DISCUSSION PAPER SERIES
2008-02
Spatial competition with home bias
Takanori Ago and Ikuo Ishibashi

October 26, 2008

Discussion Papers can be downloaded:
http://www1.tcue.ac.jp/home1/c-gakkai/dp/dp08-02

Spatial competition with home bias
Ikuo Ishibashiy

Takanori Ago

October 26, 2008

Abstract
This paper analyzes the location equilibrium for duopolistic …rms in which consumers prefer a good produced by a nearer …rm when the prices of both goods are equal (home bias, orvertical di¤erentiation). We consider a linear space, and the …rms choose their locations priorto quantity (price) competition. The nearer a …rm is to a market, the higher the quality of its
good for the market becomes. As a result, we compute numerically to show that there exists
a symmetric dispersed location equilibrium. When transport costs are decreasing, the …rms
approach toward the center each other in quantity competition. However, in price competition,
the result is reversed: The …rms separate each other. This contrastive result is because expanding the market share takes precedence over avoiding the competition in quantity competition.
Meanwhile, in price competition, it more matters to avoid keen price competition.

JEL Classi…cation: L13, R12, R30.

Keywords: home bias, location, vertical product di¤erentiation, mismatch cost
Faculty of Regional Policy, Takasaki City University of Economics. E-mail: ago@tcue.ac.jp
y Department of Economics, Aoyama Gakuin University, Tokyo, Japan. E-mail: ishibash@econ.aoyama.ac.jp1

1

Introduction

This paper develops a spatial competition model à la Hotelling (1929) with home bias in which
consumers prefer a good produced by a nearer …rm when the prices of both goods are equal.
In other words, we deal with a model of vertical di¤erentiation (henceforth VPD) endogenously
determined by locations. Hotelling (1929) analyzed a location-then-price competition model with
duopolistic …rms selling a homogeneous commodity to consumers uniformly distributed over a line
segment; then, he showed the …rms agglomerate in the center. This result is termed principle ofminimum di¤ erentiation. D’Aspremont, Gabszewicz and Thisse (1979) have corrected this principle.
They adopted a quadratic transport cost function, and showed that the two …rms mutually locate
themselves apart from each other at the edges of the market. Principle of maximum di¤ erentiationarises in this location-then-price competition model. This shows a strong tendency for …rms to
avoid keen price competition by locational dispersion.
As another trend for spatial competition, Hamilton, Thisse and Weskamp (1989) and Anderson
and Neven (1991) incorporated a quantity competition à la Cournot into spatial competition models.
They consider that each point over the line segment has a Cournot market, and …rms engage
in location-then-quantity competition. Consequently, both studies showed principle of minimumdi¤ erentiation, where all …rms agglomerate in the center. Hence, the results of price competitionand quantity one are contrastive. This paper analyzes both types of competition in order to better
understand the di¤erences.
Unlike those studies with a homogeneous good, this paper deals with a model of a di¤erentiatedproduct. When it comes to a spatial competition with product di¤erentiation, De Fraja and Norman(1993) analyzed a spatial model with a horizontally di¤erentiated good. In their model, givena di¤erentiation parameter set representing elasticity of substitution, duopolistic …rms engage inlocation-then-price competition. As a result, under a su¢ cient di¤erentiation, the …rms agglomerate
in the center. We adopt a VPD model instead of a horizontal one.
2

It is location determines product quality that is the most special feature in our model.1 Qualityof a good produced by a nearer …rm is assumed to be higher than that produced by a farther …rm foreach consumer. Note that because the consumers are distributed over a linear city, the evaluation
for the quality of a good varies in locations of the markets. In the real world, we can imagine thefollowing examples. First, we often prefer a product from the home country to that from foreigncountries; thus, we would like to buy one from the home rather than the others even if the pricesof those goods are all equal. 2 That means quality depends on location. Second, we can regard thespace as a characteristic space rather than a geographical one. In Japanese cities that have small
land and concentrated population, people there prefer a smaller car that spares space and is easily
driven in a narrow road. On the other hand, in the US cities that are relatively wider, consumers
may prefer a bigger car.
Our main result is that the duopolistic …rms locate apart each other to some extent. It is naturalthat they never agglomerate. If so, they do produce the same product (homogeneous product) to
be involved by a severe competition. As Motta (1993) has shown, when the …rms can choose thequalities of their goods, they o¤er the di¤erent qualities each other to avoid such a competition.
Further, the dispersion in location is more under Bertrand than under Cournot, which is also
consistent with Motta (1993). This is because Bertrand competition is more severe, then the …rms
have a stronger incentive to locate apart in order to avoid competition.
When transport costs are decreasing, the …rms approach toward the center each other when they
engage in quantity competition. However, in price competition, this result is reversed: The …rms
separate each other. This contrastive result is because quantity competition is less competitive. In
other words, expanding the market share takes precedence over avoiding the competition in quantity
competition. Meanwhile, in price competition, it more matters to avoid keen price competition
1 In most models of VPD, quality is enhanced by (R&D) investment with cost that is a direct strategic variable.
However, in this paper, in order to concentrate on location factors, we omit such a direct choice of quality.
2 The strict de…nition of (higher) quality in VPD is: when all consumers always buy a good rather than anothergood if the prices of them are equal, quality of the good that is bought is higher than that of another.

3caused by low transport costs.
The remainder of the paper is organized as follows. In Section 2, the two-stage game is presented
and the games (price and quantity competition) at the second stage are also analyzed. In Section
3, we evaluate the equilibrium and computes some economic values: prices, pro…ts, and consumer
surplus in the equilibrium for a parameter set. Section 4 summarizes the results.

2

The model

We consider a linear city expressed by the line segment, L = [0; 1]. There are two competing …rms
indexed by i (i = 1; 2). Let the …rms’locations be x1

x and x2

1

y, respectively (xi 2 L);

and each …rm produces a di¤erentiated good (good i). Each point in the city has a set of consumers
that have di¤erent tastes for quality and buy either of the good inelastically or do not buy it atall. Consumers at z have the same utility function (and zero utility if they do not buy the good).
When they buy good i, they get the utility as

U (z) = ui (z)vpi (z);

where ui (z) and pi (z) are the quality and the price of good i at z respectively; and v 2 [v; v], being
uniformly distributed with unit density, is a taste parameter.3 The quality of a good in a market
becomes higher when the producer locates nearer to the market. Then, we assumeui (z) = at jz

xi j ;

3 This setting is based on a familiar VPD model by Shaked and Sutton (1982).

4where a (> t > 0) is reserved quality and t is a mismatch cost parameter.4 Let m(1 + x(x1 + x2 )=2 =y)=2. Then, for the consumers over [0; m) (resp. (m; 1]), the quality of good 1 (resp. good2) is higher.
We consider a two-stage location-price game and a two-stage location-quantity one. The structure of the games are based on Motta (1993), where two …rms simultaneously choose their qualities
at the …rst stage; then they compete in quantity (or price) at the second stage. He analyzed anon-spatial, single market where the …rms directly choose their qualities with costs. Meanwhile,
our …rms only can choose their qualities through location choices. Further, we consider a continuum
mass of markets in the linear city. The other structure is the same as in Motta (1993).

2.1

Bertrand competition

2.1.1

Price equilibrium

First, we analyze Bertrand competition at the second stage.5 We consider a market at z 2 [0; m),where the quality of good 1 is higher.6 The consumer indi¤erent between buying good 1 or good 2
has a taste parameter v12 such that v12 = [p1 (z)p2 (z)] = [u1 (z)u2 (z)]. The consumer indi¤erent

between buying good 2 and not buying at all has a taste parameter v 2 = p2 (z)=u2 (z). Following
Motta (1993), we assume that the market is not covered; that is, v < v 2 for any z. Thus, we haveeach …rm’s demand qi (z) as follows:

q1 (z)= v

q2 (z)=[p1 (z)[p1 (z)p2 (z)] = [u1 (z)p2 (z)] = [u1 (z)u2 (z)] ;u2 (z)]

p2 (z)=u2 (z):

(1)
(2)

4 The assumption about a ensures a positive quality value and positive demands for both …rms for all locations.
We exclude direct quality competitions like R&D, and the generic product quality (a) is given and the same for both…rms.
5 Here, we exclude the case of x = x . In this agglomeration case, there is no di¤erentiation for the entire market,
1
2
and the prices becomes marginal cost, 0. We can readily show that this is not an equilibrium.
6 At z = m, because there is no di¤erentiation, the prices and the pro…ts clealy become zero.

5Firms choose prices to maximize their local pro…tsi (z)= pi (z)qi (z) for any quality pair(u1 (z); u2 (z)) speci…ed by their locations. From the …rst-order conditions, we have the following
equilibrium prices:

p1 (z)=

2vu1 (z) [u1 (z)u2 (z)] = [4u1 (z)u2 (z)] ;(3)

p2 (z)= vu2 (z) [u1 (z)u2 (z)] = [4u1 (z)u2 (z)] :(4)

The corresponding local pro…ts are given by:

1 (z) = 4 [u1 (z)u2 (z)] fvu1 (z)= [4u1 (z)2 (z) = u1 (z)u2 (z) [u1 (z)u2 (z)]gu2 (z)] fv= [4u1 (z)2if z 2 [0; m) ;

(5)

2

(6)

u2 (z)]g

if z 2 [0; m) :We proceed the remaining markets over (m; 1], where the quality of good 2 is higher. Due tothe symmetry, exchanging the …rms’indexes in equations (1)-(6) generates the equilibrium values.
The local pro…ts are:

1 (z) = u1 (z)u2 (z) [u2 (z)2 (z) = 4 [u1 (z)u1 (z)] fv= [4u2 (z)u2 (z)] fvu1 (z)= [4u1 (z)2

u1 (z)]gu2 (z)]g2

if z 2 (m; 1];

if z 2 (m; 1]:

(7)
(8)

Hence, we can calculate the total pro…ts in Bertrand competition as a function of the locationpair:
Bi (x; y) =

Z 1
0

where

i (z) is given by (5)-(8).

6i (z)dz;(9)2.1.2

Location equilibrium

Here, we proceed the location competition in the …rst stage. We de…ne a location equilibrium as alocation pair in which no …rm can earn a greater pro…t by relocation when the other …rm’s location
is …xed. Hereafter, let

t=a (0 << 1) as the mismatch cost measure. Unfortunately, due tothe nonlinearity and complexity of equations, we will compute equilibria approximately by Newton
method. However, some properties of equilibria can be shown analytically. First, we can showthat a candidate of corner solutions, x = 0 or y = 0 cannot be an equilibrium. Second, we readily
have the fact that @B
1 (x; y)=@x and @B
2 (x; y)=@y are symmetric with regard to x = y.

Hence,

the equilibria are, if any, to be symmetric with regard to the center (x = y). At last, we haveBi (1=2; 1=2) = 0 for i = 1; 2, and

@ B
@ B
2
1
=
=
(x; y)(x; y)@x@yx=1=2;y=1=2
x=1=2;y=1=2

5
a v 2 < 0:18

Hence, we have the following fact.

Remark 1 The central agglomeration cannot be an equilibrium.

This is straightforward because the central agglomeration leads to zero pro…ts for the …rms.
They can earn positive pro…ts by locational dispersion. Hence, we will focus on symmetric, inner
solutions. Note thatB
1 (x; y) andB
2 (x; y) are identical when x = y. We haveav 2 f B@ Bi(x; y)=@x
33750 (3 +y=x

25 x) (3 +

2;

2 x)

where

fB

32500 (2x1) 9 + 3 (2

7x) +

7

2

1

7x + 10x22

ln (3 +

5x ) + :::x
0.5

Cournot
0.4

0.3

Bertrand

0.2

0.1

0.2

0.4

0.6

0.8

1

τ

Figure 1: Location equilibrium: Quantity competition (upper line), Price competition (lower line).

Note that f B is a function of x andand the location equilibria must satisfy f B = 0. We proceed

a computational analysis; then, we …nd a unique equilibrium (x = y = xB ) as the upward sloping
connected line in Figure 1.7
As the limiting cases, we have the following values:

xB = 0:1875when! 0 (a ! 1)

xB = 0:269605

when! 1 (a ! t)

2.2

Cournot competition

2.2.1

Quantity equilibrium

Next, we consider Cournot competition. Consider a market at z 2 [0; m), where the quality of good
1 is higher. Inverting the system of demand functions given by (1) and (2), we have inverse demand
7 We can …nd that these equilbria satisfy the second-order conditions.

8

functions as follows:

p1 (z)= vu1 (z)p2 (z)=[v

q1 (z)u1 (z)q1 (z)

q2 (z)u2 (z);(10)

q2 (z)] u2 (z):The …rms choose quantities to maximize their local pro…ts(11)

i (z) = pi (z)qi (z), for any given quality

pair (u1 (z); u2 (z)). From …rst-order conditions, we have the following equilibrium quantities:

q1 (z)= v [2u1 (z)u2 (z)] = [4u1 (z)q2 (z)= vu1 (z)= [4u1 (z)u2 (z)] ;u2 (z)] :

(12)
(13)

The corresponding local pro…ts are given by:

1 (z) = u1 (z) fv [2u1 (z)2u2 (z)] = [4u1 (z)2

2 (z) = u2 (z) fvu1 (z)= [4u1 (z)u2 (z)]gu2 (z)]g

if z 2 [0; m] ;if z 2 [0; m] :

(14)
(15)

Note that these pro…ts are always positive even when u1 (z) = u2 (z). We proceed the remaining
markets over (m; 1], where the quality of good 2 is higher. Due to the symmetry, exchanging the…rms’indexes in equations (10)-(15) generates the equilibrium values. The local pro…ts are:

1 (z) = u1 (z) fvu2 (z)= [4u2 (z)2 (z) = u2 (z) fv [2u2 (z)u1 (z)]g2

if z 2 (m; 1];u1 (z)] = [4u2 (z)u1 (z)]g2

if z 2 (m; 1]:

(16)
(17)

Hence, we can calculate the total pro…ts in Cournot competition as a function of the location pair:Ci (x; y) =

Z 1
0

9

i (z)dz;(18)

wherei (z) is given by (14)-(17).

2.2.2

Location equilibrium

We proceed the location competition in the …rst stage. Again, due to the nonlinearity of equations,
we have equilibria numerically. As in the Bertrand case, corner solutions cannot be an equilibrium,and @C
1 (x; y)=@x and @C
2 (x; y)=@y are symmetric with regard to x = y. Hence, the solutions are,if any, to be symmetric with regard to the center. Further, we have

@ C
@ C
1
2
=
=
(x; y)(x; y)@x@yx=1=2;y=1=2
x=1=2;y=1=2

1
a v 2 < 0:27

Hence, the central agglomeration cannot be an equilibrium again. We will focus on symmetric,
inner solutions henceforth. From (18), we haveav 2 f C@ Ci=(x; y)@x5625 (3 +
y=x

2;

2

5 x) (3 +

2 x)

where

fC

1250 (2x1) 9 + 3 (2

7x) +

2

7x + 10x2

12

ln (3 +

5x ) + :::Note that f C is a function of x and . The location equilibria must satisfy f C = 0; then, we havea unique equilibrium (x = y = xC ) as in Figure 1. As the limiting cases, we have the following
values:xC = 0:428571 whenC

x = 0:38613

3! 0 (a ! 1)

when

Discussion

We can summarize the result as the following remarks.
10! 1 (a ! t):

Remark 2 With regard to interior location equilibria, there uniquely exists the symmetric dispersed
equilibrium for both price competition and quantity competition (no agglomeration). The …rms are
more dispersed in Bertrand competition than in Cournot competition for any mismatch costs.

This result is straightforward and is consistent with Motta (1993). Because Bertrand competition is more intense than under Cournot competition, …rms have a stronger incentive to separate
from the rival …rm in the former case. Especially, agglomeration in the center makes no di¤erentiation at all and the most severe competition. That is why both …rms never agglomerate.
However, the reactions to the changes of mismatch costs are di¤erent between the competitions.

Remark 3 The less the mismatch costs are; the more the …rms separate in Bertrand competition,
while the nearer to the center they locate in Cournot competition.

This result is not so straightforward. A decrease in the mismatch costs strengthens both thecontrastive e¤ects: (1) pro-competitive e¤ect and (2) market expanding e¤ect. The former is due
to less di¤erence in quality that leads the …rms to intense price competition, which is a centrifugale¤ect. The latter is from the competitiveness at relatively remote markets, which is a centripetale¤ect. In Bertrand competition, pro-competitive e¤ect is so dominant that the …rms separate fromthe rival. On the other hand, in Cournot competition, the competition is not so intense that the…rms rather approach each other to earn more pro…ts from the entire market.
To con…rm these intuitive explanation, we compute the prices and the total pro…ts under (a; v) =(5; 5). Figure 2 and Figure 3 show the equilibrium prices at z = 0:45 and z = 0:1 respectively. In
the …gures, the superscript shows the competition type: B is Bertrand and C is Cournot. The
subscript indicates the …rm: H is high quality (…rm 1) and L is low quality (…rm 2).
These …gure shows that the prices under Bertrand approach down to zero due to intense pricecompetition by a decrease in the mismatch costs. Meanwhile, when the mismatch costs decrease,the prices for the low quality …rm under Cournot go up and approach to a positive value with the
11

10

8

pCH

6

pCL

4

pBH

2

pBL
02
.

04
.

06
.

08
.

1

τ

Figure 2: The equilibrium price at z = 0:45.

10

p CH
8

p BH

6

4

p CL
2

p BL
02
.

0 4.

06
.

0 8.

Figure 3: The equilibrium price at z = 0:1.

12

1

τ

profit14

Cournot12
10
8

Bertrand

6
4
2
0.2

0.4

0.6

0.8

1

τ

Figure 4: Pro…ts under Bertrand and Cournot in the location equilibrium.

prices for the high quality …rm. Hence, a decrease in the mismatch costs is less welcome for the…rms under Bertrand, and they try to locate apart in order to avoid such a competition. On theother hand, under Cournot competition, a decrease in the mismatch costs does not lead a intense
price competition; thus, the …rms approach to the center to expand the market share. Figure 4
recon…rms such an explanation by computing the equilibrium total pro…ts under Bertrand andCournot.

4

Concluding Remarks

We have developed a new type of spatial competition model: Location determines quality. We also
have considered the two types of competition: Bertrand and Cournot. One of our results is that the…rms choose dispersed locations, and neither maximum nor minimum di¤erentiation arises for bothcompetition settings. Another result is somewhat surprising that the reactions to the mismatch
costs are di¤erent between the competitive nature. When mismatch costs are decreasing, the …rms
approach toward the center (resp. separate) each other when they engage in quantity (resp. price)13

competition. This is because quantity competition is less competitive; then, expanding the market
share matters more than avoiding the competition in quantity competition. Meanwhile, in pricecompetition, it is more important to avoid keen price competition caused by low mismatch costs.

References[1] Anderson, S.P. and A. de Palma (1988), “Spatial Price Discrimination with Heterogeneous
Products,” Review of Economic Studies 55, 573-592.
[2] Anderson, S. P. and D. J. Neven (1991), “Cournot competition yields spatial agglomeration,”International Economic Review 32, 793-808.
[3] d’Aspremont, C., J. J. Gabszewicz, and J.-F. Thisse (1979), “On Hotelling’s ‘Stability In Competition’,” Econometrica 47 (5), 1045-1050.
[4] De Fraja, G. and G. Norman (1993), “Product di¤erentiation, pricing policy and equilibrium,”
Journal of Regional Science 33, 343-363.
[5] Hamilton, J. H., J.-F. Thisse, and A. Weskamp (1989), “Spatial discrimination, Bertrand vs.
Cournot in a model of location choice,” Regional Science and Urban Economics 19, 87-102.
[6] Hotelling, H. (1929), “Stability in competition,” Economic Journal 39, 41-57.
[7] Motta, M. (1993), “Endogenous quality choice: price vs. quantity competition,”The Journal ofIndustrial Economics 41 (2), 113-131.
[8] Shaked, A. and J. Sutton (1982), “Relaxing price competition through product di¤erentiation,”
Review of Economic Studies 49, 3-13.

14

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