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Regional Science and Urban Economics 38 (2008) 32 – 48 www.elsevier.com/locate/regec

Welfare properties of spatial competition with location-dependent costs Hiroshi Aiura ⁎, Yasuhiro Sato Graduate School of Environmental Studies, Nagoya University, Japan Accepted 1 August 2007 Available online 4 September 2007

Abstract We analyze a two-firm spatial competition model in which firms must transport raw materials from a raw material site to their locations in order to produce. The model has two equilibrium configurations: (i) a symmetric one in which firms locates equidistantly from the raw material site, and (ii) an asymmetric one in which one firm locates at the raw material site and the other locates distantly from it. We show that these two configurations are possible as multiple equilibria, that the asymmetric equilibrium is more efficient than the symmetric one, and that the social welfare first falls then rises as transport costs decline. © 2007 Elsevier B.V. All rights reserved. JEL classification: L13; R39 Keywords: Spatial competition; Material transportation; Asymmetric equilibrium

1. Introduction Over the past century, improvements in transportation technology and decreases in the significance of heavy manufactured goods in consumption bundles have been observed. Due to these historical changes, the costs of transporting raw materials are considered to play a less and less important role in the location decisions of many retail firms. In view of this, volumes of recent studies on retail firm location have focused on the effects of the distribution of consumers (and an those of shopping costs). However, it would also be interesting to investigate the effects of the historical change mentioned above on firm location, i.e., to explore the impact of the decline in the costs of transporting raw materials on firm location decisions and on social welfare. Moreover, it is still true that some retail firms must consider the costs of the transport of the raw materials for their products or services to their places of business. In such a case, they face location-dependent costs, which would influence their location and price decisions. As an example, consider stores (or restaurants) that sell (serve) seafood. In the neighborhood of a fishing port or a fish market, we often find stores and restaurants that provide seafood at reasonable prices, whereas we also find stores that offer similar seafood at high prices in a city center that is far from a fish market. Locating in a city center enables stores to access a large consumer market, whereas locating in the neighborhood of a fish market reduces transportation costs. Combined with the shopping costs for consumers, these differences in locational characteristics would lead to different prices for ⁎ Corresponding author. Furocho, Chikusaku, Nagoya, 464-8601, Japan. Tel.: +81 52 789 4733. E-mail addresses: [email protected] (H. Aiura), [email protected] (Y. Sato). 0166-0462/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2007.08.007

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similar seafood. Other examples include stores that sell fresh vegetables and fruits, and local brand goods such as local brand beer, wine, and sake. This paper studies the co-effects of access to a consumer market and the transport costs of raw materials on the location and price decisions of firms, and reveals what will happen with a change of raw material transport costs. A large number of studies have previously investigated the location decision of firms competing within a demand space; the models described in these studies are called “spatial competition models.” In his pioneering work, Hotelling (1929) analyzed a situation in which two firms compete for consumers with respect to location and price in a linear market in which consumers are uniformly distributed. Each consumer chooses one firm to buy one unit of a good inelastically. In doing so, she/he bears the shopping cost depending linearly on the distance between her/his location and the firm's location. A consumer chooses a firm that offers the lower sum of the price and shopping cost. Therefore, due to spatial differentiation, even if two firms charge different prices, a firm with the lower price cannot necessarily obtain all the demand. It is then possible that for given locations of firms, different prices appear as an equilibrium outcome. However, in the original Hotelling model, price equilibrium does not necessarily exist for given locations of firms. d'Aspremont et al. (1979) modified the Hotelling model and assumed that the shopping cost for a consumer is a quadratic function of the distance between the firm's location and the consumer. This modification enables us to obtain a price equilibrium for any firm's location. They then analyzed a case in which firms choose location then price endogenously, and showed that, in a sub-game perfect Nash equilibrium, two firms separately locate at either end of the linear market; i.e., each firm tries to locate as far from the other firm as possible. This result is called the “maximum differentiation principle.” To this date, many studies have extended the basic spatial competition model and obtained various results.1 Alongside the literature on spatial competition, studies pioneered by Weber (1929) analyzed the effects of locationdependent costs on firms' locations. They explicitly considered the transportation of raw materials and investigated the optimal locations of firms that minimize the total costs of transporting raw materials from a raw material site to firms' locations and products from firms' locations to consumer places. By combining these two approaches, we analyze a location-then-price spatial competition model which includes location-dependent costs. To the best of the authors' knowledge, Karlson (1985) would be the most relevant to our purpose.2 Karlson (1985) considered a circular market in which consumers are uniformly distributed and two firms choose locations. A raw material site is assumed to locate in the circular market. A firm must transport raw materials from the raw material site to its location in order to produce, and the transportation cost depends linearly on the distance. Hence, the cost of transporting raw materials depends on the firm's location. Moreover, Karlson (1985) assumed that consumer's demand is elastic, that the shopping cost is linear in the distance, and that firms choose their locations and prices simultaneously. In this game, one firm can obtain all the demand by choosing the same location as the rival firm and cutting its price to slightly below the rival firm's price. This leads to the non-existence of a Nash equilibrium. Therefore, a local Nash equilibrium was examined in Karlson (1985).3 It was shown that in a local Nash equilibrium, firms locate equidistantly from the raw material site (i.e., symmetric locations with respect to the raw material site) and choose the same price. This paper modifies the model developed by Karlson (1985) by assuming that the shopping cost is quadratic in the distance, that the consumer's demand is inelastic, and that the game is two-stage (location-then-price) a la d'Aspremont et al. (1979).4 Under these modifications, we can obtain a subgame-perfect Nash equilibrium (see d'Aspremont et al., 1979). We investigate equilibrium locations and prices in a linear market model, in which a raw 1

For example, existing studies have examined the effects of changes in the shape of the market (circular, square, etc), in the demand function (elastic, inelastic), in the distribution of consumers (trigonometric, normal distribution), and in the number of firms. 2 Spatial competition which includes the costs of transporting raw materials is examined also in Heal (1980) and in Matsushima (2004, 2006). Heal (1980) considered a circular market in which consumers are uniformly distributed and firms choose their locations. However, because it is assumed that a raw material site is located at the center of a circle, the cost of transporting raw materials is the same for all firms. Matsushima (2004, 2006) considered models with upstream and downstream firms, where downstream firms engage in the location-then-price competition. Upstream firms also locate in the same space, and decide the prices of raw materials to sell to downstream firms. However, in contrast with our model, it is assumed that upstream firms must bear the costs of transporting materials. Under these settings, the effects of a vertical merger were discussed. 3 A local Nash equilibrium is a set of prices and locations such that no firm can benefit from slightly changing its location or price given the other firm's price and location. 4 We can interpret the quadratic shopping cost as that in which consumers bear not only the monetary costs of going to a store but also its psychological costs, with the latter costs increasing more than proportionally to the distance.

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material site is located at the center of the linear market. Our main focus is to examine welfare properties, none of which were investigated in Karlson (1985). We show that there are two types of equilibrium. In one type of equilibrium, firms' locations are symmetric with respect to a raw material site. In the other type of equilibrium, one firm locates at the raw material site and the other locates away from it. The former (latter) emerges when the raw material transport cost is low (high) relative to the shopping cost of consumers. Hence, the configuration in asymmetric equilibrium is consistent with the examples described above. It is also shown that these two types of equilibrium may exist at the same time (as multiple equilibria). On the one hand, firms try to differentiate geographically in order to avoid price competition. On the other hand, firms have an incentive to locate in the neighborhood of the raw material site in order to reduce the transport cost. These two incentives work equally for both firms in symmetric equilibrium. In asymmetric equilibrium, one firm benefits from locating at the raw material site, whereas the other benefits from putting as high a price as possible due to sufficient geographical differentiation. The optimal configuration is symmetric when the raw material transport cost is small, whereas it is asymmetric when the transport cost is large. By comparing the optimal with the equilibrium configurations, we show that the firm locating away from the raw material site is located too distantly from the raw material site in the equilibrium. This is because the incentive of geographical differentiation works too intensely. We also examine the welfare level in each equilibrium, and show that asymmetric equilibrium is superior to symmetric equilibrium from the welfare viewpoint. More importantly, the welfare level first falls then rises as the raw material transport cost decreases. When the transport cost falls from high levels, price competition becoming more important for firms, firms start to move apart. This will also increase the distance between the raw material site and the location of at least one firm, which raises the total transport costs and worsens the social welfare. When the transport cost becomes sufficiently small, the strategic location effect is small and the direct effect from declines in transport cost becomes prominent. Therefore, decreases in transport cost raise the welfare level. This paper is organized as follows. Section 2 gives a brief description of the model. Section 3 develops the model, and shows the equilibrium. Welfare properties are analyzed in Section 4. Section 5 concludes the paper. 2. Brief description of the model This section gives a brief description of the model described in this paper. We consider a linear market in which consumers are uniformly distributed and two firms that sell the same good choose locations in it. They choose locations then prices sequentially a la d'Aspremont et al. (1979). In order to sell products, firms have to transport raw materials from a raw material site to their locations. A raw material site exists at the center of the consumer market. The cost of transportation is linear in the distance. Consumers bear the shopping cost that is quadratic in the distance between their locations and the firm's location, and buy the good from the firm that offers the lower sum of the price and shopping cost. The demand for goods is assumed to be inelastic. Under these settings, we explore the subgame-perfect Nash equilibrium and its welfare properties. 3. Model and its equilibrium Consider a continuum of consumers of measure one. There are two firms (firms a and b) that sell the same good. Consider also a linear market (i.e., a line) on which consumers are distributed and two firms choose locations.5 We assume that consumers are uniformly distributed on the line segment [− 1/2, 1/2]. At its center, there exists a raw material site whose location is indexed as 0. Locations of firms are denoted by xa and xb, respectively. We assume that it is possible for firms to locate outside of the line segment [− 1/2, 1/2].6 Firms transport raw materials from the raw material site to their locations, produce the good, and sell it. The prices of the good are denoted by pa and pb. We can assume that xa ≤ xb without loss of generality. Consumers buy one unit of the good inelastically. In doing so, they have to bear the shopping cost that is quadratic in the distance. The location of a consumer is denoted by x. They buy the good from the firm that offers a lower sum of the 5 We also investigated an equilibrium for a linear market with an arbitrary location of a raw material site and for a circular market, and obtained results very similar to those obtained here. See Aiura and Sato (2006). 6 This assumption is made only for expositional simplicity. Even if firms can locate only on the line segment, the major results are unaltered.

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price pi(i = a, b) and shopping cost t(xi − x)2, where t is a positive constant representing the level of shopping cost. Hence, a consumer located at x buys the good from a firm that satisfies min pi þ tðxi xÞ2 :

iafa;bg

ð1Þ

On the linear market, there is one location X⁎ of a consumer who is indifferent between buying from firm a and firm b. Consumers on [− 1/2, X⁎] buy from firm a whereas those on [X⁎, 1/2] buy from firm b. Using xi and pi, X⁎ is given as X⁎ ¼

xa þ xb pa pb þ : 2 2tðxa xb Þ

ð2Þ

Because the total number of consumers is one and the demand is perfectly inelastic, the volume of sales, denoted by Qi (i = a, b), is 1 1 Qa ¼ X ⁎ þ ; Qb ¼ X ⁎: 2 2

ð3Þ

The transport cost of the raw materials necessary to produce one unit of good is assumed to be linear in the distance. Hence, the transport cost for firm i is given by TliQi, where T is a positive constant representing the level of transport cost, and li denotes the length of the distance between the raw material site and the firm's location.7 For the sake of expositional simplicity, we assume that firms must bear no cost other than the transport cost. The profit of each firm πi (i = a, b) is given as xi ; if xi b0 pi ¼ ð pi Tli ÞQi ; li ¼ : ð4Þ xi ; if xi z0 We consider a location then price game, which we solve by backward induction. In the second stage, firms determine their prices in order to maximize their profits given the locations of firms: max pi jxi :fixed ; i ¼ a;b: pi

ð5Þ

The first-order conditions for profit maximization give a price equilibrium as follows:8 1 1 p⁎a ¼ tðxb xa Þð3 þ xa þ xb Þ þ T ð2la þ lb Þ; 3 3

ð6Þ

1 1 p⁎b ¼ tðxb xa Þð3 xa xb Þ þ T ðla þ 2lb Þ: 3 3

ð7Þ

Consider an increase in li. The above result shows that two-thirds of increases in transport cost due to this change are reflected in increases in the price of firm i and that one-third of them are absorbed in the price of the rival firm j( j ≠ i). Put differently, a firm cannot shift all burdens from increases in transport cost to its own price. Substituting (6) and (7) into (2), we have X⁎ ¼

xa þ xb T ðlb la Þ þ : 6tðxb xa Þ 6

ð8Þ

We can derive a location equilibrium of the first stage by substituting (3) and (6)–(8) into (4), and solving the firstorder conditions for the following maximization: max pi ; xi

i ¼ a;b:

ð9Þ

7 In our model, firms locating at different places face different costs. Here, due to the costs’ dependency on location, our model does not suffer from the problem of non-existence of equilibrium that is observed in the model described in Ziss (1993). 8 For the derivation of the first-order conditions, see Appendix A.

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Let k denote the relative level of transport cost to shopping cost (i.e., k = T/t). A location equilibrium is described in the following proposition. Proposition 1. Consider a linear market in which the raw material site is located at the market's center. A location equilibrium (xa⁎, xb⁎) is given as follows: (i) When 0 ≤ k b 3/4, x ⁎a ¼

3 2k 3 2k ; xb⁎ ¼ : 4 4

(ii) When k = 3/4, any pair (xa⁎, xb⁎) that satisfies 3=4V xa⁎V0; xb⁎ ¼ xa⁎ þ 3=4: pﬃﬃﬃ (iii) When 3/4 b k b 3(4 2 − 5) / 2, x ⁎a ¼

3 2k ⁎ 3 2k ; xb ¼ ; 4 4

or x ⁎a ¼ 0; xb⁎ ¼

3k ; 3

or x ⁎a ¼

3k ; xb⁎ ¼ 0: 3

pﬃﬃﬃ (iv) When 3(4 2 − 5) / 2 ≤ k b 3, x ⁎a ¼ 0; xb⁎ ¼

3k ; 3

or x ⁎a ¼

3k ; xb⁎ ¼ 0: 3

(v) When k ≥ 3, x ⁎a ¼ 0; xb⁎ ¼ 0: Proof. See Appendix A. Note first that the equilibrium locations depend on the relative transport cost k. Because firms can locate outside of [− 1/2, 1/2], the absence of transport cost (i.e., k = 0) case corresponds to the case analyzed in Lambertini (1994) and Tabuchi andp Thisse (1995).9 Fig. 1 describes the equilibrium locations of firms. From this figure, we can see that when ﬃﬃﬃ 3/4 ≤ k b 3(4 2 − 5) / 2, multiple equilibria exist: one equilibrium is such that firms locate equidistantly from a raw material site (we call this “symmetric equilibrium”). The other equilibrium is such that one firmplocates at a raw ﬃﬃﬃ material site and the other locates away from it (we call this “asymmetric equilibrium”). When 3(4 2 − 5) / 2 ≤ k b 3, only asymmetric equilibrium exists. In the following, we analyze the asymmetric equilibrium in detail. Asymmetric equilibrium does not exist when k b 3 / 4. A firm can fully avoid price competition by moving away from the other firm. Therefore, when k is sufficiently small, the effect of price competition dominates the effect of transport cost and only 9 Both firms locate outside of [− 1/2, 1/2] for k b 1/2, and one of the firms in the asymmetric equilibrium locates outside of [− 1/2, 1/2] for 3/4 ≤ k b 1/2. These results may seem surprising. Regarding these results, Tabuchi and Thisse (1995) stated “It reflects the fact that price competition under quadratic transportation costs is very fierce, leading firms to set up far apart from each other. In the real world, shopping centers located away from the limits of small and medium size cities may indeed be observed, presumably because they want to be separated.” When we assume that firms can locate only on [− 1/2, 1/2], firms locate on the edges of the segment (i.e., x⁎a = − 1/2, x⁎b = 1/2) for small k.

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Fig. 1. Location equilibrium.

symmetric equilibrium exists. When both symmetric and asymmetric equilibria are possible ((iii) in Proposition 1), it can be easily shown that a firm that locates at the raw material site in asymmetric equilibrium gains higher profit than its profit in symmetric equilibrium, whereas a firm that locates distant from the raw material site obtains lower profit than that in symmetric equilibrium. Consider an asymmetric equilibrium in which firm a locates at the raw material site and firm b locates separately from it ((iii) and (iv) in Proposition 1). The prices, volumes of sales, and profits in this equilibrium are given, relatively, as: ð36 6k 2k 2 Þt ð18 þ 15k 7k 2 Þt ; pb⁎jx ¼0; x ¼x ⁎ ¼ ; a b b 27 27 6þk 3k ; Qb*jx ¼0; x ¼x ⁎ ¼ : Q ⁎a jx ¼0; x ¼x ⁎ ¼ a b b a b b 9 9 2ð3 kÞð6 þ kÞ2 t 2ð3 kÞ3 t pa⁎jx ¼0; x ¼x ⁎ ¼ ; pb⁎jx ¼0; x ¼x ⁎ ¼ : a b b a b b 243 243 p ⁎a jxa ¼0; xb ¼xb⁎ ¼

We can observe that π⁎a N π⁎b and Q⁎a N Q⁎b always hold, which represents the fact that firm b locating separately from the raw material site produces less than does firm a in order to reduce the total transport costs. Regarding prices, if 3/ 4 ≤ k b 6/5, pa⁎ N pb⁎, and if 6/5 ≤ k b 3, pa⁎ b pb⁎. As the relative level k of transport cost increases, firm b is more inclined to reduce its volume of sales and raise its price to a higher level than does firm a. Moreover, because Qa⁎ is an increasing function of the relative level k of the transport cost to the shopping cost and Qb⁎ is a decreasing function of k, the difference between the two firms' volumes of sales increases as k increases. Even if the transport cost per unit of good is high, a firm can avoid bearing large transport costs by decreasing the volume of sales. And even if the volume of sales is small, a firm can obtain sufficient profits by raising price. Therefore, when the transport cost is high and k is large, it is the best response for firm b to locate away from firm a (i.e., the raw material site) and to set a high price that is made possible by geographical differentiation. In contrast, firm a can take advantage of locating at the raw material site, leading to the result that firm a earns more than does firm b. In this equilibrium, firm b doesn't have the incentive to move closer to the raw material site because doing so leads to fiercer price competition with the rival firm. Thus, asymmetric equilibrium emerges. In Karlson (1985), the shopping cost is linear in the distance and the effect of geographical differentiation is weak, which makes it impossible for firm b to raise its price without largely reducing the volume of sales. Hence, asymmetric equilibrium is never observed in Karlson (1985). Moreover, in the model described in this paper, we also observe symmetric pﬃﬃﬃ equilibrium, in which two firms move closer to the raw material site as k becomes large. When k increases to 3(4 2 − 5) / 2, symmetric equilibrium becomes unsustainable. This result is also different from the one obtained by Karlson (1985) in which two firms move closer and

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closer to the raw material site in a symmetric configuration until both firms eventually locate at a raw material site. This difference in results comes from the difference in the assumption regarding shopping cost. For a shorter distance, the marginal shopping cost is smaller and leads to fiercer price competition and more differentiation in the case of a quadratic shopping cost, which d'Aspremont et al. (1979) assumed, than in the case of a linear shopping cost. Hence, as the two firms move closer, symmetric equilibrium collapses under the assumption of a quadratic shopping cost. 4. Welfare analysis Proposition 1 implies that our model can have multiple equilibria. We should then examine which equilibrium is preferable from the viewpoint of social welfare. We then discuss how changes in material transport costs affect the social welfare. As a criterion, we consider the social surplus, i.e., the consumer surplus plus the producer surplus. In our setting, this is equivalent to some constant minus the sum of all costs (i.e., the firms' costs of transporting raw materials plus the consumers' shopping costs). Hereafter, we call the sum of all costs the social cost. Then, when the social cost is at a minimum, the social surplus is at a maximum.10 If firms locate on the same point, the social cost SC is given as Z 1=2 SC ¼ ½tðxa xÞ2 þ Tla dx; ð10Þ 1=2

whereas when firms locate on different points, SC becomes Z X Z 1=2 SC ¼ ½tðxa xÞ2 þ Tla dx þ ½tðxb xÞ2 þ Tla dx; 1=2

ð11Þ

X

where X is such that consumers in [− 1/2, X ) buy the good from firm a and consumers in [X, 1/2] buy the good from firm b. Taking into account that it is optimal to serve each consumer by the firm with a lower sum of transport and shopping costs, the optimal X is determined as X SO ¼

xa þ xb T ðla lb Þ þ : 2tðxa xb Þ 2

ð12Þ

Substituting this into SC, we investigate xa and xb that minimize SC. The following proposition gives the optimal locations of firms: Proposition 2. Consider a linear market in which the raw material site is located at the center of the market. The optimal locations of firms are given as follows: (i) When k b 1/4, xSO a ¼

1 2k ; 4

xSO b ¼

1 2k : 4

(ii) When k = 1/4, any pair (xa⁎, xb⁎) that satisfies * * 1=4V x* a V0; xb ¼ xa þ 1=4: (iii) When 1/4 b k b 1, SO xSO a ¼ 0; xb ¼

1k ; 3

or xSO a ¼ 10

1k ; xSO b ¼ 0: 3

− in terms of money, the social surplus is given by U− - SC. If we explicitly assume that the utility from consuming one unit of the good is U

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

39

Fig. 2. Optimum location.

(iv) When k ≥ 1, SO xSO a ¼ 0; xb ¼ 0:

Proof. See Appendix B. Fig. 2 describes the optimum locations of firms. If there is no transport cost of raw materials (k = T / t = 0), the optimal locations are xa = −1/4 and xb = 1/4, implying that the two firms should disperse in order to reduce the total shopping costs. Introducing the transport cost, we can see that both firms should move closer to the raw material site. Note here that the total shopping costs are the smallest when xa = − 1/4 and xb = 1/4, whereas the total transport costs are the smallest under the full concentration of firms on the raw material site. When k b 1/4, the optimal locations are symmetric with respect to the raw material site and firms have to locate closer to the raw material site as compared with xa = − 1/4 and xb = 1/4. When the relative level k of transport cost to shopping cost is very large, the effect of the transport cost dominates the effect of the shopping cost and it becomes optimal that both firms locate at 0, i.e., the raw material site. When k takes a moderate value, it is optimal that firm b locates between 0 and 1/4. In this case, both the transport and shopping costs matter. Hence, one firm should locate on the raw material site in order to avoid the transport costs whereas the other firm should locate apart from it in order to reduce the shopping costs. Thus, the optimal locations are asymmetric. Note further that in any case, the optimal location of firm b is closer to the raw material site than its asymmetric equilibrium location; i.e., the equilibrium locations are too dispersed. In asymmetric equilibrium, firm b benefits from a geographical differentiation that gives firm b market power in the neighborhood of its location. This leads to an incentive of firm b to locate more distantly from the raw material site (where firm a locates) than in the optimum. The difference in the social cost between the optimum and the equilibrium can be seen as follows. First, SC in the optimum is given as follows: (i) When 0 ≤ k b 1/4, SC SO ¼

1 þ 12k 12k 2 t: 48

(ii) When 1/4 ≤ k b 1, SC SO ¼

5 þ 12k 12k 2 þ 4k 3 t: 108

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(iii) When 1 ≤ k, SC SO ¼

9 t: 108

In contrast, SC in the equilibrium is given by Z X⁎ Z 1=2 SC ¼ ½tðxa xÞ2 þ Tla dx þ ½tðxb xÞ2 þ Tla dx; X⁎

1=2

ð13Þ

where X⁎ is described in (8). Hence, SC in the equilibrium is as follows: (i) When 0 ≤ k b 3/4, SC ⁎ ¼

13 þ 12k 12k 2 t: 48

(ii) When k = 3/4, 43 96xa 128x2a ðfor 3=4V xa V0Þ: 192 pﬃﬃﬃ (iii) When 3/4 b k b 3(4 2 − 5) / 2, SC ⁎ ¼

SC ⁎ ¼

13 þ 12k 12k 2 t; 48

SC ⁎ ¼

189 þ 108k 108k 2 þ 20k 3 t: 972

or

pﬃﬃﬃ (iv) When 3(4 2 − 5)/2 ≤ k b 3 SC ⁎ ¼

189 þ 108k 108k 2 þ 20k 3 t: 972

(v) When k ≥ 3, SC ⁎ ¼

9 t: 108

Fig. 3 describes the social costp inﬃﬃﬃthe optimum and that in the equilibrium. Keep in mind that our model has multiple equilibria for 3/4 b k( = T / t) b 3(4 2 − 5) / 2. In this case, we can see from Fig. 3 that the social cost is smaller in

Fig. 3. Social costs.

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asymmetric equilibrium than in symmetric equilibrium. Stated differently, asymmetric equilibrium is superior to symmetric equilibrium from the welfare viewpoint. The following proposition describes the welfare properties of an equilibrium. Proposition 3. In asymmetric equilibrium, a reduction in the transport cost increases the social cost and lowers the social welfare. In symmetric equilibrium, the social cost first rises then falls as the transport cost decreases. In this figure, we see the following. We start with high relative transport cost. Very high k implies that both firms locate at 0 (see (v) in Proposition 1). Consider continuous declines in the relative transport cost k. At some moment, one firm (say, firm b) starts moving away from 0, and asymmetric equilibrium emerges. In this asymmetric equilibrium in which xa⁎ = 0, xb⁎ = 1 − k/3 and X⁎ = k/9 + 1/6 (see (iii) and (iv) in Proposition 1), the social costs can be described as Z SC ¼

X⁎

1=2

Z

¼

Z ½tx2 dx þ

1=2 X⁎

⁎ ⁎ ½tðxb xÞ2 þ Txb dx

Z 1=2 Z 1=2 ⁎ ½tx2 dx þ ½tðxb⁎ xÞ2 dx þ ½Txb dx X⁎ X⁎ 1=2 |ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} X⁎

Total shopping costs

ð14Þ

Total transport costs

! 2 2 2 1 x⁎ x⁎ x⁎ b b ⁎ ⁎ ⁎ ⁎ ⁎ b ⁎ þ þ xb X xb X xb X ¼t þ tk : 12 4 2 2 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Total shopping costs

Total transport costs

In the last line of (14), the first and second terms represent the total shopping costs and the total transport costs, respectively. We can see that changes in the relative transport cost k affect the social costs by changing the locations of firms (the strategic effect) and by changing the total transport costs for a given distance (the direct effect). In fact, differentiating SC with respect to k yields ⁎ ⁎ dSC 1 ⁎2 2x⁎X ⁎ dxb þ tð2x⁎ X ⁎ x⁎2 Þ dX ¼ t þ x⁎ b þX b b b dk 4 dk dk |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Strategic effect in the shopping costs

⁎ ⁎ 1 dxb dX ⁎ xb ⁎ ⁎ ⁎ ⁎ X tkxb xb X þ tk þt 2 dk dk 2 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Strategic effect in the transport costs

¼

Direct effect

2t ⁎ 2t t ⁎2 xb ð1 2xb⁎Þ xb⁎ð1 x⁎ xb bÞ þ 9 3 3 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄ{zﬄ} Shopping costs

Transport costs

|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

Direct effect

Strategic effect

2t t ⁎2 t xb ¼ xb⁎ð2 x⁎ ¼ xb⁎ð4 5x⁎ bÞ þ b Þb0: 9 3 9 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄ{zﬄ} Strategic effect

ð15Þ

Direct effect

As shown in Fig. 1, as k decreases in asymmetric equilibrium, firm b starts to move apart from firm a (i.e., from the raw material site). This raises the transport costs by strategic effect, whereas the total shopping costs first fall (as long as xb⁎ b 1/2, i.e., k N 3/2) then rise. The former always dominates the latter, and the overall strategic effect works in the

42

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

opposite direction to the changes in k. A decrease in k lowers SC via the direct effect. But the strategic effect always dominates the direct effect, and declines in k raise the social welfare. pﬃﬃﬃ When the transport cost declines to a certain level and k becomes smaller than 3(4 2 − 5) / 2, symmetric equilibrium is possible. In symmetric equilibrium in which xb⁎ = −xa⁎ = (3 − 2k) / 4 (see (i) to (iii) in Proposition 1), the social costs can be written as Z SC ¼ 2

1=2

2 ⁎ ½tðx⁎ b xÞ þ Txb dx

0

! 1 x⁎ 2 b ⁎ þ xb þ tkxb⁎ : ¼t |{z} 12 2 |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Total transport costs

ð16Þ

Total shopping costs

Again, SC can be decomposed into the total shopping costs and the total transport costs. The strategic and direct effects are also revealed here by differentiating SC with respect to k: ⁎ dSC 1 dxb dx⁎ b ⁎ ¼ t þ 2xb þ tk þ tx⁎ b |{z} dk 2 dk dk |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} Directeffect Strategic effect

¼

t 2 |{z}

þ

Strategic effect

tx⁎ b |{z}

¼

tð1 2kÞ : 4

ð17Þ

Direct effect

Here, decreases in k raise SC via the strategic effect and lower it via the direct effect. Notice that the marginal strategic effect is constant, whereas the marginal direct effect decreases in k because dxb⁎ / dk b 0. Therefore, the strategic effect dominates the direct effect when k is large and the opposite is true when k is small. SC is the highest when k is equal to 1/2. Thus, in symmetric equilibrium, declines in the transport cost first raise then decrease SC. 5. Concluding remarks This paper studied how the costs of transporting raw materials affect the location and price decisions of firms. In the analysis, we used a location-then-price spatial competition model involving the transport costs of raw materials as well as the shopping costs for consumers. We considered a linear market, and we observed a possibility of multiple equilibria. In one equilibrium, firms locates equidistantly from the raw material site (symmetric equilibrium), whereas in the other, one firm locates at the raw material site and the other firm locates away from it (asymmetric equilibrium). In asymmetric equilibrium, it is observed that the firm distant from the raw material site sets a higher price and sells less than does the firm at the raw material site. The following two properties regarding welfare were shown. First, asymmetric equilibrium is superior to symmetric equilibrium. Second, declines in the transport cost may reduce the social welfare: in fact, the welfare level first falls then rises as the transport cost decreases. Some extensions are worth mentioning. First, we can consider a non-uniform distribution of consumers. At the location where population (demand) is large, a firm could obtain positive profits by setting a sufficiently high price to cover the transport cost when it faces a sufficient level of geographical differentiation. Therefore, if the raw material site and the location with large demand are distant, one firm may locate at the former whereas the other firm would locate at the latter in an equilibrium. Secondly, it would be important to examine the effects of the sequential decisions of firms. This includes the sequential location decision as well as the sequential price decision. These modifications would alter the degree of price competition, and hence, may lead to changes in welfare properties. Finally, as in Lai and Tsai (2004), it would be possible to consider the effect of zoning. By limiting the possible location for firms, a local government can affect the equilibrium locations of firms and welfare. It would be interesting to explore optimal zoning. These are all topics for future investigation.

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

43

Acknowledgements We thank Richard Arnott, Ikuo Ishibashi, Toshiharu Ishikawa, Tatsuaki Kuroda, Noriaki Matsushima, Dan Sasaki, Takatoshi Tabuchi, and two anonymous referees for their helpful comments and discussion. Of course, all remaining errors are our own. The second author acknowledges the financial support from the Japan Society for the Promotion of Science through Grants-in-Aid for Scientific Research ((B)14330004 and (S)18103002), and a Grant-in-Aid for Young Scientists ((B)18730162). Appendix A. Proof of Proposition 1 From (2), (3), and (4), notice first that Api AQi ¼ Qi þ ðpi Tli Þ ; Api Api AQa AX ⁎ 1 AQb AX ⁎ 1 and : ¼ ¼ ¼ ¼ 2tðxa xb Þ 2tðxa xb Þ Apa Apa Apb Apb

ð18Þ

Hence, the first order conditions for the profit maximization with respect to prices are 1 xa þ xb pa pb pa Tla þ þ þ ¼ 0; 2 2 2tðxa xb Þ 2tðxa xb Þ

ð19Þ

1 xa þ xb pa pb pb Tlb þ ¼ 0; 2 2 2tðxa xb Þ 2tðxa xb Þ

ð20Þ

which lead to the equilibrium prices (6) and (7) given in the text.11 Substituting (3) and (6)–(8) into (4), we can rewrite the profits of firms as 1 pa ¼ 2tðxb xa Þ X * þ Qa ¼ 2tðxb xa ÞQ2a ; 2 pb ¼ 2tðxb xa Þ

1 X ⁎ Qb ¼ 2tðxb xa ÞQ2b : 2

By differentiating πa and πb with respect to xa and xb, respectively, we obtain Apa AQa ¼ 2tQa Qa þ 2ðxb xa Þ ; Axa Axa Apb AQb ¼ 2tQb Qb þ 2ðxb xa Þ : Axb Axb

ð21Þ

ð22Þ

ð23Þ

ð24Þ

Furthermore, by differentiating Qa and Qb with respect to xa and xb, respectively, we have

11

Al AQa 1 ðxb xa Þ Axaa þ ðlb la Þ ¼ þ k; 6 Axa 6ðxb xa Þ2

ð25Þ

Al AQb 1 ðxb xa Þ Axbb ðlb la Þ ¼ k: 6 Axb 6ðxb xa Þ2

ð26Þ

The second-order conditions are obviously satisfied.

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H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

Note that if the equilibrium locations are such that xa⁎ ≠ 0 and xb⁎ ≠ 0, then Qa N 0 and Qb N 0 must hold true. This is because whenever xa⁎ ≠ 0, xb⁎ ≠ 0 and Qa ≤ 0 hold, firm a can become better off if it relocates to − xb⁎ instead of xa⁎. We derive the location equilibrium by considering the following cases depending on the values of xa and xb: (i) 0 b xa b xb Substituting la = xa, lb = xb, ∂la / ∂xa = 1 and ∂lb / ∂xb = 1 into (23)–(26), we obtain Apa tQa ð3 þ 3xa xb þ kÞ ¼ 0; ¼ Axa 3

ð27Þ

Apb tQb ð3 þ xa 3xb kÞ ¼ 0: ¼ Axb 3

ð28Þ

We know that Qa N 0 and Qb N 0, leading to the following profit maximizing locations: x⁎ a ¼

3 þ 2k 3 2k and x⁎ : b ¼ 4 4

ð29Þ

These do not satisfy 0 b xa b xb, implying that there is no equilibrium in this case. (ii) xa b xb b 0 Based on arguments very similar to those given in (i), no equilibrium exists in this case. (iii) xa b 0 b xb Substituting la = − xa, lb = xb, ∂la / ∂xa = − 1 and ∂lb / ∂xb = 1 into (23)–(26), we obtain Apa tQa ½ðxa xb Þð3 þ 3xa xb Þ kðxa 3xb Þ ¼ 0; ¼ Axa 3ðxa xb Þ

ð30Þ

Apb tQb ½ð3 þ xa 3xb Þðxa xb Þ kð3xa xb Þ ¼ 0: ¼ Axb 3ðxa xb Þ

ð31Þ

Qa N 0 and Qb N 0 lead to the following first-order conditions: ðxa xb Þð3 þ 3xa xb Þ kðxa 3xb Þ ¼ 0;

ð32Þ

ð3 þ xa 3xb Þðxa xb Þ kð3xa xb Þ ¼ 0:

ð33Þ

These two equations give ðxa þ xb Þ½3 þ 4ðxa xb Þ ¼ 0:

ð34Þ

From xa b 0 b xb, the equilibrium locations must satisfy xb = − xa or xb = xa + 3/4. Substituting xb = − xa and xb = xa + 3/4 into (32) yields 2xa ð3 þ 2k 4xa Þ ¼ 0 and

1 ð3 4kÞð9 þ 8xa Þ ¼ 0; 16

ð35Þ

respectively. xa b 0 b xb requires that the solutions for the first-order conditions are xa = −(3 − 2k) / 4 and xb = (3 − 2k) / 4 when k b 3 / 2, and that any pair (xa, xb) that satisfies xb = xa + 3/4 and −3/4 ≤ xa ≤ 0 when k = 3/4.

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

45

pﬃﬃﬃ We first show that when 0 b k b 3(4 2 − 5) / 2 and k ≠ 3/4, equilibrium locations are given as x⁎ a ¼

3 2k 3 2k ; x⁎ : b ¼ 4 4

ð36Þ

pﬃﬃﬃ In doing so, we have to show that xa⁎ = −(3 − 2k)/4 is the best response to xb⁎ = (3 − 2k)/4 when 0 b k b 3(4 2 − 5) / 2. Substituting xb⁎ = (3 − 2k)/4 into (30) gives Apa tQa ½4xa þ ð3 2kÞ½12xa ð9 10kÞ: ¼ Axa 48ðxa xb Þ

ð37Þ

Because Qa N 0, ∂πa/∂xa = 0 is satisfied at xa ¼

3 2k 9 10k and xa ¼ : 4 12

ð38Þ

Moreover, we can easily see that ∂πa / ∂xa b 0 if − (3 − 2k) / 4 b xa b (9 − 10k) / 12 and that ∂πa / ∂xa N 0 if xa b − (3 −2k) / 4 or (9 − 10k) / 12 b xa. Because −(3 − 2k) / 4 b −(9 − 10k) / 12 is satisfied iff k b 9/8, xa b −(3− 2k) / 4 gives the local maximum of firm a's profit when k b 9 / 8. Moreover, it also gives the maximum for xa ≤ 0 if pﬃﬃﬃ πa| x a = − (3 − 2k) / 4, x b = (3 − 2k) / 4 N πa| x a = 0,x b = (3 − 2k) / 4 i.e., k b 3(4 2 − 5) / 2. For xa ∈(0, xb⁎), (27) implies that ∂πa|xb = (3 − 2k) / 4 /∂xa b 0, which shows that firm a's profit is smaller than πa|xa = 0,xb = (32k)/4. For xa′ N xb⁎ N 0, we can see that πa|xa = xa′, xb = x⁎b b πa|xa = x⁎b − (x′a − x⁎b ), xb =x⁎b . Therefore, there exists some xa that is smaller than xb⁎ and leads to higher firm a's profit than πa|0 b x⁎b b xa′. For xa = xb⁎, the price competition a la Bertrand leads to zero profit for both firms. Hence, for xa N 0, firm a's profit is always smaller than πa|xa = − (3 − 2k) / 4,xb = (3 − 2k) / 4. ⁎ ⁎ Summarizing pﬃﬃﬃ the above arguments, we have that xa ⁎= −(3 − 2k) / 4 is the best response to ⁎xb = (3 − 2k) / 4 when 0 b k b 3(4p2ﬃﬃﬃ − 5) / 2. Similarly, it can be shown that xb = (3− 2k) / 4 is the best response to xa = −(3 − 2k) / 4 when 0 b k b 3(4 2 − 5) / 2. Therefore, (36) represents the locations in a subgame-perfect Nash equilibrium. Next, we show and that when k = 3/4, any pair (x⁎a , x⁎b ) satisfying

⁎ ⁎ x⁎ b ¼ xa þ 3=4 and 3=4V xa V0

ð39Þ

is a location equilibrium. For that purpose, we must first show that xa⁎ = xb⁎ − 3/4 is the best response to xb⁎ ∈ [0, 3/4] when k = 3/4. Substituting k = 3/4 into (30) provides Apa tQa ð3 þ 4xa 4xb Þð3xa xb Þ: ¼ Axa 12ðxa xb Þ

ð40Þ

From Qa N 0, ∂πa / ∂xa = 0 is satisfied at xa ¼ xb

3 xb and xa ¼ : 4 3

ð41Þ

Similar arguments to those given above show (a) that when k = 3/4, xa⁎ = xb⁎ − 3/4 is the best response to any given xb⁎ ∈ [0, 3/4] and (b) that xb⁎ = xa⁎ + 3/4 is the best response to any given xa⁎ ∈ [− 3/4, 0] when k = 3/4. Hence, (36) represents a subgame-perfect Nash equilibrium when k = 3/4. (iv) xa = 0 b xb la = 0, lb = xb and ∂lb/∂xb = 1 yield Apb jxa ¼0 tQb ½3 k 3xb ¼ 0: ¼ Axb 3

ð42Þ

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H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

In a way very similar to (iii), we can show that πb (πa) attains its maximum at xb⁎ = (3 − k) / 3 (xa⁎ = 0) when xa = 0 (xb = (3 − k)/3). Therefore, a pair satisfying xa⁎ = 0 and xb⁎ = (3 − k) / 3 represents a subgame-perfect Nash equilibrium when 3/4 ≤ k b 3. (v) xa b 0 = xb Similar arguments to those given in (iv) show that a pair satisfying xa⁎ = − (3 − k)/3 and xb⁎ = 0 are the equilibrium locations when 3/4 ≤ k b 3. (vi) xa = xb Note that xa = xb always leads to zero profit for both firms due to price competition a la Bertrand. Note further that if xa = xb ≠ 0, then it is profitable for firm a to relocate to xa = −xb and that this case is not a Nash equilibrium. Hence, it is sufficient to consider the case of xa = xb = 0. We can show that xa = 0 is the best response to xb⁎ = 0 as follows. Given xb⁎ = 0, we obtain 8 1 x k > < þ a ; if xa b0 2 6 6 ð43Þ ; Qa jxb ¼0 ¼ 1 x > : a k ; if xa N0 2 6 6 which implies that Qa|xa ≠ 0, xb = 0 b 0 when k ≥ 3. Hence, firm a cannot be better off by relocation from xa = 0 to other locations and xa = 0 is the best response to xb⁎ = 0 when k ≥ 3. Similarly, it can be shown that xb = 0 is the best response to xa⁎ = 0 when k ≥ 3, leading to the result that a pair satisfying xa⁎ = xb⁎ = 0 represents a subgameperfect Nash equilibrium when k ≥ 3. Appendix B. Proof of Proposition 2 We first derive the location that minimizes the social costs (SC ) under the constraint xa = xb. When xa = xb, SC is given as Z 1=2 1 ð44Þ SC ¼ ½tðxa xÞ2 þ Tla dx ¼ tx2a þ Tla þ t: 12 1=2 Hence, SC takes its minimum value t/12 at xa = xb = 0. Next, we derive the location that minimizes SC under the constraint xa ≠ xb. In this case, SC can be written as Z SC ¼

X SO

1=2

Z ½tðxa xÞ2 þ Tla dx þ

1=2

½tðxb xÞ2 þ Tlb dx;

SO X

1 1 1 1 1 1 1 ¼ tðxa X SO Þ3 þ Tla X SO tðxa þ Þ3 Tla þ tðxb Þ3 þ Tlb 3 2 2 3 2 2 3 1 3 tðxb X SO Þ þ Tlb X SO ; 3

ð45Þ

which leads to ASC xa xb T ðla lb Þ 2 1 Ala Qa Ala Qb þ 2xa þ ¼ t þ tðxa þ Þ2 þ TQa ¼ k ; Axa 2tðxa xb Þ 2 2 Axa t Axa

ð46Þ

ASC xa x b T ðla lb Þ 2 12 Alb Qb Alb þ Qa 2xb ¼t tðxb Þ þ TQb ¼ k : Axb 2tðxa xb Þ 2 2 Axb t Axb

ð47Þ

Here, we show that in the optimum, both Qa N 0 and Qb N 0 must hold true. Assume Qa = 0. Then, all consumers buy goods from firm b. Now relocating firm a to xa = − xb does not change the total transport costs of raw material, whereas it reduces the total shopping costs. Hence, both Qa N 0 and Qb N 0 must hold true in the optimum. Moreover, (46) gives that ∂SC|0 b xa ≤ xb / ∂xa N 0, which implies that 0 b xa ≤ xb cannot be the optimum. Similar arguments show that xa ≤ xb b 0 is not the optimum. Therefore, it is sufficient to consider the case of xa ≤ 0 ≤ xb.

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

47

Here, we have two possibilities: (i) the optimal is attained as an interior solution and (ii) it is attained as a corner solution. When xa b 0 b xb, we can see that la = − xa, ∂la / ∂xa = − 1, lb = xb, and ∂lb / ∂xb = 1, which, combined with (46) and (47), lead to the first order conditions for case (i): Qb þ 2xa k ¼ 0;

ð48Þ

Qa 2xb k ¼ 0:

ð49Þ

Using these two equations, we obtain ðQb Qa Þ þ 2ðxa þ xb Þ ¼ ðxa þ xb Þ½

1 k þ 1 ¼ 0; xa xb

ð50Þ

which is satisfied if xa = − xb or xa − xb = − k. Note that X SO jxa ¼xb ¼ 0 and X SO jxa xb ¼k ¼ xa þ xb :

ð51Þ

Substituting XSO|xa = – xb = 0 into (48) and (49), we obtain xa ¼

1 2k 1 2k ; xb ¼ ; 4 4

Substituting XSO|xa k ¼ 1=4;

– x b = – k = xa + xb

ð52Þ into (48) and (49), we obtain

xa ¼ xb 1=4:

ð53Þ

Moreover, these are interior solutions only when 0 ≤ k b 1/2. Substituting these into SC gives the social cost in (i): SCjxa ¼ð12k Þ=4;xb ¼ð12k Þ=4 ¼

SCjk¼1=4;xa ¼xb 1=4

1 þ 12k 12k 2 t; 48

13 1 þ 12k 12k 2 t ¼ t : ¼ 192 48

ð54Þ

ð55Þ

In (ii), we first consider firm a as the firm locating at the raw material site and hence consider xa = 0. xa ≤ xb implies that ∂lb/∂xb = 1 and 1 1 X SO jxa ¼0;xb z0 ¼ xb þ k: 2 2

ð56Þ

Substituting this into (47), we can show that the optimal location of firm b is xb = (1 − k) / 3 when 0 ≤ k( = T / t) b 1. In this case, the social cost becomes SCjxa ¼0;xb ¼ð1k Þ=3 ¼

5 þ 12 12k 2 þ 4k 3 t: 108

ð57Þ

(54) minus (57) is equal to ð4k 1Þð11 16k 4k 2 Þ t: 432 Therefore, we have the following: d 0Vkb1=4Z SCjxa ¼ð12k Þ=4;xb ¼ð12k Þ=4 bSCjxa ¼0;xb ¼ð1k Þ=3; d 1=4bkb1=2Z SCjxa ¼ð12k Þ=4;xb ¼ð12k Þ=4 NSCjxa ¼0;xb ¼ð1k Þ=3; d k ¼ 1=4Z SCjxa ¼ð12k Þ=4;xb ¼ð12k Þ=4 ¼ SC k¼1=4;xb ¼xa þ1=4 ¼ SCjxa ¼0;xb ¼ð1k Þ=3d

ð58Þ

48

H. Aiura, Y. Sato / Regional Science and Urban Economics 38 (2008) 32–48

Moreover, SCjk¼1;xa ¼0;xb ¼ð1k Þ=3 ¼ ASCjxa ¼0;xb ¼ð1k Þ=3 Ak

¼

1 t ¼ SCjxa ¼xb ¼0 ; 12

ð k 1Þ 2 N0 9

when kp1;

ð59Þ

ð60Þ

which implies that 1 d 0Vkb1Z SCjxa ¼0;xb ¼ð1k Þ=3 b t ¼ SCjxa ¼xb ¼0 ; 12 1 d kN1Z SCjxa ¼0;xb ¼ð1k Þ=3 N t ¼ SCjxa ¼xb ¼0 ; 12 1 d k ¼ 1Z SCjxa ¼0;xb ¼ð1k Þ=3 ¼ t ¼ SCjxa ¼xb ¼0 : 12 Similar arguments are possible if we consider firm b as the firm locating at the raw material site. Summarizing the above arguments, we have Proposition 2. References Aiura, H., Sato, Y., 2006. Asymmetric configurations in spatial competition with location dependent costs. DEE Discussion Papers 06–1. Nagoya University. d'Aspremont, C., Gabszewicz, J.J., Thisse, J.-F., 1979. On Hotelling's “Stability in Competition”. Econometrica 47, 1145–1150. Heal, G., 1980. Spatial structure in the retail trade: a study in product differentiation with increasing returns. The Bell Journal of Economics 11, 565–583. Hotelling, H., 1929. Stability in competition. Economic Journal 39, 41–57. Karlson, S.H., 1985. Spatial competition with location-dependent costs. Journal of Regional Science 25, 201–214. Lai, F.-C., Tsai, J.-F., 2004. Duopoly locations and optimal zoning in a small open city. Journal of Urban Economics 55, 614–626. Lambertini, L., 1994. Equilibrium locations in the unconstrained hotelling game. Economic Notes 24, 438–446. Matsushima, N., 2004. Technology of upstream firms and equilibrium product differentiation. International Journal of Industrial Organization 22, 1091–1114. Matsushima, N., 2006. Vertical mergers and product differentiation. GSBA Kobe University Discussion Paper Series: 2006–9. Tabuchi, T., Thisse, J.-F., 1995. Asymmetric equilibria in spatial competition. International Journal of Industrial Organization 13, 213–227. Weber, A., 1929. Theory of the Location of Industries. University of Chicago Press, Chicago. Ziss, S., 1993. Entry deterrence, cost advantage and horizontal product differentiation. Regional Science and Urban Economics 23, 523–543.