International Trade in Bilateral Oligopoly

Transcription

1 International Trade in Bilateral Oligopoly (Work in progress) Alexander Konovalov Johan Stennek May 23, 2014 Abstract In many intermediate goods markets, prices are negotiated between buyers and sellers on a bilateral basis. We provide a framework for studying such markets and analyze the process of endogenous link formation and the subsequent bilateral bargaining. We show that there is a unique stochastically stable network in bilateral duopoly and identify conditions under which such a network is characterized by predominant intra-industry trade. The paper provides a rational for intra-industry trade in intermediate goods markets, based on surplus-sharing in combination with efficiency differentials between firms. Key Words: intra-industry trade; bilateral oligopoly; firm heterogeneity; link formation; pairwise stability; stochastic stability JEL classification: F12, L14, C78 We are grateful for comments from Mary Amity, Henrik Horn and Peter Neary as well as from participants in seminars at NOITS 2003 in Bergen, ETSG 2008 in Warsaw, EARIE 2011 in Stockholm, Swedish National Conference in Economics 2011 in Uppsala, ESEM 2013 in Gothenburg, Stockholm School of Economics, University of Gothenburg, University of Antwerp, Lund University and High school of Economics in St. Petersburg. University of Gothenburg, alexander.konovalov@economics.gu.se University of Gothenburg, johan.stennek@economics.gu.se 1

2 1 Introduction All the three major models of trade, perfect competition, monopolistic competition and oligopolistic competition, fail to formalize some common characteristics of intermediate goods markets that both the buyers and the sellers are concentrated, and that the contracts are negotiated on a bilateral basis. Little is thus known about the determinants of international trade in intermediate goods. This paper provides a framework for analyzing such markets, and shows how surplus-sharing among heterogeneous firms with bilateral market power may explain an important stylized fact of international trade, namely intra-industry trade in intermediate goods. Intermediate goods markets Trade in intermediate goods and services is a large and growing feature of the international economy (Johnson and Noguera, 2012). This process is often referred to as fragmentation, offshoring or task trade. And intermediate goods trade is increasingly characterized by intra-industry trade. In fact intra-industry trade is more common in intermediate goods markets than in final goods markets (see e.g. Marrewijk, 2012). All previous explanations for intra-industry trade focus on markets with perfect, monopolistic or oligopolistic competition. This focus is unfortunate since intermediate goods markets are typically organized differently. Many intermediate goods markets are concentrated both among sellers and buyers they are bilateral oligopolies in which both sides exercise market power. Contracts are typically negotiated bilaterally between pairs of buyers and sellers. These negotiations are interdependent and the contract agreed in one negotiation constitutes part of an equilibrium prevailing in the market as a whole. Even if numbers are hard to find, The Economist (2000) estimates that about 80 to 90 percent of all intermediate goods are traded through extended term contracts, and that spot markets just constitute a tiny tip of a huge market of such one-to-one contract deals. Bilateral oligopoly Following the emerging literature on bilateral oligopoly, the present paper models an intermediate goods market as a set of interdependent bilateral bargaining games. Pairs of buyers and sellers simultaneously negotiate bilateral price/quantity contracts. 1 An important characteristic of 1 The literature has taken different routs to model negotiations. A hybrid cooperative/non-cooperative bargaining model is used by Davidson (1988), Horn and Wolin- 2

3 the equilibrium is that prices are relation specific, even if the goods are perfect substitutes. Every bilateral price is set to split the extra profit created by the bilateral trade equally between the two firms. As it turns out, surplus-sharing implies that every bilateral price depends on how much the two firms trade with one another. If two firms trade a very small quantity (in relation to their total volumes) their bilateral price is close to the Walrasian price. But two firms trading a large volume will agree on a price that may deviate substantially from the Walrasian price. If the buyers are more price-sensitive than the sellers, i.e. if demand is more elastic than supply, bilateral prices will exhibit quantity discounts. Otherwise there will be quantity surcharges. It follows that the firms on one side of the market prefer concentrated trading patterns while the other side prefers a dispersed trading pattern. But even if (say) the buyers gain from concentrated trading, they are only able to achieve this, if they can find a supplier who is large enough and has sufficient spare capacity, given what it sells to other buyers, also implying that the buyers may have to compete for a favourable trading position. To illustrate this, consider the case when there are quantity discounts in equilibrium, so that downstream firms benefit from single sourcing. Also assume that the two downstream firms are equally efficient and that they produce two units each in equilibrium. The upstream firm in one country produces three times as much as its competitor. This difference may result from superior efficiency. Then there are two polar trading structures as described by figure 1. In the N-shaped network, the buyer in Country 1 enjoys single sourcing while the competitor buys equal amounts from both sellers. In the n-shaped network, it is the other way around. Clearly, if the quantity discounts are important in relation to transportation costs, both downstream firms prefer to trade exclusively with the large upstream firm and not to trade at all with the small upstream firm. Expressed differently, the downstream firm in Country 2 has an incentive to promote the n-shaped network, while the competitor has the opposite interest. As the number and identity of their trading partners matter to firms, they will want to chose which firms to negotiate with. As these choices are interdependent and as different firms have conflicting interests, the outcome of the sky (1988a), and Dobson and Waterson (1997). Non-cooperative models have been studied by Horn and Wolinsky (1986b), Hart and Tirole (1990), McAfee and Schwartz (1994), Stole and Zweibel (1996) and Björnerstedt and Stennek (2007). 3

4 Figure 1: Different trading patterns Country 1 Country 2 Country 1 Country 2 Q = 3 Q = 1 Q = 3 Q = Q = 2 Q = 2 Q = 2 Q = 2 "link-formation" problem is far from obvious. Previous analysis of bilateral oligopoly takes the structure of the negotiation network (what firms negotiate with what other firms) as exogenously given, however. In the present paper, the analysis of bilateral oligopoly is extended and includes a stochastic process of link-formation. In this process, the firms preferences over trading partners endogenously determine the link-structure of the negotiation network and the trade pattern in each period. Results First, we prove that there always exists a pairwise stable network and that all pairwise stable networks are minimally connected (as the N- and the n-network in figure 1) and almost efficient. That is, all firms produce approximately their efficient quantities, but transportation costs are not necessarily minimized (as in the n-network). This result is quite general and holds for any number of up- and downstream firms. Second, in bilateral duopoly, we prove that there is a unique stochastically stable trading network. We identify the crucial parameters which affect the architecture of the stable trading network, namely the difference in price-sensitivity of the buyers and the sellers and the difference in the sizedistribution of buyers and sellers. Assume for example that the sellers are more price-sensitive implying that there will be quantity discounts. Then, if the buyers are more heterogeneous, the stable trading network entails predominantly domestic trade and some international trade according to comparative 4

5 advantage (N-network). But if the sellers are more heterogeneous (as in figure 1), the firms will trade internationally rather than with domestic partners (as in the n-network above), implying a large level of intra-industry trade. Relation to the international trade literature Our model suggests that the main driver for international trade in intermediate goods markets (bilateral oligopoly) is surplus-sharing in combination with vertical differences in pricesensitivity and horizontal differences in efficiency. These drivers are partly different from the drivers of international trade in final goods markets (perfect, monopolistic and oligopolistic competition). First, while international trade arises partly because upstream firms are more efficient in one country and downstream firms are more efficient in the other, equilibrium trade may not be induced by specialization. Efficient upstream firms in one country trade with efficient downstream firms in the other country, but less efficient upstream firms in the second country may trade with less efficient firm in the first country. One may say that there is trade in accordance with comparative advantage, but also trade in accordance with comparative disadvantage. 2 Second, our model predicts two-way trade in the intermediate good even if commodities are identical (cross-hauling). Product differentiation does thus not drive our results. Note, however, that assuming homogenous goods is only atricktoclearlydistinguishthepresentmechanismfromthatofspecialization. Surplus-sharing may induce trade also if goods are differentiated. 3 Third, trade induced by surplus-sharing differs also from the reciprocal dumping that may occur under oligopolistic competition (Smithies, 1942; Brander, 1981; Brander and Krugman, 1983). Reciprocal dumping suggests that 2 It has also been shown that intra-industry trade may arise as a result of comparative advantage in perfectly competitive markets with constant returns to scale (Davis, 1995). 3 The most widely accepted explanation of intra-industry trade in final goos is that of Krugman (1979, 1980). Using a model of monopolistic competition, Krugman argues that trade allows countries to specialize in a limited variety of products and thus reap the advantages of economies of scale without reducing the variety of goods available for consumption. More recently it has also been demonstrated that there exists large and persistent productivity differences among establishments in the same narrowly defined industries (Melitz, 2003; Helpman, Melitz and Yeaple, 2004; Bernard, Jensen, Redding, and Schott, 2007). Then (intra-industry) trade provides additional gains due to shifting resources away from less productive firms and towards more productive firms. Ethier (1979, 1982) shows that also intra-industry trade in intermediate goods may be due to product differentiation and economies of scale, assuming that the intermediate goods market is monopolistically competitive. 5

6 all sellers wish to compete in all national markets when trade costs are small, suggesting a complete (or at least a very dense) trading network. In contrast, our model suggests a very sparse (minimally connected) network. Relation to the literature on dual sourcing Minehart et al. To be written. Also include 2 The Model Consider a bilateral duopoly in two countries. In each country there is one upstream firm and one downstream firm, trading a homogenous intermediate good. Each upstream firm has a quadratic cost function given by C u (Q u )= c u Q u + (Q u ) 2, where Q u = P 2 d=1 q ud is firm u s total production and >0 is the slope of the marginal cost curve. Each downstream firm has a quadratic revenue function R d (Q d )=(a d Q d )Q d, where Q d = P 2 u=1 q ud and >0 is the slope of the marginal revenue curve. The downstream firms revenues are net of the costs of transforming a unit of intermediate goods into a unit of final goods. Our main interpretation is that all other markets in the economy are competitive. The reason why marginal revenues are falling is then that the downstream firms have increasing marginal costs in transforming the intermediate good into a final good which is sold at given prices. An alternative interpretation for the downstream firms declining marginal revenues is that they have market power in the final goods market. For simplicity it is assumed that the downstream firms revenues are independent of each others quantities, implying that the firms must be local monopolists in their respective downstream markets. 4 One country is more efficient in the upstream sector, while the other country is more efficient in the downstream sector. In particular, the upstream firm in country two has a larger cost of producing any quantity, i.e. c 2 >c 1,andthe downstream firm in country two has a higher revenue from selling any quantity than its counterpart in country one, i.e. a 2 >a 1. A possible explanation for the latter is that the downstream firm in country two has a lower production costs. We assume that asymmetries are bounded from above and that the 4 Notice that we assume the intermediate goods to be perfect substitutes to clearly show that our analysis does not in any way rely on specialization. Also notice that our arguments do not rely on economies of scale. 6

7 inefficient firms still have sufficient incentives to trade with each other: a 1 c 2 >c 2 c 1, a 1 c 2 >a 2 a 1. If upstream firm u ships goods to downstream firm d, theupstreamfirmhas to pay a per unit trade cost t ud. If the two firms belong to the same country the trade cost is zero (t 11 = t 22 =0) while if they belong to different countries the trade cost is strictly positive (t 12 = t 21 = t>0). We concentrate on the most interesting case when t is low. 5 Acontractbetweenupstreamfirmu and downstream firm d specifies a price p ud and a quantity q ud. A contract structure is one contract for every pair of upstream and downstream firms. The profit of upstream firm u is given by 2X u = (p ud t ud )q ud C u (Q u ). d=1 The profit of downstream firm d is d = R d (Q d ) 2X p ud q ud. u=1 A negotiation network g describes which upstream firms will bargain with which downstream firms. In bilateral duopoly, four possible negotiation links can be formed and there are 2 4 =16different possible negotiation networks. Denote by g c the complete negotiation network with all four links present and by g ij the negotiation network with three links (the link between firms i and j absent). Respectively, let g ij, i 0 j 0 be a network with two links between firms i and j and firms i 0 and j 0 present. Finally, let g ij be the network with the only link between i and j present. The empty negotiation network is denoted by g e. Firms actually trading with one another can be said to form a trading network. The trading network is then a subset of the negotiation network, coinciding with it if all firms that negotiate with one another also agree to trade with one another. The timing of the model is as follows. At the first stage, a negotiation 5 The case when t is high is not very interesting since it expectedly leads to predominantly domestic trade (no international trade when t is prohibitive). 7

8 network is formed. Each firm announces a set of firms with which it would like to negotiate. A link is formed if and only if both firms agree to negotiate with each other. At the second stage partnership-specific prices and quantities are determined in the process of simultaneous bilateral bargaining. Once the agreements are reached, the agreed quantities are produced and traded, and profits are realized. Competitive outcome Before studying the bargaining and link-formation game, it is convenient to find the competitive solution as a reference point. First consider the case when international trade is prohibited and transportation costs are irrelevant. Assuming sellers to be price takers, their individual (= market) supply functions are given by the inverse of their marginal costs, i.e. S i =(p i c i ) /2. Assuming buyers to be price takers, their individual (= market) demand functions are given by the inverse of their marginal revenues, i.e. D i =(a i p i ) /2. Under autarky, the Walrasian prices are therefore given by p W i = + a i + + c i. Clearly the price in the Country 2 is higher, both as a result of higher demand and as a result of lower supply. Consider now the case when international trade is allowed, assuming contracts to include delivery. Since the price is higher in Country 2, Country 1 will export the intermediate good to Country 2. In order for the efficient upstream firm in Country 1 to be active in both markets it is required that p 1 = c 1 +2 (q 11 + q 12 ) and p 2 = t + c 1 +2 (q 11 + q 12 ) implying p 2 = p 1 + t. This price difference precludes the inefficient seller in Country 2 to sell to the inefficient buyer in Country 2. Thus, there will only be trade in accordance with advantage. Global demand, written as a function of the price in Country 1, is given by D = 1 (a a 2 ) p 1 t 2 and global supply, written as a function of the price in Country 1, is given by S = p t 1 2 (c 1 + c 2 ). The Walrasian prices are given by p W 1 = P W t/2 and p W 2 = P W + t/2 where p W a1 + a 2 c1 + c 2 =

9 would be the Walrasian price in case of zero transportation costs. By standard arguments, the competitive solution is efficient, i.e. it maximizes the sum of all profits in the market. In the present context, efficiency does not only require efficient production (MC 1 MC 2 ), efficient consumption (MR 1 MR 2 )andanefficienttotalactivitylevel(mc i MR i ), but also that transportation costs are minimized. In particular, the N-shaped trading network minimizes transportation costs since the smallest possible volume consistent with productive efficiency is traded internationally. If trade costs are zero, several different trading patterns could arise in equilibrium. In fact, any trading pattern in which the bilateral quantities sum up to the four firms total outputs (which are uniquely determined) are possible. Two polar cases are noteworthy. The difference in aggregate production between the two upstream firms is given by Q 1 Q 2 =(c 1 c 2 ) /2 >0 and the difference in between the two downstream firms is given by Q 2 Q 1 =(a 2 a 1 ) /2 >0. We will thus say that the buyers are more heterogeneous than the sellers if (a 2 a 1 ) /2 >(c 1 c 2 ) /2 and that the sellers are more heterogeneous otherwise. Note that, if the sellers are more heterogeneous, the efficient upstream firm is larger than the efficient downstream firm (i.e. Q 1 >Q 2 ). Then, the efficient upstream firm must trade with both downstream firms in equilibrium. It follows that, in addition to the N-shaped trading pattern, also the n-shaped trading pattern is an equilibrium, as well as a trade corresponding to any convex combination of the two. When the buyers are more heterogeneous, the efficient downstream firm is larger than the efficient upstream firm. Then, in addition to the N-shaped trading pattern, also the o-shaped trading pattern is an equilibrium, as well as a trade corresponding to any convex combination of the two. The reason why the n and the o trading patterns do not constitute competitive equilibria also with positive transportation costs is that they entail intra-industry trade and therefore do not minimize transportation costs. 2.1 Prices and Quantities Consider now the case when contracts are determined through negotiations. If two firms have formed a negotiation link, each firm will send out a representative to meet with a representative of the other firm to negotiate a price/quantity-contract. Even though every negotiation is a bilateral affair, 9

10 the negotiations are interdependent: The preferred agreement in one negotiation may depend on the agreement made in another negotiation. An example is that a seller who agrees to ship a large volume to one buyer will have less capacity available to trade with a second buyer. For this reason, standard Nash bargaining theory cannot be used. Instead, we adopt the natural extension, the Nash equilibrium in Nash bargaining solutions approach which has been developed to study bilateral oligopoly. 6 Björnerstedt and Stennek (2007, Prop. 1) show that there exists a Nash- Nash equilibrium for every negotiation network under quite general condition, including those assumed in the present model. The equilibrium is also unique for every negotiation network. Quantities The Nash-Nash bargaining solution implies that the two firms in every negotiation agree on the bilaterally efficient quantity, i.e. the quantity maximizing the sum of the two firms profits, taking all other contracts as given. The quantity q ud is thus the solution to the maximization problem max q ud 0 u (q ud,q ud )+ d (q ud,q ud ) (1) given quantities q ud traded through other links. The corresponding Kuhn- Tucker condition requires the two firms to either agree on the quantity q ud that equalizes the downstream firm s marginal revenues to the upstream firm s marginal cost, including transportation costs, taking all other contracts as given: or else to set q ud ud (Q ud (Q u )+t ud, Notice that, given any negotiation network, the equilibrium trading pattern is efficient from the point of view of the firms. That is, taken together, the bilaterally efficient quantities also maximize aggregate profit in the industry. The formal proof is trivial since all prices drop out from the sum of all profits and q ud therefore enters this sum only through the profits of firms u and d. 7 Thus, any inefficiency in the intermediate goods market can not arise as a 6 This approach was introduced by Davidson (1988) and Horn and Wolinsky (1988). 7 If the downstream firms have market power in the final goods markets, the equilibrium is not socially efficient. Still, market power in the intermediate goods market does not give rise to inefficiencies over and above those resulting from market power in the final goods market, as the quantities chosen with price taking and bilateral efficiency coincide. 10

11 result of the bargaining process. It has to result from the strategic formation of bargaining links, which act as a constraint on the subsequent bargaining equilibrium. This property makes it easy to develop an intuition for the equilibrium trading pattern, given any negotiation network. We will present these outcomes in order of decreasing efficiency. First, if the negotiation network is complete or N-shaped, first-best can be achieved and therefore will be agreed. Both total production and the trading pattern (the N-shaped trading network) is the same as under competitive equilibrium. So, how is efficiency achieved in a market where prices do not play a coordinating role? By equalizing its marginal valuation to the marginal costs in all negotiations (bilateral efficiency), a buyer guarantees that all its partners have the same marginal cost. Similarly, every seller guarantees that all its partners have the same marginal valuations of the good. If the negotiation network is sufficiently dense, all sellers produce at the same marginal cost (efficient production) and all buyers have the same marginal valuation (efficient consumption). The total quantity is efficient, since the common marginal valuation is equal to the common marginal cost. Second, if the negotiation network is n-shaped and sellers are more heterogeneous than the buyers, then also the trading network will be n-shaped. Similarly, if the negotiation network is o-shaped and the buyers are more heterogeneous, then also the trading network will be o-shaped. In both these cases, transportation costs are not minimized and the quantities are somewhat distorted from first best as a result of the somewhat higher costs. But since transportation costs are small, the equilibrium is still almost efficient. That is, all firms aggregate quantities (Q u and Q d )areapproximatelythesameas in the N-shaped network. Still, the equilibrium trading pattern may differ substantially between different (almost) efficient networks, as illustrated in figure 2. In the figure, it is assumed that the total volume of intermediate goods is 1 and that the efficient firms produce approximately twice as much as the inefficient firms and that the size difference is marginally larger for the upstream firms. The left panel shows the equilibrium trading volumes in case the negotiation network is complete or N-shaped. The right panel shows the equilibrium trading volumes in case the negotiation network is n-shaped. Even if the total volume is (approximately) the same when trade costs are very low, the trade pattern differs. In the N network most trade is domestic, minimizing trade costs. International 11

12 q 11 = 2ε q 22 = 1/3-ε q 11 = 1/3+ε Figure 2: Equilibrium trade in different negotiation networks Q 1 = 2/3+ε Q 2 = 1/3-ε Q 1 =2/3+ε Q 2 = 1/3-ε q12 = 1/3 q 21 = 1/3-ε q12 = 2/3-ε Q1 = 1/3+ε Q2 = 2/3-ε Q1 = 1/3+ε Q2 = 2/3-ε trade is in accordance with comparative advantage. In the n network, there is intra-industry trade and little domestic trade. Moreover, trade is much more concentrated in the n network. In the N-shaped network, both the efficient firms trade approximately equal volumes with two different partners. In the n-shaped network, they trade with essentially one partner. Third, with fewer or less well-adapted links in the negotiation network, the bargaining equilibrium is still efficient conditional on the negotiation network. But, the equilibrium will not be close to first best. For example, if the negotiation network is divided into connected components (e.g. countries), trade maximizes aggregate profit in each component, but marginal cost and revenue may differ between components. These cases are not described here in detail however. Prices The agreed quantity q ud gives rise to a bilateral surplus which is defined as the increase in the sum of the two firms profits, taking all other contracts as given. The next question is how this surplus should be shared between the two firms. The Nash-Nash bargaining solution requires that they agree on the price that yields an equal split of the bilateral surplus, i.e. p ud = 1 2 Rd (Q d ) R d (Q d q ud ) + Cu (Q u ) C u (Q u q ud ) + t ud. (2) q ud q ud 12

13 The price is thus equal to the half of the sum of the buyer s average incremental revenue and the seller s average incremental cost of producing q ud including transportation. 8 The prices are relation-specific. They depend both on the two firms characteristics (their revenue and cost functions), on how much they produce in equilibrium (Q d and Q u ) and on how much they trade with one another (q ud ). Notice, however, that every bilateral trade is evaluated at the margin. That is, p ud is determined by how q ud affects costs and revenues, given all other contractual obligations of the two firms. Clearly, if q ud is infinitesimal, the two firms average incremental cost and average incremental revenue are equal to their marginal cost and marginal revenue in equilibrium. Moreover, if the aggregate equilibrium quantities (Q d and Q u )areefficient,thismeansthat, p ud = p W + 1 t 2 ud where p W is the Walrasian price (at zero transportation cost). The intuition is straightforward. Even if one upstream firm has a higher production cost than another (i.e. C 2 (Q) >C 1 (Q) for every given quantity) they will adjust production to equalize marginal costs in (any efficient) equilibrium. When the equilibrium is efficient, but bilateral trade is not infinitesimal, the price is given by p ud = p W ( ) q ud t ud, (3) given the cost and revenue functions used here. 9 Since a bilateral price depends on how much the two firms trade with one another, two equilibria with different trading patterns give rise to different prices, even if both equilibria are efficient. Recall for example that the N- and n-networks of figure 2 are associated with different trading patterns. In the N-network, upstream firm 1 trades equal amounts with the two downstream firms while in the n-network it trades its complete volume with only downstream firm 2. As a result the price it charges to downstream firm 2 differs between the two equilibria. This fact is illustrated in figure 4a where seller u s total equilibrium quantity is indicated as Q u and the total equilibrium quan- 8 The equal split follows from the symmetric Nash bargaining solution representing equal time preferences in a Rubinstein-Ståhl bargaining game. 9 The linearity follows from the fact that both the cost and the revenue functions are assumed to be quadratic. (Otherwise, equation 3 is a second-order Taylor approximation in q ud around q ud =0.) The fact that the price is independent of the identity of the buyer and the seller follows from the fact that ( ) is the same for the two upstream (downstream) firms. 13

14 tity to all buyers except d is indicated as Q u q ud. Area IC ud is seller u s equilibrium incremental cost of supplying buyer d with the equilibrium quantity q ud. Similarly figure 4b illustrates seller u s equilibrium incremental cost Figure 3: Incremental costs C u / Q u C u / Q u IC ud IC' ud Q u -q ud Q u quantity Q u -q' ud Q u quantity (a) Relation with small quantity (b) Relation with large quantity of supplying the the same buyer d in case this particular equilibrium quantity is larger, i.e. qud 0 >q ud, while keeping u s total sales Q u constant. Clearly a seller s incremental cost is increasing in the quantity traded in the link. But since the marginal cost curve slopes upwards, i.e. >0, theaverageincremental cost is a decreasing function of the quantity traded in the link. Since parties share the bilateral surplus equally, this tends to create a quantity discount. The steeper the marginal cost curve is, i.e. the higher is, the larger the quantity discount tends to be. Similarly, the average incremental revenue is an increasing function of the quantity traded via the link. And since parties share the bilateral surplus equally, this tends to create a quantity surcharge. The steeper the marginal revenue curve is, i.e. the larger is, the larger the quantity surcharge tends to be. The net effect of buying a larger quantity on price is determined by the relative importance of the two effects which, in turn, is determined by the relative steepness of the marginal cost and revenue functions. If supply is steeper than demand, i.e. >, then there will be quantity discounts. As a result downstream firms would prefer to trade with few upstream firms while upstream firms would prefer to trade with many downstream firms. If demand is steeper than supply, i.e. <, there will be quantity surcharges. In this case upstream firms prefer to trade with few downstream firms while downstream firms prefer multiple sourcing. In any case, one side of the market has an interest in concentrating trade on a few links. 14

15 Also the identity of the trading partner matters. In case (say) upstream firms wish to concentrate their trades and only deal with one partner, then the large (i.e. efficient) upstream firm need to trade with the large downstream firms. In particular, the partner has to be bigger than it is itself. Otherwise the upstream firm will have to sell to both downstream firms. Profits The up- and downstream firms equilibrium profits are denoted u (g) and d (g). The profits depend on the negotiation network since the negotiation network determines which upstream firms that will trade with which downstream firms and also what prices they will agree upon. 2.2 Negotiation network The firms decisions on whether or not to form a negotiation link with other firms is represented as a strategic game. In this link formation game, each firm invests in a capacity to negotiate and trade with all the firms with which it wants to negotiate. A negotiation link is formed between u and d only if both firms want that link, and have made the necessary investments. The set of negotiation links constitute the negotiation network. Since the negotiation game in stage two uniquely determines each firm s profit for each negotiation network, there exists a well defined payoff function also for the link formation game in stage one, and any non-cooperative equilibrium concept can be used to analyze this game. Such an equilibrium corresponds to a sub-game perfect equilibrium of the complete model. The critical assumption is that the firms in the link-formation stage cannot enter into binding agreements of any kind, such as those relating to the investment decision or the subsequent divisions of profits Nash equilibrium Although the investment cost is assumed to be small in relation to the gains of trade, and will be suppressed in the calculations, it is clear that the firms will not invest in negotiation links that are not used in equilibrium. For example, if the complete negotiation network would be formed, the international link between the two inefficient firms would not be used. Therefore, the complete network is not an equilibrium. 15

16 In general, any negotiation network where all trades are strictly positive is anashequilibrium: nofirmbenefitsfromcuttingsuchatradelink,andno new link is formed since it requires a bilateral deviation from an equilibrium. Thus, the N network is always Nash. Moreover, the Nash networks could also include either n (if the efficient upstream firm is larger than the efficient downstream firm) or o (if the downstream firm is larger). These networks are either efficient (N) or almost efficient (n and o) asaresultofaminorlossdue to the higher transportation cost associated with intra-industry trade. It is obvious that a firm with only one link could not profit from cutting that link. But this is less obvious for a firm with two links, since trade would increase on the remaining link. Consider for example the efficient N network and assume that supply is steeper than demand, >,sothatthereare quantity discounts. Then the efficient downstream firm could e.g. cut the domestic link and buy more from its international partner. But since the two upstream firms have the same marginal cost in the N network, such a reallocation would increase production costs and thus lead to a higher price. (The logic of the quantity discount only applies when comparing the same link in two efficient equilibria or when comparing two trades in a given efficient equilibrium.) Thus, as is typical of link-formation games, there is a plethora of Nash equilibria. In addition to the networks mentioned above, any negotiation network in which every firm is connected to at most one firm on the other side is anashequilibrium Pairwise stability A pairwise stable network is a Nash equilibrium network where no two firms can benefit from making a new link. This is a classic concept in network analysis introduced by Jackson and Wolinsky (1996) stability to eliminate Nash equilibria that rely on non-credible coordination mistakes. If two firms would both gain from adding a link, then an agreement to do so would be selfenforcing even if it is not legally binding. In what follows we concentrate on the values of / which are not too extreme and belong to sufficient proximity of 1. We show that a pairwise stable network always exists. In particular, the N-shaped efficient network g c is always pairwise stable. This network implies trade according to comparative 16

17 advantage. However, there could exist two more pairwise stable networks, namely the o- and n-shaped networks, as described in Table 1. Table 1: Pairwise stable networks / < 1 / > 1 (a 2 a 1 )/ > (c 2 c 1 )/ N, o N (a 2 a 1 )/ < (c 2 c 1 )/ N N, n In these cases, there is intra-industry trade in equilibrium. 10 Proposition 1. Suppose that transportation cost t is sufficiently small and that / is sufficiently close to 1. Then the pairwise stable networks are given by Table 1. There are no other pairwise stable networks in the game. The role of pairwise stability is thus to eliminate the inefficient networks. It is obvious that two firms with no previous links have an incentive to add a link. But also firms with pre-existing links may gain from adding a link. Whenever the trading network is disconnected, the marginal cost and the marginal revenue will differ between the components. It will then be profitable for any upstream firm in the component with a lower marginal cost to form a link with any downstream firm in the component with a higher marginal revenue. For example in the II-network, the two efficient firm have an incentive to start trading. The reason is that the two efficient firms produce less than the efficient quantities in the II-network and that the two inefficient firms produce more. Thus the efficient upstream firm s marginal cost is lower than that of the inefficient upstream firm. The efficient downstream firm can then lower its price by buying part of its inputs from the efficient upstream firm. Even if pairwise stability implies (almost) efficiency, an almost efficient trading networks need not be pairwise stable. The n-networkis almostefficient whenever the sellers are more heterogenous than the buyers. But the network is pairwise stable only if it is also the case that there are quantity discounts in equilibrium. [Intuition...] 10 If / is large, then the two sellers will trade only with the efficient buyer a V-shaped stable network will arise. If / is close to zero, then buyers will shun the inefficient seller an inverted V network will be stable then. 17

18 2.2.3 Comparison of pairwise stable networks (Preliminary) Under some conditions there are two pairwise stable networks (N and either n or o). The two pairwise stable networks result in approximately the same output for all firms (Q 1,Q 2, Q 1 and Q 2 )andapproximatelythesametotalwelfare. The disadvantage of the n- ando-networks is in higher levels of international trade and, therefore, higher transportation costs. But, as transportation costs tend to zero, the relative disadvantage of the inefficient network disappears. There are, however, other important differences, both in terms of the pattern of international trade and in terms of the distribution of welfare both between firms of different efficiency and between different countries. We focus on the case with zero transportation costs and assume more heterogeneity among the sellers, implying that the efficient upstream firm is larger than the efficient downstream firm. Pattern of international trade First, consider the amount of international trade. Depending on the market, the amount of international trade in the N- network (T N = q 12 = Q 1 Q 1 ) may vary between almost no trade (when efficiency differentials both up- and downstream are small) and almost all goods being traded internationally (when efficiency differentials both up- and downstream are large). Depending on the market, the amount of international trade in the n-network (T n = q 12 + q 21 = Q 2 + Q 2 )mayvarybetweenhalf of all goods being traded internationally (large upstream heterogeneity but small downstream heterogeneity) and all goods being traded internationally (small efficiency differentials both up- and downstream). In a given market, the difference in the amount of international trade between the two networks is given by =T n T N = Q + Q 2 Q 1 =2Q 2 2 (0,Q), where Q 2 is the size of the smallest firm on the market. Corollary 1. Assume that the sellers are more heterogeneous than the buyers. If t is sufficiently small, there is less international trade in the N-shaped network than in the n-shaped network. In the extreme, when the two firms on both sides are almost equally efficient, there is almost only domestic trade in the N-shaped network and almost only international trade in the n-network. Second, consider the importance of intra-industry trade, as measured by 18

19 the standard Grubel-Lloyd (1971) index for countries 1 and 2: GL i =1 X i M i X i + M i 2 [0, 1] where X i and M i denote the country s export and import. With only two countries, both countries will have the same index. Also notice that GL i =0in the N-network since each country either imports or exports. In the n-network GL n i =1 q 12 q 12 q 12 + q 12 =1 Q 2 Q 2 Q 2 + Q 2. Thus: Corollary 2. Assume that the sellers are more heterogeneous than the buyers. The Grubel-Lloyd index of intra-industry trade is zero in the N-network, but positive in the n-network. In the extreme, when the two firms on both sides of the market are almost equally efficient, the Grubel-Lloyd index is almost equal to one. Distribution of welfare First, we compare the firms profits in equilibrium. Corollary 3. Even if the firms on one side of the market are almost equally efficient, they may end up earning substantially different profits. In fact, an inefficient firm may be more profitable than its competitor. To see this, consider the case when supply is steeper than demand and the sellers are more heterogeneous. Then the N- and n-networks are pairwise stable. The two downstream firms benefit from single sourcing. Assume that they are almost equally large, buying two units each, but that the efficient upstream firm is three times as large as the inefficient upstream firm. Then, in the N-network the inefficient downstream buys both units from its domestic partner while the efficient downstream firm buys one unit from both sellers. In this case, the inefficient downstream firm earns a higher profit than its more efficient competitor. In the n-network, the efficient downstream firm buys both units from the efficient upstream firm, while the inefficient downstream firm buys one unit from both both sellers. In this case, the marginally more efficient downstream firm earns a substantially larger profit. These differences may clearly distort the firms incentives to invest in productive efficiency. 19

20 Second, despite the total profit being the same (if t =0), firms from one country may prefer one pairwise stable network over another. This may be demonstrated by the same example as above. Since each sellers has an equally concentrated trading structure in the N- and the n-networks (the efficient firm selling 2 units to one buyer and 1 unit to the other and the inefficient firm selling 1 unit to one buyer) their profits are the same in the two equilibria. The distribution of welfare between the two downstream firms differs between the two networks. This opens the possibility for a government to intervene and influence the probability of the preferred network formation (by, for instance, subsidizing domestic or international trade and reducing or increasing in such a way firms relative efficiencies) Stochastic stability Since there are sometimes two pairwise stable networks with large differences, we need to determine which equilibrium is most plausible. For this reason, consider now a dynamic model with many periods and the following stochastic process of link formation. The process starts at any negotiation network g. In each period 2{1, 2,...} alinkij is randomly selected with some strictly positive probability P ij (g). If the link is not in the network and firms in question would like to add it to the network, that is i (g + ij) i (g), j (g + ij) j (g) with at least one inequality strict, then the link is added. If the link is already in the network and one of the two firms prefers to delete it, i (g ij) > i (g) or j (g ij) > j (g), then the link is deleted. Firms are myopic and look ahead only to the next period. With a certain probability a mistake is made, a non-profitable link is made or left undeleted. We assume that a costly mistake is less likely, that is, if creating a link causes a profit loss for one of the two firms or deleting a link is costly for both firms, a mistake occurs with probability. If adding or deleting the link does not change the profits of both firms (except maybe 20

21 paying or retrieving small cost of link maintenance l), the mistake is made with probability r > where 0 <r<1. 11 Such a process is a finite-state, aperiodic, irreducible Markov chain and, therefore, has a steady-state distribution. A network g is stochastically stable if its steady-state probability is bounded from zero as the error rate tends to zero. In words, a network is stochastically stable if the probability that the process finds itself in it tends to some strictly positive number as converges to zero. Our first main result states that there is a unique stochastically stable negotiation network g S,sothattheprocessdescribedabove,ifrunendlessly, finds itself in g S with probability converging to 1. Depending on the parameters of the model the unique stochastically stable network is either N-, n-, or o- shaped, as described by Table 2. Table 2: Stochastically stable networks / < 1 / > 1 (a 2 a 1 )/ > (c 2 c 1 )/ o N (a 2 a 1 )/ < (c 2 c 1 )/ N n Proposition 2. There is a unique stochastically stable negotiation network g S,asdescribedbyTable2 The proof is relegated to Appendix B. In words, suppose that demand is steeper than supply, <. If the efficiency difference between the upstream firms is large in relation to the efficiency difference between the downstream firms, (c 2 c 1 ) / > (a 2 a 1 )/, the unique stochastically stable network is N-shaped, while if (c 2 c 1 ) / < (a 2 a 1 )/,itiso-shaped. To see the logic of the argument, it is useful to first note that upstream and downstream firms have opposing interest in promoting a concentrated or nonconcentrated trading structure. Adding a link will thus often benefit one firm but hurt the other. Such deviations are therefore unlikely. But there is always one side of the market that has an interest in promoting a more concentrated trading structure by unilaterally cutting links. There is a caveat, however. To 11 The assumption that r is equal across different links does not play any role and is made for simplicity. 21

22 cut one of two links and trade with only one partner, the remaining partner must have the necessary capacity to increase this trade in an efficient manner. We focus on the case when demand is steeper than supply ( > )sothat the upstream firms prefer a concentrated trading structure but downstream firms prefer dual sourcing. (The opposite case can be analyzed in a perfectly analogous way.) As the power to cut links is unilateral (and adding links require consent), the upstream firms incentives will be decisive. Assume first that the difference in efficiency is larger downstream than upstream, which means that the efficient downstream firm is larger than the efficient upstream firm. In this case there are two pair-wise stable networks, namely the N-shaped and the o shaped. But only the o shaped network is stochastically stable. To see why, consider the possibility that the firms are currently in the N-shaped negotiation network implying that the inefficient upstream firm has concentrated sales. Then with some probability the link between the two inefficient firms is added by mistake to create the complete negotiation network. This is described in Figure 4. This mistake is almost Figure 4: Stochastically unstable N-shaped network. Efficent Inefficent Efficent Inefficent Efficent Inefficent Inefficent Efficent Inefficent Efficent Inefficent Efficent Almost costless addition of an unused link Efficient upstream firm profitably cuts link to domestic partner costless since the link will not be used in equilibrium. But once the complete negotiation network has been formed, the efficient upstream firm has an incentive to cut the link to its domestic partner, thereby creating the o shaped network. This is strictly profitable to the efficient upstream firm since it will then be able to sell all its produce to the efficient downstream firm as that firm is even larger. Since the firms can leave the N-shaped network without much cost, it is not stochastically stable. In contrast, leaving the o shaped network imposes a significant cost on a decisive firm. Cutting a link is obvi- 22

23 ously costly in terms of lost production and the efficient upstream firm would clearly oppose adding the missing link since it would immediately lead to the N-shaped trading network thereby reducing the firm s trading concentration. Assume next that the difference in efficiency is smaller downstream than upstream, which means that the efficient downstream firm is smaller than the efficient upstream firm. In this case, only the N-shaped network is pairwise and consequently stochastically stable. Consider the possibility that the network is N-shaped. As before, with some probability the link between the two inefficient firms is added by mistake to create the complete negotiation network. This mistake is almost costless since the link will not be used in equilibrium. Now the efficient upstream firm does not have an incentive to cut the link to its domestic partner, thereby creating the o shaped network, since the downstream firm is too small to buy all the produce from the efficient upstream firm. The inefficient upstream firm has a (weak) incentive to cut the superfluous international link, however, thereby creating the N-shaped negotiation network. 3 Many firms (to be written) In this section we consider the case with an arbitrary number of up- and downstream firms and show that pairwise stability implies that the equilibrium network is minimally connected and almost efficient (as transportation costs are approach zero). As demonstrated by Proposition 1, the reverse is not true. Some almost efficient and minimally connected networks are not pairwise stable under certain conditions. 4 Concluding Remarks In this paper, we give the complete characterization of the unique stochastically stable network in a bilateral duopoly where trade is conducted through bilateral contracts. We identify conditions that lead to predominant intra-industry trade. The paper provides a rational for intra-industry trade in intermediate goods markets, based on surplus-sharing in combination with efficiency differentials between firms. 23