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International Journal of Industrial Organization 20 (2002) 163–190 www.elsevier.com / locate / econbase Spatial competition among multi-store firms Debashis Pal a , *, Jyotirmoy Sarkar b b a Department of Economics, University of Cincinnati, Cincinnati, OH 45221, USA Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, Indianapolis, IN 46202, USA Abstract The paper analyzes spatial Cournot competition among multi-store firms. It demonstrates that the complex problem of determining equilibrium store locations for competing multi-store firms can be approximated by a simple one, in which each firm behaves as a multi-store monopolist in choosing its store locations. A firm’s equilibrium store locations often coincide with its monopoly locations, and in general, converge to its monopoly locations as the demand grows larger. When the firms have an equal number of stores, the stores belonging to competing firms agglomerate at discrete points that coincide with each firm’s monopoly store locations. 2002 Elsevier Science B.V. All rights reserved. JEL classification: D43; L13 Keywords: Spatial competition; Multi-store firms; Cournot oligopoly 1. Introduction Spatial competition has a rich and diverse literature, with its origin dating back to the seminal work of Hotelling (1929). Despite its long history, it is surprising how little attention has been paid to study competition among firms who can set up multiple stores.1 The literature on spatial competition that allows firms to choose * Corresponding author. Tel.: 11-513-556-2630; fax: 11-513-556-2669. E-mail address: [email protected] (D. Pal). 1 Note that stores can be interpreted as plants. In this paper we use the word stores, although stores and plants can be used interchangeably. 0167-7187 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0167-7187( 00 )00080-1 164 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 their locations typically assumes that each firm can set up only one store.2 Presumably, this assumption is often made to avoid the analytical complexity that would arise otherwise. It is important, however, to recognize that most firms actually set up multiple facilities. A casual look at a typical U.S. city reveals that diverse firms, such as Pizza Hut, J.C. Penney, Kroger and Circuit City, all have several stores.3 In fact, retailers with four or more outlets account for more than half of the total retail business in the United States. In the context of manufacturing, a producer often manufactures a homogeneous product at several production facilities. For example, in the United States, Lafarge Corporation, a leading cement producer, has 15 cement plants. In the natural gas liquids (NGL) industry, the industry leaders GPM Gas Corporation (formerly Philips 66 Natural Gas) and Warren Petroleum Company (a subsidiary of Chevron Corporation) have 18 and 57 gas liquid plants, respectively. In the ready-mixed concrete industry, Florida Rock Industries has 82 ready-mixed concrete plants, Texas Industries Inc. has 29 ready-mixed concrete plants and the industry leader Lafarge Corporation has as many as 450 production facilities.4 It may be argued that the assumption of single-store firm is merely a technical simplification and the results should extend to multi-store firms. However, observe that when a firm has several stores, each store’s behavior affects the decisions of all other stores, including those owned by the same firm. Consequently, each store cannot be treated independently as a single-store firm and the results obtained with multi-store firms are likely to differ from those obtained with single-store firms.5 Naturally, the study of spatial competition among multi-store firms deserves special attention, which is the objective of this paper. Although the literature on this topic is surprisingly brief, the study of location decisions by multi-store firms originates more than thirty years back. Teitz (1968) is the first to study spatial competition among multi-store firms. In the context of Hotelling’s linear city model with linear transport cost, Teitz (1968) points out that a Nash location equilibrium does not exist if the firms have multiple stores. This non-existence of a location equilibrium may have contributed to the brevity of the literature during the next two decades. Subsequently, Martinez-Giralt and Neven (1988) assume quadratic transport cost together with mill pricing and demonstrate 2 Even in the context of a spatial monopolist, only a few papers analyze the location and pricing decisions of a multi-store monopolist. Katz (1980) and Chu and Lu (1998) are among the few studies in this area. See Chu and Lu (1998) for a related discussion. 3 The telephone directory for the city of Cincinnati lists 25 stores for Pizza Hut, 7 stores for J.C. Penney, 53 stores for Kroger, and 6 stores for Circuit City. 4 Source: Gale Business Resources. Internet address: www.galenet.com / servlet / GBR /. 5 In the context of retailing, Ghosh and McLafferty (1987, Chapter 6) argue that the traditional methods of site selection with single-outlet firms are inadequate to analyze location decisions of multi-outlet retail firms. They claim that the analysis of multi-outlet retailers requires systematic evaluation of the impact of each store on the entire network. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 165 an intriguing result. In the context of a spatial duopoly where each firm may open up to two stores, they show that for both the linear city and the circular city models, the firms do not take up the opportunity of opening multiple stores; in equilibrium, each firm opens only one outlet. Observe that the results demonstrated by Teitz (1968) and Martinez-Giralt and Neven (1988) are rather fascinating given the prevalence of spatial competition involving firms with multiple outlets. Subsequent work in this area involves either pricing policies other than mill pricing or vertical product differentiation (as opposed to horizontal product differentiation). Eaton and Schmitt (1994) analyze economies of scope using a model that resembles a model of competitive spatial price discrimination among firms with multiple outlets. Thill (1997, 2000) analyzes spatial competition among multi-store firms by assuming fixed prices but allowing each outlet to choose a quality for its product, in addition to choosing a geographic location on a line. In the context of vertical product differentiation, Champsaur and Rochet (1989) analyze a duopoly where the firms may choose multiple qualities for their products.6 In this paper, we consider an entirely different approach to study spatial competition among multi-store firms. Instead of assuming an exogenously fixed price or price competition, we assume that the firms compete in quantities (a` la Cournot). The assumption of Cournot competition enables us to avoid the puzzling findings obtained in spatial models with price competition and to demonstrate results that closely resemble the real world. We allow the firms to set up multiple stores, and analyze a two-stage problem of location and quantity choices in Hotelling’s linear city model. We deviate from the previous literature on multistore firms by assuming that in the second stage, the firms behave as Cournot oligopolists and discriminate over space. In non-spatial contexts, the assumption of Cournot competition needs no further justification. The Cournot model is probably the most widely used oligopoly model. In spatial contexts, the predictions arising from a spatial Cournot model often describe the real world better than those arising from a spatial price competition model. For example, it is Cournot competition, not price competition that successfully explains the commonly observed phenomenon of overlapping geographic markets of the competing firms selling a homogeneous product.7 Anderson and Neven (1991) provide compelling arguments justifying the appro6 Anderson and dePalma (1992) depart from a standard Hotelling type location model and pioneer an alternative approach to study competition among multi-product firms. Anderson and dePalma (1992) endogenize the pricing decisions of the firms and use a nested logit model of demand to characterize a symmetric equilibrium for multi-product firms. 7 Phlips (1983) and McBride (1983) provide support for spatially overlapping markets consistent with Cournot competition. Price competition with homogeneous product always implies non-overlapping markets for the competing firms. This is true for a variety of alternative pricing strategies including mill pricing and spatial discriminatory pricing. For example, see d’Aspremont et al. (1979), Lederer and Hurter (1986). 166 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 priateness of the Cournot assumption in various spatial models.8 It is established that spatial Cournot competition is appropriate for industries where quantity is less flexible than price at each market point. Such industries would include, for example, oil, natural gas, cement and ready-mixed concrete. In fact, from an empirical perspective, the Cournot model of spatial competition is employed to analyze international oil and natural gas markets (see, for example, Salant, 1982). The predictions of the Cournot model in terms of delivered prices are confirmed by McBride (1983) in the cement industry and by Greenhut et al. (1980) in a representative sample of industries.9 In the context of single-store firms, spatial models with Cournot competition are becoming increasingly popular in recent years. Anderson and Neven (1991), and Hamilton et al. (1989) pioneer the study of spatial Cournot competition with endogenous location choice.10 They analyze a two-stage problem of location and quantity choices in Hotelling’s linear city model. Afterwards, Hamilton et al. (1994) consider a variant of the Cournot competition, where the firms choose total output and use mill pricing. Gupta et al. (1997) extend Anderson and Neven’s (1991) analysis by considering non-uniform consumer distribution in the linear city model. Mayer (2000) contributes by allowing the production costs to differ at various locations. In the present paper, we contribute to the previous literature by considering multi-store firms in the context of spatial Cournot competition with endogenous location choice. We identify the equilibrium store locations for competing multistore firms and establish the following results. First, using Cournot competition we avoid some of the puzzling results obtained in location models with price competition. Under price competition with mill pricing, each firm locates all of its outlets at the same market point (MartinezGiralt and Neven, 1988). Thus, in effect, firms do not take up the opportunity of opening multiple outlets. This result is intriguing given the prevalence of multioutlet firms in the real world. This puzzling outcome, however, does not arise with Cournot competition. Under Cournot competition, each firm always chooses distinct locations for its outlets. Furthermore, under price competition with both 8 Also, see Greenhut et al. (1991), which argues that between the Bertrand and Cournot models, the latter warrants major consideration in modelling spatial competition. 9 Furthermore, following Eaton and Schmitt (1994), a spatial model with Cournot competition can be interpreted as a non-spatial Cournot model involving firms that enjoy economies of scope. Eaton and Schmitt (1994) model economies of scope by considering firms that may produce few basic products, which can be modified to produce any other variant in the attribute space. When the firms compete in prices (a` la Bertrand), the model mirrors a spatial model with discriminatory pricing. Following this approach, therefore, if the firms compete in quantities, instead of prices, a non-spatial Cournot model with economies of scope would be equivalent to a spatial Cournot model. 10 Previous literature on spatial Cournot competition treats locations as exogenously fixed. See Greenhut and Greenhut (1975), Greenhut and Ohta (1975), Norman (1981) and Ohta (1988) for related discussions. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 167 mill pricing and spatial discriminatory pricing, outlets of competing firms never agglomerate. In contrast, under Cournot competition, outlets of competing firms may agglomerate at finitely many market points; a result that is consistent with the clustering of stores of competing firms at various shopping malls in the same city. Second, we characterize situations when, in equilibrium, each firm would simply choose its monopoly locations. In other words, each firm would locate its stores as if it were a monopolist and did not face any competition. We also characterize situations when the equilibrium store locations differ from the monopoly locations. In this situation, however, we demonstrate that as the demand becomes large, relative to the transport cost, all firms’ equilibrium store locations converge to their respective monopoly locations. In fact, numerical simulations show that even for smaller demands, the equilibrium store locations are usually quite close to their respective monopoly locations. Thus, we demonstrate that the complex problem of determining equilibrium store locations for competing multistore firms can be approximated by a very simple one, in which we treat each firm as a multi-store monopolist and determine optimal store locations for a monopolist with multiple stores. Third, we shed new light on a prevailing perception in the literature that concludes that price competition yields spatial dispersion whereas Cournot competition gives rise to spatial agglomeration of firms. In the context of price competition, coincident location of stores of competing firms offering identical products severely intensifies price competition and thus, stores belonging to competing firms never agglomerate in a location-price game. In contrast, in the context of single-store firms, pioneering work of Anderson and Neven (1991) and Hamilton et al. (1989) establishes that in Hotelling’s linear city model, Cournot competition gives rise to spatial agglomeration of firms. Subsequently, Gupta et al. (1997) relax the assumption of uniform consumer distribution in the linear city model and confirm the agglomeration result for a wide variety of consumer distributions. Mayer (2000) allows production cost to differ at various market points and finds that the firms may still agglomerate. Thus, it may indeed seem reasonable to conclude that price competition generates spatial dispersion, while Cournot competition gives rise to spatial agglomeration. In this paper, however, we establish that if the firms can have multiple stores, the conclusion drawn above is incorrect. Under Cournot competition involving multi-store firms, stores belonging to competing firms may not agglomerate; giving rise to complete dispersion of stores. In fact, depending on the number of stores, price competition with spatial price discrimination and Cournot competition generate quite similar location patterns. Thus, this paper contributes to the literature of spatial competition in several ways. First, it incorporates the commonly observed phenomenon of multi-store firms and allows the firms to choose their store locations. Second, by considering Cournot competition, it avoids some of the puzzling outcomes associated with spatial price competition involving multi-store firms. Third, it develops a mecha- D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 168 nism to identify the equilibrium locations. Finally, it characterizes the equilibrium locations and demonstrates several results that closely resemble the real world. The paper is organized as follows. Section 2 describes the model. Section 3 characterizes various properties of the location and quantity equilibria. Section 4 identifies the equilibrium locations and demonstrates the results. Section 5 concludes the paper. 2. Model We consider a spatial multi-store Cournot oligopoly serving a linear market of unit length. For expositional simplicity, we restrict the number of firms to two; the results extend to more than two firms. Without loss of generality, the interval f0,1g represents the linear market. The consumers are distributed uniformly over f0,1g. The market demand at each point j [f0,1g is given by p 5 a 2 bQ, where a . 0, b . 0 are constants, Q is the aggregate quantity supplied at j , and p is the market price at j . Without loss of generality, we assume b 5 1 / 3 for expositional simplicity. Two firms, 1 and 2, compete in quantities at each market point j [f0,1g. Firms 1 and 2 have m > 1 and n > 1 stores, respectively. Without loss of generality, assume that m < n. The firms locate their stores in f0,1g. Note that the stores may be interpreted as plants. In this paper, we use the word stores, although stores and plants can be used interchangeably. The vector x] 5 (x 1 , x 2 , . . . , x m ) denotes the locations of Firm 1’s stores. Here x i is the distance measured from the left endpoint of the market. Without loss of generality, we assume 0 < x 1 < x 2 < ? ? ? < x m < 1. Similarly, the vector y 5 ( y 1 , y 2 , . . . , y n ) denotes the locations of Firm 2’s stores, where 0 < y 1 < y 2 <] ? ? ? < y n < 1. The firms deliver the product to the consumers and thus, can discriminate across consumers. The firms have identical production and transportation technologies. Each firm produces at a constant marginal and average cost (both normalized to zero) and pays a linear transport cost of t . 0, per unit distance. Arbitrage among the consumers is assumed to be infeasible due to high transaction costs. We also assume that a > 2t. This condition ensures that both firms will always serve the whole market. Each firm serves a market point j incurring the lowest possible transport cost, and hence from the store which is nearest to the market point. Define firm i’s si 5 1, 2d effective delivered marginal cost c i ( j ) at the market point j as c 1s jd 5 minhtux 1 2 ju, tux 2 2 ju, . . . , tux m 2 juj and c 2s jd 5 minhtu y 1 2 ju, tu y 2 2 ju, . . . , tu y n 2 juj It now follows that two stores of the same firm never coincide and each store serves a contiguous market around itself. Also, on each side, a store’s market extends up to the midpoint between itself and the next store (owned by the same D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 169 firm), if such a next store exists, otherwise, it extends up to the appropriate endpoint. We study the sub-game perfect Nash equilibria (SPNE) of a two-stage game, where in stage one, the firms choose the locations of their stores along the linear market and in stage two, the firms compete in quantities. We proceed by backward induction and characterize the quantity equilibrium in the second stage for given locations. Since marginal production cost is constant and arbitrage among the consumers is not feasible, quantities set at different points by the same firm are strategically independent. Therefore, the second stage Cournot equilibrium can be characterized by a set of independent Cournot equilibria, one for each market point j [f0,1g. At each market point j [f0,1g, firm i si 5 1, 2d chooses qis jd to maximize its profit f p 2 c is jdg qis jd. By simultaneously solving the first order conditions for profit maximization of the two firms, we obtain the following equilibrium outcomes at each j [f0,1g: qis j , ]x, yd 5 a 2 2c is jd 1 c js jd ] fa 2 2cis jd 1 cjs jdg 2 ]]]]]] pis j , x, y 5 ] ]d 3 where i 5 1, 2 and j 5 1, 2, but i ± j. qis j , x, ] ]yd and pis j , ]x, ]yd denote firm i’s equilibrium quantity and equilibrium profit at market point j , given the locations y . sx, d ] ] Therefore, given the locations sx, ] y]d, firm i’s si 5 1, 2d equilibrium aggregate profit is 1 P sx,] ]yd 5Ep sj, x,] ]yd dj i i 0 Our objective is to solve for a pair of location vectors sx*, such that, given y*, ] y* ] dmaximizes Firm ]2’s x* maximizes Firm 1’s aggregate profit and given x*, y* ] ] ] aggregate profit. Thus, sx*, ] ]y*d is a sub-game perfect location equilibrium in which neither firm finds it profitable to unilaterally relocate any of its stores. 3. Properties of a location equilibrium In this section, we characterize several properties of sub-game perfect Nash equilibrium (SPNE) locations. These properties are used in Section 4 to determine the SPNE locations. We first define the notion of quantity-median of a store’s market and then in the following proposition we associate each store’s equilibrium location with the quantity-median of its market. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 170 Definition 1. The quantity-median of a store’s market is the point such that the total quantity supplied by the store to the left of that point is equal to the total quantity supplied by it to the right of that point. Since a firm never locates more than one store at the same point, each store’s market is an interval around itself, and the total quantity supplied by the store to the left of any point z is a continuous function of z on this interval. Hence, the quantity-median of each store’s market is unique and well defined. How does a store’s optimal location relate to its quantity-median? Observe that a store supplies the maximum amount to the market at its own location, and the quantity it supplies to the markets further and further away in either direction, decreases (in fact, piecewise linearly). However, there may be asymmetry in the aggregate quantities supplied to the two sides of a store and hence, a priori it is not evident how the quantity-median of a store’s market may be related to the store’s location. The following proposition establishes a useful relationship between a store’s optimal location and the quantity-median of its market. Proposition 1. Given the location vector of the other firm, a firm maximizes its profit if and only if it locates its stores in such a way that each store is located at the quantity-median of its market. Proof. Without loss of generality, consider Firm 1. Note that, to maximize profit, it never places more than one store at the same location, nor does it locate any store at 0 or 1. Hence, 0 , x 1 , x 2 , ? ? ? , x m , 1 and its profit is given by P (x,] y)] 1 x1 5 2 [a 2 2t(x 2 j ) 1 c ( j )] E ]]]]]]] dj 3 [a 2 2t( j 2 x ) 1 c ( j )] ]]]]]]] dj 1E 3 [a 2 2t(x 2 j ) 1 c ( j )] ]]]]]]] dj 1E 3 [a 2 2t( j 2 x ) 1 c ( j )] ]]]]]]] dj 1 ? ? ? 1E 3 [a 2 2t(x 2 j ) 1 c ( j )] ]]]]]]] dj 1E 3 [a 2 2t( j 2 x ) 1 c ( j )] 1 E ]]]]]]] dj 3 1 2 0 2 (x 1 1x 2 ) / 2 1 2 x1 2 x2 2 2 (x 1 1x 2 ) / 2 2 (x 2 1x 3 ) / 2 2 2 x2 2 xm m 2 (x m 21 1x m ) / 2 1 2 m xm Observe that 2 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 P ≠ ]]1 ≠x 1 x1 171 [a 1 c 2 ( j )] 2 [a 2 2t(x 1 2 j ) 1 c 2 ( j )] dj 1 ]]]] 3 F GE 4t 1F]G E [a 2 2t( j 2 x ) 1 c ( j )] dj 3 x 2x x 1x F a 2 2tF]]G 1 c S]]DG 2 2 1 ]]]]]]]]]]] 2 4t ]] 3 5 0 (x 1 1x 2 ) / 2 1 x1 2 1 2 1 2 2 2 F 6 G S DG 2 x2 2 x1 x1 1 x2 2 a 2 2t ]] 1 c 2 ]] [a 1 c 2 ( j )] 2 2 2 ]]]] 2 ]]]]]]]]]]] 3 6 P F FF G x 2x x 1x 1 1 ]Fa 1 2tF]]G 1 c S]]DGG 2 2 2 x 1x 2t 5 ] F 2 3a 1 (3x 1 x )t 2 4c (x ) 1 c S]]DG , 0 3 2 x1 1 x2 ≠2 4t ]] 5 ] 2t ]] 2 2[a 1 c 2 (x 1 ) 3 2 ≠x 21 1 2 1 2 2 1 1 2 2 1 2 2 Therefore, the second order condition for profit maximization holds everywhere, while the first order condition, P ≠ ]]1 5 0, ≠x 1 becomes equivalent to x1 E [a 2 2t(x 2 j ) 1 c ( j )] dj 5 E 1 0 2 (x 1 1x 2 ) / 2 x1 [a 2 2t( j 2 x 1 ) 1 c 2 ( j )] dj Thus, the location of store 1 coincides with its quantity-median. Similarly, the profit maximization condition of any other store implies that its location must coincide with its quantity-median. h The intuition behind Proposition 1 is as follows. Consider a small rightward movement of a store. It decreases the profit earned by this store from each market on its left and it increases its profit from each market on its right. At equilibrium, the marginal decrease in profit must be equal to the marginal increase in profit for this store. For Cournot competition, a firm’s profit in a market is proportional to the square of the quantity served by the firm in that market. Hence, the marginal profit in a market is proportional to the quantity served in that market. Therefore, the total quantity served to markets on its left must be equal to the total quantity served to the markets on its right. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 172 The following corollary follows as an immediate implication of Proposition 1. Corollary 1. (The market-median property) A pair of location vectors constitutes a sub-game perfect Nash equilibrium if and only if each store of every firm is located at the quantity-median of its own market. Usually, in the context of single-store firms, the market-median property is fairly robust. It can be verified in Anderson and Neven (1991), and in Hamilton et al. (1989), where the consumer distribution is assumed to be uniform. In Gupta et al. (1997), the market-median property is satisfied for wide varieties of nonuniform consumer distributions. Corollary 1 demonstrates that it holds for multistore firms as well. Instead of directly using the first order conditions, the market-median property suggests an alternative way to determine the SPNE locations, which often turns out to be quite useful. In the context of a multi-store monopoly, it is well known that a monopolist maximizes its profit by locating its stores at their respective quantity-medians. Since a monopolist and the duopolists all satisfy the market-median property, it is interesting to examine how each firm’s SPNE locations in a duopoly compare with its monopoly locations. Note that, if a firm is a monopolist, it locates its stores at the center of their respective markets so as to satisfy the market-median property. If a firm has a rival, however, a store located at the center of its market may not satisfy the market-median property. This is because, now the store may face different delivered marginal costs from the rival firm on its two sides, and as a result, the equality of left and right market lengths does not guarantee the equality of quantity supplied on each side. Thus, it is the asymmetry in the rival firm’s delivered marginal costs on the two sides of a store that alters a firm’s SPNE locations from its monopoly locations. As the demand parameter a grows larger, however, the rival’s cost discrepancy effect becomes less significant and consequently, a firm’s optimal store locations are likely to move closer to their respective monopoly locations. In fact, the following proposition establishes that, in the limit, a firm’s optimal location vector converges to its monopoly location vector. Proposition 2. As the demand parameter a becomes large (relative to t), each firm’s SPNE location vector converges to its monopoly /transport cost minimizing location vector. Proof. The proof follows from the first order conditions of profit maximization. Without loss of generality, consider Firm 1. Observe that the first order conditions for profit maximization imply that x1 E [a 2 2t(x 2 j ) 1 c ( j )] dj 5 E 1 0 2 (x 1 1x 2 ) / 2 x1 [a 2 2t( j 2 x 1 ) 1 c 2 ( j )] dj D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 E xi (x i 21 1x i ) / 2 5 E [a 2 2t(x i 2 j ) 1 c 2 ( j )] dj x( i 1x i 11 ) / 2 xi E 173 [a 2 2t( j 2 x i ) 1 c 2 ( j )] dj , ;i 5 2, . . . , m 2 1 xm (x m 21 1x m ) / 2 [a 2 2t(x m 2 j ) 1 c 2 ( j )] dj 5 E 1 xm [a 2 2t( j 2 x m ) 1 c 2 ( j )] dj Dividing both sides by a and then taking the limit as a → ` in each of the above equations, the proof of this proposition follows. h Proposition 2 is useful, since for large demand, it can be used to approximate the SPNE locations of multi-store firms. Also, to numerically determine the SPNE locations for any specific a, the monopoly locations can serve as an initial approximation. The market-median property and the limiting property of the SPNE locations are useful but may not be sufficient to determine the exact SPNE locations. In Proposition 3 below, we establish the existence and the uniqueness of the symmetric SPNE locations, which together with Corollary 1 facilitates the determination of the SPNE locations. Lemmas 1 and 2 are used to prove Proposition 3. Lemma 1. Given the location vector of the other firm, a firm’s profit maximizing location vector is unique. Proof. Without loss of generality, let Firm 2’s store locations be y 0 5s y 10 , y 20 , . . . , y n0d. In response to y 0 , as soon as Firm 1 selects the location of its] store 1, x 1 , the location of its store]2, x 2 , is uniquely determined. This is because, by Proposition 1, the quantity supplied by store 1 to its left must be equal to the quantity supplied to its right. Thus, given x 1 , the right endpoint of store 1’s market is uniquely determined. This point, however, must coincide with (x 1 1 x 2 ) / 2, hence x 2 is also uniquely determined. Repeated application of this argument implies that as soon as Firm 1 selects the location of its store 1, x 1 , the location of its store m, x m , is uniquely determined. Now, as soon as x m is determined, the quantity supplied by store m in [(x m21 1 x m ) / 2, x m ] is fixed. Consequently, there is a unique point u . x m such that the quantity supplied in [(x m21 1 x m ) / 2, x m ] equals the quantity supplied in [x m , u ]. On the other hand, the right end point of D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 174 store m’s market is always fixed at 1. Therefore, x 1 must be chosen in such a way that u coincides with 1. Note that u increases continuously as x 1 increases. Therefore, by the intermediate value theorem, there is a unique choice for x 1 such that u (x 1 ) 5 1. h Lemma 2. If the location vector of one firm is symmetric around 1 / 2, then so is the unique profit maximizing location vector of the other firm. Proof. The proof is by contradiction. Suppose the location vector of Firm 1 is symmetric around 1 / 2, but the unique profit maximizing location vector of Firm 2 is not symmetric around 1 / 2. Then by reflecting all locations around 1 / 2, we do not change the location vector of Firm 1, but change the location vector of Firm 2. Hence, we obtain another distinct profit maximizing vector for Firm 2, contradicting Lemma 1. h Proposition 3. There exists a sub-game perfect Nash equilibrium at which each firm locates its stores symmetrically around 1 / 2. Moreover, the symmetric SPNE is unique. Proof. To prove the existence of a symmetric SPNE, define the set G5 H sx 1 , x 2 , . . . , xmd such that 0 < x 1 < x 2 < ? ? ? < x m < 1 and sx 1 , x 2 , . . . , x md is symmetric around 1 / 2 J Observe that the set G is compact. Now, consider a vector x˜ [ G. x˜ specifies a location vector for Firm 1 that is symmetric around 1 / 2. Let R 1 and R 2 be the best response functions of firms 1 and 2, respectively. Observe that both R 1 and R 2 are continuous and by Lemma 2, y˜ 5 R 2 (x˜ ) is symmetric around 1 / 2. Now consider the function g such that g(x˜ ) 5 R 1 (y˜ ) 5 R 1 (R 2 (x˜ )). Note that g is continuous and by Lemma 2, is also symmetric around 1 / 2. Therefore, g is a continuous function that maps G into G. Consequently, by Brower’s Fixed Point Theorem, ' x* [ G such that g(x*) 5 x*. Let y* 5 R 2 (x*), then R 1 ( y*) 5 R 1 (R 2 (x*)) 5 g(x*) 5 x*. Therefore, (x*, y*) is a symmetric SPNE. The uniqueness of the symmetric SPNE is established using mathematical induction on the number of stores of the two firms. An outline of the proof is presented below. Let the notation (m, n) denote that Firm 1 has m stores and Firm 2 has n stores. The uniqueness of the SPNE for (1, 1) is already established in Anderson and Neven (1991). Assuming the uniqueness of the symmetric SPNE for (1, j); j < n, we show that the symmetric SPNE is unique for (1, n 1 1). Next, we assume the uniqueness of the symmetric SPNE for ( j, n); j < m and n > 1, and establish the uniqueness of the symmetric SPNE for (m 1 1, n);n. Hence, the symmetric SPNE is unique for all (m, n). h D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 175 Proposition 3 justifies the search for location equilibrium among vectors that are symmetric around 1 / 2.11 In the next section, we use Proposition 3 together with Corollary 1 to characterize the SPNE locations. 4. Determination of the SPNE locations In this section we determine the SPNE locations, using the properties derived in the previous section. Proposition 4 describes a mechanism by which the SPNE locations for firms with m and n stores can be used to determine the SPNE locations for firms with km and kn stores, for any positive integer k. To clarify notations used throughout this section, note that only those location points which lie in f0,1g are to be considered. If for some choice of parameter values a location point exceeds one, that location point is to be discarded. For example, suppose S D 1/2 1/2 1 1 1/2 1 d 2 1 x* 5 ], ]]], . . . , ]]]] . d d d Then for d 5 1, x* 5 (1 / 2); and for d 5 2, x* 5 (1 / 4, 3 / 4); etc. Proposition 4. Suppose there are two firms with m and n stores, respectively. Let d be the greatest common divisor ( gcd) of m and n. If the vectors xˆ and yˆ represent the equilibrium locations for firms with m /d and n /d stores facing a market demand p 5 ad 2 Q / 3 at each market point in f0,1g, then the SPNE locations for the original problem are S D S D xˆ 1 1 xˆ d 2 1 1 xˆ x* 5 ], ]], . . . , ]]] d d d and yˆ 1 1 yˆ d 2 1 1 yˆ y* 5 ], ]], . . . , ]]] . d d d Furthermore, if xˆ and yˆ are symmetric around 1 / 2, so are x* and y*. Proof. First consider the problem for firms with m /d and n /d stores facing a market demand p 5 ad 2 Q / 3 at each point in [0,1]. Refer to this problem as the reduced problem. By definition, the locations specified by the vectors xˆ and yˆ satisfy the market-median property of the reduced problem. Now consider the original problem for firms with m and n stores facing a market demand p 5 a 2 Q / 3 at each market point in [0,1]. Consider the locations specified by the vectors xˆ /d 11 In general, we are unable to rule out the possibility of SPNE locations, where both firms locate their stores asymmetrically around 1 / 2. We can rule out asymmetric SPNE for special cases and in general, conjecture that such a scenario does not arise. 176 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 and yˆ /d. Observe that the locations specified by the vectors xˆ /d and yˆ /d are located in the interval [0,1 /d]. Applying the transformation v 5 u /d to the equations that characterize the market-median property of the reduced problem note that these stores satisfy the market-median property of the original problem over the interval [0,1 /d]. Translating each store by 1 /d, we arrive at the vectors (1 1 xˆ ) /d and (1 1 yˆ ) /d which specify locations in [1 /d,2 /d]. Note that these stores satisfy the market-median property of the original problem over the interval [1 /d,2 /d]. Continuing in this manner, it can be seen that the stores located in the intervals [2 /d,3 /d], . . . , [(d 2 1) /d,1] also satisfy the market-median property. Finally, xˆ is symmetric around 1 / 2 implies xˆ 1 1 xˆ m / d 5 1. As a result, xˆ 1 d 2 1 1 xˆ m / d x *1 1 x m* 5 ] 1 ]]]] 5 1, d d and therefore, x* is symmetric around 1 / 2. h In general, it is relatively easier to determine the SPNE locations for firms with fewer numbers of stores. Thus, Proposition 4 is quite useful since it utilizes the solution for a problem with m and n stores, and uses that to solve a problem with km and kn stores, for any positive integer k.12 The following example illustrates how Proposition 4 transforms a complicated problem to a relatively simpler one. Example 1. Determination of SPNE locations for m 5 4 and n 5 2. Let Firm 1’s SPNE store locations be sx 1* , x 2* , x 3* , x 4*d and Firm 2’s SPNE store locations be s y 1* , y *2 d. In principle, it is possible to solve for the SPNE locations by using the market-median property (Corollary 1) and symmetry (Proposition 3), though it may be quite difficult. To see the complexity of this direct approach, consider Firm 1. Note that by symmetry, x 3* 5 1 2 x *1 and x 4* 5 1 2 x *2 . Thus, solutions for x 1* and x 2* are sufficient. Now, by the market-median property, the quantity supplied by store 1 to its left equals the quantity supplied to its right. This generates a quadratic equation involving both sx 1*d 2 and sx 2*d 2 , since the right endpoint of store 1’s market involves x 2* . Similarly, the market-median property for store 2 also generates a quadratic equation involving both sx 1*d 2 and sx *2 d 2 . Moreover, both equations involve y 1* . Thus, we also need to include the equation that emerges from the market-median property of Firm 2’s store 1. Therefore, to solve for x 1* , x 2* and y 1* , we must solve three equations simultaneously, two of which are quadratic equations involving both sx 1*d 2 and sx *2 d 2 . Obviously, it would involve extensive algebra. In this situation, Proposition 4 can be successfully used. Note that in this case, the greatest common divisor (gcd) is 2. Therefore, we need to solve a problem for 12 The converse of Proposition 4 is also true. That is, the solution for firms with km and kn stores can be used to determine the solution for firms with m and n stores. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 177 firms with m 5 2 and n 5 1 stores facing a market demand p 5 2a 2 Q / 3 at each market point in f0,1g, and use them to determine the solution for the original problem. Let xˆ 5sxˆ 1 , xˆ 2d and yˆ 5syˆ 1d be the solution for the sm 5 2, n 5 1d problem. Observe that by symmetry yˆ 1 5 1 / 2, and xˆ 1 and xˆ 2 are located symmetrically around 1 / 2. Thus, the right endpoint of store 1’s market is 1 / 2, which does not involve xˆ 2 . Therefore, by market-median property, xˆ 1 can be solved from a single quadratic equation without involving xˆ 2 . Note that, the quantity supplied to the left of xˆ 1 is x̂ 1 E Faˆ 2 2tsxˆ 2 jd 1 tS]12 2 jDG dj 1 0 and the quantity supplied to the right of xˆ 1 is 1/2 E Faˆ 2 2tsj 2 xˆ d 1 tS]12 2 jDG dj 1 x̂ 1 where aˆ 5 ad 5 2a. Thus, xˆ 1 can be solved from the equation Faˆ 2 2txˆ 1 ]2t G xˆ 1 ]2t sxˆ d 1 1 1 2 5 Faˆ 1 2txˆ 1 ]2t GF]12 2 xˆ G 3t 1 2F]GFS]D 2sxˆ d G 2 2 1 1 2 2 1 It can be checked that ]]]] ]]] aˆ 2œsaˆ d 2 2 aˆ 1 ]21 a 1 2 xˆ 1 5 ]]]]] 5 a 2 a 2 ] 1 ]. 2 2 8 œ Now, we can use Proposition 4 to solve for the original problem. Here, yˆ 1 1 y *1 5 ] 5 ], 2 4 1 1 yˆ 1 3 y *2 5 ]] 5 ], 2 4 and xˆ 1 xˆ 2 1 1 xˆ 1 x *1 5 ], x 2* 5 ], x 3* 5 ]] 2 2 2 and 1 1 xˆ 2 x 4* 5 ]], 2 where ]]] a 1 xˆ 1 5 a 2 a 2 2 ] 1 ] 2 8 œ D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 178 and xˆ 2 5 1 2 xˆ 1 . 4.1. SPNE locations when n 5 km Proposition 4 together with symmetry (Proposition 3) and the market-median property (Corollary 1) characterize the SPNE locations. This sub-section summarizes the results for firms with m and n stores, where n 5 km. The results for n ± km are summarized in the following sub-section. Proposition 5. Suppose there are two firms with m and km stores, respectively, where k > 1 is a positive integer. (i) In equilibrium, Firm 1’s store locations coincide with its monopoly /transport cost minimizing locations.13 That is, s2i 2 1d x *i 5 ]]] ;i 5 1, 2, . . . , m. 2m (ii) If k 5 1, then in equilibrium, each firm’s store locations coincide with its monopoly /transport cost minimizing locations. Proof. Proposition 5 follows from symmetry (Proposition 3) and Proposition 4. To see the proof of part (i), observe that if m 5 1 then at an SPNE, Firm 1 locates its single store at 1 / 2. Now, if m . 1, then from Proposition 4, we first need to solve for SPNE locations for firms with 1 and k stores facing a market demand p 5 am 2 Q / 3 at each market point in f0,1g. The solution for this problem is, x̂ 1 5 1 / 2, independent of k. Therefore, x̂ 1 1 x *1 5 ] 5 ], m 2m and in general, si 2 1d 1 xˆ 1 si 2 1d 1 ]12 s2i 2 1d x *i 5 ]]]] 5 ]]] 5 ]]] ;i 5 1,2, . . . , m. m m 2m This completes the proof of part (i). Interchanging the roles of the two firms, the proof of part (ii) follows. h The following corollary immediately follows from Proposition 5. Corollary 2. If each firm has m stores, then in equilibrium, the stores belonging to competing firms agglomerate at m discrete points. Furthermore, agglomerations 13 Observe that the monopoly location vector is also the socially optimal one. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 179 of stores occur precisely at those locations that would have been chosen by a multi-store monopolist with m stores. Corollary 2 has the following implications. First, it identifies a scenario when the complex problem of determining equilibrium locations for multi-store firms can be reduced to a simple one, where each firm simply behaves as a multi-store monopolist while choosing its locations. Moreover, it can be checked that the conclusion is true for any n > 2 firms. Thus, if there are n firms with m outlets each, then we observe clustering of n competing stores at each of the m locations that would have been chosen by a monopolist with m outlets. Observe that the outlet locations are independent of the number of firms n and depend only on the number of outlets m. Second, the location pattern is consistent with many commonly observed phenomena, such as the clustering of outlets of competing firms at various shopping malls and the clustering of qualities offered by competing firms in the context of vertical product differentiation. Contrast the result with those obtained under price competition. In a spatial model with price competition, stores belonging to competing firms never agglomerate. In the context of single store firms it is shown by d’Aspremont et al. (1979) for mill pricing and by Lederer and Hurter (1986) for spatial discriminatory pricing. In the context of multi-store firms with mill pricing, identical conclusion is drawn by Martinez-Giralt and Neven (1988). It can be checked that the conclusion holds for multi-store firms with spatial discriminatory pricing. Similar result is also demonstrated in the context of vertical product differentiation with price competition. For a duopoly choosing multiple qualities in a quality-price game, Champsaur and Rochet (1989) demonstrate that the qualities chosen by the competing firms never overlap. In fact, the highest quality chosen by one firm is lower than the lowest quality offered by its rival. Intuitively, price competition cannot give rise to a clustering result, since coincident location of competing firms severely intensifies competition and drives profits to zero. Clustering, however, is feasible under spatial Cournot competition, since coincidentally located competing firms now earn positive profits. Hence, in contrast to the results obtained with price competition, spatial Cournot competition gives rise to a result that is consistent with frequently observed clustering of outlets in the real-world.14 14 There are alternative explanations in the literature justifying the agglomeration of competing firms. For instance, firms competing in prices may agglomerate in spatial dimension if their products are differentiated by attribute(s) other than location (see, for example, dePalma et al., 1985; Anderson and dePalma, 1988; De Fraja, 1993; Irmen and Thisse, 1998). A different explanation can be found in Stahl (1982), where consumers search for optimal product characteristics and their search costs are influenced by firm locations, leading to a spatial concentration of demand where sellers find it profitable to agglomerate. In Thill (1997), multi-store firms that choose both locations and product qualities agglomerate for sufficiently high (consumer) reservation price. 180 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 The intuition behind the coincidence of the agglomeration and the monopoly locations is as follows. If two stores owned by rival firms agglomerate at a location jˆ and serve the market at j , then each store supplies a 2 2tu jˆ 2 ju 1 tu jˆ 2 ju 5 a 2 tu jˆ 2 ju amount at j . Observe that if a monopolist with identical technology has a store located at jˆ, it would supply (3 / 2)fa 2 tu jˆ 2 jug at j . Therefore, if the monopoly store locations satisfy the market-median property, the oligopoly agglomeration locations would do the same and consequently, oligopoly agglomeration locations coincide with monopoly store locations. Corollary 2 highlights a result when the firms have equal number of stores. In contrast, the following corollary highlights a result when the firms have different numbers of stores. It follows immediately from Proposition 5 and the marketmedian property (Corollary 1). Corollary 3. If the firms have m and km stores (with k . 1), then Firm 1’s equilibrium location vector coincides with its monopoly location vector. Firm 2’s equilibrium location vector, however, always differs from its monopoly location vector. Corollary 3 reiterates how the complex problem of determining equilibrium locations for a multi-store oligopolist can be reduced to a simple one. Here, Firm 1 simply behaves as a multi-store monopolist while choosing its locations. For example, if the firms have 3 and 3k stores, then irrespective of the values of k, Firm 1 locates its stores at their corresponding monopoly locations; 1 / 6, 1 / 2 and 5 / 6. To see why firm 2 does not locate all of its stores at their corresponding monopoly locations, consider m 5 1 and n 5 2. Clearly, x *1 5 1 / 2. Now, if Firm 2 had chosen its monopoly locations, then yˆ 1 5 1 / 4 and yˆ 2 5 3 / 4. However, the locations yˆ 1 5 1 / 4 and yˆ 2 5 3 / 4 cannot satisfy the market-median property. To see this, consider the store at 1 / 4. The length of the market to its left equals the length of the market to its right. However, Firm 1’s delivered marginal cost is greater in f0, 1 / 4g than in f1 / 4, 1 / 2g, and consequently, Firm 2’s store 1 supplies a larger quantity to its left than to its right. Obviously, this violates the market-median property and yˆ 1 5 1 / 4, yˆ 2 5 3 / 4, cannot be equilibrium locations for Firm 2. In fact, if s y *1 , y 2*d are Firm 2’s equilibrium locations, then y 1* must be less than ]41 and y *2 must be greater than 3 / 4, in order to satisfy the market median property. As shown in Example 1, ]]]] a y *1 5 a 2 a 2 2 ] 1 ]81 , ]41 . 2 œ SD Proposition 6. Suppose there are two firms with m and km stores (with k . 1 being a positive integer), respectively. If k is even, then no two stores agglomerate at the same location. If k is odd, then there are exactly m agglomeration points at D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 181 1 3 2m 2 1 ], ], . . . , ]]. 2m 2m 2m Proof. It follows from Proposition 5 that Firm 1 locates its stores at s2i 2 1d x *i 5 ]]] ;i 5 1, 2, . . . , m. 2m Thus, Firm 1 locates one store at the center of each of the intervals m21 F0, ]m1 G, F]m1 , ]m2 G, . . . , F]] , 1G. m Now, from Proposition 4 it follows that in each of these intervals Firm 2 locates k stores. In fact, by Proposition 3, Firm 2 locates its stores symmetrically around the midpoint in each interval. Therefore, if k is odd, Firm 2 locates one store at the center of each interval m21 F0, ]m1 G, F]m1 , ]m2 G, . . . , F]] , 1G, m yielding m agglomeration points, where each firm locates one of its stores. On the other hand, if k is even, Firm 2 does not locate any of its stores at the center of any of these intervals. As a result, the stores never agglomerate. h The following corollary follows immediately from Propositions 5 and 6. Corollary 4. Suppose the firms have m and km stores. If k is even, then the store locations exhibit complete spatial dispersion. If k is odd, the store locations simultaneously yields spatial agglomeration and dispersion unless m 5 k 5 1. If m 5 k 5 1 (that is, each firm has only one store), then the stores agglomerate at the center of the market. Corollary 4 sheds new light on a prevailing perception in the literature, which states that as opposed to price competition, Cournot competition gives rise to spatial agglomeration. Under price competition, agglomeration intensifies price competition and forces profits to zero. Each firm may earn a higher profit by choosing a different location and consequently, price competition generates dispersed locations. In contrast to the spatial dispersion obtained under price competition, the pioneering work of Hamilton et al. (1989), and Anderson and Neven (1991) demonstrates that if single-store firms compete in quantities (Cournot competition), they agglomerate at the market center. Gupta et al. (1997) relax Anderson and Neven (1991)’s assumption of uniform consumer distribution and demonstrate that the agglomeration result is robust under a wide variety of consumer distribution. Mayer (2000) allows the production cost to differ at various market points and establishes that firms may still agglomerate. Thus, it may indeed 182 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 seem reasonable to conclude that as opposed to price competition, Cournot competition yields spatial agglomeration. Corollary 4, however, shows that this notion is incorrect, if we allow the firms to have multiple stores. Similar to the location pattern obtained with price competition, Cournot competition may also yield complete spatial dispersion. In fact, depending on the number of stores, price competition with spatial price discrimination and Cournot competition generate quite similar location patterns. Furthermore, Cournot competition may simultaneously yield spatial agglomeration and dispersion; the firms may agglomerate at finitely many dispersed points. 4.2. SPNE location when n ± km Propositions 5 and 6 characterize the SPNE locations for firms with m and km stores. In fact, Propositions 4–6, together with the market-median property (Corollary 1) and symmetry (Proposition 3) can be used to determine various combinations of m and n, such as, sm 5 1, n 5 1d, sm 5 1, n 5 2d, sm 5 1, n 5 3d, sm 5 2, n 5 4d. However, we are yet to characterize SPNE locations for combinations such as sm 5 2, n 5 3d and sm 5 3, n 5 4d. In other words, we are yet to characterize the solutions if n ± km, or equivalently, 1 < d , m , n, where d is the greatest common divisor (gcd) of m and n. The following proposition contributes in this regard. In view of Proposition 4, we assume d 5 1 in Proposition 7. Lemma 3 is used to prove the Proposition 7. Lemma 3. Suppose 1 , m , n and d 5 1. Two stores may agglomerate only at 1 / 2. Proof. See Appendix A. Proposition 7. Suppose there are two firms with m and n stores, respectively. Let d be the greatest common divisor ( gcd) of m and n. If 1 5 d , m , n, then (i) The only possible agglomeration of stores may occur at 1 / 2. Stores agglomerate at 1 / 2 if and only if both m and n are odd. (ii) The relative ordering of the stores of both firms is the same as that of their monopoly ordering. That is, for all 1 < i < m and 1 < j < n, x i* _ y j* if and only if 2i 2 1 2j 2 1 ]] _ ]]. 2m 2n Proof. Proof of part (i) follows directly from Lemma 3, the symmetry of the SPNE around 1 / 2, and noting that both m and n cannot be even. Proof of part (ii) is as follows. From part (i), observe that the center-most stores of the firms agglomerate if and only if both m and n are odd. Moreover, they can D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 183 agglomerate only at 1 / 2. Two monopoly locations agglomerate if and only if both m and n are odd, and m11 n11 i 5 ]], j 5 ]]. 2 2 In this case, 2i 2 1 2j 2 1 1 ]] 5 ]] 5 ]. 2m 2n 2 Therefore, x *i 5 y *j if and only if 2i 2 1 2j 2 1 ]] 5 ]]. 2m 2n Now, suppose x i* . y j* , but 2i 2 1 2j 2 1 ]] , ]]. 2m 2n By Proposition 2, however, as a → `, x i* → (2i 2 1) / 2m and y *j → (2j 2 1) / 2n. Therefore, ' aˆ such that x i*saˆ d 5 y j*saˆ d. However, since 2i 2 1 2j 2 1 ]] ± ]], 2m 2n it follows from the discussion above that x i*saˆ d ± y j*saˆ d, a contradiction. h Remark 1. In Proposition 7, we assume d 5 1. For 1 , d , m , n, we solve a problem for firms with m /d and n /d stores facing a market demand p 5 ad 2 Q / 3 at each market point in f0,1g, and use Proposition 4 to obtain the solution for the original problem. It can be seen that for 1 , d , m , n, there are d agglomeration points if and only if both m /d and n /d are odd. Similar to Propositions 5 and 6, Proposition 7 also demonstrates that Cournot competition may yield both agglomeration and dispersion. In fact, from Propositions 5–7, it follows that Cournot competition yields complete dispersion if n 5 km with k being even, or if n ± km and m /d or n /d is even. In this scenario, the location pattern under Cournot competition resembles those under spatial price discrimination, although, in contrast to the price competition, Cournot competition generates overlapping markets. If n 5 km with k > 1 being odd or if n ± km and both m /d and n /d are odd, Cournot competition simultaneously exhibits agglomeration and dispersion. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 184 4.3. Exact SPNE locations To facilitate a better understanding of the SPNE locations, we now present the exact SPNE locations for 1 < m < 3 and 1 < n < 3. For simplicity, we assume t 5 1. Example 2 below presents the general solutions in terms of the demand parameter a. Example 3 presents the solutions for specific values of a. Example 2. General solution in terms of the demand parameter a In Table 1, observe that when the firms have equal number of stores, both firms choose their respective monopoly locations irrespective of the values of a. For sm 5 1, n 5 2d and sm 5 1, n 5 3d, only Firm 1 chooses its monopoly location. For sm 5 2, n 5 3d, however, explicit solutions in terms of a are not feasible. Table 2 (Example 3) presents solutions under specific values of a. Observe that, in this case, neither firm chooses its monopoly locations. To understand how the SPNE locations differ from their respective monopoly locations, we present the numerical solutions, under different values of a, in Example 3 below. Example 3. Numerical solutions for sm 5 2, n 5 3d under different values of a. Table 2 enables us to compare the firms’ SPNE locations to their respective monopoly locations. Note that Firm 1’s monopoly locations are s1 / 4, 3 / 4d 5s0.25, Table 1 General solutions in terms of the demand parameter a, 1 < m < 3, 1 < n < 3, m < n [m\n] n51 n52 n53 x *1 5 1 / 2 x *1 5 1 / 2 x *1 5 1 / 2 m51 ]]]] y *1 5 a 2Œa 2 2 a / 2 1 1 / 8 y *1 5 1 / 2 ]]]] y *1 5 (12a 1 5 2Œ144a 2 1 16a 1 12) / 26 y *2 5 1 / 2 y *2 5 1 2 y *1 y *3 5 1 2 y *1 x *1 5 1 / 4, x 2* 5 3 / 4 m52 y *1 5 1 / 4, y *2 5 3 / 4 Explicit solutions are not feasible See Table 2 below for solutions under specific values of a x *1 5 1 / 6, x *2 5 1 / 2, x *3 5 5 / 6 m53 y *1 5 1 / 6, y *2 5 1 / 2, y *3 5 5 / 6 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 185 Table 2 Numerical solutions for (m 5 2, n 5 3) under different values of a y *1 5 0.1586 x *1 5 0.2526 y *2 5 0.5 a52 x *2 5 0.7474 y *3 5 0.8414 y *1 5 0.1637 x *1 5 0.2508 y *2 5 0.5 a55 x *2 5 0.7492 y *3 5 0.8363 y *1 5 0.1652 x *1 5 0.25038 y *2 5 0.5 a 5 10 x *2 5 0.74962 y *3 5 0.8348 0.75d and Firm 2’s monopoly locations are s1 / 6, 1 / 2, 5 / 6d 5s0.16667, 0.5, 0.8333d. Comparing the actual locations to the firms’ monopoly locations, observe that the SPNE locations are quite close to their monopoly locations. Furthermore, as the demand parameter a increases, the SPNE locations quickly converge to their respective monopoly locations. Numerical simulations with other values of m and n confirm similar location pattern. 5. Conclusion We contribute to the literature of spatial competition by considering multi-store firms in the context of spatial Cournot competition with endogenous location choice. Previous work on spatial competition that allows the firms to choose their locations mostly assumes single-store firms. The brief literature that considers multi-outlet firms with endogenous location choice, considers either price competition (Martinez-Giralt and Neven, 1988) or fixed prices (Teitz, 1968; Thill, 1997). By using Cournot competition we demonstrate results that describe the real world better than those obtained under price competition. In the context of price competition with mill pricing, Martinez-Giralt and Neven (1988) demonstrate that each firm locates all of its stores at the same market point, and thus, the firms do not take up the opportunity of opening up multiple outlets. This result is rather 186 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 intriguing given the abundance of firms with multiple outlets. This puzzling outcome, however, does not arise in spatial Cournot competition where each firm chooses distinct locations for its stores. Moreover, under both mill pricing and spatial discriminatory pricing, locations of competing firms do not agglomerate (d’Aspremont et al., 1979; Lederer and Hurter, 1986; Martinez-Giralt and Neven, 1988). A similar result is demonstrated by Champsaur and Rochet (1989) in the context of vertical differentiation with price competition, where qualities offered by competing firms never coincide. In contrast, under Cournot competition, outlets of competing firms may agglomerate at finitely many locations; a result that is consistent with the clustering of stores of competing firms at various shopping malls and clustering of qualities offered by competing firms. Also, we identify situations when, in equilibrium, a firm simply chooses its monopoly locations. Furthermore, when a firm’s equilibrium store locations differ from its monopoly locations, we demonstrate that the equilibrium locations converge to their respective monopoly locations, as the demand becomes large. Even for smaller demands, numerical simulations indicate that the equilibrium store locations are quite close to their respective monopoly locations. Hence, we demonstrate that the complex problem of determining equilibrium locations for multi-store firms can be approximated by a simple one, where each firm simply behaves as a multi-store monopolist while choosing its locations. We also address a prevailing notion in the literature that follows by comparing spatial models with price and Cournot competitions. Previous work with single-store firms shows that price competition generates spatial dispersion of firms, while Cournot competition gives rise to spatial agglomeration (Anderson and Neven, 1991; Hamilton et al., 1989; Gupta et al., 1997, Mayer, 2000). We demonstrate that if the firms have multiple stores such a notion is incorrect; Cournot competition may as well give rise to complete spatial dispersion. In fact, depending on the number of stores, spatial price discrimination and Cournot competition generate quite similar location patterns. The paper can be extended in several ways. For example, it assumes exogenously fixed number of stores for each firm. The number of stores, however, can be determined endogenously. The fixed cost needed to set up a store is likely to be significant and can be used to determine the number of stores for each firm.15 A three stage game can be analyzed where in stage I, the firms decide on the number of stores. In stage II, they decide on store locations and in stage III, they compete in quantities. Initial investigation suggests that even if the firms have identical set-up cost per store, they may choose different number of stores. For example, suppose the demand at each market point is p 5 4 2 Q and the fixed set-up cost 15 The number of firms can be made endogenous as well. The fixed set-up cost can be used to determine the number of firms. D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 187 per store is 1.5, then there are three SPNE equilibrium outcomes for the number of stores: sm 5 2, n 5 0d,sm 5 0, n 5 2d and sm 5 1, n 5 1d. It is also possible for the firms to have positive but different number of stores. For example, if the fixed set-up cost above is decreased to 0.20, there are only two SPNE outcome: sm 5 2, n 5 1d and sm 5 1, n 5 2d. In this case, there is no equilibrium where the firms choose equal number of stores. In general, however, it is difficult to express the number of stores as a function of the fixed set-up cost, since the solutions depend critically on the specific values of the fixed set-up cost.16 Another way to extend the paper would be to relax the assumption that the transport cost is linear in quantities. The marginal cost of delivering a unit of output at a specific market point may be less than the marginal cost of delivering the previous unit at that point. Therefore, there are situations where it would be appropriate to consider transport cost that is concave in quantities at each market point. (Observe that this problem does not arise in spatial models where the consumers are assumed to buy only one unit at each market point.) Initial investigation in this regard indicates a potentially challenging problem, which remains an agenda for future research. The abundance of spatial competition among multi-store firms has motivated the present study. Certainly, the analysis would also be useful to study various other important economic phenomena. For example, any horizontal merger in a spatial model would give rise to firm(s) with multiple stores. To properly evaluate the implications of such a merger, it is essential to incorporate the possibility of relocations of stores following the merger. Obviously, such a task would involve competition among firms with multiple stores.17 Overall, further study involving spatial competition among multi-store firms is clearly worthwhile. Acknowledgements We are grateful to Simon Anderson, Wolfgang Mayer, David Sappington and two anonymous referees for valuable comments. Appendix A. Proof of Lemma 3 Suppose, if possible, two stores of different firms agglomerate at j [f0,1g with 16 From these examples it appears that entry deterrence is feasible in spatial Cournot competition. The result is similar to that obtained under spatial price discrimination (Gupta, 1992; Eaton and Schmitt, 1994). In contrast, spatial preemption may be infeasible in spatial competition involving mill pricing (Judd, 1985). 17 See Norman and Pepall (2000) for a recent analysis of horizontal mergers in a spatial Cournot model. 188 D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190 j ± 1 / 2. First, assume that the stores at j are not the leftmost or the rightmost store for the respective firms. If at least one firm has a leftmost or rightmost store at j , then the proof below can be modified suitably. Note that, since the stores agglomerate at j , each store supplies the same quantity at j and its vicinity. Therefore, to satisfy the market-median property, the market of one store must be a subset of the market served by the other store. Moreover, j is not only the quantity-median of the store with the smaller marker, but also the midpoint of its market. Thus, its neighboring stores (which are owned by the same firm) are symmetrically situated around j . This in turn implies that the quantities offered by the two firms are symmetric around j up to markets of these neighboring stores or the end points of the market of the store at j owned by the other firm, whichever is nearer to j . Proceeding in this manner, it can be seen that the neighboring stores of the other firm are also located symmetrically around j . Repeating the same argument, note that the quantities offered by the two firms and the stores of the two firms are symmetric in a closed interval I, with midpoint j and the endpoints being the market boundaries of a store of Firm 1 and another store of Firm 2. Let J be the minimal subset of I with the above two properties. If J 5f0,1g, then j 5 1 / 2 which contradicts the hypothesis. On the other hand, if J ,f0,1g, we establish below a contradiction to the hypothesis that d 5 1. Suppose that J 5fa, bg ,f0,1g. Note a . 0 and / or b , 1. Taking reflection(s) of J about a and / or b, we obtain a copy / copies of J. The interval J together with its reflected copy / copies may cover the entire interval f0,1g. Otherwise, taking successive reflection(s) about the endpoint(s) of the latest copy / copies of J, we eventually cover the entire market. 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