CSIS Discussion Paper No. 108

Agglomeration due to the imperfect information revisited

Takaaki Takahashi
Center for Spatial Information Science, University of Tokyo, 5-1-5, Kashiwa-no-ha,
Kashiwa, Chiba 277-8568, Japan

February 2011

Abstract
The purpose of this paper is to characterize all the spatial distribution patterns of retailﬁrms realized when consumers are uncertain about the characteristics of available varieties.
Emphasis is put on the explicit treatment of taste heterogeneity, based on the spatial competition theory. By constructing a two-region model, we show that only two distribution
patterns, i.e., segregation and full agglomeration, may be supported by an equilibrium. The
former is an equilibrium pattern if transport costs are high, and the latter is an equilibrium
pattern if transport costs are low. This closely resembles the main result of new economic
geography (NEG) models.
Keywords: agglomeration; break point; characteristics cost; diﬀerentiated product; Hotellingmodel; imperfect information; pro-competitive eﬀect; segregation; spatial competition; sustain point; transport cost
JEL Classiﬁcation Numbers: L13; R10; R32
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
I have beneﬁted greatly from the comments made by Masahisa Fujita, Ryusuke Ihara, Tomoya Mori, andSe-il Mun. I also appreciate the comments from anonymous referees and seminar participants at various
institutions. This research is partly supported by the Grant in Aid for Research (No. 20330045) by the
Ministry of Education, Science and Culture in Japan.

1Introduction

It has become increasingly widely recognized that economic activities tend to agglomerate
within a limited space (see, among others, Combes et al. (2008) and the literature citedthere). One of the most successful attempts to explain why agglomeration occurs is a series
of works, bracketed together as new economic geography (NEG), originating with Krugman(1991). A key element in the explanation is circular causation brought about by demand–
supply linkages. In typical models such as the standard core–periphery model (Krugman(1991)), the analytically solvable model, also known as the “footloose entrepreneurs model”(Forslid and Ottaviano (2003)), and the linear model (Ottaviano et al. (2002)), the linkages
work as follows. Demand for a ﬁnal product is larger in a region with greater expenditure;
this enables that region to attract more production activities. This is called a “backward
linkage” (Fujita et al. (1999)) or a “market-access eﬀect” (Baldwin et al. (2003)). At thesame time, the cost of living is lower in a region with more ﬁrms, or more varieties; therefore,
the owners of mobile factors and, consequently, production activities, are drawn toward that
region. This is a “forward linkage” or a “cost-of-living eﬀect”. Each of the two linkages cangenerate a snowball eﬀect, or cumulative causation, resulting in agglomeration. The building
block in this explanation is the postulate that a shift of production activities accompanies
that of income. When production is agglomerated in a particular region, all of the ownersof mobile factors migrate into that region and put their factors into the production processthat takes place there. Thus, the income of such factor-owners moves to that region at thesame time.1
Such an explanation, however eﬀective for geographical agglomeration on a large scale,for instance in a global economy or a national economy, does not work well for agglomeration
on a smaller scale, for instance that within a city. Now let us consider the agglomerationof commercial activities within a city, such as a retail concentration in a central business
district and a shopping center represented by a shopping mall at “edge cities”. It is usually
the case that such an agglomeration does not entail the spatial concentration of income. Thisis because not many of the residents in a city engage in the commercial activities there, andmany of them work for other industries, which are often located outside the city. Therefore,
the location of commercial activities has at best a minor eﬀect on the location of consumers
and thus that of income.
One demand–supply linkage that explains the agglomeration of commercial activities
within a city without relying on the co-movement of production and income involves taste
1 A “footloose capital model” shows that if a factor (e.g., capital) is mobile but its owner is not, noagglomeration will occur because both the linkages disappear (see Martin and Rogers (1995) and Baldwin
et al. (2003)).

1

heterogeneity and imperfect information.2 More speciﬁcally, we consider the situation inwhich consumers have diﬀerent tastes with regard to the characteristics of a product but
are uncertain about the characteristics of the varieties sold in each commercial area. In thiscase, they guess the characteristics of the best variety that they will ﬁnd in each area fromthe information on the number of commercial facilities located there: if more shops sell a
product in a particular commercial area, they expect to ﬁnd a more favorable variety upongoing there. Therefore, they prefer visiting a commercial area with a higher concentrationof commercial facilities, ceteris paribus. Thus, a larger number of ﬁrms or a higher levelof production results in a higher level of expenditure, which attracts further production
activities. This produces a backward linkage or a market-access eﬀect and causes the circularcausality.
The observation that taste heterogeneity together with imperfect information leads toagglomeration is not novel. On the contrary, it is widely known among researchers throughthree important pieces of research.3
First, Stahl (1982), analyzing consumers’ search behaviors and the locational choices ofsellers, derived a set of conditions for the location of all ﬁrms in one marketplace being
supported by an equilibrium. However, he abstracted away price competition by assuming
a given price of a diﬀerentiated product. Secondly, Wolinsky (1983) obtained another setof conditions for the geographical concentration of shops selling varieties. Although his
framework is general enough to allow for price competition, he, like Stahl, did not derive
an entire equilibrium conﬁguration of economic geography: all that was shown was that
agglomeration can be an equilibrium outcome.
Finally, Konishi (2005), whose interest was closest to ours, explained the concentrationof retail stores by constructing a model with a two-dimensional geographical space, in whicha given number of stores choose their locations from a set of potential shopping centers andcompete with each other over prices, and consumers decide which shopping center to visit.
The main diﬀerence between his work and ours is in the manipulation of taste heterogeneity,
which underpins the model. Konishi assumes that the utility that a consumer receives
from consumption of a given product is a random variable, which he interprets to meanthat “consumers do not know their exact preferences over commodities”. In this paper, incontrast, the utility itself is a deterministic variable and formulated within the framework ofa spatial competition theory á la Hotelling (1929) and Salop (1979). The characteristics of2 Another is the demand–supply linkage for intermediate products, not for ﬁnal products (see Krugman

and Venables (1995) and Venables (1996), among others). Although agglomeration is generated by a similarintertwinement of the backward linkage (the market-access eﬀect) and the forward linkage (the cost-of-livingeﬀect), these linkages do not necessitate the location choice of factor-owners.
3 For the other related works, see the literature cited in Konishi (2005).

2

the products are the random variables: consumers do not know what kinds of products they
can ﬁnd upon arriving at a shopping center. We believe that the latter approach is more
appropriate for studying the topic, on two grounds. First, we believe that being uncertain
about available characteristics is a much more common situation in our everyday life thanbeing uncertain about our own taste. Secondly, the spatial competition theory is one of
the most powerful tools for analyzing ﬁrms’ behaviors under consumers’ taste heterogeneity,
and there is a rich accumulation of research in that ﬁeld. Our approach can take advantageof this heritage to give an explicit foundation for consumers’ behaviors, whereas Konishi’s
approach puts taste heterogeneity in a black box.
The purpose of this paper is to characterize all the spatial distribution patterns of ﬁrms
that are realized when consumers are uncertain about the characteristics of available varieties. Emphasis is put on the explicit treatment of taste heterogeneity based on the spatialcompetition theory. For this purpose, we construct a model with two regions and a diﬀerentiated product. On the one hand, there are consumers who cannot move across regions.
Without knowing the characteristics of the varieties sold in respective regions, but by forming an expectation, they decide whether to go shopping or not, and, if the answer is yes,
which region to visit. It is necessary for them to pay transport costs to go to a shop in
a foreign region, whereas it does not cost anything to go to a counterpart in their homeregion. On the other hand, there are ﬁrms (sellers or shops) who are free to choose theirlocation, or, more precisely, they freely enter and exit from the industry in each region,
depending on the amount of demand they are to receive. They compete with each otherover prices. We explore an emerging economic geography by formulating the consumers’
and ﬁrms’ behaviors in a three-stage game.
Our simple model yields two equilibrium patterns of economic geography: segregationand full agglomeration. In segregation, all the consumers visit a shop in their home region.
Firms are dispersed over the two regions according to the distribution of consumers. Thispattern corresponds to “dispersion” or the symmetric pattern in the standard NEG models.
In full agglomeration, on the other hand, all the consumers in the economy visit a shopin the same region, i.e., a core, where ﬁrms are concentrated. This pattern corresponds
to “agglomeration” in the NEG models. We derive the conditions for each pattern being
supported by an equilibrium and show that the conditions depend on several parameters,in particular, transport costs. If the transport cost is lower than a critical value, i.e., abreak point, the segregation breaks; if it is lower than another critical value, i.e., a sustain
point, the agglomeration becomes sustainable. This closely resembles the main result of
NEG models. In other words, the main result of NEG models can alternatively be derived
from a demand–supply linkage in a partial equilibrium setting, i.e., a linkage which does not3entail the co-movement of production and income.
The rest of the paper consists of four sections. The next section describes the model.
In section 3, we derive the conditions for an equilibrium by analyzing the ﬁrms’ decisions
on prices to charge, the consumers’ decisions on the region to visit, and the number ofﬁrms. Section 4 shows that only segregation and full agglomeration can be supported as
equilibrium distribution patterns. We obtain the conditions for each of these being anequilibrium pattern and discuss the overall eﬀect of a change in transport costs. Section 5gives the conclusions.

2

Model

In this section, after describing the basic setting, we explain the structure of the game.

2.1

Basic setting

The economy consists of two regions, 1 and 2. In each region, ﬁrms produce a horizontallydiﬀerentiated product, which is formulated as in Salop (1979). Varieties are “located” in a
one-dimensional characteristics space, which is a circle of length 1. Each ﬁrm produces one
variety, and each variety is produced by one ﬁrm. The number of ﬁrms is therefore equal to
the number of varieties in each region. Furthermore, the one-to-one correspondence between
a ﬁrm and a variety implies that each ﬁrm can be identiﬁed by the location of its variety inthe characteristics space. Thus, we say that a ﬁrm is “located” at a particular point on thecircle when it produces a variety at that point.
Our attention is limited to the case where the ﬁrms, and thus the varieties, are locatedequidistantly on the circle in each region. They line up at distances 1/ni from each other inregion i, where ni ∈ N0 , the set of non-negative integers, is the number of ﬁrms in region i(i = 1, 2). Furthermore, without loss of generality, we measure the distances on the circle inthe clockwise direction. The location of the ﬁrst ﬁrm is at a distance of x̄i from the originof the circle in region i (see Figure 1). Firms are numbered in the order of their locationsin the clockwise direction, and the varieties are referred to accordingly. Firm k is therefore
located at xki ≡ x̄i + (k − 1)/ni in region i for k ∈ Ni ≡ {1, 2, · · ·, ni }.
Figure 1: Characteristics space
To produce a variety, each ﬁrm pays a constant marginal cost, which is normalized to 0,
and a ﬁxed cost f . The proﬁt of ﬁrm k in region i is therefore equal toΠki = πik − f

with

4

πik ≡ pki dki ,(1)where πik is its operating proﬁt, pki is the price of its variety, and dki is the demand for it (orthe mass (number) of consumers who buy it).
Moreover, there are L̄ consumers in the economy. They do not move across regions. The
share of consumers in region i is denoted by λi , with λ1 + λ2 = 1 (i = 1, 2). We assume,
without loss of generality, that region 1 is no smaller than region 2: λ1 ≥ 1/2.
Each consumer prefers a certain characteristic.4 In parallel with the terminology usedin the theory of spatial competition, we say that in the characteristics space, a consumer is“located” at the point of her most-preferred characteristic. To keep the analysis simple, weconcentrate on the case where the most-preferred characteristics are distributed uniformly
over the circle. Furthermore, the farther away the consumed variety is located from theconsumer’s most-preferred characteristic, the lower the level of utility she receives. Morespeciﬁcally, the decrease in utility is a quadratic function of the distance between the consumed variety and the consumer: the utility she gets by consuming variety k in region i isgiven by[()2 ]vik = r − pki + c z − xki,(2)where z is the location of the consumer, pki is the price of the variety, and r is the reservation
price of the product: consumers are willing to pay up to this amount for its consumption.
Here, c > 0 is a constant, which we call a characteristics-cost coeﬃcient, and we refer to
)2
(
)2
(
c z − xki and pki + c z − xki as the “characteristics cost” and “subjective price” of thevariety, respectively.5 The reason why we assume a quadratic characteristics cost, in contrast
to the original setting by Salop (1979), who assumes a linear cost, will be explained later.
Finally, in order to get a variety, consumers need to visit the ﬁrm, or the “shop”, that
produces the variety. To visit a shop in a foreign region, they must pay a transport cost,t. To visit a shop in their home region, on the other hand, they do not have to pay any
transport costs. The overall utility of a consumer who lives in region j and buys variety kin region i is thus represented by
Vijk = vik − τij ,(k ∈ Ni ; i = 1, 2; j = 1, 2),(3)where τij = 0 if i = j and τij = t when i 6= j.

4 In our model, each consumer consumes at most one variety.

In the NEG models, on the other hand,consumers have a “love for variety”, so each of them consumes all the varieties, however small the amount ofeach variety is. This diﬀerence is not fundamental. It is known that the aggregate demand of heterogeneous
consumers each of whom consumes only one variety coincides with its counterpart in the NEG models withthe love for variety (see Anderson et al. (1992) and Combes et al. (2008)).
5 In the spatial competition literature, the subjective price is usually called a “delivered price”. However,this paper deals with two spaces, namely a characteristics space and a geographical space. To avoid confusion
between them, we use the term subjective price whenever the characteristics space is concerned.

5

2.2

Structure of the game

We consider the following three-stage game.
In the ﬁrst stage, ﬁrms enter the industry if they will obtain a non-negative proﬁt bydoing so, and exit from it if they have been incurring a loss. At the equilibrium, therefore,
each ﬁrm earns a non-negative proﬁt, and if one additional ﬁrm enters the market, it will
suﬀer a loss. In other words, the pari of equilibrium number of ﬁrms, (n∗1 , n∗2 ), must satisfy
the following equations:6
Πi (n∗1 , n∗2 ) ≥ 0 if n∗i > 0(i = 1, 2)

(4)

and
Π1 (n∗1 + 1, n∗2 ) < 0 andΠ2 (n∗1 , n∗2 + 1) < 0.

(5)Here, the proﬁt of a ﬁrm in region i, deﬁned by (1), is expressed as a function of the numbersof ﬁrms in the two regions. (Because it turns out that ﬁrms in the same region earn an equal
proﬁt, the superscript k is dropped here.) Note that, for the second requirement, we consider
only the entry of a single ﬁrm: we do not take into account the possibility that more than
one ﬁrm sets up at the same time.7
In the second stage, ﬁrms and consumers move simultaneously.
On the one hand, ﬁrms decide the price of a variety so as to maximize their proﬁt, given
the prices charged by their competitors. To keep the analysis tractable, we focus on thesymmetric Nash equilibrium, in which all the ﬁrms in the same region charge the same
price. It then suﬃces to examine the problem of a representative ﬁrm in each region when
all the other ﬁrms in that region charge an equal price. We can write the operating proﬁtof the representative ﬁrm in region i as πi (pi , p̄i ), where pi and p̄i are the prices charged bythat ﬁrm and the competitors in that region, respectively. The equilibrium price, p∗i , is thesolution to max πi (pi , p∗i ).
pi

On the other hand, consumers decide whether to go shopping or to stay at home, andwhich region to visit if they decide to shop.8 In this decision-making process, consumers
know the number of varieties provided in each region, which is observable at the end of theﬁrst stage, as well as the fact that they line up equidistantly. However, they do not know
the locations of the provided varieties. More precisely, for them, the absolute position of theﬁrst ﬁrm, x̄i , is a random variable. Because the location of the available variety nearest to
6 The assumption that ﬁrms do enter the industry if Π (n , n ) = 0 is not important.
2
i 1

We would get

similar results if we assumed otherwise.
7 This is consistent with the concept of the Nash equilibrium.
8 In this paper, we assume that consumers can visit a shop at most once. This considerably simpliﬁes theanalysis, eliminating the possibility that they engage in search. An important agenda for future research isto relax this assumption so as to take into account consumers’ search behaviors.

6

their own location also becomes a random variable, the consumers’ decision-making is basedon their expectation of it. In relation to this point, two simplifying assumptions are imposed.
First, we suppose that the location of the ﬁrst ﬁrm is designated randomly: the probability
that it is at a given point on the circle is uniformly distributed. Secondly, consumers are
assumed to be risk neutral: they choose which region to visit by comparing the levels of
utility computed from the expected location of the nearest variety.
Let A = {0, 1, 2} be a set of consumers’ choices on a shopping trip: 1 and 2 represent
visiting a shop in regions 1 and 2, respectively, and 0 means refraining from going shopping.
We denote by Uaj the expected utility of a consumer in region j when she takes a ∈ A. Then
(3) implies that visiting a shop in region i yields[][ ]Uij = ui − τij where ui ≡ E max vik(i = 1, 2; j = 1, 2).

k∈Ni

(6)

On the other hand, refraining from going shopping yields U0j = 0. The problem of consumers
in region j is then to ﬁnd a ∈ A that maximizes Uaj . We denote the set of the solutions{[]}
by A∗j : A∗j ≡ a : a = arg max Uaj . Furthermore, let us denote the mass of region ja∈A

consumers who choose option a ∈ A as a share of the region j population by saj ∈ [0, 1]∑(s0j + s1j + s2j = 1). We must then have
saj = 1 (j = 1, 2). The share of consumersa∈A∗j

who take option a ∈ A in the entire economy is given by sa ≡ λ1 sa1 + λ2 sa2 (a ∈ A).
In the third and last stage, consumers realize x̄i upon arriving at region i, and decide
whether to buy a variety or not and, if any, which variety to buy. For the sake of simplicity,
we focus on the case where they buy at most 1 unit of the product. In this stage, furthermore,
the transport cost paid in the second stage is already sunk. Therefore, a consumer who islocated at z and visits region i[ ] buys a variety at x∗i (z) = arg max vik if buys no variety

xk
i |k∈Nimax

xk
i |k∈Ni[ k]vi ≥ 0(7)

otherwise.

To derive the demand that each ﬁrm receives in the third stage, let us consider the set ofthe locations of consumers who buy from ﬁrm k in region i, Mik ≡ {z : x∗i (z) = xki }. Since
the characteristics cost is quadratic, the set becomes a segment, which we call the “marketarea” of ﬁrm k or variety k. Since everything is symmetric, we can express the size of themarket area of a ﬁrm as a function of its own price and of that of its competitors, i.e., asmi (pi , p̄i ). Finally, since the circle has a unit length, the demand for each ﬁrm is given bydi (pi , p̄i ) = si L̄mi (pi , p̄i ) .

(8)Note that no mention is made of the ﬁrms’ choices with regard to the variety to produce.
It would be possible to incorporate this into our model by expanding the price game in7the second stage to a location–price game (here, “location” obviously refers to that in thecharacteristics space). However, it has already been established that in a location–pricegame with a circular characteristics space, there is an equilibrium with a given numberof ﬁrms being placed evenly spaced apart along the circle when the characteristics cost isquadratic (see Economides (1988)). We therefore postulate this equilibrium conﬁguration.
Furthermore, it is known that equal spacing is not necessarily supported by an equilibrium
if the characteristics cost is not quadratic. If it is linear, for instance, there is the possibilityof a ﬁrm deviating from the equal spacing conﬁguration to obtain a higher proﬁt. This is
why we assume quadratic characteristics costs in this paper.

3

Equilibrium conditions

As the solution concept, we use a subgame perfect Nash equilibrium. The equilibrium isobtained by the standard method of backward induction. Because the third-stage subgame
has already been solved (see (7)), we begin with the second-stage subgame. First, theﬁrms’ decisions on the prices to charge are discussed. Secondly, we analyze the consumers’
decisions on the shopping trip. Lastly, we turn to the ﬁrms’ decisions on entry and exit in
the ﬁrst stage.

3.1

Price

To begin with, let us examine the ﬁrms’ pricing strategies determined in the second stage.
The setup of this part is the same as that in Salop (1979), except that he assumes a linear
characteristics cost, not a quadratic cost.
As has been mentioned, we focus on a symmetric equilibrium and examine the behaviorof a representative ﬁrm, say ﬁrm k in region i, who charges pi , and all the other ﬁrms in that
region charge the same price, p̄i . From (1) and (8), the ﬁrm’s operating proﬁt is expressedasπi (pi , p̄i ) = si L̄pi mi (pi , p̄i ).

(9)

We begin the analysis with the case of ni ≥ 2.
To derive the size of the market area, let us deﬁne zi (pi , p̄i ) as the location of a marginal
consumer who is located in the interval (xk , xk+1 ) and is indiﬀerent with respect to buying//
from ﬁrm k and buying from ﬁrm k + 1: zi (pi , p̄i ) ≡ ni (p̄i − pi ) (2c) + (xk+1 + xk ) 2.
For variety k, such a marginal consumer pays a subjective price equal to qi (pi , p̄i ) ≡ pi +[]2c zi (pi , p̄i ) − xk . Depending on the relative size of this subjective price compared to the8reservation price, we can identify three types of market area,()1
r − pi 2M
ifm (pi ) ≡ 2


c



]
[ (
)
B
mi (pi , p̄i ) =
ρ
(p̄
),
p̄
−
x
if
m
(p̄
)
≡
2
z
i
i
i
k
i
i







 mC (pi , p̄i ) ≡ 2[zi (pi , p̄i ) − xk ] if

qi (pi , p̄i ) > r,qi (pi , p̄i ) = r,

(10)

qi (pi , p̄i ) < r,as long as zi (pi , p̄i ) ∈ (xk , xk+1 ). Here,[1ρi (p̄i ) ≡ r − c−ni(

r − p̄ic) 12 ]2

gives, as a function of the price charged by the competitors, ﬁrm k’s price for which theassociated subjective price equals the reservation price for the marginal consumer. In otherwords, ρi (p̄i ) is equal to the value of pi that solves qi (pi , p̄i ) = r.
The ﬁrst case in (10) describes the situation in which the market area of ﬁrm k does nottouch those of the neighboring ﬁrms (see the ﬁrst panel in Figure 2). We refer to this type ofmarket area as a monopoly type, or an M type. It follows from (9) that the operating proﬁt[]1
of ﬁrm k is equal to πiM (pi ) ≡ 2si L̄pi (r − pi )/c 2 . Furthermore, the last case describes
the situation in which the competition between ﬁrm k and its neighbors is so intense that
it will lose some of its market area when the competitors slightly lower their prices (see thelast panel in Figure 2). For this type, namely a competitive type, or a C type, the operating[]/
proﬁt of ﬁrm k becomes πiC (pi , p̄i ) ≡ si L̄pi n2i (p̄i − pi ) + c (cni ). Finally, the second caseis a borderline case between the above two cases: if ﬁrm k slightly enhances its price, its
market area becomes M type, whereas if it slightly reduces its price, its market area becomesC type (see the second panel in Figure 2). When the market area of ﬁrm k is this borderline]/[ {}type, or B type, its proﬁt is given by πiB (p̄i ) ≡ si L̄ρi (p̄i ) n2i p̄i − ρi (p̄i ) + c (cni ).

Figure 2: Three types of market area at equilibrium

A necessary condition for ﬁrm k’s proﬁt maximization is given by−∂ ln mi (pi , p̄i )= 1,∂ ln pi(11)

where the left-hand side is the price elasticity of the size of the market area. Furthermore,note that at any symmetric equilibrium, the market area of each ﬁrm must be one of theabove three types.
First, suppose that the equilibrium involves a market area of type M. Solving (11) form (pi ) yields pi = pM ≡ 2r/3. Since the market area is isolated, the optimal price doesM

9

not depend on the price charged by the other ﬁrms. Therefore, all the ﬁrms charge pM at
the equilibrium. The size of each market area and the operating proﬁt of each ﬁrm are given
)− 1
1(
1
3
by mM (pM ) = 2r 2 3c 2 and πiM (pM ) = 4si L̄r 2 c− 2 /3, respectively. At the equilibrium,qi (pi , p̄i ) > r is reduced tor < riM ≡

3c
.
4n2i

(12)

Secondly, suppose that a C-type market area is realized at the equilibrium. Computing
the ﬁxed point pi = p̄i for the proﬁt maximization prescribed by (11), we derive the equi2
librium price for type C: pCi ≡ c/ni . The size of each market area and the operating proﬁt)(C= csi L̄/n3i , respectively. We can rewriteof each ﬁrm are given by 1/ni and πiC pC
i , piqi (pi , p̄i ) < r as
r > riC ≡

5c
.
4n2i

(13)

Note that riM < riC .
Thirdly, and ﬁnally, suppose that a B-type market area is realized at the equilibrium.
2
At a symmetric equilibrium with pi = p̄i , the price is given by pBi ≡ r − c/(4ni ). The( )=size of each market area and the operating proﬁt of each ﬁrm are 1/ni and πiB pB
isi L̄(4n2i r − c)/(4n3i ), respectively.
We can now establish the following result.

Lemma 1. Suppose that ni ≥ 2. There exists a unique symmetric Nash equilibrium suchthati) every ﬁrm charges pM if r < riM ,Cii) every ﬁrm charges pC
i if r > ri , andCMiii) every ﬁrm charges pB
i if ri ≤ r ≤ ri .

The proof is tedious and is relegated to the Appendix.
This is a good place to introduce one simplifying assumption. We conﬁne our analysis
to the case where the reservation price is suﬃciently high that r > riC for any ni ≥ 2: themarket area of each ﬁrm becomes C type at equilibrium whenever there are no fewer than
two ﬁrms in a region. A suﬃcient condition is given by the following inequality:

Assumption 1. (Competitive market)

r > 5c/16.

Let us turn our attention from a regional economy with ni ≥ 2 to that with ni = 1.
Suppose that the only ﬁrm charges pM . If it is suﬃciently high for only part of the circle tobe served by the ﬁrm, pM maximizes its proﬁt. This occurs when the associated subjective
10

price at the point farthest from the location of the ﬁrm exceeds the reservation price, that¯
is, when pM + c/4 > r or r < 3c/4 = riM ¯n =1 . If pM is low enough for the ﬁrm to deliver its
iproduct to the entire market, it is not a proﬁt-maximizing price; with a price higher thanthis, the ﬁrm can still serve the entire market while raising its revenue. In this case, theﬁrm chooses a price such that the associated subjective price at the point farthest from its¯¯
location exactly reaches the reservation price. Such a price turns out to be pB
i ni =1 = r−c/4.
Assumption 1 guarantees that this is positive.
To sum up, the equilibrium market area of each ﬁrm is C type if ni ≥ 2, M type if ni = 1
and r < 3c/4, and, ﬁnally, B type if ni = 1 and r ≥ 3c/4.

Lemma 2. Given the number of ﬁrms in region i, the price, the size of the market area,
and the proﬁt of a ﬁrm in that region at the equilibrium are obtained as follows:

p∗i =




pC

 i







pM



 pB ¯¯






ni =1

, m∗i =




1/ni












π ∗C

 i

( )
, πi∗ =
mM pM
πi∗M









 1/n

 π ∗B
i

i

i

i




n ≥2

 i
if

3.2

(
) 

C

≡ πiC pC
,
p

i
i

(
)
M
M
≡ πi p( )≡ π B pB






.
ni = 1 and r < 3c/4






 n = 1 and r ≥ 3c/4 iShopping trip

In this subsection, we turn to the consumers’ decisions on the shopping trip in the second
stage. At this stage, the number of ﬁrms in each region is known and given to consumers.
From this piece of information, they can correctly infer the equilibrium prices that will be
realized at the end of the second stage. Therefore, the consumers’ decision-making in thesecond stage involves the equilibrium outcome of the ﬁrms’ decision-making in that stage,
which is summarized in Lemma 2, despite the fact that they move simultaneously.
Consider a consumer in region j who visits a shop in region i (i = 1, 2; j = 1, 2). Because
the origin is chosen at random from the uniform distribution and the varieties are located at
distances 1/ni apart from each other along the circle, the cumulative distribution functionof d ≡ min z − xki for a given z becomes Fi (d) = 2ni d. The expected utility of the consumerk∈Ni

is therefore given byui ≡∫ m∗i /2d=0[( ∗ )2 ]( ∗)cmir − pi + cd2 dFi (d) = ni m∗i r − p∗i −
12
11

(14)

(see (2) and (6)).
The right-hand side of the last equality in (14) shows that the utility level depends onfour factors. The ﬁrst is the reservation price. As it increases, the value of the product isenhanced, which raises the utility level: a positive reservation-price eﬀect. The second factor
is the price of the varieties. The higher it is, the lower the utility level is: a negative priceeﬀect. The third factor is the amount of the characteristics cost that a consumer needs to( )2 /
pay. This is captured by c m∗i
12 in (14), which represents the average characteristics-cost
payment. As this payment increases, the utility level declines: a negative characteristicscost-payment eﬀect. The last factor is the probability that a consumer can ﬁnd a variety
worth buying upon arriving at one of the two regions. This is given by ni m∗i because thelength of the circle is normalized to unity. A higher probability is associated with a higher
utility level, ceteris paribus: a positive range-of-varieties eﬀect.
The next result immediately follows from Lemma 2.

Lemma 3. The expected utility, exclusive of the transport cost, is obtained as follows:)/(C2(i = 1, 2).
i) If ni ≥ 2, then u∗i = ui ≡ r − 13c 12ni
3
1/ √
 uM ≡ 4r 2 c− 2 9 3 if r < 3c/4ii) If ni = 1, then u∗i =(i = 1, 2).
 uB ≡ c/6
if r ≥ 3c/4∗
Here, note that uC
i > 0 by Assumption 1. Therefore, ui is always positive.

Because u∗i plays an important role in the following analysis, it is worthwhile examiningMB
how various factors aﬀect each of uC
i , u , and u .

First, let us consider uC
i . When the market area of each ﬁrm is C type, the probabilityof a consumer ﬁnding a variety worth buying is always 1. There is therefore no range-ofvarieties eﬀect.
We begin by explaining that uC
i increases with ni . As ni rises, competition among ﬁrms
becomes ﬁercer, which drives them to charge lower prices. This has a favorable impact onuC
i through the price eﬀect mentioned earlier. Furthermore, as a result of the rise in ni , themarket area of each ﬁrm shrinks. So, consumers now need to pay only a smaller amount of
the characteristics cost on average, which raises uC
i through the characteristics-cost-paymenteﬀect. Because there is no reservation-price eﬀect or range-of-varieties eﬀect, we are left with
only these two positive eﬀects. Such pro-competitive eﬀects are not observed in NEG models
with a Dixit–Stiglitz type preference, and only the ﬁrst eﬀect, namely the price eﬀect, exists
in NEG models with a quasi-linear utility. In addition, because uC
i is concave in ni , thepro-competitive eﬀects are weaker when there are more ﬁrms.
Furthermore, uC
i increases with r and decreases with c. The reason for the former is the
positive reservation-price eﬀect. For a change in c, two forces are at work. First, a higher c12shields ﬁrms more from competition; this gives them a greater monopoly power. This tempts
them to charge a higher price. Secondly, the rise in c boosts the expected characteristics-cost
payment since the size of each market area is ﬁxed. Both of these have an adverse impact
on the utility level through the price eﬀect and the characteristics-cost-payment eﬀect.
Secondly, uM increases with r and decreases with c. On the one hand, as r increases, theprice rises and the market area expands. The utility level rises because of the reservationprice eﬀect and the range-of-varieties eﬀect, but falls because of the price eﬀect and thecharacteristics-cost-payment eﬀect. Because it turns out that the ﬁrst two eﬀects dominate
the last two, the utility level rises. On the other hand, an increase in c directly raises theaverage characteristics-cost payment but indirectly reduces it by narrowing each market
area. Because the direct eﬀect more than oﬀsets the indirect one, the characteristics-costpayment eﬀect is negative. At the same time, as a result of the shrinkage of the market
area, the probability of a consumer ﬁnding a variety worth buying decreases, i.e., a negative
range-of-varieties eﬀect. In the absence of a reservation-price eﬀect and a price eﬀect, theoverall eﬀect of a rise in c is negative.
Thirdly, and lastly, uB increases with c but does not depend on r. Note that when themarket area is B type, its size is constant. There is therefore no range-of-varieties eﬀect.
A rise in c has two eﬀects. First, because it scales down the competitors’ market areas, aﬁrm now needs to charge a lower price to keep its market area B type. Secondly, it directly
raises the characteristics-cost payment. Thus, the utility level increases through the priceeﬀect and decreases through the characteristics-cost-payment eﬀect. It turns out that theformer dominates the latter. A rise in r, on the other hand, boosts the price by the sameamount. The price eﬀect therefore just oﬀsets the reservation-price eﬀect and thus uB does
not change.
We are now ready to examine the consumers’ decision-making.
First of all, suppose that there are some ﬁrms in each region. Since Ujj = u∗j > 0 implies[]
max U1j , U2j > 0, no consumer chooses the option to stay at home, 0 ∈/ A∗j , that is, s0j = 0
for j = 1, 2. Consequently, if U1j > U2j , all the consumers in region j visit a shop in region
1 (A∗j = {1}), and s1j = 1 and s2j = 0. Similarly, if U2j > U1j , all the consumers in region
j visit a shop in region 2 (A∗j = {2}), and s1j = 0 and s2j = 1. Finally, if U1j = U2j , someconsumers visit a shop in region 1 and the others visit a shop in region 2 (A∗j = {1, 2}), ands1j + s2j = 1.
Next, suppose that ﬁrms are concentrated in region i. Then, if Uij ≥ 0 for j 6= i, boththe consumers in region 1 and those in region 2 visit a shop in region i (A∗1 = A∗2 = {i}),and si1 = si2 = 1 and s01 = sj1 = s02 = sj2 = 0 for j 6= i. Otherwise, only the consumers
in region i visit a shop in that region (A∗i = {i} and A∗j = {0}), and sii = s0j = 1 and13

sij = sj1 = sj2 = s0i = 0.
With some manipulation, we can establish the following result.
Lemma 4. i) Suppose that there are some ﬁrms in both regions. Then, the equilibrium
demand shares must satisfy(λ1 , λ2 )if t > u∗1 − u∗2 and t > u∗2 − u∗1





 (1, 0)
if t < u∗1 − u∗2() s1 , s2 =(0, 1)
if t < u∗2 − u∗1 (λ1 + µ2 , λ2 − µ2 ) if t = u∗1 − u∗2 (λ − µ , λ + µ ) if t = u∗ − u∗
1
1
2
1
1
2
for µj ∈ [0, λj ] (j = 1, 2).
ii) Suppose that ﬁrms are concentrated in region i. Then, the equilibrium demand sharesmust satisfy()

s1 , s2 = (1, 0)

if t ≤ u∗i

 (λ , 0) if t > u∗ .
iiHere, µj is the ratio of consumers in region j who visit a shop in the foreign region.
Three observations follow. First, the ﬁve cases in i) are mutually exclusive and exhaustive, as well as the two cases in ii). Secondly, at the equilibrium, there is no possibility ofa “cross trip”: the consumers in region 1 and those in region 2 never visit a shop in theirrespective foreign regions at the same time.9 Finally, if ni rises and the rise involves a change()
in the pattern of s1 , s2 , then si increases or remains constant. For example, as n1 rises,()
the pattern may switch from (λ1 , λ2 ) to λ1 + µ2 , λ2 − µ2 or to (1, 0); in either case, s1()
increases or remains constant. However, it never switches from (λ1 , λ2 ) to λ1 − µ1 , λ2 + µ1
or to (0, 1): s1 never decreases. Consequently, si is a non-decreasing function of ni .

3.3

Number of ﬁrms

In this subsection, we examine the number of ﬁrms in each region, which is determinedthrough free entry at the ﬁrst stage, given the demand shares obtained in Lemma 4. Let us
denote the level of proﬁt corresponding to the operating proﬁt derived above as a functione T (si , ni ) ≡ π ∗T − f for T ∈ {C, M, B} (i = 1, 2).10of si and ni : Π
i

9 To see this, note that U

11> U12 and U22 > U21 because t > 0. If U21 ≥ U11 , therefore, we have

U22 > U12 . Consequently, U21 ≥ U11 and U12 ≥ U22 cannot hold at the same time.
10 π ∗M − f does not actually depend on n . For the sake of convenience, however, we express it as aiie M (si , ni ).
function of not only si , but also of ni , as Π14

To begin with, we ask how many ﬁrms the regional economy can accommodate when thee C (si , ni ) = 0 for
market area of each ﬁrm is C type. To answer the question, let us solve Π1
1()
ni to obtain ni = n̄si3 . Here, n̄ ≡ cL̄/f 3 denotes the potential of the economy, which gives
the number of ﬁrms that earn zero proﬁt when all the consumers in the economy visit a shopin the same region (si = 1), with the integer constraint being disregarded. The potential
increases with total population, the characteristics-cost coeﬃcient, and the inverse of theﬁxed cost, all of which measure the proﬁtability of a market (recall that the characteristicscost coeﬃcient is positively related to the spatial monopoly power of a ﬁrm). Now, let us
1deﬁne nC (si ) as the maximum integer that is no greater than n̄si3 :( 1]
1nC (si ) ∈ n̄si3 − 1, n̄si3 ∩ N0 .

(15)

e C (si , ni ) decreases with ni , it represents the maximum number of ﬁrms region i’s
Since Π()
e C si , nC (si ) ≥ 0
market can accommodate for a given share of demand.11 In other words, Π()
e C (si , ni ) < 0 for any ni ∈ nC (si ), ∞ ∩ N0 .
and Π
Using this result, we can fully characterize the equilibrium number of ﬁrms. For this()()
e C sbC , 2 = 0 and Πe T sbT , 1 = 0, respecpurpose, let us deﬁne sbC and sbT as solutions to Π
tively, for T ∈ {M, B}. These are the demand shares that make the proﬁt of a ﬁrm or ﬁrms
in a region equal to 0. For sbC , we consider the situation in which there are two ﬁrms in that
region, and for sbM and sbB , we consider the situation in which there is only one ﬁrm there.
)
[
]
3
3 (
It turns out that sbC ≡ 8/n̄3 , sbM ≡ (3c) 2 r− 2 / 4n̄3 , and sbB ≡ 4c/ (4r − c)n̄3 .
e C (si , ni ) increases
Now, suppose that the demand share is so high that si ≥ sbC . Since Πe C (si , 2) ≥ 0. Therefore, at least two ﬁrms can, and indeed do,with si , this implies that Π
because of free entry (see (5)), operate in region i. Their market area is therefore C type,which implies that the number of ﬁrms is equal to nC (si ) ≥ 2. Next, suppose that thedemand share is not suﬃciently high, that is, si < sbC . Because the inequality implies thate C (si , 2) < 0, no more than one ﬁrm can operate in region i: either one ﬁrm operates orΠno ﬁrms operate. If, on the one hand, r < 3c/4, the market area of the one ﬁrm must bee M (si , 1) ≥ 0 if and only if si ≥ sbM . In this case, therefore, one ﬁrm
M type. However, Π
operates if si ≥ sbM and no ﬁrms operate otherwise. On the other hand, if r ≥ 3c/4, themarket area of the one ﬁrm must be B type. By similar reasoning, we conclude that one ﬁrm
operates if si ≥ sbB and no ﬁrms operate otherwise. (Note that sbB > 0 as long as r ≥ 3c/4.)
The following lemma summarizes these ﬁndings.

11 ∂ Π
e C (si , ni )‹∂ni < 0 follows from the fact that the entry of a ﬁrm makes the market more competitive.

This corresponds to the “price-cutting eﬀect” described by Konishi (2005). However, as ni rises, si maye C (si , ni ) increases or remains
change. Since si is a non-decreasing function of ni , as mentioned above, Πunchanged as a result of the change in si , which is what Konishi calls the “market-size eﬀect”.

15

Lemma 5. The equilibrium numbers of ﬁrms must satisfy

( )
C


 n si ≥ 2

ni =

1



 0

if si ≥ sbC[)
if si ∈ sbT , sbC
if si < sbT ,where T = M if r < 3c/4 and T = B otherwise (i = 1, 2).

We focus on a non-trivial case where the ﬁxed cost per capita is suﬃciently small for atleast two ﬁrms to operate in a region whenever no fewer than λ2 L̄ consumers visit a shope C (si , ni ) increases with si and decreases within that region, that is, nC (λ2 ) ≥ 2. Because Πe C (λ2 , 2) ≥ 0. A necessary and suﬃcient condition is that λ2 ≥ sbC ,ni , it is equivalent to Π
which we assume.

Assumption 2. (Proﬁtable market)

λ2 ≥ sbC .

1

Note that this assumption implies that n̄ > 16 3 > 2 since λ2 ≤ 1/2.
One comment is worth adding. The above assumption guarantees that more than one
ﬁrm operates in a region at the equilibrium if at least λ2 L̄ consumers visit a shop in that
region. Oﬀ the equilibrium, however, it may be the case that only one ﬁrm operates, evenif λ2 L̄ consumers or more visit a shop in the region. We can show that this ﬁrm can indeed
earn a positive proﬁt.12 In other words,e T (si , 1) > 0 for any si ≥ λ2 ,Π(16)where T = M if r < 3c/4 and T = B otherwise (i = 1, 2).

4

Equilibrium distribution patterns

To recapitulate, we have so far obtained two relationships. Lemma 4 gives the equilibrium
demand share in each region as a function of the equilibrium numbers of ﬁrms since u∗idepends on ni . To articulate this dependence, we hereafter denote u∗i and uC
i as functionsof ni , i.e., as u(ni ) ≡ u∗i and uC (ni ), respectively. Lemma 5 gives the equilibrium number
12 On the one hand, suppose that the ﬁrm’s market area is M type.
3

3

Because Assumption 1 implies that332(r/3c) 2 > 1, 32si (r/3c) 2 > λ2 for any si ≥ λ2 . This and Assumption 2 give 32si (r/3c) 2 > sbC , whiche M (si , 1) > 0. On the other hand, suppose that the market area is B type. Because
is equivalent to Π

this appears only if r ≥ 3c/4, we have r > 3c/8, which is equivalent to 2(4r − c)/c > 1. Consequently,
2si (4r − c)/c > λ2 for any si ≥ λ2 . Hence, we conclude from Assumption 2 that 2si (4r − c)/c > sbC , whiche B (si , 1) > 0.
is equivalent to Π16

of ﬁrms in each region as a function of the equilibrium demand share in that region. To
complete the analysis, we need to “solve” these relationships simultaneously.
Note that Lemma 5 indicates that ni = 0 whenever si = 0 because sbM > 0 and sbB > 0.
Therefore, neither (s1 , s2 ) = (1, 0) nor (s1 , s2 ) = (0, 1) is supported by an equilibrium if
there are some ﬁrms in both regions. Combining the two lemmas, consequently, we canreadily verify the following result.

Lemma 6. At the equilibrium,

((

) (
))
si , sj , ni , nj
must be given by one of the following

patterns (i = 1, 2; j = 1, 2; j 6= i).
(())i) segregation, (λi , λj ), nC (λi ), nC (λj ) , only if()()
t > u nC (λ1 ) − u nC (λ2 ) ; 13

(17)

(
(
))
ii) full agglomeration at region i, (1, 0), nC (1), 0 , only if()
t ≤ u nC (1) ;

(18)

(
(
))
iii) partial agglomeration at region i, (λi , 0), nC (λi ), 0 , only if()
t > u nC (λi ) ;

(19)

(
)
(
)
iv) incomplete agglomeration at region i, (si , sj ) = λi + µj , λj − µj , ni = nC λi + µj , and nC (λ − µ ) if µ ∈ (0, λ − sbC ]
i
j
j
j
nj =
(
]
 1
if µ ∈ λ − sbC , λ − sbT
j

j

j

(
]
for µj ∈ 0, λj − sbT , only if()( )t = u nC (λi + µj ) − u n∗j .

(20)Here, T = M if r < 3c/4 and T = B otherwise.

In what follows, we elucidate the necessary conditions and suﬃcient conditions for eachof the four patterns being supported by an equilibrium; this is done by examining the freeentry conditions, (4) and (5), in addition to whichever one of (17)–(20) is relevant.

13 The necessary condition for segregation is that both (17) and t > u

` C
´
`
´
n (λ2 ) − u nC (λ1 ) are satisﬁed.

However, (17) implies the latter inequality since u(ni ) increases with ni when ni ≥ 2 and nC (·) is increasing
monotonically.

17

4.1

Segregation

First, let us examine segregation.
For free entry conditions, it is obvious that (4) is always satisﬁed as a result of Assumption()
e C λi , nC (λi ) ≥ 0 for i = 1, 2. Thus, what we need to examine is (5). Suppose that one
2: Π()
e C si , nC (λi ) + 1 .
additional ﬁrm enters the industry in region i to earn a proﬁt equal to Π()
e C (si , nC λi ) + 1 < 0 by the deﬁnition of nC (λi ),On the one hand, if si remains at λi , Πand, therefore, (5) is satisﬁed. On the other hand, if si changes, it increases because si isa non-decreasing function of ni (recall the explanation after Lemma 4). The entrant mightthen succeed in obtaining a positive proﬁt. A suﬃcient condition for (5) is, therefore, that(the entry of an additional ﬁrm does not alter si . This is the case if both t > u nC (λi ) +
)
(
)
(
)
(
)
1 − u nC (λj ) and t > u nC (λi ) − u nC (λj ) + 1 hold for both i = 1 and i = 2 (j 6= i).
The suﬃcient condition for this is()()
t > u nC (λ1 ) + 1 − u nC (λ2 ) .

(21)

Since (17) is satisﬁed whenever (21) holds, (21) is a suﬃcient condition for the segregation
being supported by an equilibrium. This means the following: consumers in the smaller
region patronize a shop in their home region not only when their foreign region boasts only
as many ﬁrms as free entry admits, but even when one additional ﬁrm enters the latterregion.
Roughly speaking, the necessary condition, (17), and the suﬃcient condition, (21), indicate that if t is suﬃciently high, the segregation is supported by an equilibrium; otherwise, it“breaks”.14 More precisely, we can obtain two critical levels, or “ break points”, of transport
costs, which can be used to judge whether the segregation breaks or not.

Proposition 1. (Break points for segregation)
There are critical values t BR and t̄BR with tBR < t̄BR such that the segregation breaks ift ≤ tBR , but it does not break if t > t̄BR . The values are given by the following equations:15
( 1( 1 ) 13c)1
 1
tBR ≡ uC n̄λ13 − 1 − uC n̄λ23 =

)2 
2 − (
1
12 n̄2 λ 3
3n̄λ1 − 1
2


(22)
( 1
( 1
)
) 13c
1
1


t̄BR ≡ uC n̄λ13 + 1 − uC n̄λ23 − 1 =( 1)2 − ( 1)2  .
12n̄λ23 − 1n̄λ13 + 1
14 The segregation in our model and the dispersion in the NEG models have the common feature that

there are some ﬁrms in both regions. This is why we use the term “break” here, which is widely used in the
NEG literature to express the situation in which dispersion fails to be supported by a stable equilibrium.
1

1

15 Because of Assumption 2, n̄λ 3 − 1 > 0 and n̄λ 3 − 1 > 0.
1
2

18

The proof is tedious and is relegated to the Appendix.
It is important to note that the reason for the segregation breaking is that the consumers
in the smaller region are tempted to visit a shop in their foreign region if the transport cost is
too low. This is a clue to understanding how various factors aﬀect the break point transport
costs.
First, an inspection of (22) reveals that both tBR and t̄BR increase with λ1 for λ1 ∈[]1/2, 1 − sbC ; that is, the segregation becomes more likely to break as the population distribution becomes more biased. The reason is simple. As λ1 rises, the distribution of ﬁrms
also becomes more biased toward the larger region; this, through the pro-competitive eﬀect,
makes the option for the consumers in the smaller region to visit a shop in their foreign
region relatively more attractive compared to the option of visiting a shop in their homeregion.
The result can be illustrated by a ﬁgure. In Figure 3, the two solid lines depict the utility)( 1levels associated with tBR as functions of λ1 : the u1 curve represents uC n̄λ13 − 1 , and( 1)
the u2 curve represents uC n̄λ23 . Note that the former expression is the lower limit of theutility level that a consumer receives by visiting a shop in region 1, and the latter is theupper limit of the utility level that she receives by visiting a shop in region 2. Because theutility level increases with the number of varieties provided, the u1 curve slopes upward andthe u2 curve slopes downward. Furthermore, tBR is equal to the vertical distance between
the two curves, i.e., u1 − u2 . Therefore, tBR increases with λ1 . We can similarly explaint̄BR as the vertical distance between the two dotted lines that represent the utility levels
associated with it.
Figure 3: Break point
Secondly, the eﬀects of n̄ on tBR and t̄BR are, in general, ambiguous. However, they
are unambiguously negative in the special case where n̄ is suﬃciently large. There are two
reasons why we pay attention to that case. First, in the context of our discussion of thelocation of commercial activities within a city, n̄ corresponds to the number of stores at, say,a central commercial district in a city when retail stores are concentrated there. That number
is usually quite large. Secondly, a large n̄ alleviates the cumbersome eﬀects of the integer
constraint: by focusing on an economy with a large n̄, and, therefore, large nC (si ), one canobtain more or less “smooth” or “well-behaved” results. Now, we can show that when n̄ is
suﬃciently large and λ1 > λ2 , the break point transport costs are inversely related to n̄.16
and K̄ are positive constants, Γ ≡
„ 1
«3
„ 1
«3
«3
„ 1
1
¯ ≡
, Γ̄ ≡ λ1 n̄λ23 − 1− λ23 n̄λ13 + 1 , ∆ ≡ n̄3 n̄λ13 − 1≥ 0, and ∆16 Note that ∂tBR /∂ n̄ = KΓ/∆ and ∂ t̄BR /∂ n̄ = K̄ Γ̄/∆,¯ where K
13−2n̄3 λ1 − λ2 3

„

1
3

«3

1
3

n̄λ1 − 1„ 1«3 „ 1«3n̄λ13 + 1n̄λ23 − 1
≥ 0. Suppose that λ1 > λ2 . Then, as n̄ tends to positive inﬁnity, Γ and Γ̄

19This is explained as follows. As n̄ grows, the numbers of ﬁrms in the respective regions
at segregation tend to increase, which raises the utility levels through the pro-competitiveeﬀect. Recall that this eﬀect is stronger when there are fewer ﬁrms. The utility level is,therefore, improved more in region 2 than in region 1 unless λ1 = λ2 . Consequently, asmaller transport cost suﬃces to discourage consumers in region 2 from visiting a shop in
region 1. Thus, the break points decrease. In Figure 3, the change is represented by arelatively small upward shift of the u1 curve and a relatively large upward shift of the u2
curve; this shortens the vertical distance between the two curves, u1 − u2 , and therefore
reduces tBR .
Finally, the impact of a change in c is also ambiguous. The direct eﬀect is one thatwould be observed if n̄ remained unchanged. As is shown in (22), this is positive. To
understand the reason, remember that a rise in c makes consumers worse oﬀ through theprice eﬀect and the characteristics-cost-payment eﬀect. However, it has a more devastating
impact on consumers who visit a shop in the smaller region than it does on those who visit
a counterpart in the larger region.17 In other words, the larger region becomes relatively
more attractive for consumers. To counteract this eﬀect, the break points increase. We also
have an indirect eﬀect that is transmitted through a change in n̄. As c rises, n̄ increases.
We have seen that this has a negative impact on the break point transport costs as long asn̄ is suﬃciently large. In this case, therefore, the direction of the indirect eﬀect is opposite
to that of the direct eﬀect; consequently, the direction of the overall change depends on therelative magnitudes of the two eﬀects.
These results are summarized in the following proposition.

Proposition 2. (Factors aﬀecting the break points)
The break points rise, and the segregation becomes more likely to break, asi) the population distribution gets more biased, and/orii) the potential of the economy declines (given that it is suﬃciently high and the population
distribution is not symmetric).

4.2

Full agglomeration

We next examine full agglomeration, in which ﬁrms are concentrated in one region, i.e., a“core”, and all the consumers in the economy visit a shop in that region. Let region i be
the core. Region j 6= i is referred to as a “periphery”. The pattern is supported by anapproach negative inﬁnity. Consequently, ∂tBR /∂ n̄ < 0 and ∂ t̄BR /∂ n̄ < 0 for suﬃciently large n̄.
17 Note that d2 uC (n )/dc dn = 13/(6n3 ) > 0.
i
i
i

20

equilibrium, or is “sustainable” only if (18) is satisﬁed. If this condition did not hold, theconsumers in the periphery would stop going shopping.
For free entry conditions, (4) is satisﬁed because si = 1 > sbC by Assumption 2. Next,()()
e C 1, nC (1) + 1 < 0 by
let us examine (5). Since si = 1 for (ni , nj ) = nC (1) + 1, 0 and Π
the deﬁnition of nC (·), the ﬁrst inequality of (5) is satisﬁed. The second inequality, on theother hand, demands that no single ﬁrm be willing to launch a business in the periphery.
To begin with, suppose that()
t < u nC (1) − u(1).

(23)

Then, even if one ﬁrm starts a business in the periphery, the consumers in that region, as
well as those in the core, continue to visit a shop in the core. Therefore, (23) is a suﬃcientcondition for no single ﬁrm to be willing to operate in the periphery. On the other hand, anecessary condition is that()
t ≤ u nC (1) − u(1).

(24)

Suppose that this is not the case. If a ﬁrm began a business in the periphery, consumers
in that region would prefer visiting it to visiting a counterpart in the core. The shop in
the periphery would therefore attract at least λ2 L̄ consumers and obtain a positive proﬁt
because of (16). Hence, (24) is a necessary condition.
Now, note that (23) implies (18) because u(1) > 0. Therefore, (23) is a suﬃcient condition for full agglomeration being sustainable. However, (24) is a necessary condition for
this. These conditions indicate that if t is suﬃciently low, full agglomeration is sustainable;
otherwise, it is not. The following proposition gives two critical levels of transport costs, i.e.,“sustain points”, which are yardsticks in judging the sustainability of full agglomeration.
Proposition 3. (Sustain points for full agglomeration)
There are critical values tSS and t̄SS with tSS < t̄SS such that full agglomeration is sustainable if t ≤ tSS , but it is not sustainable if t ≥ t̄SS . The values are given by the following
equations:tSS≡ u (n̄ − 1) − u(1)

=

t̄SS

≡

=

T
tSS
T ≡ r − Ψ − Φ,

(25)
T
t̄SS
T ≡ r − Ψ − Φ̄,3
1/ √where T = M if r < 3c/4 and T = B otherwise; ΨM ≡ 4r 2 c− 2 9 3; ΨB ≡ c/6; Φ ≡
/[
]
/(
)
13c 12(n̄ − 1)2 ; and Φ̄ ≡ 13c 12n̄2 .
u (n̄) − u(1)The proof is tedious and is relegated to the Appendix.
Figure 4 depicts u (n̄ − 1), u (n̄), and u(1) as functions of r.18 tSS is measured by thevertical distance between the u (n̄ − 1) curve and the u(1) curve, whereas t̄SS is measured
18 It is readily veriﬁed that both the u (n̄ − 1) curve and the u (n̄) curve cut u(1) at r < 3c/4 since n̄ > 16 1
3

from Assumption 2.

21

by that between the u (n̄) curve and the u(1) curve. Full agglomeration is sustainable if t issmaller than the former distance, and only if it is smaller than the latter distance.

Figure 4: Sustain points

Equation (25) enables us to examine how various factors aﬀect the sustain points. The
key observation is that the sustain points give the maximum transport cost that induces
consumers in the periphery to visit a shop in the core when one ﬁrm happens to launch abusiness in the periphery.
First, tSS and t̄SS , or the two vertical distances mentioned above, increase with r. Thisis explained as follows. Suppose that r rises. The utility level of a consumer who visits ashop in the core increases only through the reservation-price eﬀect; it therefore increases by
the same amount as r. Consequently, the slopes of the u(n̄ − 1) curve and the u(n̄) curve are
equal to 45◦ . Now, suppose that a ﬁrm opens a shop in the periphery. The utility level ofa consumer who visits that shop may or may not increase. Two cases can be distinguished.
On the one hand, when the market area of the only ﬁrm is M type, the utility level increases
since uM increases with r. In addition, it can be shown that the amount of the increase issmaller than that of r. Therefore, in the range of r associated with the M type (r < 3c/4),
the u(1) curve slopes upward but is ﬂatter than the 45◦ line. On the other hand, whenthe market area of the only ﬁrm is B type, the utility level remains unchanged since uB is
independent of r. Therefore, the u(1) curve is horizontal in the range of r associated with
the B type (r ≥ 3c/4). These arguments establish that the vertical distance between theu(n̄ − 1) curve and the u(1) curve, and that between the u(n̄) curve and the u(1) curve,
increase with r.
Secondly, tSS and t̄SS increase with n̄. As n̄ grows, more ﬁrms come to operate in thecore. This raises the utility level of a consumer who visits a shop in that region through the
pro-competitive eﬀect. Thus, the u(n̄ − 1) curve and the u(n̄) curve shift upward. In the
periphery, in contrast, nothing is aﬀected by the change in n̄ because only one ﬁrm operates
in that region. Consequently, the u(1) curve does not move. Hence, the vertical distances
increase with n̄.
SS
SSSS
Thirdly, and ﬁnally, both tSS
B and t̄B decrease with c, but its eﬀects on tM and t̄M areambiguous.19 We can explain this with the help of Figure 5. On the one hand, as c rises,
the u(n̄ − 1) curve and the u(n̄) curve shift downward. In the ﬁgure, the u(n̄ − 1) curves
before and after the change are depicted by the solid line and the dotted line, respectively‹‹
c < 0, dt̄SS
M /dc =

19 Tedious computation yields dtSS /dc = (ΨM − 2Φ) (2c), dtSS /dc = −(ΨB + Φ)
M
B
‹
‹
B
(ΨM − 2Φ̄) (2c), and dt̄SS
B /dc = −(Ψ + Φ̄) c < 0 (note that n̄ depends on c).

22(the u(n̄) curves are not shown). On the other hand, the quadratic part of the u(1) curve
rotates clockwise since uM decreases with c, whereas the horizontal part shifts upward sinceuB increases with c. In the ﬁgure, the u(1) curves before and after the change are also
depicted by the solid line and the dotted line, respectively. Hence, for r > 3c/4, the risein c necessarily shortens the vertical distances: it lowers the sustain points. For r ≤ 3c/4,
on the other hand, it may shorten or strengthen the vertical distances because the u(n̄ − 1)
curve and the u(1) curve (and u(n̄) curve and the u(1) curve) move in the same direction:
the direction of the change in the sustain points is ambiguous.

Figure 5: Eﬀect of a change in c on the sustain points

We have established the following proposition.

Proposition 4. (Factors aﬀecting the sustain points)
The sustain points rise, and full agglomeration becomes more likely to be sustainable, asi) the reservation price goes up,ii) the potential of the economy grows, and/oriii) the characteristics-cost coeﬃcient declines when r ≥ 3c/4.

4.3

Partial agglomeration

We also examine partial agglomeration, where ﬁrms are concentrated in region i, i.e., thecore, but attract only consumers living in that region (i = 1, 2), that is, the consumers in
the other region, the periphery, refrain from going shopping. It is straightforward to see, by
reasoning similar to that used in the previous subsections, that the necessary condition for no()
single ﬁrm having an incentive to set up in the periphery is given by t ≤ u nC (λi ) − u(1).
However, this contradicts (19) since u(1) > 0. Hence, there is no possibility of partial
agglomeration being supported by an equilibrium.
Proposition 5. (Partial agglomeration)
Partial agglomeration is not an equilibrium pattern.

4.4

Incomplete agglomeration

Finally, let us consider incomplete agglomeration. All the consumers in region i, i.e., thecore, go to a shop in their home region, and some of the consumers in the periphery visit a23

shop in their home region but the rest go to the counterpart in the core (i = 1, 2). We do
not pay any further attention to this pattern for two reasons.
First, it is “unstable”.20 As the number of ﬁrms in one region diverges slightly from theequilibrium number by accident, the equality in (20) breaks: consumers in the periphery
are now not indiﬀerent between visiting a shop in the core and visiting a counterpart in
the periphery. The economic geography therefore turns to either full agglomeration orsegregation. There is no force that is able to restore it to incomplete agglomeration.
More importantly, equilibrium with incomplete agglomeration does not exist, except fora singular case. Here, one needs to recall that the equilibrium number of ﬁrms must be an
integer. Now, suppose that (n01 , n02 ) satisﬁes (20). Then, both n01 and n02 are integers only
in a singular case: at least either of them is almost always a non-integer, in which case theequilibrium does not exist. In addition, suppose that n01 is not an integer, without loss ofgenerality. Even if n1 is an integer that is quite close to n01 , incomplete agglomeration cannot
brestorere realized. This is because, however close the integer is to n01 , the consumers in the
periphery are not indiﬀerent between visiting a shop in the core and visiting a counterpart
in the periphery, which results in either full agglomeration or segregation.

4.5

Eﬀects of parameter changes on economic geography

We are now ready to study how economic geography depends on various parameters. The
most important parameter is transport cost. In Figure 6, the three thick lines represent theequilibrium patterns, i.e., segregation (s1 = λ1 ), full agglomeration at region 1 (s1 = 1),
and full agglomeration at region 2 (s1 = 0). These show that segregation does not breakfor t > t̄BR , and that full agglomeration is sustainable for t ≤ tSS . The ﬁgure shows aremarkable resemblance to that in conventional NEG models. When the transport cost is
too high, only segregation, corresponding to dispersion in the NEG models, is supported byan equilibrium. When the transport cost is medium, both segregation and full agglomeration
become equilibrium patterns. Finally, when it is suﬃciently low, only full agglomeration is
supported by an equilibrium. Thus, we can reiterate a common narrative in the NEG
literature: as the transport cost declines over a long period of time, the economic geography
shifts from segregation (dispersion) to (full) agglomeration.

Figure 6: Transport costs and economic geography
20 In order to discuss the stability in a rigorous manner, it would be necessary to formulate the behaviors

of ﬁrms and consumers oﬀ the equilibrium as a dynamical system. However, expanding the model in sucha direction does not add much insight, although it involves considerable elaboration. Therefore, we believethat it is better to treat this matter informally.

24

One might wonder if the sustain points are greater than the break points, as the ﬁgure
describes. We can show that this is indeed the case, as long as n̄ is suﬃciently large.

Proposition 6. (Relationship between break points and sustain points)
If n̄ is suﬃciently large, t̄BR < tSS .

The proof is tedious and is relegated to the Appendix.
This result guarantees that as long as n̄ is suﬃciently large, there is an overlap in therange of t where both segregation and full agglomeration are supported by an equilibrium. In
this range, there are three multiple equilibria, and which of them is selected is indeterminate.
This may be determined by a historical accident, in which case hysteresis is present, or by
the expectations of ﬁrms and consumers. As is well known in the NEG literature, theexistence of this overlap has many important implications, especially on economic policies(see Baldwin et al. (2003), for example).
The pitchfork diagram also shows by a thin dotted line the locus of t̄BR when λ1 , measured by the vertical axis, changes. The line, which we refer to as a BR line, slopes upward
since t̄BR increases with λ1 . Thus, segregation is more likely to break when the population
distribution is more biased, which we have already seen in i) of Proposition 2.
It is also straightforward to examine the eﬀects on the economic geography of changes inother parameters. First, as r rises, the sustain points increase (see i of Proposition 4), but
it makes no impact on the break points. Therefore, full agglomeration becomes more likely
to be sustainable and nothing changes for segregation. Secondly, as c declines, the sustain
points increase as long as c is suﬃciently low (see iii of Proposition 4). Because the breakpoints remain unchanged, in this case, the change only makes full agglomeration more likely
to be sustainable. Thirdly, and lastly, the eﬀect of a change in n̄, when n̄ is suﬃciently large,
is a little more complicated. As it rises, the break points decrease and the sustain points
increase. Therefore, the rise makes segregation less likely to break and full agglomeration
more likely to be sustainable. Thus, both patterns become more likely to be an equilibrium
pattern. Consequently, the range of the overlap expands and more indeterminacy is involvedin the economic geography.

5

Concluding remarks

This paper has studied the economic geography realized when consumers are heterogeneousin their tastes and are uncertain about the characteristics of the available varieties in respective commercial areas. We have veriﬁed that such a combination of taste heterogeneity25and imperfect information can be a source of spatial agglomeration. Thus, it can substitutefor the role played by the co-movement of production and income in NEG models. Furthermore, it has been shown that only two distribution patterns, i.e., segregation and full
agglomeration, can be supported by an equilibrium. We have identiﬁed critical values in thetransport cost: if the transport cost is higher than the (upper) break point, the segregation
does not break; if it is lower than the (lower) sustain point, full agglomeration is sustained.
Thus, the results exhibit a remarkable resemblance to those of standard NEG models; thisindicates that the NEG models provide merely one of several, or possibly many, explanations
for agglomeration phenomena.

26

References[1] Anderson, Simon P.; André de Palma; and Jacques-François Thisse (1992), Discrete
Choice Theory of Product Diﬀerentiation, The MIT Press, Cambridge, MA.
[2] Baldwin, Richard; Rikard Forslid; Philippe Martin; Gianmarco Ottaviano; and Frederic
Robert-Nicoud (2003), Economic Geography and Public Policy, Princeton University
Press, Princeton, NJ.
[3] Combes, Pierre-Philippe; Thierry Mayer; and Jacques-François Thisse (2008), Economic
Geography, Princeton University Press, Princeton, NJ.
[4] Economides, Nicholas (1988), “Symmetric equilibrium existence and optimality in differentiated product markets,” Journal of Economic Theory, 47, 178-94.
[5] Forslid, Rikard and Gianmarco Ottaviano (2003), “An analytically solvable coreperiphery model,” Journal of Economic Geography, 3, 229-40.
[6] Fujita, Masahisa; Paul Krugman; and Anthony J. Venables (1999), The Spatial Economy:
Cities, Regions, and International Trade, The MIT Press, Cambridge, MA.
[7] Hotelling, Harold (1929), “Stability in competition,” The Economic Journal, 39, 41-57.
[8] Konishi, Hideo (2005), “Concentration of competing retail stores,” Journal of Urban
Economics, 58, 488-512.
[9] Krugman, Paul (1991), “Increasing returns and economic geography,” Journal of Political Economy, 99, 483-99.
[10] Krugman, Paul and Anthony J. Venables (1995), “Globalization and the inequality ofnations,” Quarterly Journal of Economics, 110, 857-80.
[11] Martin, Philippe and Carol A. Rogers (1995), “Industrial location and public infrastructure,” Journal of International Economics, 39, 335-51.
[12] Ottaviano, Gianmarco; Takatoshi Tabuchi; and Jacques-François Thisse (2002), “Agglomeration and trade revisited,” International Economic Review, 43, 409-36.
[13] Salop, Steven C. (1979), “Monopolistic competition with outside goods,” The Bell
Journal of Economics, 10, 141-56.
[14] Stahl, Konrad (1982), “Diﬀerentiated products, consumer search, and locational
oligopoly,” The Journal of Industrial Economics, 31, 97-113.

27

[15] Venables, Anthony J. (1996), “Equilibrium locations of vertically linked industries,”
International Economic Review, 37, 341-59.
[16] Wolinsky, Asher (1983), “Retail trade concentration due to consumers’ imperfect information,” The Bell Journal of Economics, 14, 275-82.

28

Appendix
Proof of Lemma 1.
First, we show that it is an equilibrium that every ﬁrm charges pM if r < riM . Notethat r < riM guarantees that the market area of each ﬁrm is indeed M type when every
ﬁrm charges pM . Furthermore, suppose that a ﬁrm charges p0 but all the other ﬁrms charge
pM . The market area of the deviating ﬁrm can be no greater than mM (p0 ), whatever p0
is. Therefore, its proﬁt does not exceed πiM (p0 ). However, since pM maximizes πiM (·),( )πiM (p0 ) ≤ πiM pM . Hence, there is no incentive for such a deviation.
C
Next, we show that it is an equilibrium that every ﬁrm charges pC
i if r > ri . Notethat r > riC guarantees that the market area of each ﬁrm is indeed C type when every
ﬁrm charges pC . Furthermore, suppose that all the ﬁrms except ﬁrm k charge pC
i . Byconstruction, ﬁrm k also charges pC
,
given
that
its
marketarea
becomes
C
type.
Thus,
i
0
what we need to show is that the ﬁrm cannot obtain a greater proﬁt by charging p that
does not make its market area C type. First, suppose that ﬁrm k charges such a low pricethat the subjective price of its variety becomes lower than pC
i at the location of ﬁrm k + 1.
0
Because p0 + c/n2i > pC
for
any
p
>
0,
no
such
undercutting
is feasible. Secondly, supposei2that ﬁrm k charges such a high price that its market area becomes M type. If pC
i + c/ni ≥ r,
no such over-cutting is proﬁtable because ﬁrm k has to charge p0 ≥ r. Then, suppose that2pC
i + c/ni < r. Three observations follow: i) The market area of ﬁrm k becomes M typeC
2
CM
if and only if p0 > ρi (pCi ). However, pi + c/ni < r implies that ρi (pi ) > p , as long asMr > riC . Because function πiM (·) is concave and pM gives its maximum, p0 > ρi (pCi ) > p
)
(
)
(
)
(
C
C
C
C
M
C
M
M 0
implies that πi (p ) ≤ πi ρi (pi ) . ii) πi ρi (pi ) = πi ρi (pi ), pi by construction. iii)
)
(
C
C
C
C
C
≤ πiC (pC
πiC ρi (pC
i , pi ) since pi maximizes πi (·, pi ). These three observations implyi ), pi
C C C
M 0that πi (p ) ≤ πi (pi , pi ). Hence, no such over-cutting is proﬁtable. Thirdly, and lastly,
suppose that ﬁrm k charges a price equal to ρi (pCi ), so that its market area becomes B type.
)
(
(
)
C C C
C
C
C
≤
π
(p
ρ
(p
),
p
)
=
π
Because πiB ρi (pCi iii , pi ), such over-cutting is not proﬁtableiiieither.
Furthermore, we show that it is an equilibrium that every ﬁrm charges pBif
i if and only
¯
<
M
C
B
M
¯
ri ≤ r ≤ ri . Suppose that all the ﬁrms except ﬁrm k charge pi . Note that dπi (p)/dp p=pB = 0
>
i
>
M
M
if r = ri . If r ≥ ri , the ﬁrm does not beneﬁt from a slight increase in its price, which will<change its market area to the M type. If r < riM , in contrast, the ﬁrm beneﬁts from such
¯
>
<
¯
=
= C
over-cutting. Similarly, we can see that dπiC (p, pBi )/dp p=pB < 0 if r > ri . Therefore, ifir ≤ riC , the ﬁrm does not become better oﬀ as a result of a price reduction that makes
its market area C type. Otherwise, it does become better oﬀ. Hence, riM ≤ r ≤ riC is anecessary and suﬃcient condition for pB
i constituting an equilibrium.
Now, suppose that r < riM . We have seen that pM is an equilibrium price but pB
i is
M
C
C
not. Furthermore, since r < ri implies that r < ri , pi is not an equilibrium price either.
Consequently, pM is a unique equilibrium price. Similarly, pC is a unique equilibrium priceif r > riC . Lastly, if riM ≤ r ≤ riC , neither the M type nor the C type is realized (recall the
discussion regarding (12) or (13)), and, therefore, pB
i is a unique equilibrium.
Proof of Proposition 1.
1
1
First, suppose that t ≤ t BR . Because nC (λ1 ) > n̄λ13 − 1 and nC (λ2 ) ≤ n̄λ23 (see (15)),
29

(
)
(
)
(
)
(
)
(
)
uC nC (λ1 ) −uC nC (λ2 ) ≥ tBR . Therefore, t ≤ uC nC (λ1 ) −uC nC (λ2 ) = u nC (λ1 ) −()
u nC (λ2 ) since nC (λ1 ) ≥ 2 and nC (λ2 ) ≥ 2 by Assumption 2. Consequently, the necessary
condition, (17), does not hold. Hence, the pattern breaks. Secondly, let us suppose that
1
1
(
)
(
)
t > t̄BR . Because nC (λ1 ) ≤ n̄λ13 and nC (λ2 ) > n̄λ23 − 1, uC nC (λ1 ) + 1 − uC nC (λ2 ) ≤
(
)
(
)
(
)
(
)
t̄BR . Therefore, t > uC nC (λ1 ) + 1 − uC nC (λ2 ) = u nC (λ1 ) + 1 − u nC (λ2 ) and,
consequently, the suﬃcient condition, (21), holds. Hence, the pattern does not break.
Proof of Proposition 3.
()
First, suppose that t ≤ tSS . Because nC (1) > n̄ − 1, u nC (1) − u(1) > tSS . Therefore,()
t < u nC (1) − u(1) and, consequently, the suﬃcient condition, (23), holds. Hence, full
agglomeration is sustainable. Secondly, let us suppose that t ≥ t̄SS . Because nC (1) ≤ n̄,()()
u nC (1) − u(1) ≤ t̄SS . Therefore, t ≥ u nC (1) − u(1) and, consequently, the necessary
condition, (24), does not hold. Hence, full agglomeration is not sustainable.
Proof of Proposition 6.

[
3

BR
If r < 3c/4, lim tSS − t̄BR = lim tSS= r2
M − t̄

1

1 −

4

]

> 0, because r < 3c/4
1r2
9(3c) 2BR
implies that r < 273c/16. Instead, if r ≥ 3c/4, lim tSS − t̄BR = lim tSS= r − c/6 >B − t̄
n̄→∞

n̄→∞

n̄→∞

0. By continuity, therefore, t̄BR < tSS for suﬃciently large n̄.

30

n̄→∞

x
O

｝

｝

1/ni
1/ni

firm 1

firm 2
firm 3｝firm 4

Figure 1. Characteristics space

｝

1/ni

r
pM

firm kfirm k+1

firm k+2(a) Monopoly typer

pCi

firm kzik(pi , pi)

firm k+1

firm k+2(b) Competitive type

r
pBi

｝

1/(2ni )firm kfirm k+1

firm k+2(c) Borderline typeFig. 2. Three types of market area

u1

t

u1

BR

t BR

u2

u2
0.5
Figure 3. Break pointsλ1

u( n )u( n - 1)

t SS t SS

5 c
16

u(1)

uB

uM

r

3c
4
Figure 4. Sustain pointsu( n - 1)

uM’

uM

5 c
16

5 c’
16

3c
4

u( n - 1)’

uB’
uB

u(1)’
u(1)

3 c’
4

Figure 5. Effect of a change in c on the sustain points

r

s1

1
λ1

not
sustainable

sustainable

BR linebreak

not break

0.5

O

t BR

t BR

t SS

t SS

Figure 6. Transport cost and economic geographyt