ECONOMIC RESEARCH CENTERDISCUSSION PAPER
E-Series

No.E17-7

Price Elasticity and Pass-Through in the Nash
Bargaining Solution in a Distribution Channel
byTakanori Adachi

September 2017

ECONOMIC RESEARCH CENTERGRADUATE SCHOOL OF ECONOMICSNAGOYA UNIVERSITY

Price Elasticity and Pass-Through in the Nash
Bargaining Solution in a Distribution Channel
Takanori AdachiySeptember 12, 2017

Abstract
This paper analyzes the role of the industry’s price elasticity and the determinants ofpass-through in the Nash bargaining solution in a distribution channel. It is shown thatthe division of the upstream and downstream pro…ts is characterized by the price elasticity
and the distribution of bargaining powers. I also show that the demand curvature plays
an important role in the determination of pass-through rates.

Keywords: Distribution Channels; Bargaining; Price Elasticity; Pass-Through; DemandCurvature.
JEL Classi…cation: L13; L49; L66.

I am grateful to Germain Gaudin for helpful discussions. I also thank a Grant-in-Aid for Scienti…c Research(C) (15K03425) from the Japan Society for the Promotion of Science. All remaining errors are mine.
ySchool of Economics, Nagoya University, 1 Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan. E-mail:adachi.t@soec.nagoya-u.ac.jp

1

Introduction

In vertical relationships, a retailer is usually a multi-product …rm which sells competing brands(Choi, 1991; 1996). In this paper, I study the properties of the Nash bargaining solution(Nash, 1950) in such a distribution channel. First, I analyze the role of the price elasticity inthe determination of the division of the upstream and downstream pro…ts. It is shown thatthe common retailer’s pro…t share becomes lower if the market demand becomes less elastic,
holding the division of upstream and downstream bargaining powers is …xed. This is becausethrough a negotiation process, an upstream …rm can aggressively charge a higher wholesale
price because it loses less from a sales reduction when the demand is not elastic.
Second, I show that an upstream cost increase is partly absorbed by the common retailer,
and how much it is absorbed is determined by the demand curvature: if the market demandis “very convex,”then the pass-through at the wholesale level is almost passed through to the…nal price, irrespective of the division of upstream and downstream bargaining powers. In
this way, I argue the importance of the …rst- and the second-order demand characteristics aswell as the division of bargaining powers in characterizing the Nash bargaining solution in a
distribution channel.
In a similar vein, Aghadadashli, Dertwinkel-Kalt, and Wey (2016) study a model of one
upstream …rm and N downstream …rms which bargain over the input price, produce outputs,
and quantity-compete, and show that if the downstream …rm’s demand becomes less elastic,
then ceteris paribus, the upstream monopolist earns a higher pro…t margin from the input
price bargaining. Despite the di¤erences in vertical structure and the mode of downstream
competition, Aghadadashli, Dertwinkel-Kalt, and Wey (2016) and this paper’s model of one
downstream …rm and N upstream …rms share a similar intuition.

2

Model

Suppose that there are N

1 symmetric upstream …rms (manufacturers) whose marginal costof production is constant, cU

0. They transact with a downstream …rm (a common retailer)

who is a monopolist in the geographical market. Each manufacturer produces one type of
product, and for each manufacturer’s product, the retailer incurs a constant marginal cost of1

sales, cD

0. The common retailer sells the manufacturers’products to the …nal market bychoosing the prices p = (p1 ; p2 ; :::; pN ). Then, the demand (in terms of a market share) for
product j 2 N

f1; 2; ::; N g is written as sj = sj (p). Here, N upstream …rms are horizontallydi¤erentiated. The common retailer pays the unit price wj to manufacturer j. Thus, thePDD(pj wj cD )sj (p).
common retailer’s total pro…t is written as Djj , where
j2N

The …rst-order condition for pj is given bysj (p) + (pj

wj

cD )

@sj X
+
(pl
@pj
l6=j

wl

cD )

@sl
= 0.
@pj

(1)

On the other hand, if each manufacturer can directly distribute its product to the …nal
market, then manufacturer j’s pro…t is eUj

cU

(pj

condition for pj is given bysj (p) + (pjcU , thenHence, if wj

P

l6=j (pl

wl

cU

cD )

@sl
cD ) @p> (wjjcD )sj (p), and thus the …rst-order

@sj
= 0.
@pj
@s

cU ) @pjj because @sl =@pj > 0 for l 6= j and@sj =@pj < 0. Thus, the …nal equilibrium in a distribution channel, p = p(w), is higher thanthe equilibrium price without such a distribution channel, p0 . This is because the monopolistic
retailer internalizes the e¤ects of changing pj on not only its own demand sj but also on the
demands for the other products, sl ’s.
Following Aghadadashli, Dertwinkel-Kalt, and Wey (2016) and many others, I employ thesimplifying assumption that each bargaining is played by an upstream …rm and one of the N
delegates from the common retailer, and each bargaining is unobservable from other upstream…rms and other delegates. Additionally, it is also assumed that players hold “passive beliefs”
in the sense that even if a player in one bargaining process observes out-of-equilibrium priceo¤er, the player still holds the belief that the equilibrium is played (in the bargaining and thepricing decisions) by the players outside of this bargaining process.
Under these assumptions, I focus on the bargaining process over wj . Given the players’
belief that the symmetric equilibrium fw; pg is played, it is determined to maximize the Nashproduct, [

D

D

wj

cD )s(p) + (N

] [ Uj ]1, where2 (0; 1) is the common retailer’s bargaining power,

D

= (p

D

is its disagreement pro…t that it obtains when the bargaining with manufacturer j breaks

down, andUj = (wj1)(pcD )s(p) is the common retailer’s total pro…t,wcU )s(p) is manufacturer j’s pro…t from the wholesale bargaining. Here,2note that the bargaining game is played by manufacturer j and the delegate of the common
retailer, and they have a passive belief that the equilibrium is still played, in particular, thesymmetric retail price p will still be chosen by the common retailer. Accordingly, the common
Dretailer’s disagreement payo¤ is perceived as
D

3

D

= (p

wj

= (N

1)(p

w

cD )s(p), which implies thatcD )s(p).1

Analysis

Now, I de…ne the demand for each product under symmetric pricing by s(p)@sThen, the relationship, s0 (p) = @pjj +(N0of demand by (p)3.1

sj (p; p; :::; p).

@s1) @pjl holds. I also de…ne the industry’s price elasticity

ps (p)=s(p) > 0.

The Role of the Industry’s Price Elasticity

Then, the following proposition shows how the bargaining relationship is related to the demandconditions in the …nal market.
Proposition 1. The upstream …rm’s share of the total pro…ts, measured in terms of the …nal
price, is expressed as
cU

w
p
for

1

=

1
(p)

(2)

2 (0; 1).

Proof. The …rst-order condition for wj is given by
sD )s(p)]

[(p

wj

+(1

)[(wj

1

cU )s(p)]1[ s(p)][(wjcU )s(p)][s(p)][(pcD )s(p)] = 0,

wj

1
If consumers realize that the products are lost for the …rms with which the negotiation breaks down while
the retail prices remain the same even if such a breakdown happens, then the resulting market share for each…rm is written as se s(ep). Then,
D

D

=

(p

wj

(N
=

(p
+(N

cD )s(p) + (N

1)(p
wj

w

1)(p

w

cD )e
s

D

c )s(p)1)(pwwhich would not admit a tractable analysis as below.

3cD )fs(p)seg,cD )s(p)which determines the symmetric w:
cU ) = (1

(w

)(p

w

cD ).

Accordingly, the …rst-order condition for pricing (1) can rewritten ass(p) + (pw, s(p) + (p

w

@sj
+ (N
@pj
cD )s0 (p) = 0,
cD )

1)

@sj
=0
@pl

where the common retailer takes care of the “industry” as a whole: it takes into account not
only its own price e¤ect, @sj =@pj , but also the aggregate spillover e¤ects on other products,(N1)(@sj =@pl ). Equation (2) is then obtained by combining these two equations.
Equation (2) shows that the upstream …rm’s bargaining power becomes larger (i.e., (1)

becomes larger), then, with other things unchanged, the upstream …rm’s share of pro…ts also
becomes larger. More importantly, if the industry’s price elasticity of demand becomes lesselastic (i.e., becomes smaller), then the upstream pro…t share becomes, ceteris paribus, larger.
Intuitively, this is because the upstream …rm loses less from a sales reduction by aggressively
charging a high wholesale price to common retailer through the bargaining process.
The result above is even clearer if Holmes’ (1989) decomposition is used. As Holmes(1989) shows, under symmetric pricing, the industry’s price elasticity is the …rm’s own priceelasticity, subtracted by the cross price elasticity:(p=s(p))@sj (p)=@pj jp=(p;:::;p) and

C (p)

(p) =

F (p)

C (p),

where

F (p)

1)(p=s(p))@sl (p)=@pj jp=(p;:::;p) for any distinct(Npair of indices j and l. Thus, an increase in the degree of competition (due to less productdi¤erentiation and/or an increase in N ) raises the upstream …rm’s pro…t share, holding the…nal price p …xed. This result is, again, in line with the intuition above: the upstream …rm
loses less from an aggressive attitude in the bargaining because it already faces a severe level ofcompetition. The total e¤ect is less unambiguous, though, because the …nal price p would also
be lower by a higher level of competition. In contrast, the e¤ect of F is exactly the opposite: if
the own price elasticity is very elastic, then the upstream …rm has to be less aggressive because
an increase in w, and thus the associated increase in p, signi…cantly reduces the demand.
Now, from Equation (2) above, the wholesale price is obtained by
w=

s

1

s0
4

+ cU ,

which leads to the following equation:p = cU + cD +

s
s0

1

.

(3)

Here, as N increases, p goes down. How does the margin ratio moves as N increases? The
answer is that it is independent of N simply because
p

cU

w

3.2

cD

w

=

.

1

Pass-ThroughNext, I de…ne three di¤erent types of cost pass-through: the wholesale and the …nal price passthroughs of the upstream cost, @w=@cU andof the downstream cost,DU@p=@cU , respectively, and price pass-through@p=@cD . I also de…ne the demand curvature by (p)ss00 =[s0 ]2 .

Among others, Adachi and Ebina (2014), Chen and Schwartz (2015), and Gaudin (2016) showthat for the optimum it is necessary that 2 >, and furthermore, s(p) should not be “tooconvex,”that is, s00 is su¢ ciently small that 1 > . As Chen and Schwartz (2015) argue, many
classes of demand functions satisfy this condition. Therefore, I also assume this restriction.
Then, the following proposition is obtained.
Proposition 2. The wholesale pass-through is larger than the upstream cost pass-through andthe downstream cost pass-through, which are equal:
@w
= (2
@cU
Proof. Let F (p; w; cD )s(p) + (p

w

) U>

U

=

D

.

cD )s0 (p) and G(p; w; cU )(w

cU ) (p)(1)p.

Essentially, our simplifying assumptions make it unnecessary to consider the dependence of p
on w: the retail prices and the wholesale prices are simultaneously determined. Then, these
two equilibrium conditions, F (p; w; cD ) = 0 and G(p; w; cU ) = 0, can be utilized to develop
implications for the three types of pass-through. First, the wholesale and the …nal price passthroughs of the upstream cost, @w=@cU and
2 @F
4

@p

@F
@w

@G
@p

@G
@w

32
54

U

@p=@cU , satisfy
U

@w
@cU

5

3

5=

2 @F 3
4

@cU
@G
@cU

5.

@F
@p

Now, let the determinant be de…ned by D
D = s02 (0; 1), because0

fs0 s + p [s00 s

=

p0

@F
@w

@G
@p

. Then

ss00
p0
+
(1
)
1
[s0 ]2) + (1
)(1
) g<0

2

= s0 f (2for all@G@w(s0 )2 ]g=s2 so that
ps0
s

1 =
=

(1

1

ss00
[s0 ]2).

Now, it is proceeded as
3
@G @F
@F @G
U
@w @cU 7
167
67
6 @w @c
=
7,
6
4
5
@w
D 4 @G @F@F @G 5+@cU
@p @cU@p @cU2which implies that

U

2

3

U

s0
D

=
=

(2

) + (1

s0
D

(2

)(1

)

> 0,

and
@w
@cU

=
=

(2

)

(2
) + (1))(1

Similarly, the price pass-through of the downstream cost,
D==

1 @G @F
D @w @cD
s0
D

@F @G
@w @cD(2)(1=
Since 2

> 1, @w=@cU >

U

=

D

) + (1

.
6

> 0.

)

)

D

@p=@cD , is obtained by

=

U

.

This proposition shows that the common retailer absorbs the upstream cost shocks by
100

[(1

)=(2

)]%. Here, the demand curvature plays an important role: if the industry’s

demand becomes “very convex” ( becomes close to one), then the wholesale price increaseis almost passed through to the …nal price, irrespective of the common retailer’s bargaining
power, .2 However, as

U

increases, both

@ U
@ D
=
=
@
@
[ (2
and

@

@w
@cU

(1
)2
) + (1
)(1

(2

=

@
[ (2
Again, the latter is larger than the former.
Next, the direct e¤ects ofwhich indicates thatand @w=@cU also increase because

)2
)(1

)(1
) + (1

) ]2

) ]2

> 0,

> 0.

on the retail and the wholesale prices are obtained by
3
2
3
2
@F @G
@p
7
6 @ 7
16
7
6 @w @ 7
6
=
7
6
7,
6
D 4 @F @G 5
4 @w 5
@p @
@
s0
D

@p
=
@

p

<0

and

@w
s0
p
=
< 0.
(2
)
@
D
Interestingly, as the common retailer’s bargaining power increases for all upstream manufacturers, the …nal price decreases. This is because the common retailer can lower the wholesale pricethough the bargaining, still maintaining product competition. If the common retailer loses its
bargaining power, the loss from less product competition becomes larger than the gain from
elimination of double marginalization.
Finally, imposing symmetry on Equation (3) yields:p(p)p = cU + cD +2

On the other hand, an increase in the retailer’s marginal cost cD lowers the wholesale price:
@w
=
@cD

(1
(2

) (1
) + (1

7

)
)(1

)

< 0.

, p=

cU + cD
,
1
1
(p)

U

=

which implies that
=

D

1
1

1

.

As a summary, a higher bargaining power of the common retailer ( ) raises the upstream
cost and the downstream cost pass-throughs ( U andD), and lowers the …nal price (p), sup-pressing the degree of double marginalization.

4

Concluding Remarks

In this paper, I characterize the Nash bargaining division of the pro…ts between upstream…rms and a common retailer in terms of the industry’s price elasticity (Proposition 1) and thedi¤erent types of cost pass-through (Proposition 2). How do these results change if downstream
competition is introduced? This is an important issue for future research.

References
Adachi, T., Ebina, T., 2014. Double marginalization and cost pass-through: Weyl-Fabinger and
Cowan meet Spengler and Bresnahan-Reiss. Economics Letters, 122(2), 170–175.
Aghadadashli, H., Dertwinkel-Kalt, M., Wey, C., 2016. The Nash bargaining solution in vertical
relations with linear input prices. Economics Letters, 145, 291–294.
Chen, Y., Schwartz, M., 2015. Di¤erential pricing when costs di¤er: A welfare analysis. RAND
Journal of Economics, 46(2), 442–460.
Choi, S.C., 1991. Price competition in a channel structure with a common retailer. Marketing
Science, 10(4), 271–296.
Choi, S.C., 1996. Price competition in a duopoly common retailer channel. Journal of Retailing,72(2), 117–134.
Gaudin, G., 2016. Pass-through, vertical contracts, and bargains. Economics Letters, 139, 1-4.
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Holmes, T.J., 1989. The e¤ects of third-degree price discrimination in oligopoly. American
Economic Review, 79(1), 244–250.
Nash, J.F., Jr., 1950. The bargaining problem. Econometrica, 18(2), 155–162.

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