Discussion Paper Series
2006-01
Central Agglomeration
Of Monopolistically Competitive firms
Takanori Ago

June 7, 2006

Discussion Papers can be downloaded:
http://www1.tcue.ac.jp/home1/c-gakkai/dp/dp06-01

Central Agglomerationof Monopolistically Competitive Firms1
Takanori Ago
Faculty of Regional Policy, Takasaki City University of Economics2
June 7, 2006

1

I am grateful to Yoshitsugu Kanemoto, Masahiro Okuno-Fujiwara, Dan Sasaki, Daisuke Shimizu,
Takatoshi Tabuchi and Takaaki Takahashi for their helpful comments and suggestions. Remaining
errors are my own.
2
Faculty of Regional Policy, Takasaki City University of Economics, 1300 Kaminamie, Takasaki-shi,
Gumma 370-0801, Japan. E-mail: ago@tcue.ac.jp

Abstract
This paper analyzes the location equilibrium for …rms under monopolistic competition in a
linear market à la Hotelling. The central agglomeration of …rms is uniquely achieved withsu¢ ciently low transport costs under three pricing policies: discriminatory, mill, and uniform
delivered pricing. Welfare analysis shows that the agglomeration is also the …rst-best outcome.
Moreover, the use of mill pricing by all …rms is better for social surplus than is the use of any
other pricing. Nevertheless, there is an incentive for the use of discriminatory pricing by each…rm when all …rms apply mill pricing.
JEL Classi…cation: L13, R12, R30.
Keywords: Central Agglomeration, Linear City, Monopolistic Competition, Pricing Policy.

1

Introduction

Since Hotelling’s (1929) landmark work, research involving spatial factors in competition models has shown considerable development. Hotelling’s seminal result is that, as a consequence oflocation competition performed by two …rms in a linear city, both …rms assemble in the center.
This phenomenon is termed as the “principle of minimum di¤erentiation.” It is a pioneering
conclusion that as a result of reasonable decisions made by …rms, economic activity concentrates geographically; this is an agglomeration phenomenon. Further, it should be noted thatthe works of Hotelling and his followers consider not just a mere geographical interpretationof space but also— among others— an interpretation of characteristic space and policy space(Downs, 1957), providing a vast source of insights.
On the other hand, d’Aspremont et al. (1979) have raised doubts on the appropriateness of
this principle. This research has indicated that in Hotelling’s (1929) similar linear city model,
when considering location and price de…ned by spatial competition, as long as two …rms are
not located su¢ ciently apart from each other, price equilibrium does not exist. Further, thechange to the quadratic function of linear transport costs, which is the original setting, led tothe conclusion that the existence of the equilibrium price can be guaranteed, while at the sametime, the two …rms will mutually locate themselves apart from each other at the edges of themarket.
In line with this research, Tabuchi (1994) and Irmen and Thisse (1998) have each analyzed
the space’s dimension expanded to 2 and m (2) dimensions. Consequently, they achieved

surprising results: although the di¤erentiation is maximized in one dimension, it does not occur
in the other (m1) dimensions. This indicates that with regard to product di¤erentiation,

if su¢ cient advantage over other …rms is achieved in one aspect, it is already su¢ cient for allother aspects.
What must be taken into consideration concerning the abovementioned research is that
there are alternative interpretations about space: geographical and product quality. For instance, regarding the Irmen and Thisse (1998) model, 2 of m dimensions are considered to begeographical, while it is rather di¢ cult to regard the remaining (m2) dimensions, as charac-

teristic spaces. Nevertheless, models that can simultaneously accommodate both geographicaland characteristic spaces— especially in terms of the agglomeration of …rms handling di¤erentiated goods— are more realistic. Hence, an analysis that clearly distinguishes geographical andcharacteristic spaces is required.
When considering linear cities such as those in Hotelling (1929) as geographical spaces,
research such as Anderson and de Palma (1988) and De Fraja and Norman (1993, henceforth“DN”) can be cited as including di¤erentiation in terms of characteristic spaces within such1

a framework. In both research, in cases where di¤erentiation in terms of characteristic spaces
occurs, it has been concluded that di¤erentiation in terms of geographical spaces disappears(agglomeration in the center of the linear city). In comparison with the Irmen and Thisse (1998)
model, it could be interpreted as multidimensional where di¤erentiation of the geographicaldimension does not occur but that of the characteristic dimension does take place.
Nevertheless, the research handle oligopolies (duopolies) and analyze a relatively small
number of …rms.1 In this sense, this work performs an analysis with an increased number of…rms: a monopolistic competition model. In this setting, the present work used the “OTT
Model” monopolistic competition framework, which leads to a linear demand function, as
developed by Ottaviano et al. (2002). (See Appendix A for a brief explanation in this regard.)
Consequently, we have the bene…t of performing a clear comparison with DN, which alsouses a linear demand function. Further, in the monopolistic competition model, there is the
additional advantage of allowing analytical solutions that would be di¢ cult to achieve using
the demand function with constant elasticity in the widely used Dixit and Stiglitz (1977) work.
In compliance with these researches, this work treats the number of …rms as a continuum(as opposed to discrete), and each …rm is considered to be atomic. Therefore, although the
behavior of each …rm does not imply in a direct and strategic in‡uence upon other …rms, thespatial distribution, average price level.
In this two-stage game, all …rms decide their location in the …rst stage and their price in thesecond stage (monopolistic competition). Consequently, the type of location-price equilibria
determined by this two-stage game is analyzed. As a conclusion, based on the limitationof all goods being supplied to the entire linear city (su¢ ciently small transport costs), it
is indicated that only the type characterized by agglomeration in the center can realize alocation equilibrium. This is similar to the result achieved by Anderson and de Palma (1988)
and DN. Here, we can interpete the case of su¢ ciently small transport costs as the case ofsu¢ ciently di¤erentiated products becase these are inversely related.2 As indicated by DN,“Product di¤ erentiation, no matter whether it arises naturally or is chosen strategically, actively
encourages the agglomeration of noncooperative oligopolists” (p. 349). On the other hand,
this is also valid for monopolistically competitive …rms, indicating that if there is a constantdi¤erentiation in the characteristic space, there is no “di¤erentiation” in the geographicalspace. This may be an expression of the highly di¤erentiated and varied …rm agglomeration
phenomenon found in cities.
On the other hand, as another trend concerning spatial competition of homogeneous goods,
1 See Rydell (1967, 1971), Snyder (1971), and Greenhut et al. (1986) for the case of monopolies.
2 Although the limitation of practical parameters is explained later (12), we can state that if transportation
costs T (1) are …xed, the di¤erentiation is su¢ ciently great (c is the inverse index of di¤erentiation).

2

we should also refer to Cournot’s quantity competition model. The pioneering researches
Hamilton et al. (1989) and Anderson and Neven (1991) have included this quantity competitionin the spatial competition model occurring in the linear space à la Hotelling. These works
analyze a two-stage game comprised of an existing market where quantity competition takes
place in each point of the linear space, as well as where each …rm chooses one location and, whilebearing transport costs, chooses the quantity to be supplied to each market. Consequently,
Hamilton et al. (1989) achieved the “principle of minimum di¤erentiation,” where both …rms
agglomerate in the center of the linear space. Anderson and Neven (1991) generalized thenumber of …rms (n …rms), all …rms choose the center of the linear space.3
This resulting agglomeration is also found in the present work. In addition, when considering linear demand, the hypothesis of …rms supplying to the entire linear market is also similar
to this work. The great di¤erence between this work, DN, and other quantity competition
models, and works such as d’Aspremont et al. (1979), is that in the latter, in the case of pricecompetition with homogeneous goods, consumers only purchase goods from one …rm. Hence,in price competition models with homogeneous goods, …rms can e¤ectively obtain a local monopolistic market. However, in the former models, irrespective of where other …rms locate
themselves, the market overlaps and local monopoly is not achieved. This process weakens
the incentive for dispersion; in fact, it indicates that the center of the city, as the point wheretransport costs are most economical, is the most advantageous.
In the main model developed in this study, …rms bearing transport costs are analyzed aspracticing discriminatory pricing. As in DN, the most representative of these pricing policies—
mill and uniform delivered pricing— are considered. We can con…rm that when agglomerationin the center is considered as given, as in DN, discriminatory pricing maximizes …rm pro…t,while mill pricing maximizes consumer surplus. We can also con…rm that while social surplus
is maximized with mill pricing, there are incentives for …rms to choose discriminatory pricing.
Thus, a voluntary choice of mill pricing, and the subsequent social surplus maximizing, is
impossible.
DN indicates that, considering the sequential choice for a pricing strategy, when di¤erentiation is small, the …rst mover choose mill pricing while the second mover select discriminatory
pricing (asymmetric pricing policy). In this process, the …rst mover choose mill pricing and
set it higher, appeasing the reaction from the second mover, thereby enabling the …rst mover
to commit to mill pricing. This result is extremely interesting, but only possible in a duopoly.
In a setting such as in this study, where there are various …rms that cannot place a direct
strategic e¤ect on other …rms, this commitment is impossible and discriminatory pricing must
3 These results are realized under the condition that as goods are supplied to every point in the market,
transportation costs are su¢ ciently low.

3

be applied by …rms.
Further, welfare analysis is performed. In the …rst-best analysis where price and location
can be controlled, prices must be the unit transport costs (marginal cost), and agglomerationin the center must be achieved.
The remainder of this paper is organized as follows: in Section 2, the two-stage game forlocation-price decision is presented; in Section 3, price equilibrium is analyzed; in Section 4,
location equilibrium is analyzed and the main results are presented; in Sections 5 and 6, theissues concerning pricing policies and welfare analysis are analyzed; Section 7 summarizes theresults.

2

The Model

We consider the linear city expressed by the line segment L = [0; 1], where consumers are
uniformly and continuously distributed. The total population of consumers is normalized to
one so that we obtain a density function of f (x) = 1 (x 2 L). They are identical (in tastes
for exogenously given goods and income) except for their locations as described later in the
demand function.
There is a continuum of …rms of which the total mass is n, of which each …rm only supplies
one type of horizontally di¤erentiated products.4 Further, each …rm is treated as in…nitesimallysmall, and its action has no impacts on any other …rm and the entire economy; that is, they
engage in monopolistic competition à la Chamberlin. As in Dixit and J. Stiglitz (1977), the…rms have the same cost conditions and the di¤erentiation among their goods are symmetric;
therefore, they can be e¤ectively distinguished only by their locations. Hence, let the …rm’s
index be expressed by a real number, i 2 [0; n].
We now hypothesize on the location-price two-stage game. In the …rst stage, all …rms
choose their locations simultaneously (location competition), and in the second stage, given
the spatial distribution of …rms, prices are simultaneously chosen (price competition). The
spatial distribution of …rms is expressed in a density function g(x) such thatZ 1g(x)dx = n:0

Figure 1 illustrates an example of spatial distribution.
The demand qi for good i for consumers with the same preferences is taken as in thefollowing linear function.5
4 We regard n as a …xed number. Usually, this is endogenously decided by pro…ts becoming zero by freeentry in the market. Here, the interpretation is that, implicitly, the pro…ts at n …rm’s equilibrium match …xed
costs.
5 Namely, q = maxf0; a(b + cn)pi + cnP g; however, later, it will be assumed as qi > 0 unless otherwisei
speci…ed.

4

g(x)
g(x1)

0

x1

x

1

Figure 1: Example of spatial distribution: The mass of …rms at x1 is given by g(x1 ). The
shaded area is equal to n.

qi = a(b + cn)pi + cnP;(1)where pi is the price of good i. a, b, c are parameters, and when c ! 0, the substitutability is low(the degree of di¤erentiation is high), while when c ! 1, perfect substitution (homogeneous
products) is achieved. Hence, c is the indicator expressing the level of substitutability.
Further, the average price (price index) of all goods is given as
Z n
1
P =
pi di :n0

As part of pro…t-maximizing behavior, …rms under monopolistic competition take the priceindex P as given. Now, this formulation is calculated as in Ottaviano et al. (2002), and a brief
explanation on this can be found in Appendix A.
Each …rm i exists in L and supplies goods to all consumers, earning pro…t. We assume
that …rms bear transport costs (shipping model) and can price discriminate for each consumer(market). Let the distance between a …rm and a consumer be d

0. Then, the transport costs

function is linear to the supply amount, and the unit transport costs are only dependent on
the distance and are expressed as T (d)6 . This function monotonically increases in distance;
further, it is convex and satis…es
T (0) = 0; T 0 (d) > 0; T 00 (d)0:The simplest function satisfying this property is the linear function. If we consider xi 2 L as
the location of …rm i and y 2 L as the location of the consumer, this function becomesT (d) = td = t jxi

yj

(2)

6 We have dropped index i from the function due to the symmetry ot …rms. We will repeat this in pro…tfunctions.

5(t > 0 is the parameter expressing the transport costs per quantity, per distance). This linear
function is assumed in many studies (DN, for example); however, this limitation is not necessary
for the results in this paper.
The marginal cost necessary for the production of goods is constant and standardized to 0.
In this case, the pro…t (xi ; y) earned from y by …rm i located at xi is written as(xi ; y) = [p(xi ; y)T (jxiyj)] q(xi ; y);(3)where p(xi ; y) is the price set by the …rm at y, while q(xi ; y) is the linear demand given in (1).
The pro…t earned from each point is totaled for the entire market L = [0; 1] and expressedin the following, which represents …rm i’s total pro…t.
Z 1(xi ) =
(xi ; y)dy

(4)0Resale between consumers are assumed to be prohibitively costly. In this case, as therelationship among markets is assumed as independent, …rms can set prices so as to maximize
their pro…ts in each distinct point (market). Also, …rms are symmetrical with regard to costconditions; therefore, the …rm’s index i is accordingly abbreviated except where it is confusing
to do so.
With regard to the solution concept, subgame perfection is used: the two-stage game isanalyzed through backward induction. Hence, given the …rm’s spatial distribution, the twostage game’s price equilibrium is calculated in Section 3, and based on this, the location
equilibrium of the …rst stage is analyzed in Section 4.

3

Price Equilibrium

Considering the …rm’s spatial distribution as given from the …rst stage, we search for the priceequilibrium set by each …rm. This spatial distribution is expressed as g(x)Z 1

0 whereg(x)dx = n:0

We assumed that the …rm’s density function at x is shown.
Suppose that the …rm i is located at x (index i is omitted), the pro…t at y given by (3) is maximized by the price p(x; y). As a given price index, the …rst-order condition (@ (x; y)=@p(x; y) =0) is achieved through the following (Figure 2):p(x; y) =a + cnP (y) T (jx yj)+;
2(b + cn)
2

6

(5)

pi
demand curve
marginal revenueEp(x,y)

T(.)

qi

q(x,y)

Figure 2: Price equilibrium: point E shows equilibrium price and quantity supplied.

where P (y) is the price index at y, and is given by the following:
Z 1
1
P (y) =
p(x; y)g(x)dx :
n
0

(6)

Substituting (6) into (5) and summarizing, we have
P (y) =

b + cn1a+D(y) ;
2b + cn
n

where
D(y)

Z 1

T (jx

yj) g(x)dx

(7)

(8)

0

expresses the total transport costs for all …rms to y. In other words, it is the total of transport
costs from (consumers in) y to all …rms, and the greater the relative concentration of …rms at y,
the lower is D(y). For instance, when all …rms are concentrated in point 0 (g(0) = n; g(s) = 0
otherwise), D(0) = 0 and D(1) = n, the value in point 0 close to the …rms will become smaller,
and that in point 1 distant from the …rms will increase. Therefore, in markets where …rms areagglomerated, the price index tends to decrease, and price relatively decreases (pro-competitivee¤ect).
Substituting (7) into (5), price equilibrium p (y) given as spatial distribution is analyticallyachieved as in the following (the superscript asterisk expresses equilibrium):p (x; y) =2a + cD(y) T (jx yj)+:
2(2b + cn)
2(9)Further, from the supply amount q (x; y) in equilibrium obtained in (1), we can con…rm
the following:q (x; y) = (b + cn) [p (x; y)
7

T (jx

yj)] :(10)

Inserting the above in (3), we obtain(x; y) = [p (x; y)T (jx= (b + cn) [p (x; y)yj)] q (x; y)

(11)
2

T (jx

yj)] = (b + cn)

12[q (x; y)] :Hence, the pro…t obtained from each point is proportional to the square of the supply amount.
The total pro…t in equilibrium is expressed in the following from (11):
(x) =

Z 1

(x; y)dy = (b + cn)

Z 1[p (x; y)

T (jx

2

yj)] dy:00Moreover, to guarantee the service of the market for all the points, the greatest value fortransport costs, T (1), is limited by the following:7
T (1) < Ttrade

2a:2b + cn(12)

For the purpose of later analysis, the spatial distribution of full agglomeration is de…ned
here.
De…nition 1 We call the spatial distribution g(s) “full agglomeration”when there exists k 2 Lsuch thatg(x) =

n0

if x = k:otherwise(13)

Otherwise, in all other cases, spatial distribution is called “non-full agglomeration.”In thiscase, the support for g(x) is not singleton, and in L, …rms with positive density exist at least in
two points.8 Henceforth, we …rst consider a non-full agglomeration type of spatial distribution
as an object of analysis, and we consider the case of full agglomeration in subsection 4.2.

4Location Equilibrium

The location equilibrium in the …rst stage in the location game is de…ned as a situation where
there is no incentive for a …rm to change its location, when all other …rms’locations are given.
We refer to such a spatial distribution as the equilibrium spatial distribution. As shown later(in Appendix B), …rms never locate themselves at the edges (x = 0; 1); therefore, all we have
7 In order to serve the market, p (x; y) > T (jxyj) must be satis…ed with regard to all x; y. After adjusting
this condition under (x; y) = (0; 1) as well as g(1) = n; g(s) = 0; s 6= 1, we acquire the conditional equation(12). This condition states even if all other …rms agglomerate at 1, and the …rm locate itself at 0, which is thefarthest location from 1, it serves the most competitive location 1.
8 In this case, D(y) > 0 for all y 2 L can be guaranteed, which simpli…es the analysis. On the other hand,in case of full agglomeration, clearly D(y) = 0 in agglomeration point (k), and D(y) > 0, (y 6= k) for all othercases.

8

to focus on here is the inner solution. Since a …rm’s location, x , is the location equilibrium,
it implies that the following equation is satis…ed given g(x):(x )(x) for 8x 2 L:

(14)

Due to the symmetric properties of …rms, the above equation must be equally satis…ed forall …rms. If there exists a …rm that does not satisfy it, it means that the given g(x) is not
the equilibrium spatial distribution. Each …rm is negligibly small and a¤ects neither the other…rms’pro…ts nor the spatial distribution; thus, the equilibrium spatial distribution, g (x), musthave the following properties:
(x) =
(x)
Here,

if g (x) > 0:
if g (x) = 0> 0 (constant) represents the gross pro…ts earned by each …rm.9 In other words, inan area where …rms exist with positive density, the pro…ts earned by each …rm are the same,
while in an area where no …rms exist, if a …rm located itself there, the …rm would make smaller
pro…ts. From (14), a …rm’s optimal location, x , must satisfy the following conditions:
@
@x

= 0;
x=x

@2
@x2

0:
x=x

We will analyze the properties of the optimal location based on this condition below.

4.1

Some properties of location equilibrium

Let us consider the properties of location equilibrium in detail. We derive the following bypartially di¤erentiating the total pro…ts,

@

(x)
1
=
@x
b + cn
=

where
Q(x)

@
@x

Q(x);

Z 1

(x), with respect to x:

Z x@[q(x; y)] dy +@x0q(x; y)T 0 (jx2Z xyj) dyx

A …rm’s optimal location, x , satis…es

Z 12[q(x; y)] dyxq(x; y)T 0 (jx(15)yj) dy:(16)0Q(x ) = 0 from the …rst-order condition, @(x)=@x =0; thus, the following is satis…ed:
Z xq(x ; y)T 0 (jxyj) dy =0

Z 1

q(x ; y)T 0 (jxyj) dy(17)x

9 At long-run equilibrium with free entry and exit,
must be zero. In this chapter, the …xed costs for each…rm are implicitly assumed to be K = ; then, the excess pro…ts disappear.

9In order to understand what this implies, we now consider a case where the transport costs
are linear as indicated in equation (2). In this case of T 0 (d) = t, the …rst-order condition (17)can be reduced toZ xq(x ; y)dy =Z 1

q(x ; y)dy:x0This implies that a …rm’s optimal location is the point where aggregate supply by the …rm
to the left of that point in the entire market (left-hand side) is equal to that by the …rm to
the right of that point (right-hand side). The marginal increase in pro…t earned by a …rm
making a small leftward movement must be balanced with the marginal decrease in pro…t at
the optimal location. In a case where the transport costs are linearly proportional to distance,
the change in aggregate supply is proportional to the change in distance.10 From (11), pro…ts
are proportional to the square of the quantity supplied, and it is obvious that the marginal
pro…ts are proportional to the quantity supplied. Hence, the marginal increase in pro…t earned
by a …rm making a small leftward movement is aggregate supply to the left of the point in
the entire market, while the marginal decrease in pro…t is aggregate supply to the right of thepoint. At the optimal location, the aggregate supply on the left-hand side must be equal tothat on the right-hand side.
Now let us consider (17) without the assumption that the transport costs are proportional
to distance. The optimal location should be the point at which the weighted quantity supplied
to the left is equal to that supplied to the right, after weighting the degree of increase of
the transport costs in relation with the increase in distance. In a case where such a degree isincremental, T 00 (d) > 0, the greater the distance between a …rm and a market, the sharper is theincrease in the transport costs; thus, the advantages in coming closer to and the disadvantages
in growing apart from a market that is distant from a …rm’s location are greater than therespective advantages and disadvantages at a nearby market. The advantages of approaching
and the disadvantages of distancing from farther markets are greater than those in the nearer
markets. Therefore, …rms must select their locations such that the di¤erences between them
are adjusted. In a case where the transport costs are linearly proportional to distance, when a…rm relocates, the resulting changes in advantages and disadvantages at both a farther market
and a nearer market are the same, and such a relocation will not favor either side. We obtainthe following lemma by elaborating on this optimal location:
Lemma 1 The optimal location, x , uniquely exists in (0; 1) given by (17).
Proof. See Appendix B.
1 0 We readily have @q(x; y)=@x =(b + cn)T 0 (jxyj) =2 if x 7 y from (9) and (10).

10This optimal location is determined solely on the basis of …rms’spatial distribution. Each…rm is negligibly small, and the changes in its location do not a¤ect the spatial distribution;
therefore, the optimal location for all the …rms is the same (xi = x ; for 8i). Hence, it is clearthat the spatial distribution of non-full agglomeration cannot be the equilibrium because when
there are more than two supports of the spatial distribution, there exist …rms that situatethemselves at locations other than x , and they have an incentive to change their locations.
We will address some properties of the optimal dispersion and then present a detailed analysis
on agglomeration in subsection 4.2.
The optimal location, x , depends on the …rms’spatial distribution and has the following
properties (derivation in Appendix C):
Remark 1
0 < x < 1=2
x = 1=2

if and only if

if and only if

Z 1=2

Z 1=2

D(y)T 0 (j1=20

D(y)T 0 (j1=2yj) dy =0

1=2 < x < 1

if and only ifyj) dy >

Z 1=2

Z 1

Z 1

D(y)T 0 (j1=2yj) dy;1=2

D(y)T 0 (j1=2yj) dy;1=2

D(y)T 0 (j1=20yj) dy <Z 1

D(y)T 0 (j1=2yj) dy:1=2

For example, consider the …rst case when the following is satis…ed:
Z 1=2
Z 1
D(y)T 0 (j1=2 yj) dy >D(y)T 0 (j1=2 yj) dy:
0

(18)

1=2

D(y), given by (8), is an inverse indicator of a …rm’s agglomeration and proximity with regard
to the location, y, and indicates that the larger the value of D(y), the smaller is the numberof …rms around the location. (18) indicates that there are relatively more …rms located in theright half of a city, [1=2; 1]

L.

In this manner, that the optimal location depends on the …rms’ distribution comes fromthe substitutability between goods, that is, competition among …rms.

In a case of non-substitutability (i.e., a case where the degree of product di¤erentiation is considerably large),
each …rm practically becomes monopolistic. Thus, in this model, c = 0, and we acquire thefollowing:11
Corollary 1 When c is zero (a case of non-substitutability), the optimal location for all the…rms is the center of the city (1=2).
Proof. See Appendix D.
1 1 In the other extreme case of perfect substitution, c ! 1, we cannot analyze the equilibrium since T
trade
becomes 0 in the equation of (12).

11In other words, …rms determine their locations such that their total transport costs areminimized. This is virtual monopoly. In a case of non-substitutability, …rms determine theirlocations on the basis of the geographical element, i.e., the center of the market, rather thantheir competitors’distributions.

4.2

Full agglomeration

We limit the …rms’spatial distribution, g(x), to the function of (13) in De…nition 1, that is, we
focus on the case where all the …rms fully agglomerate at location k. Assume that the locationis the right half of a city, k 2 [1=2; 1]

L, without loss of generality. From (8), the following

is true:D(y) = nT (jkyj) :From (9), the equilibrium price at location y of a good produced by a …rm at location x is
given by:pf (x; y) =2a + cnT (jk yj) T (jx yj)+:
2(2b + cn)
2(19)Now, the superscript f represents “full agglomeration.”From (11), the pro…ts a …rm earns
at x are given byf(x; y) = (b + cn) pf (x; y)

T (jx

yj)

2

:

(20)

Its total pro…ts are given by
f

(x) =

Z 1

f

(x; y)dy:(21)0

For this full agglomeration to be an equilibrium, all we have to do is to check whether or notthere exists an incentive for a …rm to deviate from k. Here, we have the following proposition.
Proposition 1 Given T (1) < Ttrade , the central agglomeration is the unique equilibrium spatial distribution.
Proof. See Appendix E.
This result is based on the following factors. First, the transport costs are convex withregard to distance. Due to such convexity, the greater the distance between a …rm and amarket, the greater is the drop in the quantity supplied. Considering the above reasons, there
is a strong incentive for …rms to select the center as their location. On the other hand, if thetransport costs were concave with regard to distance, the greater the distance between a …rm
and a market, the lesser the drop in the quantity supplied; thus, it is possible that the central

12location might not be an equilibrium. Note that the proof of Proposition 1 is based on thisassumption of convexity.
The second factor is the uniform distribution of consumers. For instance, in an extreme
case where all consumers exist at location 0, all …rms will also locate themselves at location 0.

5

Mill and Uniform Delivered Pricing

As DN analyze, the other pricing policies that …rms employ in the spatial competition model
include mill and uniform delivered pricing. In this section, we conduct the same analysis, except
with di¤erent pricing policies. Here, we do not consider a case where one …rm (group) and theother …rm (group) employ di¤erent pricing policies, or a case of mixed strategy; our concern
is a case where all the …rms employ the same pricing policy. We will show that the central
agglomeration is the unique equilibrium in the case of mill and uniform delivered pricing forlow transport costs as we have shown in the discriminatory pricing case. We will compare thepro…ts of the pricing policies and brie‡y analyze issues in selecting the policies.

5.1

Central agglomeration under mill and uniform delivered pricing

Mill pricing charges consumers the full transport costs. Thus, the full price for a consumer isthe mill price plus the transport costs from the location of the …rm to that of the consumer.
Firms make the same pro…ts per unit of supply irrespective of where they ship the product. In
other words, the buying price faced by the consumer at y is given by pm (x) + T (jxmyj), wherem

the mill price of a …rm at x is p (x). Then, the demand, q (x; y) (wherein the superscript m
represents the mill pricing, while the index i, which represents …rms, is omitted), is given by :q m (x; y) = a(b + cn) [pm (x) + T (jxyj)] + cnP (y):(22)

Also, the total pro…ts earned by a …rm are given bymm(x) = p (x)Z 1

q m (x; y)dy:

(23)

0

On the other hand, the uniform delivered pricing is a policy where …rms sell their products
to consumers at the same price, regardless of their location. In other words, the demand ofeach consumer, q u , is independent of his location and each …rm’s location, and it is given by
the following:qu = a(b + cn)pu + cnP

13where the uniform delivered price set by a …rm located at x is pu and the total pro…ts earned
by the …rm are given byu(x) = q

u

Z 1

[pu

T (jx

yj)] dy(24)

0

Firms set a price in the second stage so that their total pro…ts are maximized, and taking
that as given, they select their locations in the …rst stage. As in the case of discriminatory
pricing, we assume that the transport costs are low, such that every consumer purchases a
good, i.e.,
T (1) < Ttrade2a2b + cn(25)Note that since Ttrade2 < Ttrade , (25) is more demanding than (12). Here, we have the same
spatial equilibrium as follows.
Proposition 2 Assume that (25) holds. Then, the central agglomeration is the unique spatial
equilibrium under mill and uniform delivered pricing.
Proof. See Appendix F.

5.2

Full prices and pro…ts among three pricing policies

When the central agglomeration is achieved under each pricing policy, we compare the full
price pj (y) for the consumer at y, where j 2 fdisc; mill; unig represents a pricing policy. Here,“disc”represents discriminatory pricing, while “mill”and “uni”are mill and uniform delivered
pricing, respectively. The prices under these pricings are calculated from (19) with k = x = 1=2
and from Appendix F as follows12 :11ya + (b + cn)T
2b + cn2
11pmill (y) =a bT + Ty
2b + cn2
1puni (y) =a + (b + cn)T ;2b + cnpdisc (y) =

where
T

Z 1

1
2

T

0

y

(26)

dy

is the aggregate transport costs from the center. In other words, this value can be interpreted
as the average transport costs from the center for consumers in the city because the mass of
1 2 We can derive these prices from outputs in Appendix F as follows. Solving simultaneous equations (43) andP (y) = pm (1=2) + T (j1=2 yj) with x = 1=2, we easily derive the mill price. Solving simultaneous equations(47) and P = pu with x = 1=2, we can derive the uniform delivered price.

14

consumers is unity. The following relationships hold from (26):puni (y)pdisc (y)puni (y)b + cnT
2b + cnbpmill (y) =T
2b + cn1pmill (y) = T T
2

pdisc (y)

1
2
1
2

T

=

T

y
y

:

y

De…ne y by the solution to13
1
y = T:
2
Clearly, there exist two solutions of y which are symmetric to the center. We rede…neTy 2 (0; 1=2) as the smaller solution of this equation. For example, suppose that the transport
cost is linear, T (d) = td, then y = 1=4. If the transport costs are given by a power function,hi1=
T (d) = td (1), then we easily derive y = 1 ( + 1)
=2, which is decreasing in .

In the limiting case of! 1, we get y ! 0.

We therefore obtain the relation between the prices in the three cases as given by thefollowing expressions:pmill (y) > pdisc (y) > puni (y) if y 2 [0; y) [ (1 y; 1]
pmill (y) = pdisc (y) = puni (y)if y = y; 1 y:pmill (y) < pdisc (y) < puni (y)if y 2 (y; 1 y)
Figure 3 illustrates these prices in the linear case, T (d) = td, with the parameters: (t; a; b; c; n) =(1; 4; 1; 1; 1). This is the same result as DN (see Figure 1 on page 349 of their paper). Figure
4 shows the case of the quadratic transport cost, T (d) = td2 , with the same parameter values.
p
In this case, y = (33)=6 0:21.
From (20) as well as (44) and (48) in Appendix F, the total pro…ts,j, under each pricing

policy are given by
disc

uni

b + cn=2(2b + cn)
mill

=

=

Z 1

2

[A(y)] dy;

(27)

0

b + cn2(2b + cn)

Z 1

A(y)dy

(28)

y

:

(29)

2

0

where
A(y)

a

bT

1
2

Using the Cauchy-Schwarz inequality, we get14
Z 1
Z 1
2
2
[A(y)] dy >A(y)dy00

1 3 Clearly, y always exists inside the city due to the intermediate value theorem.
1 4 Since A(y) is not constant with respect to y, the strict inequality holds in the present case.

15

1.8
pmill(y)

1.7

1.6

disc
p
(y)

1.5

uni
p (y)

1.4

1.3
1.2

y
0

0.25

0.5

0.75

1

discriminatory pricemill priceuniform delivered price

Figure 3: Full prices under three pricing policies with linear transport costs.

1.6
pmill(y)

1.5

p

disc
(y)

1.4
p

uni
(y)

1.3

y
0

0.21

0.5

0.79

1

discriminatory pricemill priceuniform delivered price

Figure 4: Full prices under three pricing policies with quadratic transport costs.

16

Hence, we have
disc

uni

>

=

mill

In other words, the discriminatory pricing is the most desirable for …rms. This result is thesame as that in the duopoly case (DN) as well as that in the monopoly case (Beckmann, 1970).

6

Welfare Analysis

We now consider social welfare by de…ning the social welfare function as the sum of producer
surplus (P S) and consumer surplus (CS) (social surplus, SS = P S + CS). P S and CS aregiven by
PS
CS

Z n

i di
Z n0
:
CSi di0

i represents the pro…ts earned by …rm i given by (4), while the following is consumer surplus

from …rm i’s products, where pi (y) and qi (y) are the price and demand of product i faced byconsumers at y, respectively:15
Z 1

1 ha
0 2 b

CSi

i
pi (y) qi (y)dy:(30)

Figure 5 shows these surpluses in a diagram. Note that the “perceived demand ” for a …rm
given by (1) and the “true demand ” are di¤erent in this monopolistic competition model.

6.1

First-best policy

We now consider the …rst-best policy where a social planner can control the prices of products
and locations. In this subsection, we assume that the transport costs increase monotonicallyand are convex. Obviously, the optimal price must be equal to the marginal cost; therefore,
the price that a …rm located at x should set for products sold at y, po (x; y), is equal to theunit transport costs as follows:po (x; y) = T (jx

yj)

(31)

where the superscript o means “optimum.” Since no …rms earn gross pro…ts in this case, wehave SS = CS. Hence, we will maximize the consumer surplus. Since each …rm is identical,
maximizing CSi is equivalent to maximizing CS. LetTopta(2b + cn)> Ttrade2b(b + cn)(32)

be the threshold of transport costs. We obtain the following proposition by solving this problem:
1 5 See Appendix G for the derivation of (30).

17

pi
a/b

true demand
perceived demand

CSi
p*i
.

T( )

E
PSi
qi

q*i

Figure 5: When the price and consumption are given by point E, then consumer surplus andproducer surplus (P Sii ) which are generated by the products i are represented by theshaded areas above. (The former is shown by the darker area.)

Proposition 3 If T (1) < Topt is satis…ed, then the central agglomeration is the optimal spatial
distribution.
Proof. See Appendix H.
As in the case of the …rms’ optimal distributions, the optimal spatial distribution under
the optimal price is the central agglomeration, where aggregate supply can be maximized (thetotal transport costs are minimized).16

6.2

Comparison among three pricing policies

We now compare the three pricing policies studied in the previous section in terms of welfare.
We focus on the di¤erence resulting from the pricing policies, assuming the spatial distribution
is the central agglomeration in any case. P S j of the pricing policy, j 2 fdisc; mill; unig, isP Sj = nj

due to the symmetric property, so we can easily derive
P S disc > P S mill = P S uni

as in the previous section. Further, we can easily calculate CS and SS, using the equilibrium
prices for the pricing policies obtained in the previous section. CS and SS under pricing policy
j are CS j and SS j , respectively, and we have the following:
1 6 Proposition 3 also holds in the case of the second-best problem, in which the social planner can only control
each …rm’s location, whereas its price is determined by a competitive equilibrium.

18

Proposition 4
CS mill > CS disc > CS uni:
SS mill > SS disc > SS uni
Proof. See Appendix I.
This is the same result as in the monopoly case (Beckmann, 1976). Both CS and P S are
the lowest in the uniform delivered pricing and are therefore the least attractive in terms of thesocial surplus. The discriminatory pricing enjoys the largest P S, while its CS is lower thanthat under the mill pricing. In terms of SS, mill pricing is better than discriminatory pricing.
Therefore, mill pricing is the most desirable pricing policy. In a case where all …rms employ
mill pricing, however, there always exists an incentive for …rms to change their pricing policy
to discriminatory pricing.17

7Concluding Remarks

In this paper, by applying the framework of monopolistic competition to the spatial competition
model in a linear city, we have observed that the central agglomeration is supported as thelocation equilibrium. In relation to the spatial price competition model and the spatial quantity
competition model with homogeneous goods, the …ndings in this paper conform to those of thelatter. The outstanding characteristic of the spatial price competition model is that …rms canenjoy “local monopoly” situations since consumers purchase products from only one of the…rms. Therefore, they can make pro…ts by specializing with their own customers, since they
are located at a distance from their competitors. On the other hand, in the spatial quantity
competition model, the market areas of each …rm always overlap with those of other …rms,
and it is impossible for …rms to acquire local monopoly through dispersion. Hence, …rms must
supply products to the entire city from the center and attempt to save transport costs.
Further, with regard to the roles of the characteristic space and the geographical space,
we obtained the same results as those in the duopolistic model with product di¤erentiation(Anderson and de Palma, 1988, and DN). That is, when goods are su¢ ciently di¤erentiated in
the characteristic space, they no longer try to di¤erentiate in the geographical space. Di¤erentiation is a crucial strategy to avoid a severe price competition. However, it also has a negativee¤ect where …rms move away to a less competitive market. The tendency for dispersion andagglomeration to coexist also conforms to the results of the analysis on homogeneous goods(Tabuchi, 1994, and Irmen and Thisse, 1998), namely, …rms employ the di¤erentiating policy
in a spatial dimension, if at all.
In this paper, welfare losses do not appear in location decisions that are governed by
1 7 See Appendix J for the deviation of a …rm.

19

competitive selection. In the homogeneous-product model, agglomeration results in losses
in terms of transport costs, while in the monopolistic competition model where products aredi¤erentiated, agglomeration has desirable properties that eliminate such losses.
Finally, with regard to the pricing policies, we have observed the properties of mill anduniform delivered pricing along with discriminatory pricing. Discriminatory pricing is desirable
for …rms, but mill pricing is more desirable for consumers and the society. Firms do not selectmill pricing on their own. Hence, enforcement of mill pricing by social planners will enhance
the social surplus.

20References[1] Ago, T., I. Isono, and T. Tabuchi (2006), “Locational disadvantage of the hub," forthcoming in the Annals of Regional Science.
[2] Anderson, S.P. and A. de Palma (1988), “Spatial Price Discrimination with Heterogeneous
Products," Review of Economic Studies 55, 573-592.
[3] Anderson, S.P., A. de Palma, and J.-F. Thisse (1992), Discrete Choice Theory of ProductDi¤ erentiation, Cambridge (Mass.), MIT Press.
[4] Anderson, S. P. and D. J. Neven (1991), “Cournot competition yields spatial agglomeration," International Economic Review 32, 793-808.
[5] d’Aspremont, C., J. J. Gabszewicz, and J.-F. Thisse (1979), “On Hotelling’s ‘Stability In
Competition’," Econometrica 47 (5), 1045-1050.
[6] Beckmann, M. (1970), “Spatial price policies revisited," Bell Journal of Economics 7,
619-630.
[7] Chamberlin, E. (1933), The Theory of Monopolistic Competitionconomic, Harvard University Press.
[8] Cournot, A. (1838), Recherches sur les Principes Mathematiques de la Theorie desRichesses. English edition (ed. N. Bacon) : Researches into the Mathematical Principlesof the Theory of Wealth, New York, Macmillan.
[9] De Fraja, G. and G. Norman (1993), “Product di¤erentiation, pricing policy and equilibrium," Journal of Regional Science 33, 343-363.
[10] Dixit, A. and J. Stiglitz (1977), “Monopolistic competition and optimum product diversity," American Economic Review 67, 297-308.
[11] Downs, A. (1957), An Economic Theory of Democracy, Harper and Row.
[12] Greenhut, M. L., C. C. Mai, and G. Norman (1986) “Impacts on Optimum Location ofDi¤erent Pricing Strategies, Market Structures and Customer Distributions over Space,"
Regional Science and Urban Economics 16, 329-351.
[13] Gupta, B., D. Pal, and J. Sarkar (1997), “Spatial Cournot competition and agglomeration
in a model of location choice," Regional Science and Urban Economics 27, 261-282.

21

[14] 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.
[15] Hotelling, H. (1929), “Stability in competition," Economic Journal 39, 41-57.
[16] Irmen, A. and J.-F. Thisse (1998), “Competition in Multi-characteristics Spaces: Hoteling
Was Almost Right," Journal of Economic Theory 78, 76-102.
[17] Krugman, P. (1991), “Increasing returns and economic geography," Journal of Political
Economy 99, 483-499.
[18] Matsushima, N. (2001), “Cournot competition and spatial agglomeration revisited," Economics Letters 73, 175-177.
[19] Ottaviano, G., T. Tabuchi, and J.-F. Thisse (2002), “Agglomeration and trade revisited,"
International Economic Review 43, 409-436.
[20] Pal, D. (1998), “Does Cournot competition yield spatial agglomeration?," Economics Letters 60, 49-53.
[21] Pal, D. and J. Sarkar (2002), “Spatial competition among multi-store …rms," International
Journal of Industrial Organization 20, 163-190.
[22] Rydell, C. P. (1967), “A Note on a Location Principle: Between the Median and theMode," Journal of Regional Science 7, 185-192.
[23] Rydell, C. P. (1971), “A Note on the Principle of Median Location: Comment," Journalof Regional Science 11, 391-393.
[24] Salop, S.C. (1979), “Monopolistic competition with outside goods," Bell Journal of Economics 10, 141-156.
[25] Shy, O. (1995), Industrial Organization: Theory and Applications, Cambridge (Mass.),
MIT Press.
[26] Snyder, R. (1971), “A Note on the Principle of Median Location," Journal of Regional
Science 11, 395-396.
[27] Tirole, J. (1988), The Theory of Industrial Organization, Cambridge (Mass.), MIT Press.

22

APPENDIX
Appendix A: On “OTT model”
Here, we present a supplementary explanation on the derivation of the linear demand of (1).
The formulation of our model is similar to that of the model in Ottaviano et al. (2002).
Each consumer has the same preference that is expressed by the utility function as follows:
Z n

Z n

Z n

qi2 di2+ qA ;22
00where qi is the quantity of goods i; qA , the quantity of the numéraire good; and > 0;
U (qA ; qi ; i 2 [0; n]) =

qi di

0

(33)

qi di

>

>

0, parameters. ! 0 represents no substitutes (very high degree of di¤erentiation), whereas!

represents the perfect substitutes (homogeneous goods). Each consumer maximizes theutility under the budget constraint:Z npi qi di + qA = M;

(34)

0

where M is an exogenous income18 and pi is the price of good i.
Substituting (34) into (33), the …rst-order condition yields the following inverse demand
function:
pi =

(

Z n

)qi

qi di:(35)

0Integrating this with respect to i yields the demand function
qi =

1
+ (n

1)

pi +

(

) [ + (n1) ]Let the new parameters bea;+ (n 1)Z n
1
P =pi di :
n
0

b

1
+ (n

1)

;

c

(

Z n

pi di:

0) [ + (n1) ];Thus, the linear demand of (1) is obtained.

Appendix B: Proof of Lemma 1
Proof. From (16),Z 1
2 (2b + cn)
Q(x) =[2a + cD(y) (2b + cn) T (jx yj)] T 0 (jxb + cnx
Z x[2a + cD(y) (2b + cn) T (jx yj)] T 0 (jx yj) dyyj) dy0= f1 (x) + f2 (x);

1 8 The OTT model is a general equilibrium model with endogenous income.

23

(36)

where1[T (x) T (12c D(x);

f1 (x)
f2 (x)
where

D(x)

Z 1

x)] f 4a + (2b + cn) [T (x) + T (1x)]g ;(37)(38)

D(y)T 0 (jxyj) dyx

Z x

D(y)T 0 (jxyj) dy:(39)0Consider the properties of these functions. f1 (x) is a decreasing in x since@f1 (x)= [ 2a + (2b + cn) T (x)] T 0 (x) + [ 2a + (2b + cn) T (1@xx)] T 0 (1x) < 0;where the inequality holds because T 0 > 0; (12) yields
T (x); T (1x)

T (1) <

2a:2b + cnMoreover,
1
2

f1

=0

(40)

holds. From these relationships, f1 (x) is a monotonically decreasing in x over the entire range
0x

1, and f1 (x) = 0 if and only if x = 1=2. Clearly,
f1 (0) > 0;

Next, consider

f1 (1) < 0

D(x).

@
D(x)
@x
Z 1
=
D(y)T 00 (jxZ xyj) dyx

D(y)T 00 (jxyj) dy2D(x)T 0 (0) < 0

(41)

0

holds, where the inequality is a result of T 0 > 0; T 00 0; D(x) > 0. Moreover,
Z 1
Z 1
D(0) =
D(y)T 0 (jyj) dy > 0;
D(1) =
D(y)T 0 (j1 yj) dy < 000

are met. From these,@ Q(x)< 0; Q(0) > 0; Q(1) < 0@xhold. Due to the continuity ofcondition(42)

Q(x), x uniquely exists in (0; 1) such that the …rst-orderQ(x ) = 0 is satis…ed. Moreover, from (15) and (41), the second-order condition
@2

(x)
@x2

=

@
Q(x) < 0@x

is globally satis…ed.19
1 9 Therefore, we can conclude that the corner solutions of x24

= 0; 1 do not exist.

Appendix C: Derivation of Remark 1
We utilize the results of Appendix B. Substituting x = 1=2 into (36) and (40) yields
2 (2b + cn)
Qb + cn

1
2

= f1

1
2

+ f2

1
2

sgn

Q

1
2

= sgn

1
2

= f2

=c D

1
2

:

Thus,

holds. From (42) and the continuity ofQQQ
are met, where

1
2

D

Q(x),

1
2
1
2
1
2

< 0 () 0 < x <= 0 () x => 0 ()

1
;
2

1
;
2

1
<x <1
2

Q(x ) = 0.

Appendix D: Proof of Corollary 1
Proof. When c ! 0, from (38),
lim f2 (x) = 0

c!0

holds. Since
4a + (2b + cn) [T (x) + T (1holds from (12), the …rst-order condition
T (x)x)] < 0Q(x) = 0 is satis…ed if and only if
T (1x) = 0from (37). Since T 0 > 0, this equation is satis…ed only at x = 1=2.

Appendix E: Proof of Proposition 1
Proof. Let 1=2 < k 5 1. From the total pro…t (21), we obtain
@

f

(x)
b + cn=[T (1@x xi =k
2 (2b + cn)k)

T (k)] f2a

From (12),2ab [T (k) + T (1k)] > 0:Since T 0 > 0, when 1=2 < k 5 1,
T (1

k)

T (k) < 0:25

b [T (k) + T (1k)]g :

Thus, each …rm has an incentive to move toward the center. Hence, the full agglomeration at
1=2 < k 5 1 cannot be the equilibrium spatial distribution.
Next, let k = 1=2. In this case, we obtain
2 (2b + cn)
b + cn@@x

1[T (x) T (12
cn Df (x);x)] f 4a + (2b + cn) [T (x) + T (1f(x) k=1=2 = f3 (x) + f4 (x);where
f3 (x)
f4 (x)x)]g ;

where
Z 1

f

D (x)

1
2

y

T

x

Z x

T

y

0

T 0 (jx
12yj) dyT 0 (jxyj) dyf3 (x) is the same expression as f1 (x) in (37) in Appendix B, and has the same property.
This implies that f3 (x) is also a monotonically decreasing in x over the interval 0x

1, andf3 (x) = 0 if and only if x = 1=2.
Moreover, f4 (x) is similar to f2 (x) in (38), andZ 1
11= 0; f4 (0) = cn
T yf4
T 0 (jyj) dy > 0;
2
2
0
Z 1
1
f4 (1) = cn
T y
T 0 (j1 yj) dy < 020hold. We also obtain
Z 1@ Df (x)
=
@x

T

y

x

2T

x

1
2

1
2

T 00 (jxT 0 (0)yj) dy

Z x
0

T

y

1
2

T 00 (jxyj) dy0:Then, f4 (x) is a monotonically non-increasing in x. (The equality is satis…ed if and only if
T 00 = 0 and x = 1=2.)
From these results, the pro…t function at k = 1=2 (f(x) k=1=2 ) is maximized if and only

if x = 1=2. Hence, each …rm has no incentive to change its location when all the …rms arelocated at the center. In other words, central agglomeration is the only location equilibrium.

Appendix F: Proof of Proposition 2
Proof. We divide the proof into two cases: (i) mill pricing and (ii) uniform delivered pricing.
(i) Mill pricing case.
26

First, let us consider the price equilibrium in the second-stage game. We temporarily
assume that each …rm serves the entire market; in other words, q m (x; y) > 0 holds for all x andmy. From (23), the …rst-order condition @1
p (x) =
2 (b + cn)m

Z 1=@pm = 0 yields the equilibrium price as follows:[a + cnP (y)(b + cn)T (jxyj)] dy:(43)0Note that the term in the brackets is always positive under the assumption that q m (x; y) > 0
for all x and y.
Next, we will proceed to the optimal location for the …rm in the …rst-stage game. Substituting (43) into (23), we obtain the pro…t function as follows:
m

1
(x) =
4 (b + cn)

Z 1

2[a + cnP (y)(b + cn)T (jx

(44)

0

Partially di¤erentiating it with respect to x yields
Z 1
1@ m (x)=[a + cnP (y) (b + cn)T (jx@x20
From this, we can readily obtainandyj)] dyyj)] dy [T (1x)

T (x)] :@ m= 0;@x x=1=2
1@ m? 0 if x 7 :@x2Hence, the optimal location for the …rm is x = 1=2. This is true for the other …rms as wellbecause they are mutually symmetric and the decision of one …rm does not in‡uence thedecision of another …rm.
Before concluding that central agglomeration is the unique equilibrium, we characterize asu¢ cient condition for positive supply in any market. The most di¢ cult situation under whicha …rm will supply a positive amount of goods is one in which all the other …rms are located at
the endpoint of the city; in such a case, the …rm would be located at the other endpoint andwould be supplying its goods to the market of the agglomerated endpoint. For example, when
all the …rms are located at location 1, even if a deviating …rm is located at 0, it must serve themost distant and competitive market at 1.
Let the mill price of each …rm located at 1 be pm
1 ; then, from (43) and x = 1, the following
equations must hold:
P (y)
pm
1

= pm
1 + T (1=1
2 (b + cn)

y) ;
Z 1
[a + cnP (y)027(b + cn)T (1y)] dy:Solving these simultaneously, we have
Z 1
1
1
m
T (1 y) dy ; P (y) =
p1 =
a b
a
2b + cn
2b
+
cn
0

b

Z 1

T (1

y) dy + T (1

0

y) :
(45)

Suppose that a deviating …rm changes its location to location 0. From (43) and x = 0, thedeviating …rm sets its mill price pm
0 as
Z 1
1
fa + cn [pmy)] (b + cn)T (y)g dypm=
1 + T (10
2 (b + cn) 0
Z 1
1
m
=
a + cnp1b
T (y) dy ;2 (b + cn)
0(46)

where pm
1 is given as in (45). Substituting (45) and (46) into (22), the demand of a consumer
at 1 for the goods of the deviating …rm is given as follows:
Z 1
1b2
T (y) dy + (b + cn) [aq m (0; 1) =2b + cn0Since T 0 (y) > 0 and T 00 (y)

0,

Z 1

(2b + cn) T (1)]T (y) dy > 00holds. Therefore, q m (0; 1) > 0 holds if

a2b + cnT (1)is satis…ed. Under this condition, the central agglomeration is clearly the unique location
equilibrium.
(ii) Uniform delivered pricing case.
First, let us consider the price equilibrium in the second-stage game. Again, we temporarily
assume that each …rm serves the entire market; in other words, q u > 0 holds for all theulocations of all consumers and …rms. From (24), the …rst-order condition @the equilibrium pricepu =1 a + cnP+
2 b + cn

Z 1

T (jx

=@pu = 0 yieldsyj) dy :(47)0

Next, we proceed to the optimal location of a …rm in the …rst-stage game. Substituting(47) into (24), the pro…t function is obtained as follows:
u

(x) =

1
a + cnP
4 (b + cn)(b + cn)

Z 1

2

T (jx

yj) dy(48)0Note that the term in the brackets is always positive under the assumption that q u > 0.
Partially di¤erentiating it with respect to x yields
Z 1@ u (x)1=
a + cnP (b + cn)
T (jx
@x
2
0
28

yj) dy [T (1x)

T (x)] :From this, we can readily obtain

and

@ u
= 0;
@x x=1=2@ u
1? 0 if x 7 :@x2Hence, the optimal location of the …rm is x = 1=2. This is true for the other …rms as wellbecause they are symmetric with each other and the decision of one …rm does not in‡uence
the decision of any other …rm.
Similar to the case of mill pricing, we require the su¢ cient condition that each …rm serves
the entire market. Since P = pu under the uniform delivered pricing, from (47), we can readily
obtain1
a + (b + cn)
P =p =
2b + cnuSince T 0 ( ) > 0 and T 00 ( )

0,

Z 1

T (jx

Z 1

T (jx

yj) dy :0yj) dy > 00holds. Hence, the following condition is met:pu >a2b + cn

q u > 0 always holds if pu > T (1). Therefore, the su¢ cient condition to be obtained is asfollows:a2b + cn

T (1)Under this condition, the central agglomeration is clearly the unique location equilibrium.

Appendix G: On consumer surplus
Here, we show that (30) can be interpreted as consumer surplus resulting from goods i. Appendix A already shows our utility function and the demand system of the OTT model. Those
results can be used mutatis mutandis.
Notice that

= a=b. By substituting the inverse demand function in (35) into the expression
1 ha
2 b

i
p(i; y)in (30), this expression can be rewritten as
1 na
2 b

[

(

o 1
Q] = [(
2)q(i; y)

where
Q

Z n

q(i; y)di:029)q(i; y) + Q] ;(49)

By substituting (34) into (33), the representative consumer’s utility V is rewritten byZ nV =V (i; y)di + M;0where
V (i; y)q(i; y)2Qq(i; y)2p(i; y) :Substituting (35) into the above expression in the square bracket yields
2

Q

=

2
1
[(
2

=

q(i; y)2Q2p(i; y)q(i; y)[()q(i; y)Q])q(i; y) + Q] :By comparing this with (49), we can con…rm that
V (i; y) =

1 ha
2 b

i
p(i; y) q(i; y):Hence, (30) is the sum of (indirect) utilities obtained from goods i. The summation canbe allowed since the utility function is quasi-linear, which implies transferable utility. CS
represents the sum of the utilities of each good. Thus, CS is an appropriate measure for
consumer welfare. Note that this expression does not include income M . We ignore M since
it is exogenous and constant, and hence has no crucial impact on the analysis.

Appendix H: Proof of Proposition 3
Proof. Assuming …rst that the optimal spatial distribution is of non-full agglomeration type,
we shall then show that this is a contradiction.
Substituting (31) into (30), the consumer surplus obtained from the goods of the …rm
located at x is rewritten as
CSi =Note that

Z 1

1
P (y) =
n

1 ha
0 2 b
Z 1

T (jx

o

i
yj) [ap (x; y)g(x)dx0(b + cn)T (jx

1
=
n

Z 1

T (jx

yj) + cnP (y)] dy:yj) g(x)dx=0

The …rst-order condition for the maximization of CSi requires xi to satisfy
2b@CSi= f5 (x) + f6 (x) = 0;
@x

30

1
D(y):
n

whereandf5 (x)[T (x)T (1f6 (x)

bc D(x);x)] f a (2b + cn) + b (b + cn) [T (x) + T (1x)]g ;

D(x) is given by (39). We apply a similar calculation as that used in Appendix B; then,we can show that f5 (x) and f6 (x) are monotonically decreasing functions when (32) holds, and12

f5 (0) > 0;

f5

= 0;

f6 (0) > 0;

f6 (1) < 0:f5 (1) < 0;(50)

Hence, there uniquely exists an optimal location xo 2 [0; 1] that satis…es the …rst-order condition @CSi =@x = 0. Since each …rm is negligibly small, the movement of a single …rm has
no impact on its spatial distribution function, the above calculation is valid for the other…rms as well. This implies that an optimal location is common for all …rms. Thus, a non-full
agglomeration distribution cannot be optimal.
Next, we consider full agglomeration. Without any loss of generality, let us suppose thatthe agglomeration point is k 2 [1=2; 1]. Then, an evaluation of f6 (x) at x = k yields
f6 (k) =n1bcn [T (12Hence, f6 (k) < 0 holds when 1=2 < k
1=2 < k2k)]

2

[T (k)]o:

1. Moreover, from (50), f5 (k) < 0 also holds when

1. Thus, the agglomeration at k 2 (1=2; 1] cannot be optimal since 1=2 < k1 =) @CSi =@x < 0 holds.
Finally, the agglomeration at k = 1=2 (central agglomeration) is considered. A similarcalculation yields
f6

1
2

= 0;
Z 1

@f6 (x)
=
@x

T

y

x

2T

x

1
2

1
2

00

T (jx

yj) dy

Z x
0

T

y

1
2

T 00 (jxyj) dyT 0(0)

0;where the equality holds only when T 00 ( ) = 0 and x = 1=2. Hence, when k = 1=2, the optimal
location is xo = 1=2. This implies that no rearrangements are required for an increase in social
surplus. We can conclude that the central agglomeration is optimal.

31

Appendix I: Consumer Surplus of each pricing policy
A tedious calculation yields
CS disc =CS mill =CS uni =n(b + cn)22

2b (2b + cn)n2

2b (2b + cn)n(b + cn)22

2b (2b + cn)

Z 12[A(y)] dy;

(51)

0

Z 1

2

(b + cn)2 [A(y)] + b2 T

T

0

Z 1

2

1
2

y

dy;

2

A(y)dy:0where A(y) is de…ned in (29).
We can readily establish the following by Cauchy-Schwarz inequality:
(Z
Z 1
2
1
b2 n(3b + 2cn)
1
1
mill
disc
CS
CS
=
dy
T y
T y
2
2
2
2 (2b + cn)
0
0
(Z
)
Z
2
1
1
n(b + cn)22
CS disc CS uni =[A(y)] dy
A(y)dy> 0:2
2b (2b + cn)
00Note that these strict inequalities arise because T (jy

2

dy

)

> 0;

1=2j) and A(y) are not constantwith respect to y.
Furthermore, from (27) and (51), we easily obtain
(Z
2
1
b3 n
1
mill
disc
SS
SS
=
y
dy
T
2
2
2 (2b + cn)
0(Z1n (b + cn) (3b + cn)
2
disc
uni
SS
SS =
[A(y)] dy3
2b (2b + cn)
0

Z 1

y

T

0

Z 1

2

1
2
2

A(y)dy

0

dy
)

)

> 0;

> 0;

where the inequalities hold due to the Cauchy-Schwarz inequality.

Appendix J: Deviation of …rms from mill pricing to discriminatory
pricing
Let us suppose that all the …rms are located at the center. Then, we apply the mill pricing.
From (26), the full price and price index for the consumer at y, pmill (y), and P mill (y) are thesame and are given bypmill (y) = P mill (y) =1a2b + cn12

bT + Ty:

Similar to the manner of the analysis in our model in Section 2, let us suppose that thelocations of the …rms are …xed (central agglomeration) and that only the pricing policy ischanged to discriminatory pricing. Then, from (5), the discriminatory price p~(y) for this …rm
is given by
p~(y) =

1
1
a + cnP mill (y) + T
2 (b + cn)
2
32

1
2

y

:

Let ~ denote the total pro…t for this deviating …rm. Then, we have
Z 1
2
~ = 1
[~
q (y)] dy;b + cn 0
where
q~(y) =12 (2b + cn)

2(b + cn) abT + bcn T

1
2

y

T

is the equilibrium demand of this …rm at y. A tedious calculation yields
)
(Z
Z 1
2
2
1
2
1
b
1
mill
~
T y
=
dy
> 0;
T y
dy
4(b + cn)
2
2
0
0
where

mill

is the pro…t for maintaining mill pricing, which is a result of (28). Thus, each …rm

can earn a higher pro…t by applying the discriminatory pricing instead of the mill pricing. Thissuggests that the case in which all the …rms apply the mill pricing cannot be considered anequilibrium state when the game of simultaneously choosing pricing policies before the pricegame is considered for the …rms.

33

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