Abstract

Introduction
11
A recent paper by Ghemawat et al. (2010) poses an interesting "productivity puzzle": there is "an apparent lack of any relationship" between productivity growth rates in Catalan manufacturing and changes in international trade position. This finding seems to be inconsistent with the standard results in the international trade literature. 1 However, they also point out that replacing international trade with both interregional and international trade seems to take care of this puzzle; that is, there is a positive 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 is sometimes "no." Specifically, we take into account intermediate retailers and multimarket linkages which have not been considered in the existing literature dealing with productivity and exports. We find that a 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 is natural 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 which is "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 volume of exports, even though there exists a monotonically positive relationship between the man- ufacturer's productivity improvement and its volume of total production. 2  The key lies in the two simultaneous bargaining games for the wholesale prices. An efficiency improvement also raises the man- ufacturer's outside options, and the outside option of the bigger market (i.e., the profit from the smaller market) is raised more, which enables it t to o 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 the merged firm. We find that the horizontal merger is unprofitable if the bargaining power of the merged firm 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 of the 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 share of the supplier's business. Moreover, the supplier is s 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 important ways. 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.
The Model
2
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 is linear:
where a i is a positive constant, p i is the market price, Q i 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 than that 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:
where c is a positive constant.  Retailers incur no additional costs except for the wholesale price, w i ( i = D, F ), which is bargained between the retailer and the manufacturer. The market structure in this model is summarized by Figure 1. The wholesale prices w D and w F are determined by Nash bargaining. We have in mind two simul- taneous bargaining games: the manufacturer negotiates simultaneously but independently with the two downstream retailers. Bargaining occurs before actual production takes place. Given w j (i,j = D,F, j̸= i), the bargaining problem between manufacturer M and retailer i is described by the payoff pairs B} = {[(w,Q; + w;Q;) − C(Q;,Q;)), π;||w; ≥ 0} and the disagreement point (w;Q; − C(0,Q;)), 0), where π; is the profit of retailer i given by π; = (p; − w;)Q;. The solution to the bargaining problem can be written as:
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 the wholesale prices w i ( i = D,F ).  Second, given the wholesale prices, each retailer sets its quantity supplied to consumers. The game is s solved by backward induction. 6
3
The Basic Results
Given the negotiated wholesale price w i , the maximization problem of retailer i ( i = D , F ) is
The first-order conditions lead to (Second order conditions are satisfied)
Substituting (2) into (1), we can rewrite the negotiated wholesale prices as
These lead to the following reaction functions:
From the above, we obtain the following lemma: Lemma 1 (Pricing Strategy): For any c (> 0), the wholesale prices are strategic substitutes, and the degree of strategic substitutability increases in c. This Lemma arises because the reaction functions become steeper as the value of c increases (see Figure 2). [Figure 2 here] Lemma 1 says that when w i is low, w j 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 the linearity of the demand functions. In the Appendix, we show that this relationship holds under general conditions. We now use this insight to examine how c affects w i and Q i (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. Next, the wholesale prices and quantities in equilibrium can be straightforwardly calculated as, for all i, j = D, F; i ≠ j
The difference between wD and wF and its partial derivative with respect to c are respectively
Note we have assumed a F > 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, the manufacturer's threat-point payoff is higher when negotiating with retailer D than with F , since if the former (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 this intensified competition, due to its larger market. We now look into how the quantity supplied in each market changes with an increase in c. Differen- tiation yields
Both the domestic quantity and the world total quantity decrease as c increases, 7 which is as ex- pected. However, foreign sales may decrease or increase. We find ∂Q F / ∂c > 0 if and only if c > c F , where
if and only if
The relation between c F and c is as follows: Lemma 3 c F < c if and only if 1 < a F < 1 + β . From the above, we obtain the following proposition (see alsoFigure 3). Proposition 1 Q F is increasing in c if c F < c < c and 1 < a F < 1 + β .
Intuitively, Proposition 1 can be understood as follows. Consider a decrease in c, which increases the total quantity supplied by the manufacturer Q D + Q F . However, the composition of it, Q D and Q F , is also affected (see Figure 4). And how the composition changes depends on the strategic substitutability of the two wholesale prices determined in the bargaining games. By Lemma 1, an increase in Q D (i.e., a decrease in wD ) has a negative impact on Q F . 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 decrease in c can reduce Q F when c is large.
[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 importance of taking into account the vertical relationship between the manufacturer and retailers and their strategic interactions through which profit shifting is carried out across multimarkets. Remark 1  The Choice of Retailers It has been assumed that the manufacturer must negotiate with 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 is larger 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 retail price. We can straightforwardly show that for any c < c, the manufacturer always decides to trade with both retailers simultaneously. Remark 2  The Necessity of both Retailers The existence of the retailers in two simultaneous bargaining 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 im- provement (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, an efficiency improvement also raises the manufacturer's outside options. Specifically, the outside option of 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. One might wonder what would happen if parallel trading is allowed. Remark 3   Parallel Trade 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.
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 the context 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 manufac- turer A and the other from manufacturer B . Neither retailer procures from both manufacturers, nor do both 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,
where Q i,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 be given as β i ∈ (0,1) ( i ∈ { D,F} ), we can write the solution to the bargaining problem as:
In market i ∈ { D, F } , the inverse demand is given as:
where a i is a positive constant, p i 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 and foreign markets. Given the negotiated wholesale price w i ,j , the maximization problem of retailer i,j (i = D, F, j = A, B) is
The first-order conditions lead to (second order conditions are satisfied)
Substituting (15) into (14), we can rewrite the negotiated wholesale prices as
Notice that there are three other wholesale prices on the RHS: w i,-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 directly competes with retailer i , − j in market i , indirectly competes with retailer − i , j through manufacturer j , and is also indirectly related to retailer − i , − j through manufacturer j . Let us now examine their respective relationships with w i j , which yields interesting comparisons. Lemma 4  In the reaction function of w i,j, the coefficients of w i,-j, w-i,j, and w-i,-j are respectively positive, negative, and positive. The coefficient of w i, – j is positive, so that the wholesale prices are complements across retailers in market i (i = D, F ). That is, in each market, retailer i , j 's wholesale price increases in retailer i , − j 's wholesale 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 retail market. And anticipating retailer i,j's action, manufacturer j bargains for a higher wholesale price, resulting in the positive coefficient of v j , _ j . Note that this effect diminishes as the retailer's bargaining power β i increases, because wholesale prices and profits will be depressed by a higher β i , which in turn reduces the intensity of the strategic interaction between the upstream suppliers (this possibility is also mentioned 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 w i.j . Note that this effect becomes stronger as the values of β i and c increase. A high retailer bargaining power β i depresses the wholesale price, enabling the retailer to procure more. The increase in Qi,j through the increase in β i enhances the marginal impact of the increase in Q – i,j ( the decrease in w – i,j ). A higher c tightens the production 'capacity' of the manufacturer. Therefore, the absolute value of the coefficient of w_i,j increases as the values of β i and c increase. Further, the coefficient of v − 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 reduce its supply to retailer i, j , again driving up the wholesale price w i,j . Using the reaction functions w ɨ j = w ɨ j ( w ɨ –j, w –i,j, w –i,–j), we also obtain the following lemma: Lemma 5  At the equilibrium, the wholesale price wij, the quantity Qij, each retailer's profit, and the consumer surplus in each market are respectively
β i )(1 − β − i )γ 4  and H b ≡ 16(2 + (2 + β i )c j )(2 + (2 + β − i )c j ) − 4(5 − β i − β − i + β i β − i + (2 + β i )(1 − β−i)cj + (2 + β−i)(1 − βi)c−i)γ2 + (1 − βi)(1 − β−i)γ4. Using these results, we can check whether or not a non-monotonic relationship exists between Q and c j . To simplify the analysis, we now assume that a D = 1, β D = β F = β , and c A = c B = c . Then
Since Q i,j > 0 for any i,j , the upper bound of c is given by
First note that ∂Q F,A /∂c is negative when c = 0. Next, we check whether there exists some range of c such that ∂Q F,A/∂c > 0. Substituting c = c d into ∂Q F,A/∂c to give
From the calculation we have the following result. Proposition 2  Suppose that a D = 1, βD = βF = β, and c A = c B = c.  Then Q F.j is increasing in c if CF < c < ca and 1 < a r < 1 + β . 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 c A = c B = c . It is easy to find an example that an increase in heterogeneous c i ( i = A, B ) can increase the quantities supplied in market F . Also, if aD = a F = 1, calculations give the following properties of w i,j and Q i,j:

4.2
Downstream merger
We are particularly interested in the case when the retailers in one market (say F ) merge. Such a case 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 power over 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 a D = a F = 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
Profit maximization gives the first-order conditions as (Second order conditions are satisfied)
Similarly as before, we can obtain the first-order conditions for the negotiation between retailer i,j and manufacturer i as
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 w i , j = w i , j ( w i , − j , w − i , j , w − j , − 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 w i.j , the equilibrium quantity Q i.j , the profit of each
4c² ( βd + βf + βdβf). Also, calculations give the following properties of w i,j and Q j,j :
15 We now discuss the relation between c and w i j ( Q i 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.
Therefore, if the bargaining power is strong and the cost parameter c is large enough, an increase in c can increase the quantities supplied by the retailers in market D , which again confirms the robustness of Proposition 1.
4.3
The effect of downstream merger
The merger is unprofitable if π F,M < 2π i,j , where π i,j and π F,M are given in (17) and (21). To simplify the exposition, we assume that β D = β F = β . Although calculation becomes more complicated, we can depict the difference between π f,M and 2π i,j, Δπ f,j ≡ π f,M − 2π i,j (see Figure 6). The figure leads to the following proposition: Proposition 3 Suppose that βD = βF = β. An increase in either β  or c decreases ΔπF.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 is high (see Lemma 8). The logic is as follows. The merger allows the merged retailers to change retail quantities flexibly, that is reflected in Q j j in (15) and Q F j in (19), the latter of which is more sensitive to w F j than the former. This in turn affects the reaction function of w D j (the wholesale price for the non-merged retailers). With the merger in market i , the reaction function of wD, becomes more elastic with respect to w F j . 10 This arises since an increase in w F j decreases Q F j significantly, which relaxes the production 'capacity' of manufacturer j . Moreover, Q i,j monotonically increases in β i . Therefore, if the bargaining power of retailers in market i is strong , the merger in market i allows retailers in market — i to procure more inputs from the manufacturers. It in turn increases the marginal costs of the 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 in- creases w F.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 the bargaining 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 have already exerted their strong bargaining power. As in standard oligopoly theory, the elimination of the downstream competition between the merged entities worsens welfare.
5
Conclusion
This paper analyzed the relationship between a firm's productivity improvement and its volume of exports. 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 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 corre- sponding anges 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 strategic interactions through which profit shifting is carried out across multimarkets horizontally. We also extended the model to examine the relationship between buyer power, profitability and social surplus. We find that the horizontal merger is unprofitable if the bargaining power of the merged firm is strong, but such a merger may improve welfare. Our result is quite different from those in the literature (Lommerud et al. (2005) and Symeonidis (2010)) which show that the horizontal merger is unprofitable if the bargaining power of the upstream supplier is strong.
Appendix
Here we show that Lemma 1 holds under general cost and demand conditions. We add the following notation: π D ( WD ) is the profit of the domestic retailer with π D ( WD ) < 0, Q D ( WD ) is the demand in the domestic market (Q ′ D ( WD ) < 0), Q F ( 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 Max β log[π D ( WD )]
which leads to the first-order condition
The numerator of the second fraction, wdQ'D(wd) + Qd(wd) − c'(Qd(wd) + Qf(wf))Q'D(wd), must be positive because the denominators of the fractions are positive and π ρ ( wD ) < 0. The total differential of the first-order condition is
Both the first and second fractions are negative ( c ′′ ( Q D + Q F ) > 0, Q D < 0, Q F < 0, W D Q D − c ( Q D + QF) + c(QF) > 0), and (c'(QD + QF) − c'(QF) > 0, QF < 0, WDQ'D + QD − c'(QD + QF)Q'D > 0). Thus the coefficient of dw F 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 general conditions.