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International Journal of Industrial Organization
20 (2002) 163–190
www.elsevier.com / locate / econbase
Spatial competition among multi-store firms
Debashis Pal a , *, Jyotirmoy Sarkar b
b
a
Department of Economics, University of Cincinnati, Cincinnati, OH 45221, USA
Department of Mathematical Sciences, Indiana University Purdue University Indianapolis,
Indianapolis, IN 46202, USA
Abstract
The paper analyzes spatial Cournot competition among multi-store firms. It demonstrates
that the complex problem of determining equilibrium store locations for competing
multi-store firms can be approximated by a simple one, in which each firm behaves as a
multi-store monopolist in choosing its store locations. A firm’s equilibrium store locations
often coincide with its monopoly locations, and in general, converge to its monopoly
locations as the demand grows larger. When the firms have an equal number of stores, the
stores belonging to competing firms agglomerate at discrete points that coincide with each
firm’s monopoly store locations.  2002 Elsevier Science B.V. All rights reserved.
JEL classification: D43; L13
Keywords: Spatial competition; Multi-store firms; Cournot oligopoly
1. Introduction
Spatial competition has a rich and diverse literature, with its origin dating back
to the seminal work of Hotelling (1929). Despite its long history, it is surprising
how little attention has been paid to study competition among firms who can set up
multiple stores.1 The literature on spatial competition that allows firms to choose
* Corresponding author. Tel.: 11-513-556-2630; fax: 11-513-556-2669.
E-mail address: [email protected] (D. Pal).
1
Note that stores can be interpreted as plants. In this paper we use the word stores, although stores
and plants can be used interchangeably.
0167-7187 / 02 / $ – see front matter  2002 Elsevier Science B.V. All rights reserved.
PII: S0167-7187( 00 )00080-1
164
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
their locations typically assumes that each firm can set up only one store.2
Presumably, this assumption is often made to avoid the analytical complexity that
would arise otherwise.
It is important, however, to recognize that most firms actually set up multiple
facilities. A casual look at a typical U.S. city reveals that diverse firms, such as
Pizza Hut, J.C. Penney, Kroger and Circuit City, all have several stores.3 In fact,
retailers with four or more outlets account for more than half of the total retail
business in the United States. In the context of manufacturing, a producer often
manufactures a homogeneous product at several production facilities. For example,
in the United States, Lafarge Corporation, a leading cement producer, has 15
cement plants. In the natural gas liquids (NGL) industry, the industry leaders GPM
Gas Corporation (formerly Philips 66 Natural Gas) and Warren Petroleum
Company (a subsidiary of Chevron Corporation) have 18 and 57 gas liquid plants,
respectively. In the ready-mixed concrete industry, Florida Rock Industries has 82
ready-mixed concrete plants, Texas Industries Inc. has 29 ready-mixed concrete
plants and the industry leader Lafarge Corporation has as many as 450 production
facilities.4
It may be argued that the assumption of single-store firm is merely a technical
simplification and the results should extend to multi-store firms. However, observe
that when a firm has several stores, each store’s behavior affects the decisions of
all other stores, including those owned by the same firm. Consequently, each store
cannot be treated independently as a single-store firm and the results obtained with
multi-store firms are likely to differ from those obtained with single-store firms.5
Naturally, the study of spatial competition among multi-store firms deserves
special attention, which is the objective of this paper.
Although the literature on this topic is surprisingly brief, the study of location
decisions by multi-store firms originates more than thirty years back. Teitz (1968)
is the first to study spatial competition among multi-store firms. In the context of
Hotelling’s linear city model with linear transport cost, Teitz (1968) points out that
a Nash location equilibrium does not exist if the firms have multiple stores. This
non-existence of a location equilibrium may have contributed to the brevity of the
literature during the next two decades. Subsequently, Martinez-Giralt and Neven
(1988) assume quadratic transport cost together with mill pricing and demonstrate
2
Even in the context of a spatial monopolist, only a few papers analyze the location and pricing
decisions of a multi-store monopolist. Katz (1980) and Chu and Lu (1998) are among the few studies
in this area. See Chu and Lu (1998) for a related discussion.
3
The telephone directory for the city of Cincinnati lists 25 stores for Pizza Hut, 7 stores for J.C.
Penney, 53 stores for Kroger, and 6 stores for Circuit City.
4
Source: Gale Business Resources. Internet address: www.galenet.com / servlet / GBR /.
5
In the context of retailing, Ghosh and McLafferty (1987, Chapter 6) argue that the traditional
methods of site selection with single-outlet firms are inadequate to analyze location decisions of
multi-outlet retail firms. They claim that the analysis of multi-outlet retailers requires systematic
evaluation of the impact of each store on the entire network.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
165
an intriguing result. In the context of a spatial duopoly where each firm may open
up to two stores, they show that for both the linear city and the circular city
models, the firms do not take up the opportunity of opening multiple stores; in
equilibrium, each firm opens only one outlet. Observe that the results demonstrated
by Teitz (1968) and Martinez-Giralt and Neven (1988) are rather fascinating given
the prevalence of spatial competition involving firms with multiple outlets.
Subsequent work in this area involves either pricing policies other than mill
pricing or vertical product differentiation (as opposed to horizontal product
differentiation). Eaton and Schmitt (1994) analyze economies of scope using a
model that resembles a model of competitive spatial price discrimination among
firms with multiple outlets. Thill (1997, 2000) analyzes spatial competition among
multi-store firms by assuming fixed prices but allowing each outlet to choose a
quality for its product, in addition to choosing a geographic location on a line. In
the context of vertical product differentiation, Champsaur and Rochet (1989)
analyze a duopoly where the firms may choose multiple qualities for their
products.6
In this paper, we consider an entirely different approach to study spatial
competition among multi-store firms. Instead of assuming an exogenously fixed
price or price competition, we assume that the firms compete in quantities (a` la
Cournot). The assumption of Cournot competition enables us to avoid the puzzling
findings obtained in spatial models with price competition and to demonstrate
results that closely resemble the real world. We allow the firms to set up multiple
stores, and analyze a two-stage problem of location and quantity choices in
Hotelling’s linear city model. We deviate from the previous literature on multistore firms by assuming that in the second stage, the firms behave as Cournot
oligopolists and discriminate over space.
In non-spatial contexts, the assumption of Cournot competition needs no further
justification. The Cournot model is probably the most widely used oligopoly
model. In spatial contexts, the predictions arising from a spatial Cournot model
often describe the real world better than those arising from a spatial price
competition model. For example, it is Cournot competition, not price competition
that successfully explains the commonly observed phenomenon of overlapping
geographic markets of the competing firms selling a homogeneous product.7
Anderson and Neven (1991) provide compelling arguments justifying the appro6
Anderson and dePalma (1992) depart from a standard Hotelling type location model and pioneer an
alternative approach to study competition among multi-product firms. Anderson and dePalma (1992)
endogenize the pricing decisions of the firms and use a nested logit model of demand to characterize a
symmetric equilibrium for multi-product firms.
7
Phlips (1983) and McBride (1983) provide support for spatially overlapping markets consistent
with Cournot competition. Price competition with homogeneous product always implies non-overlapping markets for the competing firms. This is true for a variety of alternative pricing strategies
including mill pricing and spatial discriminatory pricing. For example, see d’Aspremont et al. (1979),
Lederer and Hurter (1986).
166
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
priateness of the Cournot assumption in various spatial models.8 It is established
that spatial Cournot competition is appropriate for industries where quantity is less
flexible than price at each market point. Such industries would include, for
example, oil, natural gas, cement and ready-mixed concrete. In fact, from an
empirical perspective, the Cournot model of spatial competition is employed to
analyze international oil and natural gas markets (see, for example, Salant, 1982).
The predictions of the Cournot model in terms of delivered prices are confirmed
by McBride (1983) in the cement industry and by Greenhut et al. (1980) in a
representative sample of industries.9
In the context of single-store firms, spatial models with Cournot competition are
becoming increasingly popular in recent years. Anderson and Neven (1991), and
Hamilton et al. (1989) pioneer the study of spatial Cournot competition with
endogenous location choice.10 They analyze a two-stage problem of location and
quantity choices in Hotelling’s linear city model. Afterwards, Hamilton et al.
(1994) consider a variant of the Cournot competition, where the firms choose total
output and use mill pricing. Gupta et al. (1997) extend Anderson and Neven’s
(1991) analysis by considering non-uniform consumer distribution in the linear
city model. Mayer (2000) contributes by allowing the production costs to differ at
various locations.
In the present paper, we contribute to the previous literature by considering
multi-store firms in the context of spatial Cournot competition with endogenous
location choice. We identify the equilibrium store locations for competing multistore firms and establish the following results.
First, using Cournot competition we avoid some of the puzzling results obtained
in location models with price competition. Under price competition with mill
pricing, each firm locates all of its outlets at the same market point (MartinezGiralt and Neven, 1988). Thus, in effect, firms do not take up the opportunity of
opening multiple outlets. This result is intriguing given the prevalence of multioutlet firms in the real world. This puzzling outcome, however, does not arise with
Cournot competition. Under Cournot competition, each firm always chooses
distinct locations for its outlets. Furthermore, under price competition with both
8
Also, see Greenhut et al. (1991), which argues that between the Bertrand and Cournot models, the
latter warrants major consideration in modelling spatial competition.
9
Furthermore, following Eaton and Schmitt (1994), a spatial model with Cournot competition can be
interpreted as a non-spatial Cournot model involving firms that enjoy economies of scope. Eaton and
Schmitt (1994) model economies of scope by considering firms that may produce few basic products,
which can be modified to produce any other variant in the attribute space. When the firms compete in
prices (a` la Bertrand), the model mirrors a spatial model with discriminatory pricing. Following this
approach, therefore, if the firms compete in quantities, instead of prices, a non-spatial Cournot model
with economies of scope would be equivalent to a spatial Cournot model.
10
Previous literature on spatial Cournot competition treats locations as exogenously fixed. See
Greenhut and Greenhut (1975), Greenhut and Ohta (1975), Norman (1981) and Ohta (1988) for related
discussions.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
167
mill pricing and spatial discriminatory pricing, outlets of competing firms never
agglomerate. In contrast, under Cournot competition, outlets of competing firms
may agglomerate at finitely many market points; a result that is consistent with the
clustering of stores of competing firms at various shopping malls in the same city.
Second, we characterize situations when, in equilibrium, each firm would
simply choose its monopoly locations. In other words, each firm would locate its
stores as if it were a monopolist and did not face any competition. We also
characterize situations when the equilibrium store locations differ from the
monopoly locations. In this situation, however, we demonstrate that as the demand
becomes large, relative to the transport cost, all firms’ equilibrium store locations
converge to their respective monopoly locations. In fact, numerical simulations
show that even for smaller demands, the equilibrium store locations are usually
quite close to their respective monopoly locations. Thus, we demonstrate that the
complex problem of determining equilibrium store locations for competing multistore firms can be approximated by a very simple one, in which we treat each firm
as a multi-store monopolist and determine optimal store locations for a monopolist
with multiple stores.
Third, we shed new light on a prevailing perception in the literature that
concludes that price competition yields spatial dispersion whereas Cournot
competition gives rise to spatial agglomeration of firms. In the context of price
competition, coincident location of stores of competing firms offering identical
products severely intensifies price competition and thus, stores belonging to
competing firms never agglomerate in a location-price game. In contrast, in the
context of single-store firms, pioneering work of Anderson and Neven (1991) and
Hamilton et al. (1989) establishes that in Hotelling’s linear city model, Cournot
competition gives rise to spatial agglomeration of firms. Subsequently, Gupta et al.
(1997) relax the assumption of uniform consumer distribution in the linear city
model and confirm the agglomeration result for a wide variety of consumer
distributions. Mayer (2000) allows production cost to differ at various market
points and finds that the firms may still agglomerate. Thus, it may indeed seem
reasonable to conclude that price competition generates spatial dispersion, while
Cournot competition gives rise to spatial agglomeration.
In this paper, however, we establish that if the firms can have multiple stores,
the conclusion drawn above is incorrect. Under Cournot competition involving
multi-store firms, stores belonging to competing firms may not agglomerate;
giving rise to complete dispersion of stores. In fact, depending on the number of
stores, price competition with spatial price discrimination and Cournot competition
generate quite similar location patterns.
Thus, this paper contributes to the literature of spatial competition in several
ways. First, it incorporates the commonly observed phenomenon of multi-store
firms and allows the firms to choose their store locations. Second, by considering
Cournot competition, it avoids some of the puzzling outcomes associated with
spatial price competition involving multi-store firms. Third, it develops a mecha-
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
168
nism to identify the equilibrium locations. Finally, it characterizes the equilibrium
locations and demonstrates several results that closely resemble the real world.
The paper is organized as follows. Section 2 describes the model. Section 3
characterizes various properties of the location and quantity equilibria. Section 4
identifies the equilibrium locations and demonstrates the results. Section 5
concludes the paper.
2. Model
We consider a spatial multi-store Cournot oligopoly serving a linear market of
unit length. For expositional simplicity, we restrict the number of firms to two; the
results extend to more than two firms. Without loss of generality, the interval f0,1g
represents the linear market. The consumers are distributed uniformly over f0,1g.
The market demand at each point j [f0,1g is given by p 5 a 2 bQ, where a . 0,
b . 0 are constants, Q is the aggregate quantity supplied at j , and p is the market
price at j . Without loss of generality, we assume b 5 1 / 3 for expositional
simplicity. Two firms, 1 and 2, compete in quantities at each market point
j [f0,1g. Firms 1 and 2 have m > 1 and n > 1 stores, respectively. Without loss of
generality, assume that m < n. The firms locate their stores in f0,1g. Note that the
stores may be interpreted as plants. In this paper, we use the word stores, although
stores and plants can be used interchangeably. The vector x] 5 (x 1 , x 2 , . . . , x m )
denotes the locations of Firm 1’s stores. Here x i is the distance measured from the
left endpoint of the market. Without loss of generality, we assume 0 < x 1 < x 2 < ?
? ? < x m < 1. Similarly, the vector y 5 ( y 1 , y 2 , . . . , y n ) denotes the locations of
Firm 2’s stores, where 0 < y 1 < y 2 <] ? ? ? < y n < 1. The firms deliver the product
to the consumers and thus, can discriminate across consumers. The firms have
identical production and transportation technologies. Each firm produces at a
constant marginal and average cost (both normalized to zero) and pays a linear
transport cost of t . 0, per unit distance. Arbitrage among the consumers is
assumed to be infeasible due to high transaction costs. We also assume that a > 2t.
This condition ensures that both firms will always serve the whole market.
Each firm serves a market point j incurring the lowest possible transport cost,
and hence from the store which is nearest to the market point. Define firm i’s
si 5 1, 2d effective delivered marginal cost c i ( j ) at the market point j as
c 1s jd 5 minhtux 1 2 ju, tux 2 2 ju, . . . , tux m 2 juj
and
c 2s jd 5 minhtu y 1 2 ju, tu y 2 2 ju, . . . , tu y n 2 juj
It now follows that two stores of the same firm never coincide and each store
serves a contiguous market around itself. Also, on each side, a store’s market
extends up to the midpoint between itself and the next store (owned by the same
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
169
firm), if such a next store exists, otherwise, it extends up to the appropriate
endpoint.
We study the sub-game perfect Nash equilibria (SPNE) of a two-stage game,
where in stage one, the firms choose the locations of their stores along the linear
market and in stage two, the firms compete in quantities. We proceed by backward
induction and characterize the quantity equilibrium in the second stage for given
locations.
Since marginal production cost is constant and arbitrage among the consumers
is not feasible, quantities set at different points by the same firm are strategically
independent. Therefore, the second stage Cournot equilibrium can be characterized
by a set of independent Cournot equilibria, one for each market point j [f0,1g.
At each market point j [f0,1g, firm i si 5 1, 2d chooses qis jd to maximize its
profit f p 2 c is jdg qis jd. By simultaneously solving the first order conditions for
profit maximization of the two firms, we obtain the following equilibrium
outcomes at each j [f0,1g:
qis j , ]x, yd 5 a 2 2c is jd 1 c js jd
]
fa 2 2cis jd 1 cjs jdg 2
]]]]]]
pis j , x,
y
5
] ]d
3
where i 5 1, 2 and j 5 1, 2, but i ± j. qis j , x,
] ]yd and pis j , ]x, ]yd denote firm i’s
equilibrium quantity and equilibrium profit at market
point j , given the locations
y
.
sx,
d
] ]
Therefore, given the locations sx,
] y]d, firm i’s si 5 1, 2d equilibrium aggregate
profit is
1
P sx,] ]yd 5Ep sj, x,] ]yd dj
i
i
0
Our objective is to solve for a pair of location vectors sx*,
such that, given y*,
] y*
] dmaximizes Firm ]2’s
x*
maximizes
Firm
1’s
aggregate
profit
and
given
x*,
y*
]
] ]
aggregate profit. Thus, sx*,
] ]y*d is a sub-game perfect location equilibrium in which
neither firm finds it profitable to unilaterally relocate any of its stores.
3. Properties of a location equilibrium
In this section, we characterize several properties of sub-game perfect Nash
equilibrium (SPNE) locations. These properties are used in Section 4 to determine
the SPNE locations. We first define the notion of quantity-median of a store’s
market and then in the following proposition we associate each store’s equilibrium
location with the quantity-median of its market.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
170
Definition 1. The quantity-median of a store’s market is the point such that the
total quantity supplied by the store to the left of that point is equal to the total
quantity supplied by it to the right of that point.
Since a firm never locates more than one store at the same point, each store’s
market is an interval around itself, and the total quantity supplied by the store to
the left of any point z is a continuous function of z on this interval. Hence, the
quantity-median of each store’s market is unique and well defined.
How does a store’s optimal location relate to its quantity-median? Observe that
a store supplies the maximum amount to the market at its own location, and the
quantity it supplies to the markets further and further away in either direction,
decreases (in fact, piecewise linearly). However, there may be asymmetry in the
aggregate quantities supplied to the two sides of a store and hence, a priori it is not
evident how the quantity-median of a store’s market may be related to the store’s
location. The following proposition establishes a useful relationship between a
store’s optimal location and the quantity-median of its market.
Proposition 1. Given the location vector of the other firm, a firm maximizes its
profit if and only if it locates its stores in such a way that each store is located at
the quantity-median of its market.
Proof. Without loss of generality, consider Firm 1. Note that, to maximize profit, it
never places more than one store at the same location, nor does it locate any store
at 0 or 1. Hence, 0 , x 1 , x 2 , ? ? ? , x m , 1 and its profit is given by
P (x,] y)]
1
x1
5
2
[a 2 2t(x 2 j ) 1 c ( j )]
E ]]]]]]]
dj
3
[a 2 2t( j 2 x ) 1 c ( j )]
]]]]]]] dj
1E
3
[a 2 2t(x 2 j ) 1 c ( j )]
]]]]]]] dj
1E
3
[a 2 2t( j 2 x ) 1 c ( j )]
]]]]]]] dj 1 ? ? ?
1E
3
[a 2 2t(x 2 j ) 1 c ( j )]
]]]]]]] dj
1E
3
[a 2 2t( j 2 x ) 1 c ( j )]
1 E ]]]]]]] dj
3
1
2
0
2
(x 1 1x 2 ) / 2
1
2
x1
2
x2
2
2
(x 1 1x 2 ) / 2
2
(x 2 1x 3 ) / 2
2
2
x2
2
xm
m
2
(x m 21 1x m ) / 2
1
2
m
xm
Observe that
2
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
P
≠
]]1
≠x 1
x1
171
[a 1 c 2 ( j )] 2
[a 2 2t(x 1 2 j ) 1 c 2 ( j )] dj 1 ]]]]
3
F GE
4t
1F]G E
[a 2 2t( j 2 x ) 1 c ( j )] dj
3
x 2x
x 1x
F
a 2 2tF]]G 1 c S]]DG
2
2
1 ]]]]]]]]]]]
2 4t
]]
3
5
0
(x 1 1x 2 ) / 2
1
x1
2
1
2
1
2
2
2
F
6
G S
DG
2
x2 2 x1
x1 1 x2
2
a 2 2t ]] 1 c 2 ]]
[a 1 c 2 ( j )]
2
2
2 ]]]] 2 ]]]]]]]]]]]
3
6
P
F
FF G
x 2x
x 1x
1
1 ]Fa 1 2tF]]G 1 c S]]DGG
2
2
2
x 1x
2t
5 ] F 2 3a 1 (3x 1 x )t 2 4c (x ) 1 c S]]DG , 0
3
2
x1 1 x2
≠2
4t
]]
5 ] 2t ]] 2 2[a 1 c 2 (x 1 )
3
2
≠x 21
1
2
1
2
2
1
1
2
2
1
2
2
Therefore, the second order condition for profit maximization holds everywhere,
while the first order condition,
P
≠
]]1 5 0,
≠x 1
becomes equivalent to
x1
E [a 2 2t(x 2 j ) 1 c ( j )] dj 5 E
1
0
2
(x 1 1x 2 ) / 2
x1
[a 2 2t( j 2 x 1 ) 1 c 2 ( j )] dj
Thus, the location of store 1 coincides with its quantity-median. Similarly, the
profit maximization condition of any other store implies that its location must
coincide with its quantity-median. h
The intuition behind Proposition 1 is as follows. Consider a small rightward
movement of a store. It decreases the profit earned by this store from each market
on its left and it increases its profit from each market on its right. At equilibrium,
the marginal decrease in profit must be equal to the marginal increase in profit for
this store. For Cournot competition, a firm’s profit in a market is proportional to
the square of the quantity served by the firm in that market. Hence, the marginal
profit in a market is proportional to the quantity served in that market. Therefore,
the total quantity served to markets on its left must be equal to the total quantity
served to the markets on its right.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
172
The following corollary follows as an immediate implication of Proposition 1.
Corollary 1. (The market-median property) A pair of location vectors constitutes
a sub-game perfect Nash equilibrium if and only if each store of every firm is
located at the quantity-median of its own market.
Usually, in the context of single-store firms, the market-median property is
fairly robust. It can be verified in Anderson and Neven (1991), and in Hamilton et
al. (1989), where the consumer distribution is assumed to be uniform. In Gupta et
al. (1997), the market-median property is satisfied for wide varieties of nonuniform consumer distributions. Corollary 1 demonstrates that it holds for multistore firms as well. Instead of directly using the first order conditions, the
market-median property suggests an alternative way to determine the SPNE
locations, which often turns out to be quite useful.
In the context of a multi-store monopoly, it is well known that a monopolist
maximizes its profit by locating its stores at their respective quantity-medians.
Since a monopolist and the duopolists all satisfy the market-median property, it is
interesting to examine how each firm’s SPNE locations in a duopoly compare with
its monopoly locations.
Note that, if a firm is a monopolist, it locates its stores at the center of their
respective markets so as to satisfy the market-median property. If a firm has a
rival, however, a store located at the center of its market may not satisfy the
market-median property. This is because, now the store may face different
delivered marginal costs from the rival firm on its two sides, and as a result, the
equality of left and right market lengths does not guarantee the equality of quantity
supplied on each side. Thus, it is the asymmetry in the rival firm’s delivered
marginal costs on the two sides of a store that alters a firm’s SPNE locations from
its monopoly locations. As the demand parameter a grows larger, however, the
rival’s cost discrepancy effect becomes less significant and consequently, a firm’s
optimal store locations are likely to move closer to their respective monopoly
locations. In fact, the following proposition establishes that, in the limit, a firm’s
optimal location vector converges to its monopoly location vector.
Proposition 2. As the demand parameter a becomes large (relative to t), each
firm’s SPNE location vector converges to its monopoly /transport cost minimizing
location vector.
Proof. The proof follows from the first order conditions of profit maximization.
Without loss of generality, consider Firm 1. Observe that the first order conditions
for profit maximization imply that
x1
E [a 2 2t(x 2 j ) 1 c ( j )] dj 5 E
1
0
2
(x 1 1x 2 ) / 2
x1
[a 2 2t( j 2 x 1 ) 1 c 2 ( j )] dj
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
E
xi
(x i 21 1x i ) / 2
5
E
[a 2 2t(x i 2 j ) 1 c 2 ( j )] dj
x( i 1x i 11 ) / 2
xi
E
173
[a 2 2t( j 2 x i ) 1 c 2 ( j )] dj , ;i 5 2, . . . , m 2 1
xm
(x m 21 1x m ) / 2
[a 2 2t(x m 2 j ) 1 c 2 ( j )] dj 5
E
1
xm
[a 2 2t( j 2 x m ) 1 c 2 ( j )] dj
Dividing both sides by a and then taking the limit as a → ` in each of the above
equations, the proof of this proposition follows. h
Proposition 2 is useful, since for large demand, it can be used to approximate
the SPNE locations of multi-store firms. Also, to numerically determine the SPNE
locations for any specific a, the monopoly locations can serve as an initial
approximation.
The market-median property and the limiting property of the SPNE locations are
useful but may not be sufficient to determine the exact SPNE locations. In
Proposition 3 below, we establish the existence and the uniqueness of the
symmetric SPNE locations, which together with Corollary 1 facilitates the
determination of the SPNE locations. Lemmas 1 and 2 are used to prove
Proposition 3.
Lemma 1. Given the location vector of the other firm, a firm’s profit maximizing
location vector is unique.
Proof. Without loss of generality, let Firm 2’s store locations be y 0 5s y 10 , y 20 , . . . ,
y n0d. In response to y 0 , as soon as Firm 1 selects the location of its] store 1, x 1 , the
location of its store]2, x 2 , is uniquely determined. This is because, by Proposition
1, the quantity supplied by store 1 to its left must be equal to the quantity supplied
to its right. Thus, given x 1 , the right endpoint of store 1’s market is uniquely
determined. This point, however, must coincide with (x 1 1 x 2 ) / 2, hence x 2 is also
uniquely determined. Repeated application of this argument implies that as soon as
Firm 1 selects the location of its store 1, x 1 , the location of its store m, x m , is
uniquely determined. Now, as soon as x m is determined, the quantity supplied by
store m in
[(x m21 1 x m ) / 2, x m ]
is fixed. Consequently, there is a unique point u . x m such that the quantity
supplied in
[(x m21 1 x m ) / 2, x m ]
equals the quantity supplied in [x m , u ]. On the other hand, the right end point of
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
174
store m’s market is always fixed at 1. Therefore, x 1 must be chosen in such a way
that u coincides with 1. Note that u increases continuously as x 1 increases.
Therefore, by the intermediate value theorem, there is a unique choice for x 1 such
that u (x 1 ) 5 1. h
Lemma 2. If the location vector of one firm is symmetric around 1 / 2, then so is
the unique profit maximizing location vector of the other firm.
Proof. The proof is by contradiction. Suppose the location vector of Firm 1 is
symmetric around 1 / 2, but the unique profit maximizing location vector of Firm 2
is not symmetric around 1 / 2. Then by reflecting all locations around 1 / 2, we do
not change the location vector of Firm 1, but change the location vector of Firm 2.
Hence, we obtain another distinct profit maximizing vector for Firm 2, contradicting Lemma 1. h
Proposition 3. There exists a sub-game perfect Nash equilibrium at which each
firm locates its stores symmetrically around 1 / 2. Moreover, the symmetric SPNE
is unique.
Proof. To prove the existence of a symmetric SPNE, define the set
G5
H
sx 1 , x 2 , . . . , xmd such that 0 < x 1 < x 2 < ? ? ? < x m < 1
and sx 1 , x 2 , . . . , x md is symmetric around 1 / 2
J
Observe that the set G is compact. Now, consider a vector x˜ [ G. x˜ specifies a
location vector for Firm 1 that is symmetric around 1 / 2. Let R 1 and R 2 be the best
response functions of firms 1 and 2, respectively. Observe that both R 1 and R 2 are
continuous and by Lemma 2, y˜ 5 R 2 (x˜ ) is symmetric around 1 / 2. Now consider
the function g such that g(x˜ ) 5 R 1 (y˜ ) 5 R 1 (R 2 (x˜ )). Note that g is continuous and
by Lemma 2, is also symmetric around 1 / 2. Therefore, g is a continuous function
that maps G into G. Consequently, by Brower’s Fixed Point Theorem, ' x* [ G
such that g(x*) 5 x*. Let y* 5 R 2 (x*), then R 1 ( y*) 5 R 1 (R 2 (x*)) 5 g(x*) 5 x*.
Therefore, (x*, y*) is a symmetric SPNE.
The uniqueness of the symmetric SPNE is established using mathematical
induction on the number of stores of the two firms. An outline of the proof is
presented below.
Let the notation (m, n) denote that Firm 1 has m stores and Firm 2 has n stores.
The uniqueness of the SPNE for (1, 1) is already established in Anderson and
Neven (1991). Assuming the uniqueness of the symmetric SPNE for (1, j); j < n,
we show that the symmetric SPNE is unique for (1, n 1 1). Next, we assume the
uniqueness of the symmetric SPNE for ( j, n); j < m and n > 1, and establish the
uniqueness of the symmetric SPNE for (m 1 1, n);n. Hence, the symmetric SPNE
is unique for all (m, n). h
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175
Proposition 3 justifies the search for location equilibrium among vectors that are
symmetric around 1 / 2.11 In the next section, we use Proposition 3 together with
Corollary 1 to characterize the SPNE locations.
4. Determination of the SPNE locations
In this section we determine the SPNE locations, using the properties derived in
the previous section. Proposition 4 describes a mechanism by which the SPNE
locations for firms with m and n stores can be used to determine the SPNE
locations for firms with km and kn stores, for any positive integer k. To clarify
notations used throughout this section, note that only those location points which
lie in f0,1g are to be considered. If for some choice of parameter values a location
point exceeds one, that location point is to be discarded. For example, suppose
S
D
1/2 1/2 1 1
1/2 1 d 2 1
x* 5 ], ]]], . . . , ]]]] .
d
d
d
Then for d 5 1, x* 5 (1 / 2); and for d 5 2, x* 5 (1 / 4, 3 / 4); etc.
Proposition 4. Suppose there are two firms with m and n stores, respectively. Let
d be the greatest common divisor ( gcd) of m and n. If the vectors xˆ and yˆ
represent the equilibrium locations for firms with m /d and n /d stores facing a
market demand p 5 ad 2 Q / 3 at each market point in f0,1g, then the SPNE
locations for the original problem are
S
D
S
D
xˆ 1 1 xˆ
d 2 1 1 xˆ
x* 5 ], ]], . . . , ]]]
d
d
d
and
yˆ 1 1 yˆ
d 2 1 1 yˆ
y* 5 ], ]], . . . , ]]] .
d
d
d
Furthermore, if xˆ and yˆ are symmetric around 1 / 2, so are x* and y*.
Proof. First consider the problem for firms with m /d and n /d stores facing a
market demand p 5 ad 2 Q / 3 at each point in [0,1]. Refer to this problem as the
reduced problem. By definition, the locations specified by the vectors xˆ and yˆ
satisfy the market-median property of the reduced problem. Now consider the
original problem for firms with m and n stores facing a market demand p 5 a 2 Q /
3 at each market point in [0,1]. Consider the locations specified by the vectors xˆ /d
11
In general, we are unable to rule out the possibility of SPNE locations, where both firms locate
their stores asymmetrically around 1 / 2. We can rule out asymmetric SPNE for special cases and in
general, conjecture that such a scenario does not arise.
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D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
and yˆ /d. Observe that the locations specified by the vectors xˆ /d and yˆ /d are
located in the interval [0,1 /d]. Applying the transformation v 5 u /d to the
equations that characterize the market-median property of the reduced problem
note that these stores satisfy the market-median property of the original problem
over the interval [0,1 /d]. Translating each store by 1 /d, we arrive at the vectors
(1 1 xˆ ) /d and (1 1 yˆ ) /d which specify locations in [1 /d,2 /d]. Note that these
stores satisfy the market-median property of the original problem over the interval
[1 /d,2 /d]. Continuing in this manner, it can be seen that the stores located in the
intervals [2 /d,3 /d], . . . , [(d 2 1) /d,1] also satisfy the market-median property.
Finally, xˆ is symmetric around 1 / 2 implies xˆ 1 1 xˆ m / d 5 1. As a result,
xˆ 1 d 2 1 1 xˆ m / d
x *1 1 x m* 5 ] 1 ]]]] 5 1,
d
d
and therefore, x* is symmetric around 1 / 2. h
In general, it is relatively easier to determine the SPNE locations for firms with
fewer numbers of stores. Thus, Proposition 4 is quite useful since it utilizes the
solution for a problem with m and n stores, and uses that to solve a problem with
km and kn stores, for any positive integer k.12 The following example illustrates
how Proposition 4 transforms a complicated problem to a relatively simpler one.
Example 1. Determination of SPNE locations for m 5 4 and n 5 2.
Let Firm 1’s SPNE store locations be sx 1* , x 2* , x 3* , x 4*d and Firm 2’s SPNE store
locations be s y 1* , y *2 d. In principle, it is possible to solve for the SPNE locations by
using the market-median property (Corollary 1) and symmetry (Proposition 3),
though it may be quite difficult. To see the complexity of this direct approach,
consider Firm 1. Note that by symmetry, x 3* 5 1 2 x *1 and x 4* 5 1 2 x *2 . Thus,
solutions for x 1* and x 2* are sufficient. Now, by the market-median property, the
quantity supplied by store 1 to its left equals the quantity supplied to its right. This
generates a quadratic equation involving both sx 1*d 2 and sx 2*d 2 , since the right
endpoint of store 1’s market involves x 2* . Similarly, the market-median property
for store 2 also generates a quadratic equation involving both sx 1*d 2 and sx *2 d 2 .
Moreover, both equations involve y 1* . Thus, we also need to include the equation
that emerges from the market-median property of Firm 2’s store 1. Therefore, to
solve for x 1* , x 2* and y 1* , we must solve three equations simultaneously, two of
which are quadratic equations involving both sx 1*d 2 and sx *2 d 2 . Obviously, it would
involve extensive algebra.
In this situation, Proposition 4 can be successfully used. Note that in this case,
the greatest common divisor (gcd) is 2. Therefore, we need to solve a problem for
12
The converse of Proposition 4 is also true. That is, the solution for firms with km and kn stores can
be used to determine the solution for firms with m and n stores.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
177
firms with m 5 2 and n 5 1 stores facing a market demand p 5 2a 2 Q / 3 at each
market point in f0,1g, and use them to determine the solution for the original
problem. Let xˆ 5sxˆ 1 , xˆ 2d and yˆ 5syˆ 1d be the solution for the sm 5 2, n 5 1d
problem. Observe that by symmetry yˆ 1 5 1 / 2, and xˆ 1 and xˆ 2 are located
symmetrically around 1 / 2. Thus, the right endpoint of store 1’s market is 1 / 2,
which does not involve xˆ 2 . Therefore, by market-median property, xˆ 1 can be
solved from a single quadratic equation without involving xˆ 2 . Note that, the
quantity supplied to the left of xˆ 1 is
x̂ 1
E Faˆ 2 2tsxˆ 2 jd 1 tS]12 2 jDG dj
1
0
and the quantity supplied to the right of xˆ 1 is
1/2
E Faˆ 2 2tsj 2 xˆ d 1 tS]12 2 jDG dj
1
x̂ 1
where aˆ 5 ad 5 2a. Thus, xˆ 1 can be solved from the equation
Faˆ 2 2txˆ 1 ]2t G xˆ 1 ]2t sxˆ d
1
1
1
2
5
Faˆ 1 2txˆ 1 ]2t GF]12 2 xˆ G
3t
1
2F]GFS]D 2sxˆ d G
2
2
1
1
2
2
1
It can be checked that
]]]]
]]]
aˆ 2œsaˆ d 2 2 aˆ 1 ]21
a 1
2
xˆ 1 5 ]]]]] 5 a 2 a 2 ] 1 ].
2
2 8
œ
Now, we can use Proposition 4 to solve for the original problem. Here,
yˆ 1 1
y *1 5 ] 5 ],
2
4
1 1 yˆ 1 3
y *2 5 ]] 5 ],
2
4
and
xˆ 1
xˆ 2
1 1 xˆ 1
x *1 5 ], x 2* 5 ], x 3* 5 ]]
2
2
2
and
1 1 xˆ 2
x 4* 5 ]],
2
where
]]]
a 1
xˆ 1 5 a 2 a 2 2 ] 1 ]
2 8
œ
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
178
and
xˆ 2 5 1 2 xˆ 1 .
4.1. SPNE locations when n 5 km
Proposition 4 together with symmetry (Proposition 3) and the market-median
property (Corollary 1) characterize the SPNE locations. This sub-section summarizes the results for firms with m and n stores, where n 5 km. The results for
n ± km are summarized in the following sub-section.
Proposition 5. Suppose there are two firms with m and km stores, respectively,
where k > 1 is a positive integer. (i) In equilibrium, Firm 1’s store locations
coincide with its monopoly /transport cost minimizing locations.13 That is,
s2i 2 1d
x *i 5 ]]] ;i 5 1, 2, . . . , m.
2m
(ii) If k 5 1, then in equilibrium, each firm’s store locations coincide with its
monopoly /transport cost minimizing locations.
Proof. Proposition 5 follows from symmetry (Proposition 3) and Proposition 4. To
see the proof of part (i), observe that if m 5 1 then at an SPNE, Firm 1 locates its
single store at 1 / 2. Now, if m . 1, then from Proposition 4, we first need to solve
for SPNE locations for firms with 1 and k stores facing a market demand
p 5 am 2 Q / 3 at each market point in f0,1g. The solution for this problem is,
x̂ 1 5 1 / 2, independent of k. Therefore,
x̂ 1
1
x *1 5 ] 5 ],
m 2m
and in general,
si 2 1d 1 xˆ 1 si 2 1d 1 ]12 s2i 2 1d
x *i 5 ]]]] 5 ]]] 5 ]]] ;i 5 1,2, . . . , m.
m
m
2m
This completes the proof of part (i).
Interchanging the roles of the two firms, the proof of part (ii) follows. h
The following corollary immediately follows from Proposition 5.
Corollary 2. If each firm has m stores, then in equilibrium, the stores belonging to
competing firms agglomerate at m discrete points. Furthermore, agglomerations
13
Observe that the monopoly location vector is also the socially optimal one.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
179
of stores occur precisely at those locations that would have been chosen by a
multi-store monopolist with m stores.
Corollary 2 has the following implications. First, it identifies a scenario when
the complex problem of determining equilibrium locations for multi-store firms
can be reduced to a simple one, where each firm simply behaves as a multi-store
monopolist while choosing its locations. Moreover, it can be checked that the
conclusion is true for any n > 2 firms. Thus, if there are n firms with m outlets
each, then we observe clustering of n competing stores at each of the m locations
that would have been chosen by a monopolist with m outlets. Observe that the
outlet locations are independent of the number of firms n and depend only on the
number of outlets m.
Second, the location pattern is consistent with many commonly observed
phenomena, such as the clustering of outlets of competing firms at various
shopping malls and the clustering of qualities offered by competing firms in the
context of vertical product differentiation. Contrast the result with those obtained
under price competition. In a spatial model with price competition, stores
belonging to competing firms never agglomerate. In the context of single store
firms it is shown by d’Aspremont et al. (1979) for mill pricing and by Lederer and
Hurter (1986) for spatial discriminatory pricing. In the context of multi-store firms
with mill pricing, identical conclusion is drawn by Martinez-Giralt and Neven
(1988). It can be checked that the conclusion holds for multi-store firms with
spatial discriminatory pricing. Similar result is also demonstrated in the context of
vertical product differentiation with price competition. For a duopoly choosing
multiple qualities in a quality-price game, Champsaur and Rochet (1989)
demonstrate that the qualities chosen by the competing firms never overlap. In
fact, the highest quality chosen by one firm is lower than the lowest quality offered
by its rival.
Intuitively, price competition cannot give rise to a clustering result, since
coincident location of competing firms severely intensifies competition and drives
profits to zero. Clustering, however, is feasible under spatial Cournot competition,
since coincidentally located competing firms now earn positive profits. Hence, in
contrast to the results obtained with price competition, spatial Cournot competition
gives rise to a result that is consistent with frequently observed clustering of
outlets in the real-world.14
14
There are alternative explanations in the literature justifying the agglomeration of competing firms.
For instance, firms competing in prices may agglomerate in spatial dimension if their products are
differentiated by attribute(s) other than location (see, for example, dePalma et al., 1985; Anderson and
dePalma, 1988; De Fraja, 1993; Irmen and Thisse, 1998). A different explanation can be found in Stahl
(1982), where consumers search for optimal product characteristics and their search costs are
influenced by firm locations, leading to a spatial concentration of demand where sellers find it profitable
to agglomerate. In Thill (1997), multi-store firms that choose both locations and product qualities
agglomerate for sufficiently high (consumer) reservation price.
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D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
The intuition behind the coincidence of the agglomeration and the monopoly
locations is as follows. If two stores owned by rival firms agglomerate at a location
jˆ and serve the market at j , then each store supplies a 2 2tu jˆ 2 ju 1 tu jˆ 2 ju 5 a 2
tu jˆ 2 ju amount at j . Observe that if a monopolist with identical technology has a
store located at jˆ, it would supply (3 / 2)fa 2 tu jˆ 2 jug at j . Therefore, if the
monopoly store locations satisfy the market-median property, the oligopoly
agglomeration locations would do the same and consequently, oligopoly agglomeration locations coincide with monopoly store locations.
Corollary 2 highlights a result when the firms have equal number of stores. In
contrast, the following corollary highlights a result when the firms have different
numbers of stores. It follows immediately from Proposition 5 and the marketmedian property (Corollary 1).
Corollary 3. If the firms have m and km stores (with k . 1), then Firm 1’s
equilibrium location vector coincides with its monopoly location vector. Firm 2’s
equilibrium location vector, however, always differs from its monopoly location
vector.
Corollary 3 reiterates how the complex problem of determining equilibrium
locations for a multi-store oligopolist can be reduced to a simple one. Here, Firm 1
simply behaves as a multi-store monopolist while choosing its locations. For
example, if the firms have 3 and 3k stores, then irrespective of the values of k,
Firm 1 locates its stores at their corresponding monopoly locations; 1 / 6, 1 / 2 and
5 / 6.
To see why firm 2 does not locate all of its stores at their corresponding
monopoly locations, consider m 5 1 and n 5 2. Clearly, x *1 5 1 / 2. Now, if Firm 2
had chosen its monopoly locations, then yˆ 1 5 1 / 4 and yˆ 2 5 3 / 4. However, the
locations yˆ 1 5 1 / 4 and yˆ 2 5 3 / 4 cannot satisfy the market-median property. To see
this, consider the store at 1 / 4. The length of the market to its left equals the length
of the market to its right. However, Firm 1’s delivered marginal cost is greater in
f0, 1 / 4g than in f1 / 4, 1 / 2g, and consequently, Firm 2’s store 1 supplies a larger
quantity to its left than to its right. Obviously, this violates the market-median
property and yˆ 1 5 1 / 4, yˆ 2 5 3 / 4, cannot be equilibrium locations for Firm 2. In
fact, if s y *1 , y 2*d are Firm 2’s equilibrium locations, then y 1* must be less than ]41
and y *2 must be greater than 3 / 4, in order to satisfy the market median property.
As shown in Example 1,
]]]]
a
y *1 5 a 2 a 2 2 ] 1 ]81 , ]41 .
2
œ SD
Proposition 6. Suppose there are two firms with m and km stores (with k . 1 being
a positive integer), respectively. If k is even, then no two stores agglomerate at the
same location. If k is odd, then there are exactly m agglomeration points at
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
181
1 3
2m 2 1
], ], . . . , ]].
2m 2m
2m
Proof. It follows from Proposition 5 that Firm 1 locates its stores at
s2i 2 1d
x *i 5 ]]] ;i 5 1, 2, . . . , m.
2m
Thus, Firm 1 locates one store at the center of each of the intervals
m21
F0, ]m1 G, F]m1 , ]m2 G, . . . , F]]
, 1G.
m
Now, from Proposition 4 it follows that in each of these intervals Firm 2 locates k
stores. In fact, by Proposition 3, Firm 2 locates its stores symmetrically around the
midpoint in each interval. Therefore, if k is odd, Firm 2 locates one store at the
center of each interval
m21
F0, ]m1 G, F]m1 , ]m2 G, . . . , F]]
, 1G,
m
yielding m agglomeration points, where each firm locates one of its stores. On the
other hand, if k is even, Firm 2 does not locate any of its stores at the center of any
of these intervals. As a result, the stores never agglomerate. h
The following corollary follows immediately from Propositions 5 and 6.
Corollary 4. Suppose the firms have m and km stores. If k is even, then the store
locations exhibit complete spatial dispersion. If k is odd, the store locations
simultaneously yields spatial agglomeration and dispersion unless m 5 k 5 1. If
m 5 k 5 1 (that is, each firm has only one store), then the stores agglomerate at
the center of the market.
Corollary 4 sheds new light on a prevailing perception in the literature, which
states that as opposed to price competition, Cournot competition gives rise to
spatial agglomeration. Under price competition, agglomeration intensifies price
competition and forces profits to zero. Each firm may earn a higher profit by
choosing a different location and consequently, price competition generates
dispersed locations. In contrast to the spatial dispersion obtained under price
competition, the pioneering work of Hamilton et al. (1989), and Anderson and
Neven (1991) demonstrates that if single-store firms compete in quantities
(Cournot competition), they agglomerate at the market center. Gupta et al. (1997)
relax Anderson and Neven (1991)’s assumption of uniform consumer distribution
and demonstrate that the agglomeration result is robust under a wide variety of
consumer distribution. Mayer (2000) allows the production cost to differ at various
market points and establishes that firms may still agglomerate. Thus, it may indeed
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D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
seem reasonable to conclude that as opposed to price competition, Cournot
competition yields spatial agglomeration.
Corollary 4, however, shows that this notion is incorrect, if we allow the firms
to have multiple stores. Similar to the location pattern obtained with price
competition, Cournot competition may also yield complete spatial dispersion. In
fact, depending on the number of stores, price competition with spatial price
discrimination and Cournot competition generate quite similar location patterns.
Furthermore, Cournot competition may simultaneously yield spatial agglomeration
and dispersion; the firms may agglomerate at finitely many dispersed points.
4.2. SPNE location when n ± km
Propositions 5 and 6 characterize the SPNE locations for firms with m and km
stores. In fact, Propositions 4–6, together with the market-median property
(Corollary 1) and symmetry (Proposition 3) can be used to determine various
combinations of m and n, such as, sm 5 1, n 5 1d, sm 5 1, n 5 2d, sm 5 1, n 5 3d,
sm 5 2, n 5 4d. However, we are yet to characterize SPNE locations for combinations such as sm 5 2, n 5 3d and sm 5 3, n 5 4d. In other words, we are yet to
characterize the solutions if n ± km, or equivalently, 1 < d , m , n, where d is the
greatest common divisor (gcd) of m and n. The following proposition contributes
in this regard. In view of Proposition 4, we assume d 5 1 in Proposition 7. Lemma
3 is used to prove the Proposition 7.
Lemma 3. Suppose 1 , m , n and d 5 1. Two stores may agglomerate only at
1 / 2.
Proof. See Appendix A.
Proposition 7. Suppose there are two firms with m and n stores, respectively. Let
d be the greatest common divisor ( gcd) of m and n. If 1 5 d , m , n, then (i) The
only possible agglomeration of stores may occur at 1 / 2. Stores agglomerate at
1 / 2 if and only if both m and n are odd. (ii) The relative ordering of the stores of
both firms is the same as that of their monopoly ordering. That is, for all
1 < i < m and 1 < j < n, x i* _ y j* if and only if
2i 2 1 2j 2 1
]] _ ]].
2m
2n
Proof. Proof of part (i) follows directly from Lemma 3, the symmetry of the
SPNE around 1 / 2, and noting that both m and n cannot be even.
Proof of part (ii) is as follows. From part (i), observe that the center-most stores
of the firms agglomerate if and only if both m and n are odd. Moreover, they can
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
183
agglomerate only at 1 / 2. Two monopoly locations agglomerate if and only if both
m and n are odd, and
m11
n11
i 5 ]], j 5 ]].
2
2
In this case,
2i 2 1 2j 2 1 1
]] 5 ]] 5 ].
2m
2n
2
Therefore, x *i 5 y *j if and only if
2i 2 1 2j 2 1
]] 5 ]].
2m
2n
Now, suppose x i* . y j* , but
2i 2 1 2j 2 1
]] , ]].
2m
2n
By Proposition 2, however, as a → `, x i* → (2i 2 1) / 2m and y *j → (2j 2 1) / 2n.
Therefore, ' aˆ such that x i*saˆ d 5 y j*saˆ d. However, since
2i 2 1 2j 2 1
]] ± ]],
2m
2n
it follows from the discussion above that x i*saˆ d ± y j*saˆ d, a contradiction. h
Remark 1. In Proposition 7, we assume d 5 1. For 1 , d , m , n, we solve a
problem for firms with m /d and n /d stores facing a market demand p 5 ad 2 Q / 3
at each market point in f0,1g, and use Proposition 4 to obtain the solution for the
original problem. It can be seen that for 1 , d , m , n, there are d agglomeration
points if and only if both m /d and n /d are odd.
Similar to Propositions 5 and 6, Proposition 7 also demonstrates that Cournot
competition may yield both agglomeration and dispersion. In fact, from Propositions 5–7, it follows that Cournot competition yields complete dispersion if n 5 km
with k being even, or if n ± km and m /d or n /d is even. In this scenario, the
location pattern under Cournot competition resembles those under spatial price
discrimination, although, in contrast to the price competition, Cournot competition
generates overlapping markets. If n 5 km with k > 1 being odd or if n ± km and
both m /d and n /d are odd, Cournot competition simultaneously exhibits agglomeration and dispersion.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
184
4.3. Exact SPNE locations
To facilitate a better understanding of the SPNE locations, we now present the
exact SPNE locations for 1 < m < 3 and 1 < n < 3. For simplicity, we assume
t 5 1. Example 2 below presents the general solutions in terms of the demand
parameter a. Example 3 presents the solutions for specific values of a.
Example 2. General solution in terms of the demand parameter a
In Table 1, observe that when the firms have equal number of stores, both firms
choose their respective monopoly locations irrespective of the values of a. For
sm 5 1, n 5 2d and sm 5 1, n 5 3d, only Firm 1 chooses its monopoly location. For
sm 5 2, n 5 3d, however, explicit solutions in terms of a are not feasible. Table 2
(Example 3) presents solutions under specific values of a. Observe that, in this
case, neither firm chooses its monopoly locations. To understand how the SPNE
locations differ from their respective monopoly locations, we present the numerical solutions, under different values of a, in Example 3 below.
Example 3. Numerical solutions for sm 5 2, n 5 3d under different values of a.
Table 2 enables us to compare the firms’ SPNE locations to their respective
monopoly locations. Note that Firm 1’s monopoly locations are s1 / 4, 3 / 4d 5s0.25,
Table 1
General solutions in terms of the demand parameter a, 1 < m < 3, 1 < n < 3, m < n
[m\n]
n51
n52
n53
x *1 5 1 / 2
x *1 5 1 / 2
x *1 5 1 / 2
m51
]]]]
y *1 5 a 2Œa 2 2 a / 2 1 1 / 8
y *1 5 1 / 2
]]]]
y *1 5 (12a 1 5 2Œ144a 2 1 16a 1 12) / 26
y *2 5 1 / 2
y *2 5 1 2 y *1
y *3 5 1 2 y *1
x *1 5 1 / 4, x 2* 5 3 / 4
m52
y *1 5 1 / 4, y *2 5 3 / 4
Explicit solutions are not feasible
See Table 2 below for solutions
under specific values of a
x *1 5 1 / 6, x *2 5 1 / 2, x *3 5 5 / 6
m53
y *1 5 1 / 6, y *2 5 1 / 2, y *3 5 5 / 6
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
185
Table 2
Numerical solutions for (m 5 2, n 5 3) under different values of a
y *1 5 0.1586
x *1 5 0.2526
y *2 5 0.5
a52
x *2 5 0.7474
y *3 5 0.8414
y *1 5 0.1637
x *1 5 0.2508
y *2 5 0.5
a55
x *2 5 0.7492
y *3 5 0.8363
y *1 5 0.1652
x *1 5 0.25038
y *2 5 0.5
a 5 10
x *2 5 0.74962
y *3 5 0.8348
0.75d and Firm 2’s monopoly locations are s1 / 6, 1 / 2, 5 / 6d 5s0.16667, 0.5,
0.8333d. Comparing the actual locations to the firms’ monopoly locations, observe
that the SPNE locations are quite close to their monopoly locations. Furthermore,
as the demand parameter a increases, the SPNE locations quickly converge to their
respective monopoly locations. Numerical simulations with other values of m and
n confirm similar location pattern.
5. Conclusion
We contribute to the literature of spatial competition by considering multi-store
firms in the context of spatial Cournot competition with endogenous location
choice. Previous work on spatial competition that allows the firms to choose their
locations mostly assumes single-store firms. The brief literature that considers
multi-outlet firms with endogenous location choice, considers either price competition (Martinez-Giralt and Neven, 1988) or fixed prices (Teitz, 1968; Thill, 1997).
By using Cournot competition we demonstrate results that describe the real world
better than those obtained under price competition. In the context of price
competition with mill pricing, Martinez-Giralt and Neven (1988) demonstrate that
each firm locates all of its stores at the same market point, and thus, the firms do
not take up the opportunity of opening up multiple outlets. This result is rather
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D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
intriguing given the abundance of firms with multiple outlets. This puzzling
outcome, however, does not arise in spatial Cournot competition where each firm
chooses distinct locations for its stores. Moreover, under both mill pricing and
spatial discriminatory pricing, locations of competing firms do not agglomerate
(d’Aspremont et al., 1979; Lederer and Hurter, 1986; Martinez-Giralt and Neven,
1988). A similar result is demonstrated by Champsaur and Rochet (1989) in the
context of vertical differentiation with price competition, where qualities offered
by competing firms never coincide. In contrast, under Cournot competition, outlets
of competing firms may agglomerate at finitely many locations; a result that is
consistent with the clustering of stores of competing firms at various shopping
malls and clustering of qualities offered by competing firms. Also, we identify
situations when, in equilibrium, a firm simply chooses its monopoly locations.
Furthermore, when a firm’s equilibrium store locations differ from its monopoly
locations, we demonstrate that the equilibrium locations converge to their
respective monopoly locations, as the demand becomes large. Even for smaller
demands, numerical simulations indicate that the equilibrium store locations are
quite close to their respective monopoly locations. Hence, we demonstrate that the
complex problem of determining equilibrium locations for multi-store firms can be
approximated by a simple one, where each firm simply behaves as a multi-store
monopolist while choosing its locations. We also address a prevailing notion in the
literature that follows by comparing spatial models with price and Cournot
competitions. Previous work with single-store firms shows that price competition
generates spatial dispersion of firms, while Cournot competition gives rise to
spatial agglomeration (Anderson and Neven, 1991; Hamilton et al., 1989; Gupta et
al., 1997, Mayer, 2000). We demonstrate that if the firms have multiple stores such
a notion is incorrect; Cournot competition may as well give rise to complete
spatial dispersion. In fact, depending on the number of stores, spatial price
discrimination and Cournot competition generate quite similar location patterns.
The paper can be extended in several ways. For example, it assumes exogenously fixed number of stores for each firm. The number of stores, however, can be
determined endogenously. The fixed cost needed to set up a store is likely to be
significant and can be used to determine the number of stores for each firm.15 A
three stage game can be analyzed where in stage I, the firms decide on the number
of stores. In stage II, they decide on store locations and in stage III, they compete
in quantities. Initial investigation suggests that even if the firms have identical
set-up cost per store, they may choose different number of stores. For example,
suppose the demand at each market point is p 5 4 2 Q and the fixed set-up cost
15
The number of firms can be made endogenous as well. The fixed set-up cost can be used to
determine the number of firms.
D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
187
per store is 1.5, then there are three SPNE equilibrium outcomes for the number of
stores: sm 5 2, n 5 0d,sm 5 0, n 5 2d and sm 5 1, n 5 1d. It is also possible for the
firms to have positive but different number of stores. For example, if the fixed
set-up cost above is decreased to 0.20, there are only two SPNE outcome: sm 5 2,
n 5 1d and sm 5 1, n 5 2d. In this case, there is no equilibrium where the firms
choose equal number of stores. In general, however, it is difficult to express the
number of stores as a function of the fixed set-up cost, since the solutions depend
critically on the specific values of the fixed set-up cost.16 Another way to extend
the paper would be to relax the assumption that the transport cost is linear in
quantities. The marginal cost of delivering a unit of output at a specific market
point may be less than the marginal cost of delivering the previous unit at that
point. Therefore, there are situations where it would be appropriate to consider
transport cost that is concave in quantities at each market point. (Observe that this
problem does not arise in spatial models where the consumers are assumed to buy
only one unit at each market point.) Initial investigation in this regard indicates a
potentially challenging problem, which remains an agenda for future research.
The abundance of spatial competition among multi-store firms has motivated the
present study. Certainly, the analysis would also be useful to study various other
important economic phenomena. For example, any horizontal merger in a spatial
model would give rise to firm(s) with multiple stores. To properly evaluate the
implications of such a merger, it is essential to incorporate the possibility of
relocations of stores following the merger. Obviously, such a task would involve
competition among firms with multiple stores.17 Overall, further study involving
spatial competition among multi-store firms is clearly worthwhile.
Acknowledgements
We are grateful to Simon Anderson, Wolfgang Mayer, David Sappington and
two anonymous referees for valuable comments.
Appendix A. Proof of Lemma 3
Suppose, if possible, two stores of different firms agglomerate at j [f0,1g with
16
From these examples it appears that entry deterrence is feasible in spatial Cournot competition.
The result is similar to that obtained under spatial price discrimination (Gupta, 1992; Eaton and
Schmitt, 1994). In contrast, spatial preemption may be infeasible in spatial competition involving mill
pricing (Judd, 1985).
17
See Norman and Pepall (2000) for a recent analysis of horizontal mergers in a spatial Cournot
model.
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D. Pal, J. Sarkar / Int. J. Ind. Organ. 20 (2002) 163 – 190
j ± 1 / 2. First, assume that the stores at j are not the leftmost or the rightmost
store for the respective firms. If at least one firm has a leftmost or rightmost store
at j , then the proof below can be modified suitably. Note that, since the stores
agglomerate at j , each store supplies the same quantity at j and its vicinity.
Therefore, to satisfy the market-median property, the market of one store must be
a subset of the market served by the other store. Moreover, j is not only the
quantity-median of the store with the smaller marker, but also the midpoint of its
market. Thus, its neighboring stores (which are owned by the same firm) are
symmetrically situated around j . This in turn implies that the quantities offered by
the two firms are symmetric around j up to markets of these neighboring stores or
the end points of the market of the store at j owned by the other firm, whichever is
nearer to j . Proceeding in this manner, it can be seen that the neighboring stores of
the other firm are also located symmetrically around j . Repeating the same
argument, note that the quantities offered by the two firms and the stores of the
two firms are symmetric in a closed interval I, with midpoint j and the endpoints
being the market boundaries of a store of Firm 1 and another store of Firm 2. Let J
be the minimal subset of I with the above two properties. If J 5f0,1g, then j 5 1 / 2
which contradicts the hypothesis. On the other hand, if J ,f0,1g, we establish
below a contradiction to the hypothesis that d 5 1. Suppose that J 5fa, bg ,f0,1g.
Note a . 0 and / or b , 1. Taking reflection(s) of J about a and / or b, we obtain a
copy / copies of J. The interval J together with its reflected copy / copies may cover
the entire interval f0,1g. Otherwise, taking successive reflection(s) about the
endpoint(s) of the latest copy / copies of J, we eventually cover the entire market.
Note that 0 s1d must coincide with the endpoint of some copy of J; otherwise, the
leftmost (rightmost) store of at least one firm would violate the market-median
property. Let k . 1 be the number of copies of J necessary to cover f0,1g. Since
the number of stores of each firm within each copy of J are the same, both m and
n are multiples of k. This contradicts the hypothesis that the greatest common
divisor of m and n equals one.
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