GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.160

Multimarket Linkages, Buyer Power, and the Productivity Puzzle

Noriaki Matsushima and Laixun Zhao

October 2010

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Multimarket Linkages, Buyer Power, and the Productivity
PuzzleNoriaki Matsushima∗
Institute of Social and Economic Research, Osaka University
Laixun Zhao
Research Institute of Economics and Business, Kobe University
October 12, 2010

Abstract
This paper examines the relationship between ﬁrms’ productivity improvement and the volume
of exports, and shows that it can be sometimes negative. Speciﬁcally, we simultaneously take into
account intermediate retailers (i.e., vertically) and multimarket linkages (i.e., horizontally). We ﬁndthat an improvement of the manufacturing productivity aﬀects the bargained wholesale prices in
opposite directions in asymmetric markets, causing retailers to make corresponding changes thatlook surprising. This result can explain for the empirical “productivity puzzle” found in Ghemawat
et al. (2010). Related to this issue is the relationship between buyer power (caused by a retailmerger) and proﬁtability. Contrary to the existing literature, in an extended setup, we ﬁnd that themerger between the downstream duopolists does not improve their proﬁts if their bargaining poweris strong vs. upstream suppliers.

JEL Classiﬁcation Codes: L12, L14, F12, F23
Key Words: Vertical Structure, Market power, Productivity, Multi-Market Bargaining.

∗ 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: +81-8-6879-8583. E-mail: nmatsush@iser.osakau.ac.jp1

1

Introduction

A recent paper by Ghemawat et al. (2010) poses an interesting “productivity puzzle”: there is “anapparent lack of any relationship” between productivity growth rates in Catalan manufacturing andchanges in international trade position. This ﬁnding seems to be inconsistent with the standard resultsin the international trade literature.1 However, they also point out that replacing international tradewith both interregional and international trade seems to take care of this puzzle; that is, there is apositive correlation between productivity growth rates and changes in interregional and international
trade position.
The present paper is motivated by their empirical ﬁnding. We ask the following question: does
productivity improvement always enhance exports? Our theoretical model shows that the answer issometimes “no.” Speciﬁcally, we take into account intermediate retailers and multimarket linkages whichhave not been considered in the existing literature dealing with productivity and exports. We ﬁnd thata productivity improvement aﬀects the bargained wholesale prices in opposite directions in markets
that are asymmetric, causing retailers to make corresponding changes that look surprising. Our setup isnatural because many ﬁnal product makers do not directly sell to consumers but sell through retailers.
For instance, automobile manufacturers contract with car dealers, and consumer-electronics makers
supply to retailers, who then sell to consumers.
The detailed basic model is as follows. Consider two independent downstream markets, one of whichis “domestic” and the other “foreign.” To closely follow Ghemawat et al. (2010), the latter market(foreign market) is assumed to be bigger than the former one (Spanish market). In each market, there
is a monopolistic retailer that sells an identical product, which is produced by a common upstream
manufacturer. Retailers incur no additional costs except the wholesale price which is bargained between
the retailer and the manufacturer. Parallel trading is prohibitively costly. The manufacturer’s marginal
production cost is increasing in total output.
We ﬁnd a non-monotone relationship between a manufacturer’s productivity improvement and its
1 On the relationship between productivity and exports, recent theoretical papers include for instance Melitz (2003)

and Melitz and Ottaviano (2008), and for recent empirical studies, please see Bernard and Jensen (1995, 1999), Bernard
et al. (2003), and Helpman et al. (2004). Wagner (2007) provides an excellent survey on this topic.

2

volume of exports, even though there exists a monotonically positive relationship between the manufacturer’s productivity improvement and its volume of total production.2 The key lies in the twosimultaneous bargaining games for the wholesale prices. An eﬃciency improvement also raises the manufacturer’s outside options, and the outside option of the bigger market (i.e., the proﬁt from the smallermarket) is raised more, which enables it to obtain a higher wholesale price in the bigger market. As
a consequence, the retailer in this market buys less. This result can probably explain for the puzzle
found in Ghemawat et al. (2010), and it highlights the importance of taking into account the vertical
relationship between the manufacturer and retailers, as well as their strategic interactions through which
proﬁt shifting is carried out across multimarkets horizontally.3
We then extend the model to examine (i) the robustness of the results in the basic model and (ii) the
relationship between buyer power and proﬁtability. By incorpotating direct downstream competition,
we ﬁrst show that the “productivity puzzle” still holds in the extended setting, then we examine how
the buyer power caused by a downstream merger aﬀects the social surplus and the proﬁtability of themerged ﬁrm. We ﬁnd that the horizontal merger is unproﬁtable if the bargaining power of the mergedﬁrm is strong, which is starkly diﬀerent from that in the literature, for instance, Lommerud et al. (2005)
and Symeonidis (2010), who show that the horizontal merger is unproﬁtable if the bargaining power ofthe merged ﬁrm is weak. We also ﬁnd that the merger improves welfare if the bargaining power of the
upstream suppliers is weak.
In the literature, Horn and Wolinsky (1988), Dobson (1994), Zhao (2001) and Marshall and Merlo(2004) also adopt simultaneous bargaining structures. However, the main concerns of these studies
are bargaining procedures among the labor union and ﬁnal-product ﬁrms. They do not consider the
relationship between productivity and exports, and there are no intermediate traders in these studies.
Our model setting is closely related to Inderst and Wey (2007), who assume n downstream ﬁrms
located in n independent markets. The ﬁrms procure inputs from a single supplier with a convex
production function. They show that a larger buyer can get a discount, which is higher the larger his

2 The result holds even when there is more than one manufacturer.
3 Rey and Tirole (2007) and Inderst (2010) include surveys of recent progress on the study of vertical relations.

3

share of the supplier’s business. Moreover, the supplier is strictly worse oﬀ after the creation of a larger
buyer through a merger.4 However, our concerns are on merger and buyer power. Their interactions
under multimarket linkages generate results that complement Inderst and Wey (2007) in importantways.
The remainder of this paper is organized as follows. Section 2 formulates the basic model. Section 3
presents the main results and also extends the basic model. And Section 4 includes concluding remarks.
Detailed calculations are delegated to the Appendix.

2

The Model

Consider two independent downstream markets in two countries. In each market, there is a monopolistic
retailer (D in one country and F in the other country) that sells an identical product, which is produced
by a common upstream manufacturer M . In each downstream market, the demand for the product islinear:pi = ai − Qi ,where ai is a positive constant, pi is the market price, Qi is the output supplied by retailer i (i = D, F ).
To simplify the analysis, we assume that aD = 1 and aF > 1, i.e., the market of retailer F is larger thanthat of retailer D. In the basic model we assume the two markets are segmented, and parallel trading
is not allowed. These restrictions will be relaxed later.
The production technology of manufacturer M is denoted by:C(QD , QF ) = c(QD + QF )2 .
where c is a positive constant. Retailers incur no additional costs except for the wholesale price, wi(i = D, F ), which is bargained between the retailer and the manufacturer. The market structure in thismodel is summarized by Figure 1.
[Figure 1 here]
4 In a similar vein, Inderst and Valletti (2009) consider a case in which a large buyer controls several downstreamﬁrms in separate markets. This buyer’s wholesale price is then strictly smaller than the wholesale price in the case ofsymmetric buyers, which is in turn lower than the wholesale price of competing smaller ﬁrms. Eső et al. (2010) consider
a downstream industry where ﬁrms compete to buy capacity in an upstream market which allocates capacity eﬃciently.

4

The wholesale prices wD and wF are determined by Nash bargaining. We have in mind two simultaneous bargaining games: the manufacturer negotiates simultaneously but independently with the two
downstream retailers. Bargaining occurs before actual production takes place. Given wj (i, j = D, F ,j ̸= i), the bargaining problem between manufacturer M and retailer i is described by the payoﬀ pairs
BiI ≡ {[(wi Qi + wj Qj ) − C(Qi , Qj ), πi ]|wi ≥ 0} and the disagreement point (wj Qj − C(0, Qj ), 0), where
πi is the proﬁt of retailer i given by πi = (pi − wi )Qi .
The solution to the bargaining problem can be written as:
wi = arg max β log[πi ] + (1 − β) log[(wi Qi + wj Qj ) − C(Qi , Qj ) − (wj Qj − C(0, Qj ))],wi

(1)

where β ∈ [0, 1] measures the bargaining power of retailer i relative to that of manufacturer M .5
We consider a game sequence as follows. First, the manufacturer and the retailers negotiate thewholesale prices wi (i = D, F ). Second, given the wholesale prices, each retailer sets its quantity
supplied to consumers. The game is solved by backward induction.6

3

The Basic Results

Given the negotiated wholesale price wi , the maximization problem of retailer i (i = D, F ) ismax πi = (ai − Qi − wi )Qi .
Qi

The ﬁrst-order conditions lead to (Second order conditions are satisﬁed)
Qi =ai − wi(ai − wi )2, πi =.
2
4

5 This bargaining structure is often used in the literature of industrial organization.

(2)

An important property of the

Nash bargaining solution is that it can be implemented as the outcome of a dynamic non-cooperative alternating-oﬀers
bargaining game (Rubinstein, 1982 and Binmore et al., 1986).
6 We do not consider non-linear contracts. As documented in Iyer and Villas-Boas (2003, p.81), in practice, both themagnitude and the incidence of two-part tariﬀs may be insigniﬁcant. The fee structure employed here is diﬀerent fromthat in Inderst and Wey (2007), where each trading partner uses a lump-sum transfer. This diﬀerence in fee structures
leads to quite diﬀerent results.

5

Substituting (2) into (1), we can rewrite the negotiated wholesale prices as
[

wD

wF

]
(1 − wD )2= arg max β log
wD
4
[
(
)2
(
)2 ]
1 − wDaF − wF
a − wFwD (1 − wD )
−c
+
+c
,
+(1 − β) log
2
2
2
2
[
]
(aF − wF )2= arg max β log
wF
4
[
(
(
)2
)2 ]
1 − wD
1 − wDaF − wF
wF (aF − wF )
+c
.
−c
+
+(1 − β) log
2
2
2
2

(3)

(4)

These lead to the following reaction functions:
1 − β + (1 + aF (1 + β))c − (1 + β)cwF,
2+c(1 − β)aF + (1 + aF + β)c − (1 + β)cwD
wF =
.
2+c
wD =

(5)
(6)

From the above, we obtain the following lemma:
Lemma 1 (Pricing Strategy): For any c (> 0), the wholesale prices are strategic substitutes, and thedegree of strategic substitutability increases in c.
This Lemma arises because the reaction functions become steeper as the value of c increases (seeFigure 2).
[Figure 2 here]
Lemma 1 says that when wi is low, wj must be high. This stems from the two bargaining games. The
downstream retailers “indirectly” compete with each other through their independent but simultaneous
negotiations with the upstream manufacturer. This strategic substitutability does not depend on thelinearity of the demand functions. In the Appendix, we show that this relationship holds under generalconditions.
We now use this insight to examine how c aﬀects wi and Qi (i = D, F ). A higher c implies a higher
and faster-rising marginal cost. Then it becomes increasingly diﬃcult for the manufacturer to produce
a larger quantity. In turn, competition between the retailers for the manufacturer’s product becomes
more ﬁerce, shown by the increasing slopes of the reaction curves as c rises in Figure 2.
6

Next, the wholesale prices and quantities in equilibrium can be straightforwardly calculated as, for
all i, j = D, F ; i ̸= j
2(1 − β)ai + ((3 − β)ai + (1 + β)2 aj )c − ai β(2 + β)c2,(2 − cβ)(2 + (2 + β)c)(1 + β)(2ai − ((1 + β)aj − ai )c)Qi =.
2(2 − cβ)(2 + (2 + β)c)
wi =

(7)
(8)

The diﬀerence between wD and wF and its partial derivative with respect to c are respectively
wD − wF =(aF − 1)(βc − (1 − β)),(2 − cβ)∂(wD − wF )(aF − 1)β(1 + β)
=
≥ 0.
∂c
(2 − cβ)2(9)Note we have assumed aF > 1.
From Lemma 1, when c becomes higher, not only competition is intensiﬁed, but also the procurement
condition of the foreign retailer becomes better than that of the domestic retailer, due to the former’s
larger market size. This enables the foreign retailer to supply more ﬁnal output than the domestic
retailer. In fact, we have
Lemma 2 When c is suﬃciently large, the manufacturer supplies only to the foreign retailer.
In Lemma 2, c aﬀects the manufacturer’s decision as follows. In the two bargaining games, themanufacturer’s threat-point payoﬀ is higher when negotiating with retailer D than with F , since if theformer (latter) negotiation broke down, the remaining market for the manufacturer is large (small). And
as c increases, the diﬀerence between the threat-point payoﬀs becomes larger (see the last terms in (3)and (4)). The quantities supplied in the two markets (see (8)) show that the foreign retailer wins thisintensiﬁed competition, due to its larger market.
We now look into how the quantity supplied in each market changes with an increase in c. Diﬀerentiation yields(1 + aF )(1 + β)(2 + β)∂(QF + QD )
=−
< 0,
∂c
2(2 + (2 + β)c)2∂QD(1 + β)K=−< 0,∂c2(2 − cβ)2 (2 + (2 + β)c)2∂QF
β(2 + β)[4aF − (1 + β − aF )c]c − 4(1 + aF + β)=.
∂c2(2 − cβ)2 (2 + (2 + β)c)2

7

(10)
(11)
(12)

Both the domestic quantity and the world total quantity decrease as c increases,7 which is as expected. However, foreign sales may decrease or increase. We ﬁnd ∂QF /∂c > 0 if and only if c > cF ,
where


√

2[aF β(2 + β) − (1 + β) (a2F − 1)β(2 + β)]
β(2 + β)(1 + β − aF )
cF ≡

2



β(2 + β)if aF ̸= 1 + β,
(13)
if aF = 1 + β.

Note that we must also take into account the boundary condition. Using QD in (8), we ﬁnd that QD > 0
if and only if
c < c̄ ≡2.
(1 + β)aF − 1

The relation between cF and c̄ is as follows:
Lemma 3 cF < c̄ if and only if 1 < aF < 1 + β.
From the above, we obtain the following proposition (see alsoFigure 3).
Proposition 1 QF is increasing in c if cF < c < c̄ and 1 < aF < 1 + β.
[Figure 3 here]
Intuitively, Proposition 1 can be understood as follows. Consider a decrease in c, which increases thetotal quantity supplied by the manufacturer QD + QF . However, the composition of it, QD and QF , isalso aﬀected (see Figure 4). And how the composition changes depends on the strategic substitutabilityof the two wholesale prices determined in the bargaining games. By Lemma 1, an increase in QD (i.e.,a decrease in wD ) has a negative impact on QF . When c is large, the degree of strategic substitutability
is high, and the resulted sales substitution between the two retailers is also large. As a result, a decreasein c can reduce QF when c is large.
7 In the equation for ∂Q /∂c,DK ≡ 4(1 + (1 + β)aF ) − 4β(2 + β)c + β(2 + β)(aF (1 + β) − 1)c2
We can easily show that K is minimized when c = 2/((1 + β)aF − 1), and the minimized value isKmin =

4(a2F − 1)(1 + β)2> 0.
(1 + β)aF − 1

Therefore, for any c, ∂QD /∂c is negative.

8

[Figure 4 here]
This proposition can perhaps explain the stylized fact that in some markets, the volume of exports
may fall as a ﬁrm’s production eﬃciency improves (Ghemawat et al., 2010). It highlights the importanceof taking into account the vertical relationship between the manufacturer and retailers and their strategicinteractions through which proﬁt shifting is carried out across multimarkets.

Remark 1 The Choice of Retailers

It has been assumed that the manufacturer must negotiatewith both retailers. We can also consider the case in which it chooses to deal with only one or both
retailers. Suppose only one retailer is chosen, then it must be foreign because the foreign market islarger than the domestic one. In the revised setting, the game becomes a three-stage game. First,
the manufacturer determines the number of trading partners. Second, it negotiates with the retailers
independently and simultaneously. Finally, given the wholesale prices, each retailer determines its retailprice. We can straightforwardly show that for any c < c̄, the manufacturer always decides to trade withboth retailers simultaneously.

Remark 2 The Necessity of both Retailers

The existence of the retailers in two simultaneousbargaining games is essential to derive our main results. When the manufacturer directly supplies to
consumers instead, the quantity supplied in each market monotonically increases with the eﬃciency improvement (the decrease in c).8 Intuitively, without the retailers, the monopolist can increase production
at its discretion when its eﬃciency is improved. However, under two simultaneous bargaining games, aneﬃciency improvement also raises the manufacturer’s outside options. Speciﬁcally, the outside optionof the bigger domestic market is raised more, which enables it to obtain a higher wholesale price, as
shown in (9) and Lemma 2. As a consequence, the retailer in this (foreign) market buys less.

Remark 3 Parallel Trade

One might wonder what would happen if parallel trading is allowed.

We ﬁnd that it does not matter much when the market-size diﬀerence is small. That is, our main results
hold even when we incorporate parallel traders into the model.
8 The detail of this discussion is upon request.

9

4

Separate Downstream Duopolists

So far we have considered only one manufacturer. Now we extend the basic model to examine the cases
where retailers coexist in each market. There are still two retailers, but in case (i), the retailers are
independent of each other in both the domestic and foreign markets; and in case (ii), the retailers are
independent in one market but merged in the other market.
The ﬁrst case enables us to check the robustness of Proposition 1, and we ﬁnd that the additional
retailers and manufacturers do not aﬀect the result that the volume of exports may fall as ﬁrms’
production eﬃciency is improved (Ghemawat et al., 2010).
With the second case, we investigate the eﬀects of retail merger and how the merged large retailer
inﬂuences market performance. The problem caused by large retailers is an important topic in thecontext of industrial organization (see for instance, Inderst and Shaﬀer, 2007).9 Some large retailers (e.g,Tesco in the UK and Metro in Germany) can exert countervailing powers against upstream suppliers.
Such powers on the one hand reduce wholesale prices which may increase welfare, but on the other hand
they exert market power on ﬁnal consumers which tends to lower welfare.
For our purpose, the simplest setup is as follows. In each market, a retailer procures from manufacturer A and the other from manufacturer B. Neither retailer procures from both manufacturers, nor doboth retailers procure from the same manufacturer. For details, please see Figure 5.
[Figure 5 here]
The assumptions concerning the manufacturers are similar to those in the previous section. With
two manufacturers j ∈ {A, B}, their cost functions are also diﬀerent as follows,
Cj (Qi,j , Q−i,j ) = cj (Qi,j + Q−i,j )2 ,where Qi,j is the output of ‘retailer i, j’. We denote ‘retailer i, j’ a retailer in market i ∈ {D, F } who
procures from manufacturer j ∈ {A, B}. Assuming the bargaining power of each retailer in market i to

9 Inderst and Shaﬀer (2007) analyze the impact of retail mergers on product variety.

They show that, following amerger, a retailer may want to enhance its buyer power by committing to a ‘single-sourcing’ purchasing strategy.

10

be given as βi ∈ (0, 1) (i ∈ {D, F }), we can write the solution to the bargaining problem as:
wi,j = arg max βi log[πi,j ]+wi,j(1 − βi ) log[(wi,j Qi,j + w−i,j Q−i,j ) − Cj (Qi,j , Q−i,j ) − (w−i,j Q−i,j − Cj (0, Q−i,j ))], (14)where πi,j is the proﬁt, wi,j is the wholesale price, and Qi,j is the output, all related to retailer i, j.
In market i ∈ {D, F }, the inverse demand is given as:pi,j = ai − Qi,j − γQi,−j ,where ai is a positive constant, pi,j is the market price for product j, and γ ∈ (0, 1) is the degree of
product diﬀerentiation between the two products.

4.1

Downstream duopoly

We ﬁrst consider the case of the retailers being independent of each other in both the domestic andforeign markets. Given the negotiated wholesale price wi,j , the maximization problem of retailer i, j
(i = D, F , j = A, B) is
max πi = (ai − Qi,j − γQi,−j − wi,j )Qi,j .
Qi,jThe ﬁrst-order conditions lead to (second order conditions are satisﬁed)
Qi,j =(2 − γ)ai − 2wi,j + γwi,−j, πi,j = (Qi,j )2 .
4 − γ2(15)

Substituting (15) into (14), we can rewrite the negotiated wholesale prices as[]((2 − γ)ai − 2wi,j + γwi,−j )2
wi,j = arg max βi log
wi,j(4 − γ 2 )2[
wi,j ((2 − γ)ai − 2wi,j + γwi,−j )+(1 − βi ) log
4 − γ2()2(2 − γ)a−i − 2w−i,j + γw−i,−j(2 − γ)ai − 2wi,j + γwi,−j+−c
4 − γ24 − γ2()2 ](2 − γ)a−i − 2w−i,j + γw−i,−j+c.(16)
4 − γ2
These yield the following reaction functions (i ∈ {D, F }, j ∈ {A, B}):wi,j =

1× [(2 − γ)(ai (1 − βi )(4 − γ 2 ) + 4(ai + (1 + βi )a−i )cj )4(4 − γ 2 + 2cj )− 4cj (1 + βi )(2w−i,j − γw−i,−j ) + γ((1 − βi )(4 − γ 2 ) + 4cj )wi,−j ].
11Notice that there are three other wholesale prices on the RHS: wi,−j , w−i,j and w−i,−j , which
represent respectively the wholesale prices for retailers i, −j, −i, j, and −i, −j. Retailer i, j directlycompetes with retailer i, −j in market i, indirectly competes with retailer −i, j through manufacturerj, and is also indirectly related to retailer −i, −j through manufacturer j. Let us now examine theirrespective relationships with wi,j , which yields interesting comparisons.
Lemma 4 In the reaction function of wi,j , the coeﬃcients of wi,−j , w−i,j , and w−i,−j are respectively
positive, negative, and positive.
The coeﬃcient of wi,−j is positive, so that the wholesale prices are complements across retailers inmarket i (i = D, F ). That is, in each market, retailer i, j’s wholesale price increases in retailer i, −j’swholesale price. These can be understood as follows. A higher wholesale price for retailer i, −j increases
its cost, giving a competitive edge to its rival retailer i, j, enabling the latter to supply more in the retailmarket. And anticipating retailer i, j’s action, manufacturer j bargains for a higher wholesale price,
resulting in the positive coeﬃcient of wi,−j . Note that this eﬀect diminishes as the retailer’s bargaining
power βi increases, because wholesale prices and proﬁts will be depressed by a higher βi , which in turn
reduces the intensity of the strategic interaction between the upstream suppliers (this possibility is alsomentioned in Naylor (2002)).
In contrast, the coeﬃcient of w−i,j is negative. As explained in the previous section with one
manufacturer, retailers i, j and −i, j indirectly compete with each other through manufacturer j. If
retailer −i, j procures more, the manufacturer must supply less to retailer i, j because the marginal
cost of the manufacturer rises by the increase in Q−i,j , which forces up the wholesale price wi,j . Notethat this eﬀect becomes stronger as the values of βi and c increase. A high retailer bargaining power βidepresses the wholesale price, enabling the retailer to procure more. The increase in Qi,j through theincrease in βi enhances the marginal impact of the increase in Q−i,j (the decrease in w−i,j ). A higher ctightens the production ‘capacity’ of the manufacturer. Therefore, the absolute value of the coeﬃcientof w−i,j increases as the values of βi and c increase.
Further, the coeﬃcient of w−i,−j is positive. The logic can be explained analogously as before. A
higher w−i,−j increases the quantity supplied by manufacturer j in market −i, which tends to reduce12its supply to retailer i, j, again driving up the wholesale price wi,j .
Using the reaction functions wi,j = wi,j (wi,−j , w−i,j , w−i,−j ), we also obtain the following lemma:∗, the quantity Qi,j , each retailer’s proﬁt, and theLemma 5 At the equilibrium, the wholesale price wi,jconsumer surplus in each market are respectively(a−i − ai )[(4 − γ 2 )(4(1 + βi ) + (1 + β−i )(1 − βi )γ) − 8β−i (1 + βi )c−j − 4γβi (1 + β−i )cj ]Ha
2(a−i + ai )[(4 − γ )(4(1 + βi ) + (1 + β−i )(1 − βi )γ),−Hb4(a−i + ai )[2(2 + β−i )(1 + βi )c−j + γ(2 + βi )(1 + β−i )cj ]
,
−
Hb
∗
∗
(2 − γ)ai − 2wi,j+ γwi,−jQi,j =, πi,j = (Qi,j )2 , CSi = (1 + γ)(Qi,j )2 .
(17)
4 − γ2
∗
wi,j
=ai +

where Ha ≡ 16(2 − βi cj )(2 − β−i c−j ) − 4(5 − βi − β−i + βi β−i − βi (1 − β−i )cj − β−i (1 − βi )c−j )γ 2 + (1 −βi )(1 − β−i )γ 4 and Hb ≡ 16(2 + (2 + βi )cj )(2 + (2 + β−i )c−j ) − 4(5 − βi − β−i + βi β−i + (2 + βi )(1 −β−i )cj + (2 + β−i )(1 − βi )c−j )γ 2 + (1 − βi )(1 − β−i )γ 4 .
Using these results, we can check whether or not a non-monotonic relationship exists between Q and
cj . To simplify the analysis, we now assume that aD = 1, βD = βF = β, and cA = cB = c. Then
∂QF,A
8(1 + β)[(1 + β + aF )(2 + γ)2 (4 − γ + βγ)2 − 8aF β(2 + β)(2 + γ)(4 − γ + βγ)c]=−∂c((2 + γ)(4 − γ + βγ) − 4βc)2 ((2 + γ)(4 − γ + βγ) + 4(2 + β)c)28(1 + β)[16β(2 + β)(1 + β − aF )c2 ]−.
((2 + γ)(4 − γ + βγ) − 4βc)2 ((2 + γ)(4 − γ + βγ) + 4(2 + β)c)2Since Qi,j > 0 for any i, j, the upper bound of c is given by
QD,j > 0 ⇔ c <(2 + γ)(4 − γ + βγ)≡ c̄d .
4((1 + β)aF − 1)

First note that ∂QF,A /∂c is negative when c = 0. Next, we check whether there exists some range of csuch that ∂QF,A /∂c > 0. Substituting c = c̄d into ∂QF,A /∂c to give¯
∂QF,A ¯¯
8(1 + β − aF )(aF − 1 + βaF )= 2.
¯∂c c=c̄d(aF − 1)(1 + β)(2 + γ)2 (4 − γ + βγ)
From the calculation we have the following result.
Proposition 2 Suppose that aD = 1, βD = βF = β, and cA = cB = c. Then QF,j is increasing in c if
cF < c < c̄d and 1 < aF < 1 + β.
13Proposition 2 shows that the result in Proposition 1 is robust even when the number of manufacturers
and retailers increase; that is, output (exports) can still fall when a ﬁrm’s eﬃciency increases.
Note that for algebraic simplicity, we have assumed cA = cB = c. It is easy to ﬁnd an example thatan increase in heterogeneous ci (i = A, B) can increase the quantities supplied in market F . Also, if
aD = aF = 1, calculations give the following properties of wi,j and Qi,j :
∂wi,j
∂wi,j
∂Qi,j
∂Qi,j
∂Qi,j
∂wi,j
> 0,
< 0,
> 0,
< 0,
> 0,
< 0.
∂c
∂βi
∂β−i
∂c
∂βi
∂β−i

4.2

Downstream merger

We are particularly interested in the case when the retailers in one market (say F ) merge. Such acase is related to several prior researches (Naylor (2002), Lommerud et al. (2005), Matsushima (2006),
Symeonidis (2010)), who explicitly or implicitly show that a merger between downstream duopolists
does not always improve their proﬁts when upstream suppliers have relatively strong bargaining powerover downstream retailers, because the downstream merger eliminates the competition between the
upstream suppliers, making them stronger to counter and oﬀset the downstream merger.
In this subsection, we show that a downstream merger which leads to a monopoly does not improve
the total proﬁt of the merged entities if the bargaining power of the merged ﬁrm is strong, which is quite
diﬀerent from those in the literature.
For simplicity, we now assume aD = aF = 1, i.e., the market size is the same for the two markets D
and F . Then the objective function of the merged ﬁrm is
πF,M ≡ (1 − QF,A − γQF,B − wF,A )QF,A + (1 − QF,B − γQF,A − wF,B )QF,B .

(18)

Proﬁt maximization gives the ﬁrst-order conditions as (Second order conditions are satisﬁed)
QF,j =(1 − wF,j )QF,j(1 − γ) − 2wF,j + γwF,−j, πF,j =.
2(1 − γ 2 )2(19)Similarly as before, we can obtain the ﬁrst-order conditions for the negotiation between retailer i, j

14and manufacturer i as
wD,j =

14(1 − γ 2 )(4 − γ 2 + 2c)

× [(2 − γ)(1 − γ)((1 − βD )(4 − γ 2 )(1 + γ) + 2(4 + 2βD + 3γ + βF γ)c)

− 2c(1 + βD )(4 − γ 2 )(wF,j − γwF,−j ) + (1 − γ 2 )γ((1 − βD )(4 − γ 2 ) + 4c)wD,−j ],wF,j =

1× [(2 − γ)(1 − γ)((1 − βF )(1 − γ 2 )(2 + γ) + (4 + 2βF + 3γ + 2βD γ)c)(4 − γ 2 )(2(1 − γ 2 ) + c)
− 2c(1 + βF )(1 − γ 2 )(2wD,j − γwD,−j ) + (4 − γ 2 )γ((1 − βF )(1 − γ 2 ) + c)wF,−j ].

The basic properties of the reaction functions are similar to that in the non-merged case. That is,
Lemma 6 Suppose that aD = aF = 1 and cD = cF = c. When the downstream retailers in market
F horizontally merge, in the reaction functions of wD,j and wF,j , the coeﬃcients of wi,−j , w−i,j , and
w−i,−j are respectively positive, negative, and positive.
Using the reaction functions wi,j = wi,j (wi,−j , w−i,j , w−i,−j ), we further obtain the following lemma:
Lemma 7 Suppose that aD = aF = 1 and cD = cF = c. When the downstream retailers in market F
horizontally merge, the equilibrium wholesale price wi,j , the equilibrium quantity Qi,j , the proﬁt of eachretailer, and the consumer surplus in each market are respectively(1 − βD )(4 − γ 2 )(1 + γ)(2 − (1 − βF )γ)HMc(4(4 + βF (1 + βD )) + 2(3 + βD )(1 + βF )γ − (5 − βD − 4βF )γ 2 ) − 4c2 (βD + βF + βD βF )+,HM
(1 − βF )(1 − γ 2 )(2 + γ)(4 − (1 − βD )γ)wF,j =HMc(4(4 + βD (1 + βF )) + 2(3 + 2βF )(1 + βD )γ − (5 − βD − 4βF )γ 2 ) − 4c2 (βD + βF + βD βF )+,HM2(1 + βD )((1 + γ)(2 − γ + βF γ) − βF c)
QD,j =, πD,j = (QD,j )2 , CSD = (1 + γ)(QD,j )2 ,(20)
HM
(1 + βF )((2 + γ)(4 − γ + βD γ) − 4βD c), πF,M = 2(1 + γ)(QF,j )2 , CSF = (1 + γ)(QF,j )2 .
QF,j =
2HM(21)
wD,j =

where HM ≡ (1 + γ)(2 + γ)(2 − (1 − βF )γ)(4 − (1 − βD )γ) + c(16 + 2(3 + βD + 2βF ) − (5 − βD − 4βF )γ 2 ) −4c2 (βD + βF + βD βF ).
Also, calculations give the following properties of wi,j and Qi,j :
∂wi,j
∂wi,j
∂Qi,j
∂Qi,j
< 0,
> 0,
> 0,
< 0, for all exogenous parameters.
∂βi
∂β−i
∂βi
∂β−i
15

We now discuss the relation between c and wi,j (Qi,j ). To simplify the exposition, we again assume
βD = βF = β. For a large c, we have the following lemma:
Lemma 8 Suppoase that βD = βF = β. The wholesale price wi,j can be increasing in c.
Note that the details for this lemma can be shown as,
when β ≥3γ − 2,3γ + 2∂wF,j∂wD,j> 0 for all exogenous parameters and<0∂c∂c2(2 + β)(1 + γ)(2 − γ + βγ)
if and only if c >2β(2 + β)√γ(1 + γ)(1 + β)(2 + β)(2 − γ + βγ)(2 + 2β − 3γ + 3βγ);−
2β(2 + β)

3γ − 2
when β ≤,3γ + 2∂wD,j∂wF,j> 0 for all exogenous parameters and
<0
∂c
∂c

(22)

(2 + β)(2 + γ)(4 − γ + βγ)
if and only if c >4β(2 + β)√−γ(2 + γ)(1 + β)(2 + β)(4 − γ + βγ)(2 + 2β − 3γ + 3βγ)−.
4β(2 + β)
Therefore, if the bargaining power is strong and the cost parameter c is large enough, an increase inc can increase the quantities supplied by the retailers in market D, which again conﬁrms the robustnessof Proposition 1.

4.3

The eﬀect of downstream merger

The merger is unproﬁtable if πF,M < 2πi,j , where πi,j and πF,M are given in (17) and (21). To simplify
the exposition, we assume that βD = βF = β. Although calculation becomes more complicated, we candepict the diﬀerence between πF,M and 2πi,j , ∆πF,j ≡ πF,M − 2πi,j (see Figure 6). The ﬁgure leads tothe following proposition:
Proposition 3 Suppose that βD = βF = β. An increase in either β or c decreases ∆πF,j .
If the wholesale prices did not change, the merger would increase the proﬁts of the merged retailer
since downstream competition is removed. However, a counter-intuitive result arises here: the merger
can increase the wholesale prices if the value of c is large and the bargaining power of the retailer ishigh (see Lemma 8). The logic is as follows. The merger allows the merged retailers to change retail16quantities ﬂexibly, that is reﬂected in Qi,j in (15) and QF,j in (19), the latter of which is more sensitive
to wF,j than the former. This in turn aﬀects the reaction function of wD,j (the wholesale price for the
non-merged retailers). With the merger in market i, the reaction function of wDj becomes more elastic
with respect to wF,j .10 This arises since an increase in wF,j decreases QF,j signiﬁcantly, which relaxes
the production ‘capacity’ of manufacturer j. Moreover, Qi,j monotonically increases in βi . Therefore,
if the bargaining power of retailers in market i is strong, the merger in market i allows retailers inmarket −i to procure more inputs from the manufacturers. It in turn increases the marginal costs ofthe manufacturers, resulting in worse procurement conditions for the merged retailer and lowering its
proﬁts.
[Figure 6 here]
The merger also aﬀects the threat points of the manufacturers. The decrease in the quantities
supplied to the merged ﬁrm lowers the proﬁts of the manufacturers, which also raises the non-merged
retailer’s bargaining position when negotiating with the manufacturer, lowering wD,j . This then increases wF,j , since the wholesale prices are strategic substitutes as shown earlier.
Finally, we brieﬂy discuss the welfare implication of a downstream merger, using Figure 6. When thebargaining power of the retailers is weak (small β), the downstream merger improves welfare, because
the merger generates an outside value for the merged ﬁrm, which improves its procurement conditions.
However, when β is large, this positive eﬀect is weak because the pre-merged downstream ﬁrms havealready exerted their strong bargaining power. As in standard oligopoly theory, the elimination of thedownstream competition between the merged entities worsens welfare.

5

Conclusion

This paper analyzed the relationship between a ﬁrm’s productivity improvement and its volume ofexports. Speciﬁcally, we simultaneously take into account intermediate traders (i.e., vertically) and
10 The diﬀerence between the coeﬃcients of w
F,j in the post-merger and the pre-merger is−

2c(1 + βD )(4 − γ 2 )−4(1 − γ 2 )(4 − γ 2 + 2c)„«2c(1 + βD )3c(1 + βD )γ 2
−
< 0.
=−
2
4 − γ + 2c2(1 − γ 2 )(4 − γ 2 + 2c)

That is, the coeﬃcients of wF,j in the post-merger is steeper that that in the pre-merger.

17

multimarket linkages (i.e., horizontally) which have not been considered in the existing literature. We
found that the manufacturer will adjust exports when facing productivity changes, by making corresponding changes in other related markets, to take advantage of its improved outside options. Our
result might be a simple explanation to the interesting “productivity puzzle” posed by Ghemawat et al.
(2010): there is “an apparent lack of any relationship” between productivity growth rates in Catalan
manufacturing and changes in international trade position. It also highlights the importance of taking
into account the vertical relationship between the manufacturer and retailers, as well as their strategicinteractions through which proﬁt shifting is carried out across multimarkets horizontally.
We also extended the model to examine the relationship between buyer power, proﬁtability andsocial surplus. We ﬁnd that the horizontal merger is unproﬁtable if the bargaining power of the mergedﬁrm is strong, but such a merger may improve welfare. Our result is quite diﬀerent from those in theliterature (Lommerud et al. (2005) and Symeonidis (2010)) which show that the horizontal merger isunproﬁtable if the bargaining power of the upstream supplier is strong.

18Appendix
Here we show that Lemma 1 holds under general cost and demand conditions. We add the following′
notation: πD (wD ) is the proﬁt of the domestic retailer with πD(wD ) < 0, QD (wD ) is the demand in thedomestic market (Q′D (wD ) < 0), QF (wF ) is the demand in the foreign market (Q′F (wF ) < 0), and c(·)
is the cost function of the manufacturer (c′ > 0, c′′ > 0).
The bargaining problem between M and D is
M ax β log[πD (wD )]+ (1 − β) log[wD QD (wD ) + wF QF (wF )− c(QD (wD ) + QF (wF )) − {wF QF (wF ) − c(QF (wF ))}]
⇒ β log[πD (wD )] + (1 − β) log[wD QD (wD ) − c(QD (wD ) + QF (wF )) + c(QF (wF ))],
which leads to the ﬁrst-order condition
β

′
πD
(wD )
wD Q′D (wD ) + QD (wD ) − c′ (QD (wD ) + QF (wF ))Q′D (wD )+ (1 − β)= 0.
πD (wD )wD QD (wD ) − c(QD (wD ) + QF (wF )) + c(QF (wF ))The numerator of the second fraction, wD Q′D (wD ) + QD (wD ) − c′ (QD (wD ) + QF (wF ))Q′D (wD ), must′
be positive because the denominators of the fractions are positive and πD(wD ) < 0.

The total diﬀerential of the ﬁrst-order condition is[(S.O.C.)dwD + (1 − β)−c′′ (QD + QF )Q′F Q′D [wD QD − c(QD + QF ) + c(QF )][wD QD − c(QD + QF ) + c(QF )]2](c′ (QD + QF ) − c′ (QF ))Q′F [wD Q′D + QD − c′ (QD + QF )Q′D ]+dwF = 0.
[wD QD − c(QD + QF ) + c(QF )]2

Both the ﬁrst and second fractions are negative (c′′ (QD + QF ) > 0, Q′D < 0, Q′F < 0, wD QD − c(QD +QF ) + c(QF ) > 0), and (c′ (QD + QF ) − c′ (QF ) > 0, Q′F < 0, wD Q′D + QD − c′ (QD + QF )Q′D > 0).
Thus the coeﬃcient of dwF is negative. Since the second order condition is also negative, we have
dwD /dwF < 0, which means that the prices charged to retailers are strategic substitutes under generalconditions.

19

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22

wD

M

wF

D

F

Consumers

Consumers

pD = 1 − QD

pF = aF − QF

Figure 1: The market structure

23

wF

wF

wF (wD )wD (wF )wD (wF )wD

c is large.

c is small.

Figure 2: The reaction functions

24

wF (wD )
wD

c
3
2.75
2.5
2.25
2
1.75
1.5
1.25

c
4
3.5
3


c

2.5
2

cF

1.5
1.2

1.4

1.6

1.8

2

a


c

cF
1.2

(β = 1/2)

1.4

1.6

(β = 3/4)

Figure 3: The condition that ∂QF /∂c > 0.

25

1.8

2

a

Qi ,ΠM
0.5

Qi ,ΠM
0.5

0.4
0.3

0.4

QF

QF

0.3
0.2
0.1

0.2

QD

0.1

ΠM
0.5

1

1.5

2

c

QD
ΠM
0.25 0.5 0.75

(α = 11/10)

1(α = 5/4)

Figure 4: The relation between c and QD , QF , and πM .

26

1.25 1.5

c

MA
wD,A

RD,A

MB
wF,A wD,B

RD,B

wF,B

RF,A

Consumers

RF,B

Consumers

pD,j = aD − QD,j − γQD,−jpF,j = aF − QF,j − γQF,−jFigure 5: The market structure

27

0.03
DΠF, j0.004

DW F
0.02

2.0

0.002

0.01

0.000

0.00
0.0

1.5

c

2.0
1.5
1.0

0.6

Β

0.5

0.5
Β

0.8
1.0

1.0

1.0

∆πF,j

∆SWF,jFigure 6: The values of ∆πF,j and ∆SWF,j (γ = 1/2).

28

0.0c