Discussion Paper Series
2010-01
Spatial Cournot competition with mobile consumers
Takanori Ago

February 4, 2011
(First version: December 14, 2010)

Discussion Papers can be downloaded:
http://www1.tcue.ac.jp/home1/c-gakkai/dp/dp10-01

Spatial Cournot competition with mobile consumers
Takanori Ago
February 4, 2011

Abstract
This paper deals with a variable spatial-distribution of consumers according to the location
decisions of …rms in spatial competition. Speci…cally, we present a location-then-quantity gamein a circular city in which some of the consumers are attracted to the …rms’ locations. We
show that with this scenario, all …rms agglomerate when the transport cost is low. This is in
sharp contrast to the results shown in previous studies with …xed distributions of consumers,
where such a full agglomeration never occurs in equilibrium. Welfare analysis shows excess
dispersion compared with the second-best scenario. By changing the space into a line segment,
we show that a spatial dispersion can be achieved.

JEL Classi…cation: L11; L13; R12
Keywords: Spatial Cournot competition; Mobile consumers; Transport cost

1

Introduction

The study of spatial competition has a long history, beginning with the seminal work by Hotelling(1929). Despite the accumulation of knowledge, there is an unexplored issue: variable distributionof consumers. In the research about spatial competition, the spatial distribution of consumers
is exogenously …xed throughout the analysis. The distribution could vary, however, after …rms
Faculty of Regional Policy, Takasaki City University of Economics, 1300 Kaminamie, Takasaki, Gumma 3700801, Japan. E-mail: ago@tcue.ac.jp. Phone/fax: +81-27-344-7590.

1

determined their locations. Typically, …rms attract some consumers to (or near) their locations forthe following reasons.1
1. Advertisement, awareness of the products
If …rms advertised their products, some consumers would become aware of the products’
merits. We can interpret this to mean that the e¤ective mass of consumers attracted to each…rm’s characteristic is enhanced.
2. Network externalities
If the good generates network externalities (network good), it is important for consumers that
other consumers also own the same product. In other words, consumers prefer a prevalent
product to a brand-new, di¤erent type of product that few people use. This preference alsomeans there will be an enhanced mass of consumers of the established products.
3. Shopping externalities, comparison or one-stop shopping
Suppose that there are other goods that are outside of the model. For instance, we analyze
a …sh market in the model and a meat market is behind the model. If consumers buy boththe …sh and the meat, the locations where both goods are available have a great advantage
for consumers who want to do comparison shopping or one-stop shopping to minimize their
shopping costs. Hence, consumers prefer such locations, and the population there grows
faster than at other locations.2
4. Mobile workers (employees)
Suppose that some of the consumers are employed by the …rms, and their commuting costs
are not negligible. In this case, they would love to live at or near the employing …rms.
If a …rm changes its location from town A to town B, the purchasing power at town A
shrinks, while that at town B increases, because of the job disappearance and creation,
1 If …rms generated negative externalities toward their neighbors, such as contaminated smoke, they would be
avoided by consumers. Such an opposite scenario is less interesting in our model, as seen later. That is, ourbenchmark model already yields spatial dispersion of …rms; hence, the outcome is to be unchanged with our new
element.
2 See, e.g., Stahl (1982) and Wolinsky (1983) for consumers’ search and the concentration of retail …rms as arelated issue.

2respectively. Therefore, the population (mass of demand) increases at the …rms’ locations.
Due to the multiplier e¤ect, such job creation is important for the local economy. For
example, Japanese automaker Toyota Motor Corporation announced in March 2010 that
their production operation would be moved from California to Mississippi, which means
2,000 new jobs should be created in Mississippi. Haley Barbour, the governor of Mississippi,
welcomed the decision, whereas strong opposition occurred in California.
Given these reasons why consumers are attracted to …rms’locations, the objective of the present
paper is to develop a model of spatial competition that deals with such a variable distributionof consumers according to …rms’ locations. For expository convenience, let the term “mobile
consumers” denote those who change their locations after …rms choose their locations. Speci…cally,
we assume that a constant population (mass) of mobile consumers always clings to the …rms. In
other words, to simplify the analysis we ignore an endogenous residential-choice problem for mobile
consumers throughout most of our discussion (we will discuss this issue in Section 5).
Our analysis is based on spatial Cournot competition with homogeneous good that was developed by Hamilton et al. (1989) and Anderson and Neven (1991),3 in which oligopolistic …rms
choose their locations simultaneously in the …rst stage, and they decide their supply amount simultaneously for (immobile) consumers that are distributed over the space. On the one hand, theirstudies both show that, in such a linear space, the …rms agglomerate in the centre in equilibrium.
On the other hand, Pal (1998) shows that, in a circular city, two …rms are located as far away fromeach other as possible on the circumference (e.g., at 12 a.m. and at 6 a.m.). In sum, the results
depend greatly on the form of the space, linear or circular.
In our model, in addition to immobile consumers, mobile consumers are allocated at …rms’
locations, and each …rm serves all consumers.4 The introduction of mobile consumers induces
additional agglomeration and dispersion forces. If a …rm chooses a location near a rival, there
3 Another branch of spatial competition is spatial Bertrand (price) competition (e.g., d’Aspremont et al., 1979).
In the present paper we adopt the Cournot setting, because it brings us clearer and more contrastive results.
4 Gupta et al. (1997) extend the consumer distribution to more general cases. However, the distribution isexogenously …xed to the end.

3

is a positive e¤ect in the sense that the pro…ts would increase due to lower transport costs forthe mobile consumers who normally patronize the rival (agglomeration force). Meanwhile, theoriginal …rm should lose pro…ts, because its mobile consumers are attracted by the approaching
rival (dispersion force). As a result, we …nd that agglomerated equilibrium is achieved even in acircular city when the transport cost is su¢ ciently low, whereas dispersed equilibrium is present
even in a linear city when the transport cost is high. In other words, the form of the space is not
so determinative. This is in sharp contrast to the familiar results mentioned above (Hamilton etal., 1989; Anderson and Neven, 1991; Pal, 1998).5
When it comes to mobile consumers, the new economic geography (henceforth, NEG) launchedby Krugman (1991) deals with a similar issue of spatial distribution of economic activities in generalequilibrium incorporating scale economies in production and imperfect competition (speci…cally,monopolistic competition).6 The …ndings of the NEG models and ours are somewhat similar: For
instance, lower transport cost tends to lead to agglomerated equilibrium (core-periphery structure).
Nevertheless, in monopolistic competition of the NEG, each …rm has only a negligible impact on
the economy, which does not match our awareness of the issues. Despite the di¤erence between
the NEG and our oligopoly, Picard and Tabuchi (2010) developed a model similar to ours: They
present a general equilibrium model à la NEG over a circular, continuous space. In their model,the ‡at earth equilibrium (uniform distribution of …rms) is unstable in a wide class of transport
costs. Therefore, the NEG structure puts more stress on the agglomeration force (e.g., forward andbackward linkages) than does our partial equilibrium in an oligopoly. In our model, agglomeration
is not so robust.
The remainder of the paper is organized as follows. In Section 2, the two-stage locationquantity game is presented. In Section 3, the quantity choices are analyzed. In Section 4, thelocation equilibrium is established. Section 5 is devoted to welfare analysis. Section 6 deals withsome extensions of our model. Section 7 summarizes the results.
5 Matsushima (2001), Shimizu and Matsumura (2003) and Matsumura and Matsushima (forthcoming) all showthat full agglomeration never occurs in equilibrium in a circular city with more than two …rms.
6 See Fujita et al. (1999) for an excellent comprehensive survey of NEG.

4

2

The model

Our benchmark is a spatial Cournot model with the circular city developed by Pal (1998). Thereis a circular city with perimeter equal to 1. There are two …rms indexed by i (i = 1; 2) that supply
a homogeneous good with zero marginal cost, and let xi 2 [0; 1) denote the location of …rm i.
There are two types of consumers, mobile and immobile. Immobile consumers are uniformly
and continuously distributed on [0; 1) with a unit density at each location, while mobile consumers
are able to move freely with zero relocation cost in the city after the …rms determine their locations.
As mentioned in the Introduction, we assume that mobile consumers are attracted to …rms, andeach mobile consumer immediately moves to either of the …rms’locations. To simplify the analysis,
the movement is exogenous. Moreover, because the …rms are symmetric, each …rm attracts thesame number (mass) of mobile consumers.7 Let n > 0 be the mass of mobile consumers attracted
by each …rm, and let f (z) be the density function of mobile consumers at z. Thus, we have
8
>
>
>
0
>
>
>
<
f (z) =
n
>
>
>
>
>
>
: 2n

if z 6= x1 ; x2
if z = xi 6= xj ; i 6= j :(1)

if z = x1 = x2

The third case is referred to as full agglomeration in the sense that all …rms agglomerate at a point.
Each consumer, irrespective of the types, has the same, inverse demand function as follows:

P =a

bQ;

Q = q1 + q2 ;(2)

where P is the price, qi is …rm i’s supply amount, and a and b are constants.
We consider a two-stage location-then-quantity game. In the …rst stage, the …rms choose theirlocations simultaneously. In the second stage, they determine their supply amounts simultaneously
after the mobile consumers move. Subgame perfection is adopted as the equilibrium concept.
7 See Section 5 for the discussion of endogenous residential choice and Section 6 for the asymmetric cases.

5

The …rms bear transport costs, and they can set an independent supply amount for eachlocation, because arbitrage between consumers is prohibitively costly. To ship a unit of the productfrom xi to a consumer at z, the transport cost for …rm i is given by T (xi ; z) = minftjxi
jxi

zj; t(1zj)g, where t > 0 is the transport cost parameter and is assumed to be su¢ ciently low, such

that
0<

t=a < 1:

(3)This requirement ensures that both …rms serve the entire city (all consumers), irrespective of thelocations. Leti (z) = qi (z) [P (z)T (xi ; z)](4)denote the local pro…t for …rm i earned from a consumer at z. Then, the total pro…t is given
by
i =

Z 1

i (z)dz + n i (xi ) + n i (xj )0for i; j 2 f1; 2g; i 6= j,(5)where the …rst term is the pro…ts earned from immobile consumers, and the second and the thirdterms are those from the mobile consumers.

3

Quantity equilibrium

Let us analyze the second-stage Cournot competition due to backward induction. First, we consider
immobile consumers. Recall that each local market is independent. Then, the …rst-order condition@ i (z)=@qi (z) = 0 yields the equilibrium quantity for …rm i at z as follows (the asterisk refers to
the equilibrium value):qi (z) = [a2T (xi ; z) + T (xj ; z)] =3b for i; j 2 f1; 2g; i 6= j:(6)Next, we consider mobile consumers. The …rst-order conditions @ i (xi )=@qi (xi ) = 0 and@ i (xj )=@qi (xj ) = 0 yield the equilibrium quantities for the mobile consumers at xi and xj as

6

follows:

qi (xi )=[a

2T (xi ; xi ) + T (xj ; xi )] =3b = (a + T (xj ; xi )) =3b;(7)

qi (xj )=[a

2T (xi ; xj ) + T (xj ; xj )] =3b = (a(8)

2T (xi ; xj )) =3b

for i; j 2 f1; 2g; i 6= j. Substituting (6), (7), and (8) into (4), the local pro…ts are rewritten as
2i (z) = b [qi (z)] ,2i (xi ) = b [qi (xi )] , and2i (xj ) = b [qi (xj )] , respectively. Then, from (5), thetotal pro…t is obtained byi (xi ) =Z 1i (z)dz + n i (xi ) + n i (xj ):(9)0

It is noteworthy that the pro…t from the …rm’s own mobile consumers,i (xi ), increases by

separating from the rival, whereas the pro…t from the rival’s mobile consumers,i (xj ), increases

by approaching the rival. This tradeo¤ is the key to determining the location equilibrium.

4

Location equilibrium

Here, we analyze the location equilibrium8 in the …rst stage when the quantity choices are given in
the second stage, as described in Section 3. Let us …rst consider the best response of …rm 1 against
a given location of …rm 2. Due to the symmetry with regard to the space, we can assume without
loss of generality that 0

x1

1=2 and x2 = 0. Let x1 be the best response against x2 = 0. In

other words, x1 is …rm 1’s optimal (pro…t-maximizing) distance from …rm 2. Meanwhile, becauseof the symmetry with regard to the …rms, …rm 2’s optimal distance from …rm 1 must be also x1 .
This situation constitutes an equilibrium for any location pair such that the distance between the…rms is x1 .9
Fortunately, the pro…t function given by (9) takes a simple form that allows an analytical solu8 A location pair (x9 If x

1 ; x2 ) is a Nash equilibrium if and only ifi (xi )i (xi )

1 were multivalued, multiple equilibria could arise as discussed later.

7

for 8i and 8xi 2 [0; 1).

tion: It is cubic with regard to xi and the coe¢ cient of x3i is negative. Tedious but straightforward
calculations yield the following main result:
Proposition 1 Let

T (n) =

8>>
< 64n=3(5n + 2)2>>: 12n= (15n + 2)

when 0 < n

2=15
:

(10)

when 2=15 < n

The location equilibrium is classi…ed into three groups as follows:(i) Full agglomeration (minimal di¤ erentiation): Two …rms agglomerate at a point if

T (n).

(ii) Full dispersion (maximal di¤ erentiation): Two …rms separate from each other such that the
distance between them is equal to 1=2 if

max f2=5; T (n)g.

(iii) Partial dispersion (in-between di¤ erentiation): Two …rms separate from each other such thatthe distance between them is less than 1=2 if T (n)given by 5n + 2 +p(5n + 2)2

< 2=5 under 0 < n < 2=15. The distance is

16(n= ) =8.

Proof. See Appendix A.

[Insert Figure 1 here.][Insert Figure 2 here.]

Figure 1 classi…es the location equilibria in the parameter space. Because T (n) is increasing
in n, agglomeration is likely to occur when the mass of mobile consumers is large (large n) or
the transport costs are low (small ) or both. Figure 2 shows a pitchfork diagram of the location
equilibrium when n = 1=10 with the normalization of x1 + x2 = 1. Unlike Pal (1998) with n = 0,
T (n) > 0 for any positive n, which implies that su¢ ciently small transport costs generate full
agglomeration. In sum, full agglomeration always exists, even in the circular city. This is in sharp
contrast to the model described by Pal (1998).
8

It is true that a large n tends to lead to agglomeration. Nonetheless, because T (n) is boundedfrom above (limn!1 12n= (15n + 2) = 4=5), full dispersion always exists as an equilibrium, evenwhen n is large.10n[ i (xi ) +Recall the total pro…ts in (9).

11i (xj )] when n is too large.

The function degenerates intoi (xi )=

In other words, only the sales to mobile consumers

matter. Because n is just a multiplier of the function, n is independent of the determination oflocation equilibrium. Accordingly, the pro…ts earned from the mobile consumers,i (xi ) +i (xj ),

is determinative for the equilibrium. Let us take a closer look to understand of our model.
It is helpful to borrow the ideas from a model of Cournot duopoly with asymmetric cost.12
Suppose that P = 1

Q as the inverse demand function, and the marginal cost of a …rm (superior…rm) is zero and that of the other …rm (inferior …rm) is c > 0. Then, the sum of the (equilibrium)
pro…ts of the two …rms corresponds toi (xi ) +i (xj ) above, and the location choice in our model

corresponds to the choice of c (the argument is essentially the same by setting c = =2). Tedious
calculations yield the sum of the pro…ts of the two …rms as (22c + 5c2 )=9, which is a parabola

whose minimum is given by c = 1=5. On the one hand, when c > 1=5, that sum of the pro…ts
is increasing in c and is maximized at c = 1=2 (monopolization). This suggests that when thetransport costs are high, the …rms will choose location dispersion in search of (local) monopoly.
On the other hand, when c < 1=5, the sum of the pro…ts is decreasing in c and is maximized at c = 0(cost reduction). Therefore, when the transport costs are low, the …rms will agglomerate in searchof strong competitiveness against the rival.13 Furthermore, immobile consumers are evenly spreadover the space; hence, they work as a dispersion force, as Pal (1998) has shown. To synthesize thedescriptions of these e¤ects, we present Proposition 1.
Note that there is a possibility of multiple equilibria.

For instance, when n = 1=2 and= T (1=2) = 12=19, both full agglomeration and full dispersion are equilibria. The multiplicity of equilibria, however, is restricted to a zero-measure set of parameter values. Hence, location
1 0 The no-black-hole condition, which is often assumed in NEG, always holds in our model. In other words,
agglomeration forces here are weaker than in NEG.
1 1 We can also interpret the case as the model without immobile consumers.
1 2 See, among others, Boyer et al. (2003) and Ziss (1993) for spatial competition with asymmetric cost.
1 3 See Kabiraj and Marjit (2000) for the technology transfer problem depending on the cost gap as a similar
structure. Their technology transfer corresponds to the location choice here.

9equilibrium is almost always uniquely determined. This is in contrast to the core-periphery modelin the NEG, in which the symmetry-breaking and sustain points are di¤erent and hence the multiple
equilibria arise.14 In our model these two points coincide.

5

WelfareSocial welfare is another important issue to consider. Is location equilibrium desirable or not?
We de…ne the social welfare function as the sum of consumer surplus (CS) and producer surplus(P S), which gives the social surplus (SS = CS + P S). From the demand function in (2), the local
consumer surplus derived at z, cs(z), is given by

cs(z) =

1[a
2

P (z)] Q(z) =b
Q(z)2 ;
2

(11)

where P (z) and Q(z) are the price and the total supply amount at z, respectively. Then, we have
CS =

R1
0

cs(x)dx + n [cs(x1 ) + cs(x2 )]. P S is the sum of pro…ts given by P S =1+

2 , whereiis de…ned in (5). We analyze the …rst-best situation, in which the social planner is able to control
both the supply amount and the location of each …rm, and the second-best situation, in which theplanner is only able to control the location of each …rm.

5.1

First-best situation

In the …rst-best, it is clear that the planner must set the supply amount at each market suchthat the price is equal to the transport cost (marginal-cost pricing). Hence, once the locationsof both …rms are given, each market should be served by the nearer …rm at the price of the unittransport cost. Then, total transport costs in the whole economy must be minimized for the…rst-best situation. Clearly, this will be achieved by full dispersion.
1 4 In the literature, the symmetric structure, where two regions have the same share of mobile workers, breaks

down when transport costs are less than the symmetry-breaking point. The core-periphery structure, where thecore region contains all of the mobile workers and the peripheral region only has immobile workers (farmers), issustained as long as the transport costs are less than sustain point. The multiple equilibria emerge when thesymmetry-breaking point is less than the sustain point (Krugman, 1991; Fujita et al., 1999).

10

Proposition 2 In the …rst-best situation, two …rms are located such that the distance between
them is 1=2 (full dispersion), and each market is served by the nearer …rm in which the supply
amount is ordered such that the price is equal to the transport cost there.

5.2

Second-best situation

What if the social planner were able to control only locations? In this case, the supply amount foreach market would be determined as the Cournot equilibrium, as described in Section 3. The …rst
…nding is as follows.
Lemma 1 Let the supply amount for each consumer be given by Cournot competition under agiven location. Then, consumer surplus is maximized when two …rms are located at the same point(full agglomeration).
Proof. See Appendix B.
The lemma states that the consumers as a whole prefer uneven distribution of …rms to dispersed
distribution although the total transport costs are not minimized. The intuitive process behind
this is as follows. From (11), a consumer surplus for each consumer is proportional to the squareof the consumption. Suppose that there are two consumers and two units of the good. Then, if weassign one unit for each consumer, CS _ 12 + 12 = 2. Meanwhile, if we give two units to only one
consumer, we have CS _ 22 + 02 = 4. Therefore, consumer surplus is greater by uneven allocationsof the good.
In location equilibrium, neither …rm cares about consumer surplus. Thus, we can predict theexcess dispersion in equilibrium. The formal result is as follows.
Proposition 3 Suppose that the social planner can only control location, and the supply amountis determined by Cournot competition given the location pair. Let

T T (n) =8>>< 1792n=3(22n + 7)2>>:96n= (66n + 7)

11

when 0 < n

7=66:

when 7=66 < nThe second-best situation is achieved as follows:(i) Full agglomeration: Two …rms agglomerate at a point if

T T (n).

(ii) Full dispersion: Two …rms separate from each other such that the distance between them isequal to 1=2 if

max f8=11; T T (n)g.

(iii) Partial dispersion: Two …rms separate from each other such that the distance between themis less than 1=2 if T T (n)22n + 7 +p
(22n + 7)2< 8=11 under 0 < n < 7=66. The distance is given by448(n= ) =2.

Proof. See Appendix C.

[Insert Figure 3 here.]

Figure 3 shows the con…guration of the second-best situation. A comparison between Figures 1
and 3 shows excess dispersion in equilibrium. For instance, when n = 1 and= 0:8, full dispersion

is achieved in equilibrium, whereas full agglomeration is the second-best situation. This is because
consumers bene…t by agglomerated locations (Lemma 1).

5.3

On residential choice

Thus far, we have neglected the issue of endogenous residential choice. One might think it isnatural that consumers endogenously choose their locations to maximize their consumer surplus.
From (11), we readily …nd that the consumer surplus is maximized at the location that minimizes
the sum of the transport costs to both …rms. Under the linear transport cost, the sum is constantfor any location between the two …rms, due to the constant sum of the distances. This implies
that each mobile consumer has no incentive to move away from the …rms’locations; therefore, ourarbitrary assumption is not especially harmful.
The above discussion, however, would not hold if the transport cost were convex. Supposethat the transport cost is quadratic: T (xi ; z) = minftjxi
0x2

zj2 ; t(1jxizj)2 g, x1 = 0, and

1=2. Then, the consumer surplus is maximized at z = x2 =2 2 (0; x2 ). Therefore,
12our guess is that full agglomeration should be robust, while dispersion could not be robust if weincorporate the endogenous residential-choice problem.

6

Extensions of our model

6.1

Asymmetry

The numbers of mobile consumers could be di¤erent between …rms. We …rst consider the case where
only one …rm (…rm 1) attracts mobile consumers. That is, our pro…t function (5) is rewritten asi (xi ) =R10i (z)dz + n i (x1 ). In this case, …rm 2 has a strong incentive to approach …rm 1,

whereas …rm 1 has no incentive to approach …rm 2. These asymmetric incentives lead to thefollowing outcome.
Proposition 4 When only one …rm attracts mobile consumers, no location equilibrium exists inpure strategies.
Proof. See Appendix D.

As a general case, …rms 1 and 2 attract n1 and n2 mobile consumers, respectively (n1 > n2 > 0).
Although this di¤erence may cause the nonexistence of equilibrium in some parameter sets as
Proposition 4 has shown, thanks to positive mobile consumers at both …rms, we can show a similar
result with the symmetric case described in the previous section. That is, …rms tend to agglomerate
when the transport cost is su¢ ciently low, whereas they separate when the cost is high enough.15

6.2

m-…rm oligopoly

What if there are more than two …rms? Recall that full agglomeration never occurs in equilibrium
under m(2)-…rm oligopoly in the circular city without mobile consumers (Matsushima, 2001;

Shimizu and Matsumura, 2003; Matsumura and Matsushima; forthcoming). Hence, an interesting
1 5 The detailed analysis is available from the author upon request.

13

question is whether full agglomeration is an equilibrium or not. Let us focus on a symmetric casewhere all …rms attract n mobile consumers. In that case, we have the following answer.
Proposition 5 Suppose that there are m …rms and each …rm attracts n mobile consumers. In
that case, full agglomeration is an equilibrium when the transport cost is su¢ ciently low.
Proof. See Appendix E.
With mobile consumers, full agglomeration can be achieved, irrespective of the number of …rms.
This implies that agglomerative forces become too strong under low transport costs. Note that
this is not a unique outcome.16

6.3

The linear city

What if the city is linear? Without mobile consumers, central agglomeration is achieved in thelinear city under spatial Cournot competition (Hamilton et al., 1989; Anderson and Neven, 1991).
In the linear city, the center has a location advantage in the sense that total transport costs to serve
all consumers are saved. Thus, one of the most interesting questions is whether (full) dispersion
can be an equilibrium. Let the line segment [0; 1] denote the city, and let the other settings be
unchanged in the circular model.17 Further, we analyze the duopoly in which each …rm attracts
the same number (n) of mobile consumers. Then, we have the following result.
Proposition 6 Full dispersion (a …rm is located at 0 and the other …rm is located at 1) is anequilibrium if and only if max f6n=(15n + 2); (n + 2)=(5n + 1)g< 1=2 and n > 1.

Proof. See Appendix F.
With mobile consumers, we have the dispersed equilibrium even in the linear city under spatial
Cournot competition. As our circular city has shown, mobile consumers work as an agglomeration
force under lower transport costs. Hence, we can even see full agglomeration at an edge of thelinear city instead of the center under su¢ ciently low transport costs.
1 6 See Matsushima (2001), Shimizu and Matsumura (2003), Matsumura and Matsushima (forthcoming) for avariety of equilibria without mobile consumers.
1 7 The threshold for full market coverage is revised as< 1=2.

14

7

Concluding remarks

We have analyzed how location is a¤ected when some consumers are attracted to …rms’locations
after the …rms make their location decisions (mobile consumers, variable distribution of consumers).
To do so, new kinds of agglomeration and dispersion forces have emerged: When a …rm is locatednear a rival, the pro…t increases due to the lower transport cost to the mobile consumers at therival (agglomeration forces). Meanwhile, the …rm should be punished by the approaching rival inreverse (dispersion force). Our analysis has shown that the agglomeration forces become dominant
when the transport cost is low or the mass of mobile consumers is large or both.
Without mobile consumers, previous studies have shown that the circular city leads to the dispersed location equilibria (Pal, 1998; Matsushima, 2001; Shimizu and Matsumura, 2003), whereas
the linear city exhibits the agglomerated location equilibria (Hamilton et al., 1989; Anderson andNeven, 1991). In other words, the spatial form matters, not the parameter values such as transport
costs. This is in sharp contrast with our model.
In our future research, we shall continue our work on endogenous residential choice, full analysesof multiple …rms and the linear city, general forms of transport cost function, or other spatial
motions of consumers.

Appendix
A. Proof of Proposition 1
Proof. Without loss of generality, let x2 = 0 and 0x1 = xresponse of …rm 1 against x2 = 0, and let x1 = arg max

1=2. We will compute the best

1 (x) denote the best response. Then, wehave
2
1 (x) = a

12 (2n + 1)

6 (4nx + 1) +15

2

1 + 12 (5n + 2) x2

32x3=108b;

and @2 2
gA (x)=9b, where
1 (x)=@x = 2agA (x) =Note thatof

4x2 + (5n + 2) x

3
1 (x) is cubic and the coe¢ cient of x in

1 (x), we focus on gA (x).

Due to gA (0) < 0,

n= :

(A1)

1 (x) is negative. To investigate the behavior
1 (x) is locally maximized at x = 0.
2

(A1), let the discriminant of gA (x) = 0 be dA = (5n + 2)

16n= . First, when dA

2

From

0,

2

16n= (5n + 2) , it is clear that x1 = 0. Second, when dA > 0 , 16n= (5n + 2) < , gA (x) = 0
yields two local extremizers ofA
1 (x) as x =5n + 2p

dA =8 and x^A = 5n + 2 +(0 < xA < x^A ). Here, due to the negative coe¢ cient of x3 in

p

dA =8

^A is the local maximizer.
1 (x), xThen, we can classify the solution of the maximization problem into three cases:(i) full agglomeration: x1 = 0 , x
^A
1 (0)

1=2 and

1 (0)1 (1=2);or 0 < x^A < 1=2 and

xA ).
1 (^

(ii) full dispersion: x1 = 1=2 , x^A

1=2 and1 (1=2)

1 (0).

(iii) partial agglomeration: x1 = x^A (< 1=2) , x^A < 1=2 and

xA )
1 (^

1 (0).

As we have discussed the logic in Section 4, due to the symmetry, the best response of …rm 2
must be x2 = 0 when …rm 1 is located at x1 that is derived above in each case. By reducing andarranging the above conditions, we have the result.

B. Proof of Lemma 1
Proof. Without loss of generality, we assume 0x1 = x

1=2 and x2 = 0. Substitut-ing the quantity equilibrium obtained in (6), (7) and (8) into the consumer surplus, we have
CS = a2 gB (x)=54b, where gB (x) = 4 2 x3 + 3(2n@gB (x)=@x = 6 2 2 x2 + (2nxB = 12np

der n > 0 and 0 <1) 2 xdB =4 and x^B = 1

1) 2 x2

24n x +

2

6 + 12(2n + 1) and

4n . @gB (x)=@x = 0 yields two local extremizers as
2n +p

dB =4, where dB = (12n)2 + 32(n= ) > 0. Un-< 1, we readily have xB < 0 and 1=2 < x^B . Because of the positive coe¢ cientof x3 in gB (x), gB (x) is monotonically decreasing in x 2 [0; 1=2]. Therefore, full agglomeration

16

maximizes consumer surplus.

C. Proof of Proposition 3
Proof. Without loss of generality, x2 = 0 and 0x1 = x1=2. Then, the social surplus isrewritten by SS(x) = a2 gC (x)=54b, whereSS(x) = a2

28 2 x3 + 3(22n + 7) 2 x2

48n x + 2 2

12 + 24(2n + 1) =54b;

and @SS(x)=@x = a2 2 gC (x)=9b, wheregC (x) =14x2 + (22n + 7)x8 (n= ) :(C1)

Note that SS(x) is cubic and the coe¢ cient of x3 in SS(x) is negative. Due to gC (0) < 0, SS(x)2

is locally maximized at x = 0. From (C1), let the discriminant of gC (x) = 0 be dC = (22n + 7)448n= . First, when dC20,448n= (22n + 7) , it is clear that arg max SS(x) = 0. Second,2

when dC > 0 , 448n= (22n + 7) <xC = 22n + 7p, gC (x) = 0 yields two local extremizers of SS(x) as

dC =28 and x^C = 22n + 7 +p

dC =28 (0 < xC < x^C ). Due to the negativecoe¢ cient of x3 in SS(x), x^C is the local maximizer. Then, we analyze three cases like Appendix
A.
(i) full agglomeration: arg max SS(x) = 0 , x
^C
and

1 (0)

1=2 and SS(0)SS(1=2); or 0 < x^C < 1=2

xC ).
1 (^

(ii) full dispersion: arg max SS(x) = 1=2 , x^C

1=2 and SS(1=2)

SS(0).

(iii) partial agglomeration: arg max SS(x) = x^C (< 1=2) , x^C < 1=2 and SS(^xC )By reducing and arranging the above conditions, we have the result.

17

1 (0).

D. Proof of Proposition 4
Proof. Without loss of generality, …rm 1 attracts n mobile consumers whereas …rm 2 attracts
no mobile consumers. Then, the density function of mobile consumers at z in (1) is rewritten by
f (z) = 0 if z 6= x1 ; f (z) = n if z = x1 . And the pro…t functions are also rewritten adequately.
Like Appendix A, we seek the best response of …rm 1. Without loss of generality, x2 = 0 and
0x1 = x1=2. Then, we obtain

2
1 (x) = aand @12 (1 + n) + 6 (4nx

1) +

2

1 + 12 (n + 2) x21

32x31

=108b

2 2
gD (x)=9b, where
1 (x)=@x = 2agD (x) =Note that

4x21 + (n + 2) x + n= :3
1 (x) is cubic and the coe¢ cient of x1

two local extremizers ofD
1 (x) as x

=

n+2

(D1)

is negative. From (D1), gD (x) = 0 yieldspdD =8 and x^D =dD = (n + 2)2 + 16(n= ) > 0: Because xD^D
1 < 0 < 1=2 < x
1 ,

n+2+

p

dD , where

1 (x) is increasing in x 2 [0; 1=2].

Hence, the best response of …rm 1 is always 1=2.
Next, we analyze the reaction of …rm 2. Let x1 = 0 and 0@x2 = x

4n
2 (x)
(
=
@x x=1=29b

1=2. Then, we have1) < 0:

Therefore, …rm 2 never chooses a location such that the distance between two …rms is 1=2. Thus,
there is no equilibrium in pure strategies.

E. Proof of Proposition 5
Proof. (We apply the same notations as in the duopoly case.) The total number of the …rms is
m, and let j = f1; 2; : : : ; mg and xj denote the …rm’s index and the location of …rm j, respectively.
Inverse demand function at z is revised as P (z) = a
18

b

P

j qj (z), where qj (z) is supply amount of

…rm j at z. The total pro…t of …rm i is given byi (z) = qi (z) [P (z)

i =

R1

i (z)dz + n i (xi ) + n

0

P

j6=i

i (xj ), where

T (xi ; z)] is the local pro…t for …rm i at z. Then, the similar calculations yield

equilibrium supply amount at z as follows:qi (z) =Here, we requireamT (xi ; z) +P

j6=i T (xj ; z)(m + 1) b:< 2=m for each market to be served by all …rms, irrespective of the locationsof the …rms. Substituting the above equilibrium quantity into the pro…t functions, we can rewrite
the local and the total pro…ts for …rm i as
n

P

j6=i

i (xj ), respectively.

2i (z) = b [qi (z)] andi (xi ) =R10i (z)dz + n i (xi ) +Then, we check the relocation incentive for a …rm when all …rms

agglomerate at a location. Without loss of generality, all …rms agglomerate at 0 and we analyze
the optimal location for …rm 1 with 0

x1 = x1=2. Then, we have the pro…t function of …rm

1 as follows:212b (1 + m)
a2

1 (x)

=

12 (1 + mn)

6+ 2 1 + 12 (m1 + 4(m1)2 nx1) m2 n + mn + m

n x2

16m (mNote that the pro…t function is cubic and the coe¢ cient of x31 is negative, and @Let dE be a discriminant of @When dE > 0, @1 (x1 )=@x1= 0. If dE

1=2, or1 (x)=@xjx=0 < 0.

0, then the optimal location x1 = 0.

E^E ,1 (x)=@x = 0 yields the local minimizer and the local maximizer as x and x

respectively (0 < xE < x^E ). For x = 0 to be the optimal location, we requirex^E1) x3 :

1 (0) >

1 (0) >

xE ) if x^E < 1=2. By reducing these conditions, we have
1 (^

0<

< T (m; n) =) arg max

19

1 (x) = 0;

1 (1=2) if

whereT (m; n) =For any m8>>< 32(m>>: 12(m1)mn=3(m2 n + mn + mn)2

when 0 < nm=3(m2 + m

1)
:

2

1)n= 3m n + 3mn + m

3n2

when m=3(m + m1) < n3 and for any n > 0, T (m; n) > 0. This implies that if the transport cost is su¢ ciently

low such that

< T (m; n) and< 2=m, full agglomeration is a location equilibrium.

F. Proof of Proposition 6
Proof. Without loss of generality, let …rm 2 be located at 1 (x2 = 1). The equilibrium supply
amount at z for …rm 1 is given by q1 (z) = [a2T (x1 ; z) + T (x2 ; z)] =3b. We require< 1=2 foreach market to be served by both …rms, irrespective of their locations. The total pro…t for …rm 1
is given by2
1 (x1 ) = a gF (x1 )=27b, wheregF (x1 ) = 4 2 x31 + 3(5n

4)x21

6 [(5

2] x1 + 3 (5 2

1)n +

2

2 + 2)n ++1 :Note that gF (x1 ) is cubic and the coe¢ cient of x31 is positive. If the discriminant of @gF (x1 )=@x1 =0 is zero or negative, then gF (x1 ) is non-decreasing, i.e., arg max gF (x1 ) 6= 0. Suppose that theFdiscriminant is positive, and let xF^F
1 and x1 (x1x^F1 ) denote the local minimizer and the local

maximizer of gF (x1 ). For arg max gF (x1 ) = 0 to hold, we require that
Tedious calculations yield(n + 2)=(5n + 1). When n

1 (0)

1 (1)

()

1 (0)

6n=(15n + 2) and xF
1F
1 (1) and x1

0

1, these conditions violate the full-coverage condition

other words, full dispersion is an equilibrium for any0.

()< 1=2. In

satisfying the above conditions only whenn > 1.

Acknowledgements
I am grateful to Hiroshi Aiura, Toshihiro Matsumura, Noriaki Matsushima, Dan Sasaki, Takatoshi
Tabuchi, Takaaki Takahashi, and participants in seminars at ARSC (Nagoya University) and the20

University of Tokyo for their useful comments and suggestions. Remaining errors are my own.

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[12] Pal, D., Does Cournot competition yield spatial agglomeration?. Economics Letters 60, 49-53,
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22

t/a
1.0

0.8

Full dispersion

0.6

0.4

Partialdispersion

Full agglomeration

0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

n

Figure 1: The classification of the location equilibrium

xi
1.0

0.8

0.6

0.4

0.2

0

0.4

0.2

0.8

0.6

t/a

1.0

Figure 2: The pitchfork diagram (n=1/10)

t/a
Full
dispersion

1.0

0.8

0.6

Partialdispersion

Full agglomeration
0.4

0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

n

Figure 3: The classification of the second-best situation高崎経済大学地域政策学会
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