Retail sprawl and multi-store firms: An analysis of location choice by retail chains

Regional Science and Urban Economics 39 (2009) 277–286

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Regional Science and Urban Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c

Retail sprawl and multi-store firms: An analysis of location choice by retail chains☆ Vladimir Karamychev ⁎, Peran van Reeven Erasmus University Rotterdam, The Netherlands

a r t i c l e

i n f o

Article history: Received 25 April 2008 Received in revised form 21 October 2008 Accepted 22 October 2008 Available online 5 November 2008 JEL classification: D43 L11 L13 R32

a b s t r a c t A growing number of cities and towns restrict the number of chain stores within their borders in order to prevent sprawl and increase in traffic problems. This paper verifies these concerns by analyzing endogenous location choices by multi-store firms, which sell heterogeneous products and can charge different prices at every store. It takes a generalized transportation cost approach to Salop [Salop, S.C., 1979. Monopolistic competition with outside goods. Bell Journal of Economics 10, 141–156], in which consumers' utility not only depends on real transportation costs but also on their preferences or tastes. The model confirms that firms' location strategies contribute to retail sprawl. However, the resulting locations actually minimize consumers' generalized transportation costs for given sizes of retail chains. One of the implications is that further concentration in the retail industry is welfare improving. © 2008 Elsevier B.V. All rights reserved.

Keywords: Horizontal differentiation Multi-store competition Location choice Random utility

1. Introduction Our myths, our literature, and most of our theory about retailing still perceive the industry in terms of the independent ‘mom and pop’ store. Economic theory treats the latter as the archetypical decision makers, but while there are still over a million stores of this type, they have become increasingly irrelevant to consumers and the distribution system (Jones and Simmons, 1995). The largest company in the US is a retail company with 6779 stores (Fortune, 2007; Wal-Mart Stores, 2007). The list of 50 largest US firms contains seven more retail chains. According to the latest Economic Census (2002), multi-store firms account for 65% of retail sales. In addition, based on Price Waterhouse Coopers (2004), it is estimated that those firms control another 21% through franchise contracts. Hence, retail chains are responsible for approximately 86% of US retail sales, and this percentage is increasing. A growing number of cities do not appreciate this development and try to restrict the number of chain stores within their borders by means of “formula” business restrictions, store size caps, land-use restrictions, economic impact review requirements, etc. (see, e.g., www.newrules.org/retail for an overview of the rules and policies that different authorities have adopted). They observe that the location strategies of multi-store firms have shifted the retail center of gravity away from the traditional Main Street to highway interchanges, ☆ We thank three anonymous referees and Jose-Luis Moraga for valuable comments. ⁎ Corresponding author. E-mail address: [email protected] (V. Karamychev). 0166-0462/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2008.10.002

resulting in urban sprawl, automobile dependence and traffic congestion. At the same time, cities fear to loose their unique and distinctive character, maintained by one-of-a-kind businesses, which is being replaced by a uniform retail environment. In this paper, we verify these concerns by analyzing endogenous location choices by competing multi-store firms. The analysis of multi-store competition has a troublesome history in the economic literature. The common way of modeling the product space as an interval (Hotelling, 1929) or as a circle (Salop, 1979) leads to analytical intractability if applied to endogenous location choice. Teitz (1968) introduces multi-store competition in Hotelling's (1929) original model, and shows that no pure strategy equilibrium exists. Klemperer (1992) explains that “…firms' demands and, hence, their profits are discontinuous at points at which some brand's price is low enough that it just undercuts a competing brand at the competing brand's location”. By assuming transportation costs that are quadratic in distance, Martinez-Giralt and Neven (1988) regain analytical tractability. They show that firms do not open more than a single store so that they are, in fact, single-store firms. Hence, the assumption of quadratic transportation cost is inappropriate for analyzing multi-store firms. Pal (1998), Chamorro-Rivas (2000), Yu and Lai (2003), Pal and Sarkar (2002), Gupta et al. (2004, 2006), and Li (2006) approach the issue of multi-store competition in a different way. Instead of firms that compete in prices and consumers that choose which store to visit they assume that firms compete in quantity and bring products to consumers' doors. Pal (1998) shows that single-plant firms locate equidistantly under such quantity competition. Chamorro-Rivas

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(2000) extends this result to the case of two-plant duopolists, which also applies when one of the duopolists is a welfare-maximizing (public) firm (Li, 2006). In this literature, equilibrium locations do not necessarily show equidistance when consumers perceive products from firms differently. Dropping the homogeneity assumption, Yu and Lai (2003) argue that the equidistance in locations disappears when firms deliver complements. Gupta et al. (2004, 2006) show that equilibrium patterns of locations include complete agglomeration, partial agglomeration, and complete dispersion. Peng and Tabuchi (2007) analyze location-then-variety competition, and show equilibrium patterns such as segmentation, interlacing, sandwich, and enclosure. Nevertheless, equidistance remains the only outcome in a price competition setting. Janssen et al. (2005) analyze location choices of multi-store firms where in addition to the geographical dimension a linear product characteristic space is introduced. When a consumer buys a product of a firm at a store, his utility not only depends on his real transportation cost, but also on his identity and the identity of the firm and the store. In case of two competing chains and uniform pricing within the chains, Janssen et al. (2005) show that equilibrium locations exhibit equidistance. Nwogugu (2006) criticizes all existing store location models for their practical inaccuracy and identifies eleven major problems, where the biggest issue is the treatment of the distance that consumers travel. He argues that most models erroneously assume that the distance from each community to the store is constant and remains constant for each customer and for all trips to the store. In this paper, we relax this assumption by modeling the heterogeneity of products as if consumers were travelling different distances depending on from which firm they buy. We build on Salop (1979) and further develop Janssen et al. (2005) by incorporating consumer preferences over firms' products into consumers' locations. In other words, when two consumers have the same real geographical location but get different utility levels because of different preferences (tastes), we treat them as if they travel different distances and, therefore, have locations, which we call “virtual locations”. In our model, every consumer is characterized by a set of virtual locations, one for each firm, and every virtual location fully determines a consumer's utility level in case he buys from a specific firm. Hence, even if firms can observe a real location of a consumer, they do not know his preferences and, therefore, do not observe his virtual locations. From a firm's point of view, all virtual locations of a given consumer are identically (evenly) and independently distributed over the circle. Thus, the presented model can be called a “random location” model, similar to “random utility” model of Anderson and de Palma (1992).1 In our model, firms decide on the number of stores, their locations, and prices. Thus, the number of endogenous variables is endogenous, and the short-run market dynamics might be very complex when firms move sequentially. Aguirregabiria and Vicentini (2006) present an example of a dynamic location-then-price stochastic game, and compute its Markov-perfect equilibria. To avoid unnecessary complications, we focus our analysis on the long-run location-and-price decisions of firms in a simultaneous-move game, in which consumers have unit demand, very high reservation utility (in order to ensure full market coverage), and independently and evenly distributed locations. Our main results are as follows. First, in any equilibrium, every firm locates all its stores equidistantly and charges the same price in all its stores. The equidistance result generalizes the results of Janssen et al. (2005) for more than two firms and unrestricted pricing. The economic intuition is that a firm maximizes its profit by replicating the market scope and price level of its most profitable store to all its outlets. This result 1 With the main difference that the random utility approach to multi-store competition assumes the geographical dimension away.

corresponds to the actual strategy of, e.g., Wal-Mart. Wal-Mart locates stores within a 5-mile radius in urban areas and a 25-mile radius in rural areas. With regards to pricing, the Chancery Court observed in the Arkansas Wal-Mart Case, that “Wal-Mart determines the ‘every day price’ for its products at its headquarters in Bentonville”, (Shils, 1997). Although the equidistant location strategy creates retail sprawl, it does not automatically imply increase in traffic and traffic congestion. Equidistant locations of a chain of fixed size actually minimize consumers' generalized transportation cost. As a result, retail chains bring shops, for which the consumer used to go downtown, closer to the consumer. Due to uniform pricing, consumers have no incentive to search for a lower price within a chain. The reason for equidistance and uniform pricing is the following. If a firm's stores are not located evenly over the circle, or if they charge different prices, these stores have necessarily different (non-equal) profit levels. Consequently, the firm has an incentive to withdraw its less profitable stores and replace them with stores that are more profitable. Second, each firm gets zero profit in equilibrium. In Salop (1979), the zero-profit condition is exogenous and motivated by the free entry to the market. In our model, there is no free entry “for firms” as the number of firms is given. On the other hand, free entry “for stores” alone cannot explain firms' zero profits because firms expand the number of their stores until its profit starts to decline, which can also occur at a positive profit level (observe, for example, positive profit levels in Anderson and de Palma, 1992). In our model, firms can only get zero profit in equilibrium because otherwise they have a profitable deviation. As long as firm's stores are profitable, the firm can increase its total profit by increasing the size of its retail chain. The intuition is that the firm with the largest number of stores is able to provide the nearest store to the larger part of the market, which enhances its market power. Hence, multi-store firms have a strong tendency to compete not only in prices but also in terms of the distance that consumers have to travel by building more stores. This tendency results in the sprawl of stores across geographical space, but is socially beneficial as it reduces traffic. Third, the model has multiple equilibria. The market can sustain either few chains with many outlets or many chains with few outlets. Equilibrium requires that all chains are of equal sizes and charge equal prices. With the growth of retail chains, one-of-a-kind businesses loose ground, so that the local retail environments increasingly become uniform. However, this loss of variety is socially beneficial, as is shown by the next result. Fourth, concentration of multi-outlet firms is welfare improving. When the number of chains decreases, each firm develops more outlets, so that consumers need to travel shorter distances to buy at a specific retail chain. This decrease in traffic always outweighs the loss of variety, so that it is optimal to have as few competing retail chains as possible. As a result, it is socially optimal to have only two chains in the market. Our results are quite robust to changes in the shape of the assumed transportation cost function. They hold as long as transportation cost function is less convex than quadratic. Our approach is also consistent with results in existing literature on multi-product competition. In the limit, when transportation costs become asymptotically quadratic in distance, the model confirms the result of Martinez-Giralt and Neven (1988) that no firm opens more than a single store, i.e., the maximum differentiation principle. If transportation costs are concave, our model confirms the result of Anderson and de Palma (1992) that more competing chains results in more stores in total and in higher market prices. However, this result is not robust as it reverses if transportation costs are convex. Finally, even though the model has been developed to analyze competition, it correctly predicts the number of stores that a monopolist will open. Hence, the model seems to provide a unified framework for analyzing endogenous location choices by one or many single-store or multistore firms in a large variety of settings, and confirms fundamental assumptions that are widely used in the literature.

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The policy implications of our analysis is that retail chains contribute to sprawl by dispersing stores in geographical space, and also to the uniformity of the retail environment. However, they also have a positive effect in terms of reducing mobility requirements, especially if there are only a few multi-outlet firms with large retail chains. Cities and town should not restrict the number of chain stores within their borders. Instead, if they do consider the developments in the retail sector as a concern, a better strategy is to limit the number of retail chains and to allow these selected firms to build as many stores as they want. The rest of the paper is organized as follows. Section 2 presents the model, which is then analyzed in Section 3. Section 4 concludes and Appendix contains all proofs. 2. The model There is a unit mass of consumers, indexed by ω, with unit demand. Consumers reside on a circle with unit circumference, and the location of a generic consumer ω is denoted by xω ∈ S1 ≡ [0,1). There are N firms indexed by i = 1,…, N. Each firm i builds a chain of ni 1 stores xi = {xm i }, indexed by m = 1,…, ni in such a way that 0 ≤ xi b … b ni m m xi b 1. Firm i charges a price pi in its store xi , and we denote the collection of prices by pi = {pm i }. Apart from the geographical location xω, consumer ω is characterized by a set of virtual locations {x(i) ω }, i = 1,…, N, which determine his m utility level U(ω, i, xm i , pi ) when he buys a product of firm i from its m store xm i at price pi :  
 ðiÞ m m : = U0 − pm U ω; i; xm i ; pi i − t
d xω ; xi

ð1Þ

Here, U0 is the reservation utility, t is the (constant) unit transportation cost, and d(x, y) is the distance along the circle between any two locations x and y. The difference between our model and the model (i) of Salop (1979) is that it is not his real location xω but virtual location xω that determines the utility level of consumer ω who buys from a store of firm i. In terms of virtual locations, consumers' utility is given by the very same expression as in Salop (1979). The main implication of the assumption of virtual locations is that transportation costs now constitute “generalized transportation” costs, and not only represent real (geographical) transportation costs, but also include consumers' disutilities if they buy a product which is not their most preferred product. The proposed utility specification is not the true utility function that consumers maximize,2 but it is an approximation. The usefulness of this approximation and its plausibility is proven by its analytical power and by the consistency of the results that we obtain with the results in related literature. Suppose that for given firms' locations xi and prices pi, consumer ω buys from firm j. In this case, his virtual location x(j) ω fully determines his utility level. Let us fix x(j) ω = y. Now suppose that the very same consumer decides to buy from another firm i. His virtual location x(i) ω, which determines his utility level in this case, is a random variable which distribution depends on y. We denote the spatial distribution of x(i) ω conditional on y by F(x|y): F ðxjyÞuPr



ði Þ xω

2

ð jÞ ½0; xjxω

 = y ; x; y 2 S1 :

When firms' products are very similar in consumers' eyes, and in the limit become perfect substitutes (apart from the geographical heterogeneity), the distribution F(x|y) becomes degenerate, i.e., F(x|y) = 1 for x ≥y and F(x|y) = 0 for x b y. This represents a perfectly homogeneous (i) limiting case of Salop, where all virtual locations xω of consumer ω are perfectly correlated with his geographical location xω and, consequently, with each other. 2 The exact analytical form of the utility function depends on the topology of the characteristic space, on locations of firms’ products in it, and on exact consumers’ preferences over it.

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When, to the contrary, firms' products are different from each (j) other, the distribution F(x|y) is non-trivial, so that locations x(i) ω and xω are not perfectly correlated. When the difference (from consumers' prospective) between products of different firms increase, correlation (j) between consumers' virtual locations x(i) ω and xω diminishes. In other words, correlation between virtual locations of a consumer is determined by correlation of utility levels that he gets from buying products of different firms. In this paper, we assume that consumers' virtual locations are statistically independent. This limiting case corresponds to a market where consumers' disutility is mainly determined by their preferences rather than their real transportation costs. Formally, we assume the following. Assumption 1. All the virtual locations of consumer ω are independent and uniformly distributed over the circle, i.e., F(x|y) = x for x, y ∈ S1. The rest of the model generalizes Salop (1979). Under Assumption 1, the unconditional distribution of consumers on the circle F(x) is also uniform: Z1   Z1 ði Þ F ðxÞuPr xω 2 ½0; x = FðxjyÞdFðyÞ = x dFðyÞ = x; x 2 S1 : 0

0

The fixed cost of building a store is f N 0 for every firm, and the operating costs are normalized to zero. By developing a chain of stores
 P m m ci ≡ (ni, xi, pi), firm i gets a profit πi ðci Þ = nmi = 1 pm i Di − f , where Di is m the demand faced by store xi , which consists of all consumers who opt for buying at that store. Demands Dm implicitly depend on i strategies chosen by all other firms cj, j ≠ i. We analyze a long-run Nash equilibrium of the game in which firms simultaneously choose their strategies ci in order to maximize their profits πi(ci). The other modeling assumptions are in the spirit of Salop (1979). First, we assume that consumers' reservation utility U0 is sufficiently large so that in a competitive equilibrium, the participation constraint never binds and, consequently, all consumers buy. Second, in order to avoid the problem of integers, we consider the size of retail chains as a continuous variable. One consequence of this assumption is that multiple asymmetric equilibria may emerge. In order to focus on equilibria that are not due to this assumption, we assume that if two chains are equally profitable, firms strictly prefer the chain with the largest number of stores. 3. Analysis We first analyze the model for linear transportation cost function, as is set up in Section 2, and relate the results to the existing literature. Next, we analyze the sensitivity of these results to the linearity of transportation cost by analyzing a more general, non-linear model. 3.1. Linear transportation costs First, we consider strategies of firm i for a given strategy choice of its competitors. We show that only strategies that satisfy equidistance and uniform pricing can be best responses and, therefore, can be a part of Nash equilibrium. This is the content of Proposition 1. Using this result, we redefine firms' strategies as if each firm chooses a number of stores and its uniform price. Hence, Proposition 1 allows us to translate the original location-and-price game into a variety-and-price game. Next, we assume that all firms enter the market and build (a positive number of) stores. Proposition 2 proves that in the new strategy space, there exists a unique equilibrium that is symmetric. In equilibrium, all firms build the same number of stores and charge equal prices. This unique equilibrium corresponds to a continuum of equilibria in locations, which only differ from one another by the relative positions of stores from different chains.

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Finally, we show in Proposition 3, that in addition to the symmetric equilibrium, the model has a finite number of quasi-symmetric equilibria in which a number of firms enter the market and build stores whereas all remaining firms build no stores. Proposition 1 (Necessary condition). A strategy ci = (ni, xi, pi) with positive number of stores ni N 0 is a best response only if it satisfies equidistance and uniform pricing. Formally, Proposition 1 argues that for any strategy ci = (ni, xi, pi) with ni N 0 such that d(xki − 1, xki ) ≠ 1/ni or pki − 1 ≠ pki for some k, there exists another strategy ci′ = (ni′, xi′, pi′) such that either πi(ci′) N πi(ci) or πi(ci′) = πi(ci) and ni′ N ni. We illustrate this result in Fig. 1(A) and (B), where we have dropped for simplicity all subscripts i. The horizontal axis represent a segment of the unit circle S1 with chains of stores of firm i located at xk and charging prices pk, and the vertical axis represent utility of consumer ω whose (virtual) location is xiω = x. When a consumer ω buys from store xk, his utility is given by Eq. (1) and is represented in Fig. 1 by straight sloped lines. When, to the contrary, he buys from any other firm j, his virtual location xjω is random and is independent on x under Assumption 1. Thus, the maximum utility level that ω can obtain if he buys from other firms is also random and is independent on x. Its support [U, U]̄ is shaded in Fig. 1. Consumers who are indifferent between buying at xk and xk + 1 are denoted by yk, and the dotted intervals of x around stores xk represent consumers who buy from these stores with positive probability. In Fig. 1(A), there are also consumers (located around y3 and y4) who never buy from firm i because they get strictly higher utility if they buy from other firms. First, in order to be a best response, the chain must have a zero measure of consumers who never buy from it. The chain in Fig. 1(A) is not a best response because the firm can increase its profit (or, it can increase its chain size without a reduction of its profit). If all stores generate losses, firm i is better off by withdrawing all its stores. Otherwise, let store x2 be profitable. Then, by shifting store x4 and all other consequent stores towards x3, and by filling the resulting gap in demand by replicating the profitable store x2 along with its price, firm i increases its profit (if x2 is strictly profitable), or gets equal profit but increases its chain size (if x2 generates zero profit). Thus, all indifferent consumers of a chain must get a utility level that belongs to [U, Ū].

Second, in order to be a best response, the chain must provide equal utilities to all its' indifferent consumers. The chain in Fig. 1(A) is not a best response because indifferent consumers y1 and y2 get different utility levels. By shifting store x2 to the left (dotted lines), firm i strictly increases its profit. Hence, the chain of stores that is a best response must look like the chain shown in Fig. 1(B): all indifferent consumers get equal utility U⁎ ∈ [U, Ū]. Third, we define an “expected per consumer profit of a store” ρk by dividing its profit level (pkDk − f) by the measure of consumers who buy from this store with positive probability (yk + 1 − yk). In order to be a best response, the chain must have all stores with equal expected per consumer profit. If, for example, store x3 in Fig. 1(B) is strictly less profitable than store x2, the chain is not a best response. Removing store x3 from the chain, shifting store x4 and all other consequent stores towards x2, and filling the resulting gap in demand by replicating store x2, constitutes a profitable deviation. Hence, the chain in Fig. 1(B) is a best response only if all stores are equally profitable per consumer. The last argument, which completes the proof, is that if two stores are equally per consumer profitable and provide equal utility levels to their indifferent consumers, they must charge equal prices. The reason for this is that the expected per consumer profit function has a unique global maximum. The equidistance result then trivially follows from the uniform price result and the fact that indifferent consumers get equal utility levels. In accordance with Proposition 1, by locating stores equidistantly, firms maximize the market power that their chains allow, and minimize the competitive threat from other firms. Focusing on their own chain of outlets, the distance between an individual store and the store of a competitor is irrelevant so that two stores of different firms can be located close to each other, as well as at a distance. With respect to pricing, Proposition 1 supports the uniform price assumption in Janssen et al. (2005). The importance of the results of Proposition 1 is that they allow us to consider a reduced-form game by redefining firms' strategy spaces. In the reduced-form game, the objective of firm i is to choose (ni, pi) that maximizes profit πi. On one hand, if Nash equilibrium in the original location-and-price game exists, it must be in strategies that satisfy equidistance and uniform pricing. On the other hand, if a

Fig. 1. (A) and (B). An illustration to the equidistance and uniform pricing result.

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strategy combination is not Nash equilibrium because one of the firms has a profitable deviation that does not satisfy equidistance and uniform pricing, this firm has a better deviation, which does satisfy equidistance and uniform pricing. Hence, every Nash equilibrium of the reduced-form game corresponds to a unique (up to relative positions of different chains, and up to firms' identities) Nash equilibrium of the original game, and the other way around. The utility of a consumer with virtual location x(i) who buys at store xm from firm i is U(i) = U0 − pi − t · d(x(i), xm i i ). When this consumer buys from another firm j, his virtual location will be x(j), and his utility will   ð jÞ ð jÞ k be U = U0 − pj − td mink d x ; xj . Since firm j locates all its stores at a distance 1/nj from each other, and the distribution of x(j) around the circle is uniform, this utility can now be written as U(j) = U0 − pj − tεj, where random variable εj is uniformly distributed over (0, 1 / (2nj)). Hence, consumer behavior can be characterized by a random utility model, in which the distribution of random utilities is endogenously determined by the sizes of retail chains: if a firm's competitors build more stores, the distribution of random utilities gets less variation, so that the firms' stores face demand that is more elastic. In other words, the degree of horizontal (geographical) differentiation of firms' stores explicitly depends on firms' location decisions. Using this random utility specification, firms' demands can be easily derived. This is the content of the corollary below. Corollary to Proposition 1. Let all firms follow strategies (ni, pi). The overall demand for firm i is given by Z Di ðpi ; ni Þ =

1 0

      nj  nj 2
2
j max 0; 1 − nj pi − pj − z − max 0; − nj pi − pj − z dz: ni ni t t j≠i

Using this result, we continue the analysis of multi-store competition. First, we only consider equilibria where ni N 0 for all i, i.e., all N firms build stores. In the following proposition, we show that there exists such an equilibrium, it is unique and symmetric. Proposition 2. For all generic values of t and f, there exists a unique Nash equilibrium with all ni N 0. In this equilibrium: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a) the total number of stores in the market is M ⁎ = t=ð 2f Þ, and each firm builds a chain with ni = n⁎ = M⁎ / N stores; b) each firm charges a price level pi = p⁎ = fM⁎ and gets zero profit;
 c) the social welfare generated is SW = U0 − 2 − N 1+ 1 fM⁎ , and is maximized by N = 2. The Proof of Proposition 2 is in the Appendix. Firms simultaneously maximize profits using the demand function derived in the Corollary to Proposition 1. The profit function turns out to be continuously differentiable so that the proof of the uniqueness of equilibrium is straightforward. In order to prove the existence, we explicitly show that firms have no profitable deviation from the proposed equilibrium. The resulting expressions for M⁎ and p⁎ are similar to those of Salop (1979) and can be interpreted in a similar way, although the values are different. The fact that firms' products are pimperfect ffiffiffi substitutes reduces the p equilibrium number of stores bypffiffiffiffiffiffi 2. ffi In our ffiffiffiffiffiffiffiffiffiffiffiffiffiffi model, firms build M ⁎ = t=ð 2f Þ stores in contrast to MC u t=f stores as is shown by Salop. The reason for this difference is the following. Let M be the total number of stores in the market. For a consumer, the distance between two neighboring stores is the sum of the distances between his location and the closest stores in clockwise and counterclockwise directions. In Salop (1979), this distance is constant for all consumers and equals to 1/M. In our model, due to products heterogeneity, consumes are characterized by different virtual locations when they buy from different firms. Consequently, this distance is different for different consumers, and is a random variable to the firms, with the expected value of 1 / (2M). For the decision problem of a firm, the difference between 1 /M and 1 / (2M) is equivalent to dividing transportation costs in half. pffiffiffiThe same reasoning also explains why the equilibrium price p⁎ is 1= 2 times the competitive equilibrium price

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pffiffiffiffi pC = tf in Salop (1979). Comparison of M⁎ with the monopoly number ofp stores ffiffiffiffiffiffiffiffiffiffiffiffiffiin ffi Salop (1979), shows that both are identical, i.e., M⁎ = M M = t=ð 2f Þ. Hence, for the linear transportation cost function, M⁎ can also be interpreted as the number of stores that maximizes the collusive profit of all competing firms in Salop (1979). Similar to Salop (1979), firms get zero profit in equilibrium. In Salop (1979), the zero-profit condition is exogenous and motivated by the free entry to the market. In random utility models, “free entry” does not imply zero profits. For example, in Anderson and de Palma (1992), firms get positive profits in equilibrium because a firm expands the number of stores until its profit starts to decline, which does occur at a positive profit level. Moreover, in our model, we do not have unlimited number of firms that can potentially enter until profit is zero. The reason why firms get zero profit in our model is due to the endogenous distribution of random utilities. This can be explained by using Fig. 1(B). Firm i provides different utility levels to consumers who buy from this firm. Consumers who are on the same real location as a store get the highest utility level whereas indifferent consumers have to travel the longest distance and get the lowest utility level, denoted by U⁎ in Fig. 1(B). It can be seen that U⁎ N U, i.e., some other firm, let us call it j, provides utility level U to its indifferent consumers. This implies that all consumers who get utility U ∈ (U, U⁎) from firm j never buy from j. Thus, firm j faces gaps in its demand, just like firm i in Fig. 1(A). That is why Fig. 1(B) does not represent Nash equilibrium: in equilibrium, it must be that all firms provide equal utility U to all their indifferent consumers. The rest of the argument is straightforward. As U⁎ = U, firm i sells to its indifferent consumers with probability zero. This gives a firm a profitable deviation, as long as it currently gets strictly positive profits. By marginally expanding the size of its retail chain, and moving its stores marginally closer to each other, a firm gets an additional profit from its new store, but hardly foregoes any profit from its former indifferent consumers.3 Only when each store yields zero profit, this profitable deviation disappears. Another result is that social welfare decreases when competition becomes more intense. Hence, welfare is maximized when only two firms compete in the market. The reason for this result is the following. Each consumer first identifies of each firm which store is closest to his geographical location and, next, chooses one of these stores by comparing utility levels that each store yields. With more firms in the market, each firm has fewer stores so that a randomly chosen consumer has to travel longer distances but has more firms (variety) from which to choose. The expected generalized transportation cost of a random consumer increases in the number of competing firms. The symmetry and, consequently, the uniqueness of the equilibrium have the same origin as the zero-profit result. As explained above, the lowest level of utility that firms provide to their indifferent consumers must coincide across all firms. This leads to (N − 1) extra equations in addition to 2N first order conditions for firms' chain sizes and prices. For an asymmetric equilibrium to exist, 2N endogenous variables (chain sizes and prices charged by all firm) must satisfy a system of (3N − 1) equations, which generically has no solution. Only for a symmetric equilibrium, when all these extra conditions disappear, the remaining two first order conditions yield a unique equilibrium. Its existence is then trivially verified. The zero-profit result is the most important equilibrium property. It rests on the simplifying assumption that the number of stores is a continuous variable. With a discrete (integer) number of stores, firms will get either strictly positive profit levels (generically with respect to the fixed cost parameter f), or build no stores. Therefore, with a discrete number of stores, our technical assumption that firms have lexicographical preferences over the profit level and the market share 3 The positive effect of such marginal expansion is of the first order whereas the negative effect is of the second order.

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becomes redundant (the Proof of Proposition 1 only uses this assumption when all stores of a firm generate exactly zero profit level). Another implication of positive equilibrium profits is that in this case the equilibrium number of stores will be determined not by the first-order equation but the first-order inequalities. Consequently, equilibrium multiplicity and asymmetry may arise when the number of stores is considered discrete. In addition to the symmetric equilibrium derived in Proposition 2, the game has (N − 2) quasi-asymmetric equilibria. In a quasiasymmetric equilibrium, some firms (at least two) build chains with ni N 0 stores whereas all other firms choose ni = 0 and build no stores. Similar to the symmetric equilibrium, all chains with ni N 0 stores are symmetric, as the following proposition shows. Proposition 3. Let N N 2. Then, for all generic values of t and f, and for any K = 2,…,N − 1, there exists a unique quasi-symmetric Nash equilibrium in which: pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a) the total number of stores in the market is M ⁎ = t=ð 2f Þ; b) K firms build chains with ni = nK⁎ = M⁎ / K stores each, charge equal price pi = p⁎ = fM⁎, and get zero profit; c) (N − K) firms set ni = 0 and get zero profit. In accordance with Proposition 2, the market capacity M⁎ is independent of the number of firms N that open stores. Proposition 3 adds to this by saying that equilibrium conditions do not pose any restrictions on the number of chains K. Therefore, the game has multiple equilibria. The number of chains K affects consumer generalized transportation cost and, therefore, the social welfare, which is maximal with two chains.

SW ðK Þ = U0 −

K 2M⁎ ðK Þ

Z1 0

  K ð1 − zÞ K τ′ z dz − fM⁎ ðK Þ: 2M ⁎ ðK Þ

For further analysis of how the number of chains K affects the total number of stores M⁎(K) and social welfare SW(K), and in order to relate these results to the existing literature, we assume in the rest of the section the power transportation cost function τ(d) = tdα with α N 0. For this specification, M⁎(K) and SW(K) become: M⁎ ðK Þ =

 1 +1 α K 2αt ΓðK + 1 − α Þ ; 2 fK ΓðK + 1ÞΓð2 − α Þ

and SW ðK Þ = U0 −

! Γ2 ðK + 1ÞΓð2 − α ÞΓðα Þ + 1 fM⁎ ðK Þ; ΓðK + 1 − α ÞΓðK + 1 + α Þ

R∞ where ΓðxÞ = 0 t x − 1 e− t dt is the so-called Gamma-function. Analyzing the expression for M⁎(K) shows that the model has no equilibria for α ≥ 2. In the limit, when transportation costs are asymptotically quadratic, so that α approaches 2 from below, the total number of stores M⁎(K) converges to zero: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 +1 α v u K 2αt ΓðK + 1 − α Þ tK 1 u 3 =t = 0; fK ΓðK + 1ÞΓð2 − α Þ 2f ðK − 1Þ lim Γð2 − α Þ αz2 2

lim M⁎ ðK Þ = lim

3.2. Non-linear transportation costs

αz2

In order to check the robustness of the obtained results, we assume now that transportation costs are given by an arbitrary strictly increasing transportation cost function τ(d), the inverse of which we denote by h(z). It turns out that the model is equally well suited for the analysis of multi-store competition with non-linear transportation costs. Following the same reasoning as in the Proof of Proposition 1, it appears that the equidistance and uniform pricing results, i.e., Proposition 1, continue to hold for non-linear transportation costs.4 Using the random utility specification shows that the distribution of the random component of consumers' utility now is Pr(τ(εj) b x) = 2njh(x), so that firms' demands are: Di ðpi ; ni Þ         Z 1  z z = j max 0; 1 − 2nj h pi − pj + τ − max 0; − 2nj h pi − pj + τ dz: 2ni 2ni 0 j≠i

Thus, the distribution of utilities in a random utility model can also be interpreted as a particular transportation cost function in our settings. Following the same reasoning as in the Proof of Proposition 2, it can be shown that the model generically has only quasi-symmetric equilibria, in which K chains evenly share the total number of stores M⁎, get zero profits and, therefore, charge equal prices p⁎ = fM⁎. So far, the results are identical to the linear transportation cost case. The difference appears in the total number of stores M⁎, which now depends on the number of chains K, i.e., M⁎ = M⁎(K). The total number of stores M⁎(K) is implicitly given by the following equation:  2 Z1  K ð1 − zÞ− 1 2 M⁎ ðK Þ f τ′ dzK 2M ⁎ ðK Þ

It is possible that there are multiple solutions for a given transportation cost function and, therefore, there are multiple equilibria with different total number of stores. Nevertheless, the uniqueness of equilibrium continues to hold for convex transportation cost functions. The expression for the social welfare is the following:

αz2

as lim Γð2 − α Þ = + ∞. Hence, also in our model, quadratic transportaαz2

tion costs results in maximum differentiation, and firms will not open more than a single store, as Martinez-Giralt and Neven (1988) show. Analyzing the expression for SW shows that for α ∈ (0, 2), social welfare is decreasing in the number of chains, i.e., dSW/dK b 0, as in the linear case. If the number of chains increases, each firm contracts its chain so that consumers expect to travel further. The next result is that the influence of the number of chains on the total number of stores in the market depends on α. If α ∈ (0, 1), i.e., when transportation costs are concave, more chains result in higher market prices and more stores in total, i.e., dM⁎/dK N 0, as in Anderson and de Palma (1992). However, if α ∈ (1, 2), i.e., when transportation costs are convex, this result is reversed, and dM⁎/dK b 0. The last result is that the model correctly predicts the number of stores M⁎(1) that a monopolist would open. Although the model is derived under the formal assumption of at least two chains, i.e., K≥2, the last expression that determines M⁎ can be evaluated in the limit when K↓1: 0  2 Z1  K ð1 − zÞ− 1 1 = lim@ 2 M⁎ ð1Þ f τ′ dzK KA1 2M⁎ ð1Þ 0

1 − 1A

 2   1 − 1 = 2f M⁎ ð1Þ τ′ : 2M⁎ ð1Þ

The resulting expression 2(M⁎(1))2f = τ′(1 / (2M⁎)) coincides with the first-order condition in Salop's model if the monopolist faces consumers with transportation cost τ(d). Hence, the model nicely combines monopoly and competition settings in a single framework. 4. Conclusion

−1

= 1:

0

4 Proofs of this claim and those to follow are logically identical to the corresponding proofs in the linear case. We, therefore, omit them.

The analysis of location strategies of multi-store firms shows that retail chains contribute to sprawl, but not to additional traffic. Firms choose equidistant locations that, for a given number of stores, minimize consumers' generalized transportation costs. At the same time, consumers have no incentive to shop for a lower price within a chain because each firm charges uniform prices within all its stores.

V. Karamychev, P. van Reeven / Regional Science and Urban Economics 39 (2009) 277–286

Hence, consumers' real transportation costs are also minimized. Obtained equilibria do not specify how locations of stores of different firms relate to each other. Stores from competing firms can be located close to each other, as well as far apart. Due to the aforementioned results, every retail chain is fully characterized by its number of stores and its price. This allows for a further analysis of multi-store competition in a random utility model, in which the distributions of random components of consumers' utilities are endogenously determined by decisions of a firm's competitors. Hence, from a theoretical point of view, this paper presents an alternative route to a random utility model with endogenous and, possibly, asymmetric distributions of utilities. The resulting random utility model shows that the market can sustain either few firms with large retail chains or many firms with small retail chains. However, equilibrium cannot sustain retail chains of different sizes. In any equilibrium, firms must have the same number of stores, and must charge equal prices. This implies that oneof-a-kind stores cannot survive in the presence of retail chains, and that the retail environment increasingly becomes uniform as retail chains grow larger. This concentration in the retail industry is welfare improving. The symmetry of equilibria in terms of chains' number of stores and prices is due to two assumptions. The first assumption is that we have assumed that firms have equal fixed cost of opening a store. With asymmetric costs, firms' chains will be asymmetric as well. The second assumption, which guarantees firms' zero profits in equilibrium and therefore the quasi-symmetry of equilibria, is that the number of stores is a continuous variable. When the numbers of stores in chains take only integer values, as they do in practice, the zero-profit result disappears, and other asymmetric equilibria are possible. The obtained results are quite robust. The model has been analyzed for a linear transportation cost function, but all principal results continue to hold for the power transportation cost function, as long as this function is less convex than quadratic. The results are also consistent with the monopoly case. Hence, the model seems to provide a single framework for analyzing endogenous location choices by a single or multiple multi-store firms with arbitrary transportation costs. Since multi-store competition resembles product line competition, our results immediately apply to product choice of multi-product firms.

Maximizing its profit, firm i locates its stores and charges prices so that xki will be strictly in between xki and xki + 1,
 i.e., xki 2 xki ; xki + 1 . Indeed, if it were not the case, e.g., if xki N xki + 1 , then store xki would have undercut store xki + 1 of the same firm so that the latter store would never have made sales. Having removed the store xki + 1 from the chain and saved a fixed cost f, firm
i wouldhave been strictly better-off. Thus, it must   be that xki 2 xki ; xki + 1 , and only consumers from xki − 1 ; xki buy from store xki .

  ði Þ b) Let now consumer ω with location xω 2 xki − 1 ; xki buy from xki k and get a utility U(i) = U0 − pki − td(x(i) ω , xi ). If, however, this consumer buys from another firm j, he gets utility    ð jÞ m (j) , where xω U ð jÞ = U0 − min pm is uniformly j + td xω ; xj

m = 1; N ;nj

distributed around the circle. Let the distribution of U(j) be denoted by Fj and the distribution of max U ð jÞ be denoted by Gi: j≠i

a) Let us take a consumer x(i) ∈ (xki , xki + 1). Buying from store xki of firm i yields him a utility level U0 − pki − t(x(i) − xki ) whereas buying from store xki + 1 yields him a utility level U0 − pki + 1 −t(xki + 1 − x(i)). Equating these utility levels yields us the usual Hotelling's expression for the indifferent consumer xki who is indifferent between buying from stores xki and xki + 1:   pk + 1 − pk xk + 1 + xk i i + i xki xki ; xki + 1 ; pki ; pki + 1 = i 2t 2

ðA:1Þ

 Gi ðzÞuPr max U ð jÞ Vz = j Fj ðzÞ:

 Fj ðzÞuPr U ð jÞ Vz ;

j≠i

j≠i

(i) buys from store xki can The probability P with which consumer xω be written as follows:

       ði Þ ðiÞ P pki ; d xω ; xki uPr xω buys from i = Pr max U ð jÞ bU ðiÞ    j≠i ðiÞ = Gi U0 − pki − td xω ; xki : (i) This probability only depends on the distance d(xω , xki ) and the k price pi . Due to Assumption 1, the distributions Fj and, therefore, Gi, have convex and full supports. Thus, P(p, d) strictly decreases in both arguments whenever P(p, d)∈(0,1). Thus, the probability P(p, d) represents a demand function of consumer ω for products of firm i sold at store xki .

c) The demand faced by store xki can be calculated by integrating individualconsumer  demands P(p, d) over the range of virtual locations xki − 1 ; xki : k

Zxi Dki

= xki

Part 1. This part consists of three steps. Step (a) derives the location of a consumer who is indifferent between buying from two neighboring stores of firm i. Step (b) derives the probability that a given consumer buys from the nearest store of firm i. Step (c) finally derives the profit functions.





Appendix A Proof of Proposition 1. Let us fix strategies c− i of the all other firms. If strategy ci = (ni, xi, pi) is such that πi(ci) b 0, it is not a best response because the strategy ci′ = (0, Ø, Ø) is strictly better as πi(ci′) = 0. In the rest of the proof we assume that πi(ci) ≥ 0 for given c− i. The proof consists of two parts. The first part introduces notations that we use throughout the proof and derives a “per store profit” and “total profit” functions of a firm i for ni N 0. The second part proves that only uniform prices and equidistant store locations can be best responses.

283

=

   P pki ; d x; xki dx

− 1

0


1B B t@

t xki − xki

Z

− 1






t xki − xki

+ pki

Z



+ pki

Gi ðU0 − yÞdy + pki

1 C Gi ðU0 − yÞdyC A:

pki

Finally, the profit that store xki generates is: 0
πki


 pk B xi ; pi ; xi = i B t @

t xki − xki

Z

− 1






t xki − xki

+ pki

Z

Gi ðU0 − zÞdz + pki



+ pki

1 C Gi ðU0 − zÞdzC A− f;

pki

so that the total profit of firm i πi, (ci) can be written as follows ni
 X πki ; πi xi ; pi ; xi =

ðA:2Þ

k=1

where x̄i(xi, pi) is determined by Eq. (A.1). Part 2. In this part, we assume that the strategy ci = (ni, xi, pi) is a best response to c− i. Maximizing Eq. (A.2) over (xi, pi) is equivalent to maximizing πi(xi, pi, yi) over (xi, pi, yi) subject to yi = x̄i(xi, pi). This part consists of four steps. Step (a) proves that any global maximum of πi (xi, pi, yi) satisfies xki = (yki + yki − 1) / 2, i.e., the location of store xki must k be located in the middle of its domain (yk−1 i , yi ). Step (b) derives the “expected per consumer profit” of a store. Step (c) argues that in

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order to maximize the profit, all stores of firm i must generate the same level of the expected per consumer profit. Step (d) then proves that this maximum level of the expected per consumer profit is achieved, and, at the same time, the constraint yi = x̄i(xi, pi) is satisfied if, and only if, ci = (ni, xi, pi) satisfies equidistance and uniform pricing. a) Unconditional maximization of πi is equivalent to allowing the firm to choose the location yki of the indifferent consumer. Maximizing πi(xi, pi, yi) w.r.t. xi for given pi and yi yields the following first-order condition:       0 = Gi U0 − pki − t xki − yki − 1 − Gi U0 − pki − t yki − xki : Because Gi has convex support, only three cases are possible. • Gi(U0 −pki − t(xki − yki − 1))= Gi(U0 − pki − t(yki − xki ))= 1 because at least one of the arguments of Gi lies outside of its support. In this case, firm i can increase its profit by a small increase in price pki (with the corresponding shift of xki ). This deviation maintains Gi = 1 and, therefore, does not reduce the demand faced by store xki , but strictly increases the revenue of firm i. This contradicts the assumption that ci = (ni, xi, pi) is a best response. • G i(U0 − pki − t(xki − yki − 1)) = Gi(U0 − pki − t(yki − xki )) = 0 because at least one of the arguments of Gi lies outside its support. In this case, consumers located near yki or yki never buy from firm i. Hence, there are gaps in consumer demands between some neighboring stores of the firm. In accordance with our assumption, firm i in this case strictly prefers to build more stores and locate them closer to each other. By doing so, the firm does not reduce its profit but increases the number of stores. This also contradicts the assumption that ci = (ni, xi, pi) is a best response. • U0 −pki − t(xki −
yki − 1) = U0 −pki − t(yki − xki ), so that (xki − yki − 1) = (yki − xki ), or xki = 12 yki + yki − 1 . Combining these three cases proves that if ci = (ni, xi, pi) is a best response, then each store is located in the middle of its domain (yki − 1, yki ), i.e., when xki = (yki + yki − 1) / 2. In this case, each store generates the following profit:
 max π ki x0 i ; pi ; yi xi t ððyki + yki pk B = iB t @

− 1

ÞZ=2 − yki

− 1

Þ + pki

t ðyki − ðyki + yki

Z

Gi ðU0 − zÞdz + pki

− 1

Þ=2Þ + pki

1 C Gi ðU0 − zÞdzC A− f:

pki

ðA:3Þ b) Defining by zki ≡ yki − yki − 1 the measure of consumers in the domain (yki − 1, yki ) of store xki allows us to rewrite Eq. (A.3) in terms of (zi, pi) as follows:  
 2 k π˜ i zki ; pki u max πki xi ; pi ; yi = pki xi t

pki

tzki =2

Z

+

Gi ðU0 − zÞdz − f : pki

The expected per consumer profit ρki of store xki is then defined as follows: 0 1 pki + tzki =2 Z  π˜ k
zk ; pk  B C 1 ρki uρi zki ; pki = i i k i = k B 2pki Gi ðU0 − zÞdz − tf C @ A: tzi zi 

pki

It is easy to see that ρi(z, p) b 0 for either p N U0, or p = 0, or z = 0 (in the limit). Together with the continuity of ρi (z, p), this implies that either ρi attains its maximum at an interior point, or it does not have a maximum because ρi(z, p) b 0 for all p N 0 and z N 0, and limzY∞ ρi ðz; pÞ = 0.


 c) Suppose that ρi does have an interior maximum z⁎ ; p⁎ 2 arg max ρi ðz; pÞ with ρ⁎i ≡ ρi(z⁎, p⁎) ≥ 0. Then, the retail chain p; z N 0

with pki = p and ni = 1/z⁎ equidistantly located stores yields the ni
 P z⁎ ρi z⁎ ; p⁎ = ρ⁎i . On the other hand, the profit profit π˜ i = k=1

from an arbitrary retail chain of firm i can be written as ni

 P zki ρi zki ; pki Vρ⁎i . Hence, π i(z i, pi) can achieve ρ⁎i π˜ i zi ; pi = k=1

if and only if ρ i(zki , pki ) = ρ⁎, i i.e., all stores of firm i generate the same expected per consumer profit ρ⁎. i In case ρ i has no interior maxima, any retail chain yields strictly negative profit for given c− i, and the firm's best response is to stay out of the market; we have excluded this case in the beginning of the proof of the proposition.
 d) Suppose that ρ⁎i is achievable so that zki ; pki 2 arg max p;z N 0

ρi ðz; pÞ for all k. Hence, the first-order condition ∂ρ i / ∂p = ∂ρ i/∂z = 0 must be satisfied for all k. Besides, ci = (ni, xi, p i ) must satisfy the restriction (A.1), i.e., y i = x̄ i (x i , p i ). Taking into account xki = (yki + yki − 1)/2 from Part 2 step (a) and the definition of zki yields that (pki , zki ) must satisfy the following system for all k = 1, …, ni: 8 pki + 1 − pki xki + 1 + xki > > k > + > yi = < 2t 2 k k −1 ðA:4Þ yi + yi k > = x > i > 2 > : k zi = yki − yki − 1 These equations immediately imply pki + tzki / 2 = pki + 1 + tzki + 1 / 2, so that pki + tzki / 2 ≡ λ⁎i is constant for all stores k. Using this notation, the first-order condition ∂ρi/∂z = 0 can be written as   ⁎ k ⁎ Aρi  k k  pi Gi U0 − λi − ρi pi ; zi = ; 0= Az zki which implies p ki = ρ⁎i / G i (U 0 − λ⁎). Hence, the maximum i profit level ρ⁎i can be achieved only if the price is uniform for all stores of firm i. Finally, for pki + 1 = pki the system (A.4) implies xki − xk−1 = xki + 1 − xki , i.e., equidistance. The optimal i number of stores then is n i = 1/zi. This proves that the function ρi(z, p) has a unique global maximum (pki , zki ) = (ρ⁎i / Gi(U0 − λ⁎), 1/ni). i Thus, if ρi has an interior maximum ρ⁎i = ρi(z⁎, p⁎) ≥ 0, firm's i best response is to locate its ni = 1/z⁎ stores equidistantly and sell at the price p⁎ in all stores, which yields profit ρ⁎. i Consequently, if ci = (ni, xi, pi) is a best response, it must necessarily satisfy equidistance and uniform pricing. □ Proof of Corollary to Proposition 1. Let us take a store (xki , pi) of firm i and consider a consumer x(i) who is located at a distance z ≤ 1 / (2ni) from this store, i.e., x(i) =xki ± z ∈ (xki − 1 / (2ni), xki + 1 / (2ni)). Consumer x(i) gets utility U(i) =U0 −pi −tz if he buys from (xki , pi), and he gets utility U(j) =U0 − pj −tεj if he buys from firm j. Therefore, consumer x(i) prefers firm i over firm j with probability:  
 pi − pj Pr U ðiÞ N U ð jÞ = Pr ej N +z t     p − p  pi − pj i j + z − max 0; − 2nj +z = max 0; 1 − 2nj t t Hence, consumer x(i) buys from firm i with probability n  p − p   p − p o i j i j + z − max 0; − 2nj +z : j max 0; 1 − 2nj t t j≠i
 R 1=ð2ni Þ The demand faced by store xki is Dki ðpi ; ni Þ = 2 0 Pi xðiÞ dz and the total demand is Di ðpi ; ni Þ =

R1 0

n  p jj≠i max 0; 1 − 2nj i

− pj t

  n p − z nji − max 0; − 2nj i

− pj t

n

− z nji

o

dz:

□

V. Karamychev, P. van Reeven / Regional Science and Urban Economics 39 (2009) 277–286

Proof of Proposition 2. Suppose first that such an equilibrium does exist. We denote equilibrium number of stores and equilibrium price of a firm i by n⁎i and p⁎i correspondingly. We index firms in such a way that n1⁎≤…≤n⁎i ≤ ni + 1⁎ ≤ … ≤ nN⁎. In the Proof of Proposition 1, part 2a, we have shown that U0 − p⁎i − t / (2n⁎) i must belong to the support of the distribution function Gi, which is    
 Gi ðxÞuPr max Uj Vx = j Pr Uj Vx = j Pr U0 − p⁎j − tej Vx ; j≠i

j≠i

j≠i

where εj is uniformly distributed over (0, 1 / (2n⁎)). Hence, it must be j that U0 −p⁎i −t/(2n⁎) i.e., p⁎i −p⁎j ≤t(1/n⁎j − 1/n⁎)/2. Simii ≥U0 −p⁎ j −t / (2n⁎), j i larly, U0 −p⁎−t/(2n ⁎) k k must belong to the support of the distribution function Gk, which yields pk⁎ −p⁎l ≤t(1/n⁎l − 1/nk⁎) / 2. Taking j =k and l =i implies ! t 1 1 ⁎ ⁎ pi − pk = − for all i; k = 1; N ; N: ðA:5Þ 2 n⁎k n⁎i In other words, lower ends of the supports of all Gi coincide. There are N − 1 independent equations in Eq. (A.5). Using Eq. (A.5), equilibrium demand of firm i can be written explicitly as follows
 R1 Di p⁎i ; n⁎i = 0 jj≠i

= n⁎i

k=i X k=N

( ⁎ nj

n⁎j

ni

ni

z − max 0; ⁎

1=n⁎k

Z

i

!)

1=n k=i Z k X k=N

n  o j n⁎j z − max 0; n⁎j z − 1 dz j≠i

1=n⁎k + 1

j=i+1

 Z k=i k X n⁎j z dz = j n⁎j k=N

1=n⁎k + 1

j=1

zk − 1 dz;

1=n⁎k + 1

   = N   kX  k  k 1 k ⁎ Di p⁎i ; n⁎i = j nj 1= n⁎k − 1= n⁎k + 1 ; k j=1 k=i

where we have formally defined 1/n⁎ N + 1 ≡ 0. Next, we rewrite Di(pi, ni) as Z

Di ðpi ; ni Þ = ni 2ðpi − pj Þ=t



 j max 0; 1 − nj z − max 0; − nj z dz;

n⁎ p⁎ = t=ð2N Þ fn⁎ = p⁎ =N

Solving this yields that ifffi Nash equilibrium exists, it is unique and pffiffiffiffiffiffiffiffiffiffiffiffiffi given by M⁎ un⁎ N = t=ð 2f Þ, n⁎ = M⁎ / N, and p⁎ = fM⁎. We end the proof by showing a firm i does not have a profitable deviation if all (N − 1) other firms use (p⁎, n⁎), and, therefore, the proposed strategies do form Nash equilibrium. To this end, we suppose that firm i builds ni stores and charges a price pi, whereas all the other firms follow equilibrium strategy (p⁎, n⁎). The demand for firm i in this case can be written as follows:     N p − p⁎ n⁎ p − p⁎ n⁎ max 0; 1 − 2n⁎ i − max 0; − 2n⁎ i −z −z t ni t ni 1− Δ Zn + Δp
 1 ˜ i Δp ; Δn ; ðmaxð0; 1 − zÞ − maxð0; − zÞÞN − 1 dzu D = 1 − Δn R1



8 2ðpi − pj Þ=t + 1=ni > Z > > 

 > ADi A > > ð p ; n Þ = n j max 0; 1 − nj z − max 0; − nj z dz > i i i > Api Ap > j≠i i > < 2ðp − p Þ=t − pj Þ=t + 1=ni

Z

2ðpi − pj Þ=t

dz

where we define Δp ≡ 2n⁎(pi − p⁎) / t and Δn ≡ (ni − n⁎) / ni (−∞, 1). Let us consider 4 cases. a) Δp ≥ 1. In this case the price pi is so high that Di = 0. This is not a profitable deviation as firm i gets negative profit πi(pi, ni) = −fni. b) 0 ≤ Δp b 1. In this case:
 D˜ i Δp ; Δn =

1 1 − Δn

minð1;1 − Δn + Δp Þ

Z



 j max 0; 1 − nj z − max 0; − nj z dz j≠i

Using Eq. (A.5) yields at (p⁎, i n⁎): i ( !) 8 ⁎   ⁎ N 2n 2n⁎i i − 1 nj > AD 1 1 i ⁎ ⁎ ⁎ > i > j 1 − nj max ⁎ − ⁎ ; 0 j ⁎ pi ; ni = − = − < Api t j=1 t j = 1 ni nj ni > ADi  ⁎ ⁎  1  ⁎ ⁎  > > p ; n = ⁎ Di pi ; ni : Ani i i ni Maximizing πi(pi, ni) = piDi(pi, ni) − fni yields the following firstorder conditions: 8     2n⁎ p⁎ i − 1 n⁎
 > j > i i > 0 = Di p⁎ ; n⁎ + p⁎ ADi p⁎i ; n⁎i = Di p⁎i ; n⁎i − j < i i i Api t j = 1 n⁎i     > > 0 = p ADi p⁎ ; n⁎ − f = pi D p⁎ ; n⁎ − f > : i i i i i i ⁎ Ani ni Substituting Eq. (A.6) for Di(pi⁎, ni⁎) and adding Eq. (A.5) results in the following system that equilibrium values (p⁎, i n⁎) i must satisfy: 8   ⁎ k = N     > X i −1 k k 1 k ⁎ > > n⁎ p⁎ j nj = t > j nj 1= n⁎k − 1= n⁎k + 1 ; i = 1; N ; N > i i > 2 k j = 1 n⁎ j=1 > > i k=i > >    < kX =N  k  k 1 k ⁎ : j nj 1= n⁎k − 1= n⁎k + 1 fn⁎i = pi ; i = 1; N ; N > kj=1 > k=i > ! > > > > t 1 1 > ⁎ ⁎ > > : p1 − pk = 2 n⁎ − n⁎ ; k = 2; N ; N 1 k




ð1 − zÞN

−1

dz =

1 − Δp

N

Δp



N − max 0; Δn − Δp : N ð1 − Δn Þ

c) −(1 − Δn) ≤ Δp b 0. In this case:

j

2ðpi > > > > ADi 1 A > > ðpi ; ni Þ = Di ðpi ; ni Þ + ni > > > ni Api > : Ani

−1

0

j≠i

and derive the partial derivatives of Di(pi, ni):

i



Δp

ðA:6Þ

2ðpi − pj Þ=t + 1=ni

This system has 2N unknowns and (3N − 1) independent equations. Hence, for all generic values of t and f, it does not have solutions that satisfy n1⁎ b … b n⁎i b ni + 1⁎ b … b nN⁎. Assuming that nk⁎ = nk + 1⁎ for some k reduces the number of equations by 3 and the number of unknowns by 2, because in this case pk⁎ = pk + 1⁎ as well. In order to get a generic solution, we need to make (N − 1) such assumptions. Therefore, generically only symmetric equilibria exist where (p⁎, i n⁎) i = (p⁎, n⁎) for all i = 1,…, N. Equilibrium values (p⁎, n⁎) are determined by

Di ðpi ; ni Þ =

1=n⁎k

 k −1 j n⁎j z j

j=1

⁎

dz = n⁎i

z−1 ⁎

285


 D˜ i Δp ; Δn =

1 1 − Δn

Z

!

minð1;1 − Δn + Δp Þ

ð1 − zÞN
0
N 1 − NΔp − max 0; Δn − Δp = : Nð1 − Δn Þ − Δp +

−1

dz

d) Δ p b − (1 − Δ n). In this case the price p i is so low, that D i = 1, and firm i could increase the price up to the point when Δ p = − (1 − Δ n), without loosing any consumer. Hence, the deviation with Δ p = − (1 − Δ n), which belongs to case (c), is better than with Δ p b − (1 − Δ n). Therefore, we only need to consider −(1 − Δn) ≤ Δp b 1. Combining cases (b) and (c), we can write the demand as follows:
 D˜ i Δp ; Δn =

1 N ð1 − Δn Þ

N

2 X
k CNk − Δp  1 − NΔp + max 0; Δp

−2

!

N : − max 0; Δn − Δp

k=2

Next, we define Δπ(Δp, Δn) as follows:  


 tΔp n⁎ Δπ Δp ; Δn uπi ðpi ; ni Þ − πi p⁎ ; n⁎ = π i p⁎ + − π i p⁎ ; n⁎ ; ⁎ 1− Δ 2n n    tΔp ˜
fn⁎ = p⁎ + Δ ; Δ ; D − p n i 1 − Δn 2n⁎ and show that Δπ(Δp, Δn) b 0 for any (Δp, Δn) ≠ (0,0) that satisfies Δn b 1 and −(1 − Δn) ≤ Δp b 1. We consider two cases.

286

V. Karamychev, P. van Reeven / Regional Science and Urban Economics 39 (2009) 277–286

a) If Δp N 0, then

N

N  

 tΔp 1 − Δp − max 0; Δn − Δp fn⁎ Δπ Δp ; Δn = p⁎ + − N ð1 − Δn Þ 1 − Δn 2n⁎ N

N  
tΔp 1 − Δp fn⁎ ⁎ 1 − 1 + NΔp 1 − Δp V p⁎ + = − fM − b 0: Nð1 − Δn Þ 2n⁎ Nð1 − Δn Þ 1 − Δn

b) If Δp b 0, then

N  
 tΔp 1 − NΔp − max 0; Δn − Δp fn⁎ − Δπ Δp ; Δn = p⁎ + Nð1 − Δn Þ 1 − Δn 2n⁎
2   2 tΔp 1 − NΔp fn⁎ ⁎ N Δp − b 0: V p⁎ + = − fM Nð1 − Δn Þ 2n⁎ Nð1 − Δn Þ 1 − Δn

Z

+

j≠i




  N zd 1 − 1 − 2n⁎ x

−1



z

0

=

1=ð2n⁎ Þ



  N  1 1 − 1 − 2n⁎ z : 2M⁎

Integrating this expression over the whole circle yields the overall expected distance that all consumers have to travel:

⁎

2n

Z

1=ð2n⁎ Þ 0

1=ð2n⁎ Þ     Z  N  1 ⁎ E min z; min ej 1 − 1 − 2n⁎ z dz = 2n dz ⁎ j≠i 2M 0

N : = 2M⁎ ðN + 1Þ

Therefore, the social welfare generated in the market is SW = U0 −

  Nt 1 ⁎ fM⁎ : = U − 2 − − fM 0 ðN + 1Þ 2M⁎ ðN + 1Þ □

This completes the proof.

Proof of Proposition 3. Suppose that there is an equilibrium, in which only K firms build stores in equilibrium; the other (N − K) firms open no stores and get zero profit. Then, in accordance with Proposition 2, each chain must build nK⁎ = M⁎ / K stores and charge the price p⁎. All of them get zero profit and, therefore, have no incentives to close stores and leave the market. We show that none of the other (N − K) firms can profitably enter the market, so that the proposed equilibrium does exist. Suppose that one of these (N − K) firms enters and builds n stores and charges a price p. Let Δn ≡ (n −nK⁎) /n ∈ (−∞,1) and Δp ≡ 2nK⁎(p −p⁎) / t ∈ (−(1 −Δn), 1) so that n =M⁎ / (K(1 −Δn)) and p =fM⁎(1+KΔp). Then, this entering firm gets the following demand:


 ˜ Δp ; Δn = D i

+1

2 KX
k 1 − ðK + 1ÞΔp + max 0; Δp CNk − Δp

−2

fM⁎ V

π˜ i V fM ⁎

x(i) buys from the nearest store. Hence, he travels min z; min ej , and j≠i the expected distance that x(i) travels is: −1



˜
 π˜ i Δp ; Δn = fM ⁎ 1 + KΔp D K + 1 Δp ; Δn −

fM⁎ K ð1 − Δn Þ ! ! +1


2 KX
k − 2 K +1 1 + KΔp 1 − ðK + 1ÞΔp + max 0; Δp CNk − Δp − K k=2 ðK + 1Þð1 − Δn Þ

:

If Δp ≥ 0 then:

Therefore, Δπ(Δp, Δn) b 0 for any (Δp, Δn) ≠ (0,0), so that (p⁎, n⁎) is a unique best response of firm i. We complete the proof by deriving the social welfare function. Social costs consist of generalized transportation costs and fixed costs fM⁎. Let us take a consumer x(i) ∈ (xki − 1 / (2n⁎), xki + 1 / (2n⁎)), and let z = d(x(i), xki ). If he buys from store xki of firm i, he has to travel z. If he buys from another firm j, he has to travel a random distance εj, which is uniformly distributed over the interval (0,1 / (2n⁎)). Dueto equal  prices,

   Zz   N E min z; min ej xd 1 − 1 − 2n⁎ x =

and the following profit:



K + 1 − max 0; Δn − Δp

k=2

ðK + 1Þð1 − Δn Þ ! +1

2 KX
k − 2 1 V ; CNk − Δp 1 − ðK + 1ÞΔp + max 0; Δp ðK + 1Þð1 − Δn Þ k=2


K + 1 1 + KΔp 1 − Δp − 1− ðK + 1Þð1 − Δn Þ

1 K

b 0;

as (1 + KΔp)(1 − Δp)k + 1 attains its global maximum of 1 over Δp ≥ 0 at Δp = 0. If, on the other hand, Δp b 0, then: π˜ i V fM ⁎



 1 + KΔp 1 − ðK + 1ÞΔp − 1 − ðK + 1Þð1 − Δn Þ

1 K

= − fM ⁎


2 2K ðK + 1ÞΔp + 1 + 4K + 3 4K ðK + 1Þ2 ð1 − Δn Þ

b0:

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