

product (ij)0 6= ij when product ij is removed in the case of disagreement. It is assumed that∆s(p) is strictly decreasing in the number of products for any p ≥ 0. Note here that the retailprices are not reoptimized in such an event, and thus consumers still face the same price p foreach product (except for the removed product ij). Note also that in symmetric equilibriumD∆ΠDij ≡ (p−w−c )[s(p)−(N −1)∆s(p)]: this expression indicates that an additional profit gain

for the common retailer from inviting one more upstream firm (transacting N firms instead of(N − 1) firms) comes from the increase in unit s(p), multiplied by the unit margin (p − wj − cD ).
However, this addition of an upstream firm reduces the output for each of the other (N − 1)
firms: this negative effect is captured by the term (N − 1)∆s(p). The same reasoning applies
to ∆π Uij ≡ (w − cD )[s(p) − (M − 1)∆s(p)] in symmetric equilibrium.3
Finally, we follow Weyl and Fabinger’s (2013) and introduce the conduct parameter θ ∈ [0, 1]that measures the degree of imperfect competition in the retail market: with θ = 0 being perfectcompetition and θ = 1 being monopoly. We also define the industry’s price elasticity of demand∂s∂sijfor k 6= i and l 6= j. Then, theby (p) ≡ −ps0 (p)/s(p) > 0, where s0 (p) ≡ ∂pijij + (M N − 1) ∂pkl

equilibrium pair {w, p} satisfies:θs(p) + (p − w − cD )s0 (p) = 0λ(w − cU )(p)[s(p) − (M − 1)∆s(p)] − (1 − λ)[s(p) − (N − 1)∆s(p)]p = 0.
Essentially, our simplifying assumptions make it unnecessary to consider the dependence of pon w. Figure 1 depicts the situation under the symmetry assumption. Here, Holmes’ (1989) decomposition indicates that under symmetric pricing, the industry’s price elasticity is equal to thefirm’s own price elasticity, subtracted by the cross price elasticity: (p) = F (p) − C (p), whereF (p) ≡ −(p/s(p))∂sij (p)/∂pij |p=(p,...,p) and C (p) ≡ (M N − 1)(p/s(p))∂s(ij)0 (p)/∂pij |p=(p,...,p)for any distinct pair of indices ij and (ij)0 .
(p) s(p)(p) s(p) s
We make an additional restriction on ∆s(p): it must be the case that ∆s(p) < min{ seM, M −1 , eN, N −1 }Uis assumed for any p ≥ 0, M ≥ 1, and N ≥ 1 to assure that ∆ΠDand∆π
arealways
positive.
ijij33

Figure 1: Vertical Structure with N Manufacturers and M Retailers

Note: λ ∈ [0, 1] is the retailer’s Nash bargaining weight, and θ ∈ [0, 1] is the conduct parameter in the retailmarket.

3Analysis

Let F (p, w; θ, cD ) ≡ θs(p) + (p − w − cD )s0 (p) and G(p, w; λ, cU , M, N ) ≡ λ(w − cU )(p)[s(p) −(M − 1)∆s(p)] − (1 − λ)[s(p) − (N − 1)∆s(p)]p. Then,∂F ∂F∂p∂F ∂p ∂w   ∂M   ∂M 


.
= − ∂G ∂G ∂w∂G ∂p ∂w∂M∂M{z}|≡DNow, let the determinant be defined by det(D) ≡∂F∂p∂G∂w

−∂F∂w  ∂G ∂p. Specifically, it isverified thatdet(D) = s0 λ · ((1 + θ − σ)[s − (M − 1)∆s]

U0+(w − c ) [s − (M − 1)∆s]  −p00[s − (N − 1)∆s][s − (M − 1)∆s ] − [s − (M − 1)∆s][s0 − (N − 1)∆s0 ],+[s − (N − 1)∆s]

4where, for example, ∆s0 ≡ d∆s(p) and σ(p) ≡ ss00 /[s0 ]2 is the demand curvature under symmedp
try. It is assumed that s(p) is not “too convex,” that is, σ < 1 (Adachi and Ebina 2014; Chenand Schwartz 2015; and Gaudin 2015). For det(D) to be negative, it is sufficient to assumethat 0 ≥ /p and that[s − (N − 1)∆s][s0 − (M − 1)∆s0 ] ≥ [s − (M − 1)∆s][s0 − (N − 1)∆s0 ].

Thus, it is observed that  ∂θ (−λ) [s − (M − 1)∆s] −0Us+s(w−c)∆s|{z}∂M∂p|{z}<0 ∂M ≥0,
 = −1  det(D)  ∂w  | {z } 
>0∂M∂θ0U

sH+(1+θ−σ)s(w−c)∆s−λ{z}|
∂M
| {z }
<0≤0∂θ
where it is assumed that the conduct parameter θ is decreasing in M : ∂M
≤ 0, and

1 ∂G
U0H≡
= (w − c ) [s − (M − 1)∆s]  −λ ∂pp[s − (N − 1)∆s][s0 − (M − 1)∆s0 ] − [s − (M − 1)∆s][s0 − (N − 1)∆s0 ]+> 0.
[s − (N − 1)∆s]This result indicates that while the wholesale price decreases as the number of retailers decreases∂w(i.e., ∂M> 0), the retail price can increase if∂θ∂M

is sufficiently large. It may be the case thatthe retail sector’s consolidation may induce collusive pricing; if this effect is severe, this works
as a countervailing effect for countervailing power.

5

Similarly, it is observed that∂θ0 ∂Gλ[s − (M − 1)∆s]s +s∂λ∂λ} | {z |∂p{z}<0 ∂λ R0, = −1  det(D)  ∂w  | {z } ∂θ∂G0
>0−λH s + s (1 + θ − σ)∂λ∂λ∂λ| {z } |{z}

Q0<0where∂G= (w − cU )[s − (M − 1)∆s] + [s − (N − 1)∆s]p > 0.
∂λHence, if an increase in the retailer’s bargaining power raises the conduct parameter (i.e.,∂θ∂λ≥ 0), then the retail price may go up as above. However, the wholesale price always∂θdecreases. If the opposite is the case (i.e., ∂λ≤ 0), then the retail price always decreases,

whereas the wholesale price may rise.

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