When is the city sustainable?: A unified model of von Thünen and
MelitzHiroshi Gotoaa

Graduate School of Economics, Kobe University, 2-1, Rokkodai-cho, Nada-ku, Kobe, Hyogo, 657-8501,
Japan

Abstract
We develop a continuous location-space model with heterogeneous ﬁrms and investigate
the conditions under which the spatial concentration of economic activities is sustainable.
Introducing ﬁrm heterogeneity, we can assess the diﬀerence of the number of consumable
goods among locations, and show that it makes cities more attractive and strengthens theirlock-in eﬀect. Furthermore, the present model can analyze how the spatial structure depends
on the transport technology and population size in the economy. Particularly, if transport
costs of agricultural goods are too high, the supply of agricultural goods to the single city
will never be suﬃciently large to maintain the city for any population size.
JEL classiﬁcation: R13, R14, R30
Keywords: Monocentric spatial structure, Lock-in eﬀect, Market potential function, Firmheterogeneity, Love of variety

Email address: hrs.9-12.gt.o.jp.ml@hotmail.co.jp (Hiroshi Goto)
Preprint submitted to Journal of Regional Science

January 10, 2013

1. Introduction
The purpose of this paper is to develop a monopolistic competition model of a spatial
economy with heterogeneous ﬁrms and to ascertain the conditions under which the spatialconcentration of economic activities is sustainable.
Today, perhaps, many economists can agree that the most important feature of economic
geography is the spatial concentration of economic activities in cities. Of course, even now
that many metropolitan areas exist throughout the world, the concentration of economic
activities is not only a present issue but also is a historical phenomenon.1 Therefore, it isnatural that the spatial structure of economic activities has persisted as a major concernof regional scientists and urban economists for a long time. Particularly, urban economists
have pondered two fundamental questions, i.e., Why are cities formed? and Why are thesecities maintained for such a long time?
Urban economists, such as Henderson (1974) have usually emphasized the presence of
externalities related to the agglomeration of economic activities, called the agglomeration
economies, which make cities more attractive and productive places because of denser markets, increased consumption diversity, nearly costless transportation, knowledge spillovers
and other characteristics. However even though the agglomeration economies can explain
why and how cities form in response to economic forces, the fundamental questions haveremained unanswered because agglomeration economies remain as black boxes.
Abdel-Rahman (1998) and Fujita (1988) presented pioneering studies undertaken to explain the agglomeration of economic activities without treating the agglomeration economies
as black boxes. Both introduced Dixit and Stiglitz (1977) type monopolistic competition into
Alonso (1964) and von Thünen (1826) type models featuring a continuous location-space.2
Krugman (1991) developed two symmetric region model in which no region has spatial extent. Since then, various extensions have been developed to analyze the spatial structure of
economic activities. Collectively, they are called the new economic geography (NEG).3
To address the questions of why cities are formed and maintained, the NEG model
employs two key concepts: forward linkage and backward linkage. The availability of alarge variety of consumption goods in a location increases the real income of workers there(forward linkage). Numerous consumers in a location supports numerous specialized ﬁrms
there (back ward linkage).4 Based on this circular causality, a small initial disparity between
1

For example, Rome and Pompeii of the Roman Empire, Moenjo-Daro of the Indus civilization, and Ur
of the Mesopotamian civilization. These examples are evidence that cities existed even in ancient times.
2
A continuous location-space model often used in the urban economics (see Fujita (1989) for details). In
NEG, the continuous location-space type model is classiﬁed into two types. One type resembles that of vonThünen (1826). Each region is represented by a point on the real number line, which we call point location.
This type of model, in NEG, was ﬁrst developed by Fujita (1988). Since then, some extensions have beenmade: Behrens (2007), Fujita and Hamaguchi (2001), Fujita and Krugman (1995), Fujita et al. (1999a),
Fujita and Mori (1996), and Fujita and Mori (1997). The other is like Alonso (1964). In this type model,
regions are expressed discretely, but each region has continuous space and workers commute to a central
business district in each region. See Krugman and Elizondo (1996), Murata and Thisse (2005), Ottaviano
et al (2002) and Tabuchi (1998) for examples.
3
See Baldwin et al. (2003), Combes et al. (2008), Fujita et al. (1999b), and Fujita and Thisse (2002) for
details related to the NEG model.
4
These linkages are related to the Marshallian externalities. See Marshall (1920). The explanation of2

regions tends to expand over time. Furthermore, for whatever reason, once it is formed, thecity tends to be sustainable. This mechanism, which causes the concentration of economic
activities, is plausible and NEG models provide amazing insight into determination of thespatial structure. Even so, however, it must be said that NEG models present some weak
points.
In usual NEG models, ﬁrms in the manufacturing industry are assumed as homogeneousfor simplicity. However, some empirical studies using ﬁrm and plant-level data suggest that
there are the persistent productivity diﬀerences among ﬁrms and that ﬁrm’s export statusis related closely with its productivity (e.g., see Bernard and Jensen (1999) and Roberts
and Tybout (1997)). Therefore, we must treat results obtained using NEG models carefully
because these results might depend strongly on ﬁrm homogeneity.
Another, perhaps more important, weak point is that the number of consumable goods
is the same at every region in usual NEG models. In the real world, an important diﬀerence
between cities and peripheral areas is the number of consumable goods. For example, aperson will have more opportunities to see art exhibitions, movies and plays when living
in a big city because there might be no art museum, movie theater, and playhouse in asmall town or rural area. After all, The Metropolitan Museum is only in New York City!
Consequently, if consumers have a taste for variety, then they are attracted to big cities.5
The usual model in NEG cannot explain this phenomenon. Surely, it also refers to theimportance of accessibility to consumption goods and the love of variety in consumer tastes.
However, actually, the usual type model in NEG does not consider the diﬀerence in thenumber of consumable goods between regions, and the accessibility to consumption goods is
expressed only by high-low goods prices (consumers are attracted to the location in which
goods prices are low). Therefore, it seems necessary to develop a model in which the love ofvariety in consumer tastes has more important role in city formation.6
The present investigation examines why cities are maintained for a long time, developinga NEG model permitting ﬁrm heterogeneity and diﬀerence in the number of consumable
goods between regions. To do so, we produce a continuous location-space model with themonocentric spatial structure based on Fujita and Krugman (1995) (FK model), introducing
the Melitz (2003) type ﬁrm heterogeneity. The basic structure of the present model is thefollowing: Fixed and variable transport costs are imposed on each ﬁrm to transport its

forward and backward linkages presented here is based on Fujita and Krugman (1995).
5
Since the work of Dixit and Stiglitz (1977), the concept of love of variety in consumer’s tastes has served
a central role in the theory of monopolistic competition. Usually, the theory of monopolistic competition
assumes that each consumer has a taste for variety and therefore has an incentive to consume goods of manykinds in as well-balanced a way as possible: This is the love of variety. This in turn gives each ﬁrm an
incentive to diﬀerentiate their products from other. The demand for products of each ﬁrm becomes small
when the market is crowded with competitors. Therefore, consumer’s tastes for variety has an inﬂuence on
the market outcome. Benassy (1996) produced a simple model to analyze the role of consumer taste forvariety in the theory of monopolistic competition.
6
Of course, the love of variety in consumer tastes plays a key role in the NEG model of the usual type,too. A reason that each ﬁrm can specialize in production of particular product is caused by the love ofvariety. Therefore, the love of variety closely related to the backward linkage in the usual type model ofNEG. As explained later, the present model expands the usual type model to a model in which the love ofvariety also bears a relation with the forward linkage.

3

products, and the variable transport costs increase with respect to the transport distance.
Furthermore, the productivities of ﬁrms mutually diﬀer. Consequently, less-productive ﬁrms
cannot supply products to locations further away from the production location. In addition,
because the selection of heterogeneous ﬁrms is weak in the large market, ﬁrms have an
incentive to locate in cities. Distant locations are supplied with goods of few kinds. Becauseof the love of variety, cities are more attractive for consumers and lead an inﬂow of them tocities. However, because the inﬂow of consumers increases the market size of cities, cities
will have goods of more kinds. This circular causality makes cities sustainable.
Under this structure, we investigate the conditions under which the monocetric spatial
structure is sustainable. We will present lock-in eﬀects of two types, which play importantroles in making the monocentric spatial conﬁguration sustainable. Furthermore, we will
demonstrate that the spatial structure depends strongly on the transport technology andpopulation size in the economy. The monocentric spatial conﬁguration will never be sustainable when the population size is suﬃciently small because the supply of agricultural goods
is too small to maintain the monocentric spatial conﬁguration. This phenomenon does notappear in the original FK model. Particularly, if transport costs imposed on agricultural
goods are too high, then the supply of the agricultural goods to the single city is always
insuﬃcient to maintain it.
The remainder of this paper is organized as follows. In Section 2, we present the continuous location-space model with the monocentric spatial structure and describe the production
and supply side of the model. In Section 3, we determine the monocentric equilibrium. Then
we deﬁne the (market) potential function and investigate the sustainability of the monocentric spatial conﬁguration in Section 4. The last section presents both conclusions anddiscussions of the salient results and implications.
2. The Model
We consider an economy represented as a boundless, one-dimensional location space,
X ≡ R. The land density at each x ∈ X is set to one, and the quality of land is homogeneous
everywhere.7
The economy has two sectors, the manufacturing sector and the agricultural sector, which
are designated respectively as the M -sector and A-sector. The M -sector is characterizedby Chamberlinian monopolistic competition. Firms in the M -sector produce horizontallydiﬀerentiated goods (M -goods) using increasing returns technology, with labor only. All
production of M -goods takes place in the city located at c ∈ X, and M -goods are exported
from the city to each location of the agricultural hinterland. Because no M -goods areproduced at x ̸= c, without loss of generality, we can label locations such that the city is
located at the origin of X. The A-sector is characterized by perfect competition. It produces
a homogeneous agricultural good (A-good) at each x ∈ X of the agricultural hinterland, using
constant returns technology with labor and land. Production of the A-good extends among
the agricultural hinterland until a farmer can pay land rent, and A-good is exported fromeach x ∈ X to the city.
7

Description of the present model is based on Fujita and Krugman (1995) except the production structureof the manufacturing goods.

4There is a continuum of homogeneous workers with a given size L in the economy. Eachworker is endowed with one unit of labor, and each can freely choose any location for residence. However, workers can supply labor only to their location of residence. The consumersof the economy consist of workers and landlords. All landlords are attached to their land.
They consume the revenue received from land at their location.
2.1. Preference and Demand
All consumers have the common preference represented by the following utility functionu = M αM AαA ,αM , αA > 0,

αM + αA = 1,

(1)where A denotes the amount of A-good. M is the quantity index of M -goods deﬁned as
[∫

]1/ρ

M≡

ρ

q(γ) dγ,

0 < ρ < 1,

σ≡

γ∈Γ

1
> 1,
1−ρ

where q(γ) is the consumption of each variety γ of M -goods. Γ is the set of available varieties
and the measure of the set Γ represents the number of available M -goods. σ is the elasticity
of substitution between varieties. Let pA (x) and pM (γ, x) be the prices for A-good and eachvariety γ of M -goods at x ∈ X, respectively. Moreover, let Y (x) be the income at x ∈ X.
Then, from the usual utility maximization problem with (1), we can obtain demand functions
for each variety γ and the A-good at x ∈ X as follows:q(x, γ) = αM Y (x)pM (x, γ)−σ P (x)σ−1 ,αA Y (x)A(x) =.
pA (x)γ ∈ Γ,(2)(3)

Therein, P (x) is the price index of M -goods at x ∈ X deﬁned as[∫
P (x) ≡γ∈Γ ∗ (x)pM (x, γ)1−σ dγ

1
] 1−σ

,

(4)

where Γ ∗ (x) ⊂ Γ is the subset of varieties that are supplied to location x.
2.2. Production
2.2.1. Production of A-good
We assume that the transport costs of A-good take Samuelson’s iceberg form: if one unitof A-good is shipped over a distance d, only e−τA d units actually arrive. Let the price ofA-good at the city be normalized such that pA (c) = 1. Then, for all excess A-goods produced
at x ∈ X transported to the city, the A-good price curve must hold that pA (x) = e−τA |x| .
One unit of A-good is produced using aA units of labor and one unit of land. Therefore, by
the zero proﬁt condition in the A-sector at each x ∈ X, πA (x) = e−τA |x| −w(x)aA −R(x) = 0,
the land rent R(x) at x ∈ X is obtainable as below:
R(x) = e−τA |x| − aA w(x),5x ∈ X.

(5)

Therein, w(x) represents the wage rate at x ∈ X. R is called the bid rent function which
gives the maximum rent a farmer can oﬀer and is represented as a downward-sloping curvewith respect to the distance from the city. If R (x) < 0, then no farmer can produce the
A-goods at x ∈ X. On the other hand, if R (x) > 0, then some farmers can produce A-goods
at x ∈ X. Consequently, land rents will be zero at fringe location h. From (5), therefore, it
must be true thate−τA |h|w(h) =.
(6)
aA
2.2.2. Production of M -goods
Turning to the production of M -goods, variety γ is produced by a ﬁrm indexed by γ
under increasing-returns technology, using labor only, and each ﬁrm behaves as monopolist.
To enter the market, each ﬁrm must use fE units of labor as a ﬁxed input.8 The levels of
marginal labor input, aM , diﬀer among ﬁrms. G is a distribution function of aM supportedon (0, āM ] ⊂ R. All ﬁrms know the shape of the distribution function G whether or notthey enter the market, but they cannot know the level of their own marginal labor input
until they enter the market. After ﬁrms enter the market, they draw their level of marginallabor input from G. Then they decide the prices for their own products. A ﬁrm entering themarket, must use f (x) units of labor as ﬁxed input to supply any products of the M -goods
for each location x ∈ X. If a ﬁrm that enters the market is attached to a large marginal labor
input (i.e., low productivity), then it might be not able to cover the ﬁxed cost. Therefore,
a threshold a∗M (x) exists and if a ﬁrm’s marginal labor input is higher than this threshold,a ﬁrm would not supply product for location x. Hereafter, for simplicity, we assume thatf (x) = f for any x ∈ X.9
Like A-good, the iceberg form of transport costs to transport products of M -goods isassumed: i.e., if one unit of any products of M -goods is transported from a location s ∈ X
to a location x ∈ X, only e−τM |x−s| units of them actually arrive. Therefor, each ﬁrm locatedat s must produce eτM |x−s| units of its product to supply one unit of it for a location x.
Since we assume that all production of the M -sector takes place in the city which islocated at the origin, we can express the operating proﬁt of ﬁrm γ attached to marginallabor input aM as∫hπ(γ, aM ) = π(c, γ, aM ) +−hπ(x, γ, aM )dx,

(7)

where π(c, γ, aM ) and π(x, γ, aM ) respectively represent operating proﬁts earned by supplying
the product to the city and to the location x ∈ X.10 They are deﬁned as follows
π(c, γ, aM ) ≡ max [pM (c, γ)q(c, γ) − w(c) (f + aM q(c, γ)) , 0] ,[() ]π(x, γ, aM ) ≡ max pM (x, γ)q(x, γ) − w(c) f + aM eτM |x| q(x, γ) , 0 , x ̸= c,

8

fE is often interpreted as the initial investment, product development, and production start-up costs.
f (x) is interpreted, for example, as a research cost for the market of location x ∈ X. In this context,
the assumption of f (x) = f for any x ∈ X, implies that costs of research for the market of each location
do not depend on the distance from the city.
10
The city is distinguished from location 0 ∈ X of the agricultural hinterland.
96where w(c) is the wage rate at the city. If ﬁrm γ produces an M -product to x ∈ X, it choosesa price pM (x, γ) to maximize its proﬁt at the Chamberlinian equilibrium. Consequently, by
the equality of the marginal revenue and marginal cost, the pricing rules for the ﬁrm can beobtained aspM (x, γ) = pM (x, aM ) ≡w(c)aM eτM |x|,ρx ∈ X,(8)

which represents the familiar result that each monopolistic ﬁrm will charge its price at amarkup over the marginal cost. The presence of manufacturing trade costs means that theprices of M -goods increase as we move to locations further away from the city.
From (8), any ﬁrm having the same level of marginal labor input would oﬀer equal prices
for their products. Therefore, we can use aM , instead of the index γ, to characterize thevarieties in the M -sector. Note that any entering ﬁrm that draws a level of marginal labor
input aM (> a∗M (x)) will never supply its product to location x ∈ X because they cannot
cover ﬁxed costs. Thus, let us deﬁne the conditional distribution of aM as shown bellow.11{ ′G (aM (x))
if 0 < aM ≤ a∗M (x),∗ (x))G(aMµ(x, aM ) =(9)0
if a∗M (x) < aM ≤ āM .
We then rewrite the price index (4) as[∫
1−σ

P (x) =
aM ∈(0,āM ]pM (x, aM )

1
] 1−σ

n(x)µ(x, aM )daM,x ∈ X,

where n(x), the measure of Γ ∗ (x), is interpreted as the number of ﬁrms which supply theirproducts to location x. Substitution of (8) into the equation above yields that1

P (x) = n(x) 1−σ pM (x, ãM (x)),

x ∈ X,(10)

where ãM (x) is the average level of marginal labor input among ﬁrms supplying products tolocation x ∈ X and ãM (x) is deﬁned asãM (x) ≡[∫ ∗aM (x)

1] 1−σ

aM 1−σ µ(x, aM )daM,x ∈ X.

(11)0

A simple calculation yields ∂ãM (x)/∂a∗M (x) > 0 (see Appendix A), which implies that the
average productivity of ﬁrms which are supplying products to location x ∈ X will be higher(i.e., lower ãM (x)) when the cutoﬀ level a∗M (x) is lower.
2.2.3. Zero Proﬁt Cutoﬀ Condition
Using (2) and (8), the operating proﬁts can be written asπ(x, γ, aM ) = π(x, aM ) ≡11

pM (x, aM )q(x, aM )− w(c)f,σ

We assume that the distribution function G is of class C 1 .

7x ∈ X.

(12)

For any x ∈ X, the cutoﬀ level a∗M (x) is determined by the zero proﬁt cutoﬀ condition:
π (x, a∗M (x)) = 0⇐⇒p(x, a∗M (x))q(x, a∗M (x)) = σw(c)f.

(13)

Meanwhile, from (8) and (10), we rewrite the demand function for M -goods (2) as follows:)σ−1(ãM (x)αM Y (x), x ∈ X.
q(x, aM ) =aMn(x)p(x, aM )The number of entrants, nE is given as nE = n(x)/G(a∗M (x)). Then, substituting theequation above into (13), a∗M (x) is determined implicitly by the following equation:)σ−1(
σf n(x)w(c)σf nE G(a∗M (x))w(c)ãM (x)==.
(14)a∗M (x)αM Y (x)αM
Y (x)Consequently, the distance from the city will aﬀect the cutoﬀ level a∗M (x) (hence ãM (x)) only
through the market size Y (x).
2.3. Free Entry Equilibrium
Before entry, the expected ﬁrm proﬁt ΠE is]∫ a∗M (c)∫ h [∫ a∗M (x)ΠE =π(c, aM )dG(aM ) +π(x, aM )dG(aM ) dx − w(c)fE .
−h

0

0

If ΠE were negative, no ﬁrm would enter the market. Otherwise, there would be some newentrants. In the equilibrium, as long as some ﬁrms produce, the expected proﬁt is driven tozero by the unrestricted entry of new ﬁrms. Using (12), the free entry condition obtained:
ΠE = 0 ⇐⇒]∫ a∗M (c)∫ h [∫ a∗M (x)pM (c, aM )q(c, aM )dG(aM ) +pM (x, aM )q(x, aM )dG(aM ) dx0

= σw(c)[fE + f G(a∗M (c)) +

∫ h
−h

−h

0

]

f G(a∗M (x))dx

.

(15)

In equilibrium, the total revenue of the M -sector must be equal to the total labor income of
the M -sector. The following equation is then satisﬁed:
nE =

L(c)
],
∫h
∗
∗
σ fE + f G(aM (c)) + −h f G(aM (x))dx[(16)

where L(c) represents the population size in the city.
Given the production technology referenced by G and the fringe distance of the agricultural area h, we obtain following relation.12a∗ (x)Y (x)ãM (x)⋛ 1 ⇐⇒ M⋛ 1 ⇐⇒⋛ 1,∗ãM (s)aM (s)Y (s)12

The proof is given in Appendix B.

8x, s ∈ X.

(17)

Consequently, both the cutoﬀ and average productivities of ﬁrms supplying to location x ∈ X
will be lower (i.e. higher a∗M (x) and ãM (x)) when location x has a larger market size Y (x).
Although the average productivity of supplying ﬁrms is lower, consumers will gain higher
welfare when they reside in a location that has a large market. Because a larger market
induces weaker selection (i.e. higher a∗M (·)),13 even low-productivity ﬁrms can supply theirproducts to a large market. This results implies that the number of ﬁrms supplying the
products to a large market is large (i.e. large n(·)), and that consumers can consume a large
number of M -goods in a large market. Thus, holding other conditions constant, the priceindex is lower in the large market (see (10)). This reﬂects the love of variety in consumer
tastes. Of course, higher a∗M (·) engenders lower average productivity, which engenders ahigher price index. Eﬀects of the expansion of the number of supplying ﬁrms, however, aresuﬃciently large to decrease the price index (see (B.3)). Particularly because the market size
decreases with respect to the origin (note Y (x) = e−τA |x| ), the number of the consumableM -goods decreases and the price index increases as we move away from the center.
There is a mechanism that induces agglomeration of economic activities in the presented
model. Because more varieties are supplied to the location that has a larger market, consumers gain higher welfare in such locations, reﬂecting the love of variety. This induces aninﬂow of consumers to the city. Then the market size of the location becomes larger, andeven more varieties are supplied to the location. This circulatory eﬀect induces agglomeration. This mechanism diﬀers from that of usual models in NEG. Surely, in the usual typemodel of NEG, a larger market size engenders a lower price index, as with the presented
model. However it is merely reﬂective of the eﬀect of saving the transport costs because
the number of the consumable varieties is the same everywhere. The presented model has
the eﬀect of the expanding the number of the consumable M -goods, including the eﬀect ofsaving transport costs. The mechanism of the model presented here is more appropriate foranalysis of the role of the love of variety in relation to agglomeration.
3. The Monocentric Equilibrium
We described the demand and production side of the economy in the previous section.
In this section, we characterize the location decisions of workers, and determine the spatial
equilibrium of the monocentric economy for a given population size of labor force L.
3.1. Location Decision of Workers
The indirect utility function for a worker residing in each location is given as follows:αM αAV (x) = αM
αA w(x)P (x)−αM pA (x)−αA ,x ∈ X.

We have assumed that workers can freely choose where they live. Therefore, it is natural toassume that workers move to the location with higher welfare. Then, in equilibrium, workers

13

It diﬀers with Melitz and Ottaviano (2008). In Melitz and Ottaviano (2008), larger markets induced
tougher selection. This is a result from linear demand system employed in Melitz and Ottaviano (2008).

9

must achieve the same utility levels everywhere. Using (8), (10), and (11) we have() αM ()α
1 G(a∗M (h)) σ−1 ãM (x) M
w(x) =aA G(a∗M (x))
ãM (h)× exp ((αM τM − αA τA )|x| − αM (τM + τA )|h|) , x ∈ X.
Using (14), it is satisﬁed that() 1a∗M (x) G(a∗M (x)) Y (h) σ−1ãM (x)= ∗,ãM (h)aM (h) G(a∗M (h)) Y (x)x ∈ X.

Substitution of this into equation above yields
)α (
) αM
(
1 a∗M (x) M Y (h) σ−1
w(x) =aA a∗M (h)Y (x)× exp ((αM τM − αA τA )|x| − αM (τM + τA )|h|) ,(18)x ∈ X.

3.2. The Monocentric Equilibrium
We are now ready to characterize the monocentric equilibrium. First, one unit of A-good
is produced at each location x ∈ (−h, h) ⊂ X. Then, from (3) and Y (x) = e−τA |x| , the excess
supply of the A-goods at each x becomes αM . Consequently, the total supply of the A-goodto the city is given as presented below:∫ h
QA (c) = αMe−τA |x| dx.
(19)−hFrom the market clearing condition, QA (c) = A(c), we can obtain the following condition:∫hαM −h e−τA |x| dx
A
L(c) = Lc (h) ≡
.
αA
w(c)
LA
c (h) represents the number of urban consumers that satisﬁes the A-good market clearing
condition. Because all ﬁrms locate at the city, the income is Y (c) = w(c)L(c) and Y (x) =e−τA |x| at locations c and x, respectively. In equilibrium, from the Cobb-Douglas utility
function (1), the following equation must holdY (c)w(c)L(c)== αM .
∫h∫h
Y (c) + −h Y (x)dx
w(c)L(c) + −h e−τA |x| dx
Then, using (18), we can write the wage rate at the city as shown bellow:(] −αM)−αM [
τA
1 a∗M (h)αM 2(1 − e−τA |h| ) σ−1 −αM (τM +τA + σ−1)|h|w(c) =e.
∗aA aM (c)αAτA
The number of urban consumers that satisﬁes the market clearing condition for A-good istherefore+σ−1( ∗] αMσ−1
)αM [
−τA h
τA
a
(h)
2(1
−
e
)
α
M
M
eαM (τM +τA + σ−1 )h .
LA(20)c (h) = aA∗aM (c)αAτA
10However, because aA units of labor are used in the A-sector at each x ∈ X, the urban
population size can also be written as
L(c) = LLc (h) ≡ L − 2aA h.

(21)

LLc (h) is the size of the urban workers that satisﬁes the market clearing condition for labor.
Using (20) and (21), the monocentric equilibrium is then deﬁned asLLAc (h) = Lc (h).

(22)

3.3. Parametrization
To date, any distribution function G has been permitted in the discussion. Hereafter, tosimplify and clarify the some of the subsequent analyses, we specify the distribution function.
According to earlier reports in the relevant literature, we adopt the Pareto distribution of(G(aM ) =

aM
āM

)k
aM ∈ (0, āM ] ,,k > 1,

k > σ − 1,(23)

where k is the “shape” parameter. When k converges to 1, the distribution of marginal labor
input converges to a uniform distribution on (0, āM ]. As k increases, the distribution is moreconcentrated at higher levels. Especially, the distribution will degenerate into a point āM ,when k goes to inﬁnity.
Given this parametrization, the cutoﬀ and average levels of marginal labor input and theequilibrium number of entrants are given by the following equations:κ αM L(c)σf nE
κ αM e−τA |x|
G(a∗M (x)) =,σf nE w(c)G(a∗M (c)) =1ãM (x) = κ σ−1 a∗M (x),
ρ L(c)
nE =
.
k fE

(24)
x ∈ X, x ̸= c,
x ∈ X,

(25)
(26)
(27)

where κ ≡ (k + 1 − σ)/k > 0. This parameterization also yields simple derivations for theequilibrium wage rate in (18) as[

w(c) =(−τA |h|1 αM 2 1 − eaA αA τA e−τA |h|Mκ
) ] α1−σe−αM (τM +τA )|h| ,
(28)

τA κ
1 −φ|x|−αM (τM +τA + σ−1)|h| ,
ew(x) =aAx ∈ X, x ̸= o,where φ is deﬁned as presented below:
φ ≡ αA τA − αM τM −11

καM τA .
σ−1(29)The equilibrium wage rate is a decreasing (increasing) function with respect to the distance from the city if φ > 0 (< 0). In fact, φ consists of three terms. The ﬁrst term of the
RHS of (29) represents the eﬀect of change in the A-good price on the wage rate. Because
the transport costs are imposed on shipping the A-good, the A-good price decreases withthe distance from the origin. Thus, holding other conditions constant, consumers will gainhigh welfare when the distance from the center is large. For that reason, the equilibrium
wage rate must decrease with respect to the distance from the center because all workers
must have the same utility level everywhere in the equilibrium. The second term represents
the eﬀect of change in M -goods prices. Because each product of M -goods increases when
we move away from the center, holding other conditions constant, welfare decreases withrespect to the distance from the origin. To compensate all the workers with the same utility
level, the equilibrium wage rate must increase when the distance from the center becomeslarge. The ﬁnal term of the RHS of (29) represents the eﬀect of change in the number ofthe consumable M -goods. The market size of each location decreases with respect to the
distance from the center. Consequently, as we have described, consumers can consume lessvariety when they move to locations further away from the origin. The salient implicationis that, holding other conditions constant, consumers can gain higher welfare when they arecloser to the city. Therefore, to achieve identical utility everywhere, the equilibrium wage
rate must increase as the distance from the city becomes greater.
Summarizing the discussion presented above, the ﬁrst term represents the advantage tomoving away from the city although the second and ﬁnal term represents the disadvantage.
If the advantage is larger than the disadvantage, then the equilibrium wage rate will decrease
when the distance from the city becomes large (i.e. φ > 0). Otherwise, the equilibrium wage
rate will increase (i.e. φ < 0). The reason for this is that the equilibrium wage rate isdetermined as a reservation wage rate that ensures all consumers can obtain the same utility
everywhere. The sign of φ is determined as follows.
φ ⋚ 0 ⇐⇒

κ̄ − αM
τM
⋚
,
αM κ̄
τA

κ̄ ≡

k(σ − 1)∈ (ρ, 1).
k(σ − 1) + k + 1 − σ(30)

In that equation, κ̄ converges to 1 when k converges to σ − 1. In this case, φ converges to
αA τA − αM τM . Meanwhile, κ̄ converges to ρ when k goes to inﬁnity. In this case, if αM islarger than ρ, then φ is always negative for any positive value of τA and τM . We explain in
Section 4 that condition αM < ρ is the necessary and suﬃcient condition under which themonocentric spatial structure becomes unstable when the population size in the economy
is suﬃciently large. This condition is called the no-black-hole condition in the original FK
model.
Using the parameterization, LAc (h) is written as follows.
Mκ
(
)[
(
) ] ασ−1
−τA h
−τA h
2
1
−
e
2
1
−
e
α
α
M
M
LA
eαM (τM +τA )h .
c (h) = aA
−τ
h
A
αA
τA
αA τA e(31)

The monocentric equilibrium is then given as (22), which determines the equilibrium urban
population size and fringe distance uniquely. This in turn determines all other unknowns by(24)-(28).
12

4. Sustainability of the Monocentric Spatial structure
To date, we have simply assumed that production of M -goods takes place exclusively inthe city. However, this spatial conﬁguration might be not sustainable because some ﬁrms
might have an incentive to enter the market at a location that is not the city. Consequently,
in this section, we explore the spatial structure sustainability.
4.1. The Potential Function
To check the sustainability condition, it is helpful to use the properties of the (market)
potential function of the M -sector. In this subsection, we deﬁne the market potential function
and then describe the necessary and suﬃcient condition for sustainability of the monocentric
spatial conﬁguration.
First, consider a ﬁrm entering the market at s ∈ (0, h].14 If the ﬁrm draws aM from G,
the operating proﬁt of the ﬁrm obtained by supplying product to location r ∈ X is given as[]]
w(s) [
τM |r−s|π(s, r, aM ) = max 0,aM eq(s, r, aM ) − (σ − 1)f ,(32)
σ−1
where q(s, r, aM ) is the demand for the product of this ﬁrm in location r:()σ
ãM (r)w(c)αM Y (r)q(s, r, aM ) =e−στM (|r−s|−|r|) , r ∈ X.
n(r)p(r, ãM (r))
aM w(s)If π(s, r, aM ) < 0, the ﬁrm will not supply its product to r. Because π(s, r, aM ) is a decreasing
function with respect to aM , the threshold a∗M (s, r), which satisﬁes π(s, r, a∗M (s, r)) = 0,
uniquely exists and the ﬁrm will supply its product for r iﬀ aM ≤ a∗M (s, r). The expected
operating proﬁt, π(s, h), is then given as]∫ ∗∫ [∫ ∗aM (s,c)π(s, h) =hπ(s, c, aM )dG(aM ) +−h0aM (s,r)π(s, r, aM )dG(aM ) dr.
0

To derive the necessary and suﬃcient condition for sustainability of the monocentric
spatial conﬁguration, we next deﬁne the (market) potential function, Ω. Using the equation
presented above, we deﬁne Ω at each location s ∈ (0, h] asΩ(s, h) ≡π(s, h).
w(s)fE

Therein, Ω(s, h) measures the relative proﬁtability of each location s ∈ (0, h] (for M -sector)
in comparison with that in the city location. Before entry into the market at s, the expected
proﬁt is ΠE (s, h) = π(s, h) − w(s)fE . By deﬁnition, it is readily apparent thatΠE (s) ⋚ 0 ⇐⇒ Ω(s, h) ⋚ 1.
14

Because the spatial conﬁguration is symmetric with respect to the origin, it is suﬃcient that we speciﬁcally examine the right-hand side of the origin.

13

If s = c, then Ω(c, h) = 1 and thus ΠE (c, h) = 0. Therefore, we conclude that, given
population size L (and thereby knowing the fringe location h), the monocentric equilibrium
is stable if and only if the following condition is satisﬁed.
Ω(s; h) < 1 for any s ∈ (0, h].

(33)

A simple but messy calculation rewrites the potential function as∫ he−τA |r|G(a∗M (s, r))G(a∗M (s, c))
+
α
ψ(r)
dr,
ψ(r)
≡
,
Ω(s, h) ≡ αM∫hA
G(a∗M (c))
G(a∗M (r))
e−τA |r|dr
−h

(34)

−h

where G(a∗M (s, r))/G(a∗M (r)) is
G(a∗M (s, r))=G(a∗M (r))

(

) σ k
w(c) σ−1 −kτM (|r−s|−|r|)
e|
{z
},
w(s)
b
|
{z
}

r ∈ X.

(35)

a

Therefore, the potential function is a weighted average of G(a∗M (s, r))/G(a∗M (r)).
G(a∗M (s, r))/G(a∗M (r)) > 1(< 1) implies that the possibility the ﬁrm can supply its
product to location r is high (low) when the ﬁrm enters the market at location s comparedto the city. The term a of (35) represents the eﬀect of saving production costs. By (28), if φ
are positive, then the wage rate will be lower than that of the city when the ﬁrm production
takes place in a remote area. Consequently, the incentive to enter the market at the remotearea is strong because the production cost is low. If φ are negative, then the incentive is weakbecause production costs become high in remote area. Term b of (35) represents the eﬀectof change in the transport costs by changing production location from the city to locations. If |r − s| > |r|, then the production costs increase by changing the production location.
Therefore, the incentive to change the production location is weak. In contrast, the incentive
to change the production location is strong when |r − s| < |r|.
Although (33) is a necessary and suﬃcient condition for sustainability of the monocentric
spatial conﬁguration, it is not explicit. Thus, in the following subsections, we investigate
parametrical conditions under which the monocentric conﬁguration is sustainable. For thispurpose, we present the explicit form of the potential function. Substitution of (28) and (35)
into (34) yields
]
[(
)
σ
1
+
α
α
M
A
φks
−kτ
s
kτ
s
M
M
e
+
Ψ(s, h)e,(36)Ω(s, h) = D(h)e σ−1
2
2
where
[

] αρM k+1−σ
σ−1
αA
τA
D(h) ≡,αM 2(1 − e−τ A h )]∫ s −τ x [e A 1 − e−2kτM (s−x) dx0Ψ(s, h) ≡ 1 −∈ [0, 1].
∫he−τA x dx
014

4.2. Case of Fujita and Krugman
To derive the parametrical conditions for the sustainability of the monocentric spatial
conﬁguration, ﬁrst, let us consider a speciﬁc case. The shape parameter k converges to σ − 1.
In this case, the potential function (36) converges to[()
1 + αMFK
σ(αA τA −αM τM )sΩ (s, h) ≡ ee−(σ−1)τM s
2(37)]
αA F K(σ−1)τM s+ Ψ (s, h)e
,
2
where

]
∫ s −τ x [−2(σ−1)τM (s−x)
A
e
1
−
e
dx
ΨF K (s, h) ≡ 1 − 0.
∫h−τA x dxe0

ΩF K is the potential function in the FK model. This implies that our model is a generalizedversion of the FK model. It asymptotically includes the FK model.15 In this case, we canobtain some stability conditions as shown in Table 1.16
Insert Table 1
From Table 1, when k → σ − 1, the monocentric spatial conﬁguration is sustainable only
if

τM
αA
≥
τA
(1 + ρ)αM(38)

is satisﬁed. This condition ensures that ΩF K has a non-positive gradient in the neighborhood of the origin as we move away from the center. It is a necessary condition forthe monocentric spatial conﬁguration that it be sustainable when k → σ − 1 becauseΩF K (c, h) = lims→0+ ΩF K (s, h) = ΩF K (0, h) = 1. Here, (38) implies that the relative transportation costs, τM /τA , is important for ensuring the agglomeration of the M -sector.17
4.3. Lock-in Eﬀect of the City
To reveal the properties of the potential function, it is helpful to examine the behaviorof the potential function in the limiting case. First, we deﬁne the value of the potentialfunction in the neighborhood of the origin as Ω(0; h) ≡ lims→0+ Ω(s, h).18 It is then satisﬁedthat[] αρM k+1−σ
σ−1
αA
τA
.
Ω(0; h) = D(h) ≡αM 2(1 − e−τA h )15

In fact, the equilibrium wage rate, city size, and fringe distance also converge to the case of FK model,
when k → σ − 1.
16
See Fujita and Krugman (1995) for the proof.
17
Davis (1998) shows that the home market eﬀect might disappear when transportation costs to the homogeneous goods sector is introduced in Krugman’s model (Krugman (1991)). It suggests that agglomerationof certain sector depends on the relation of transportation costs between sectors.
18
City location, c is distinguished from the location 0 ∈ X of the agricultural area and, in general,Ω(c, h) ̸= Ω(0; h).

15

Therefore, we have a necessary condition for the sustainability of the monocentric spatial
conﬁguration as follows.
αM 2(1 − e−τA h )
> 1.
(39)
αA
τA
This condition ensures that the market size of city Y (c) is the largest in X, and it implies alarger number of varieties supplied to the city than to any other location. Form (21), (22),and (31), the fringe distance h is understood to increase with respect to the population size
L andD → ∞ as h → 0 as L → 0,
holds. Consequently, the condition (39) is violated when L is suﬃciently small. Particularly,
if αA τA ≥ 2αM , then the monocentric spatial conﬁguration is always unstable for any L.
Therefore, for the monocentric conﬁguration to be sustainable, it must hold that
αA
2
< .
αM
τA

(40)

Condition (40) requires that the transport costs incurred in the A-sector be suﬃciently
small or that the demand of M -goods be large compared to that of A-good (i.e., αM /αA is
suﬃciently large). This condition is necessary to maintain the monocentric spatial structure
for the following reason: the total supply of the A-good to the city given by (19) is afunction of the fringe distance h, and 2αM /τA is the least upper bound of it. Meanwhile, thetotal demand for A-good in the city is αA Y (c). Consequently, at least, αA Y (c) < 2αM /τA
must hold to maintain the city. When condition (40) is violated, Y (c) is smaller than one.
However, Y (0) = 1 implies that some locations exist in the neighborhood of the origin andthat these locations have a larger market than that of the city has. Because the larger numberof the M -goods is supplied to these locations, the wage rate in these locations became lower
than that in the city. Therefore, ﬁrms can ﬁnd it more proﬁtable to enter the market there.
αA τA ≥ 2αM means that the supply of the A-goods to the city is insuﬃcient to maintainthe city for any population size L. For that reason, condition (40) is necessary. This factsuggests that not only the relative value of the transport costs between A-good and M -goods
but also that the absolute value of the transport costs of A-good is important to maintainthe monocentric spatial conﬁguration.
Next, we explore the gradient of the potential function in the neighborhood of the origin.
It is readily found by diﬀerentiation as follows:
(
)
∂Ω
σ
lim
(s, h) = D(h)kφ − αM τM .
s→0+ ∂sσ−1Thus, by the deﬁnition of κ̄ (see (30)), we have
lim+

s→0

κ̄ − αMτM∂Ω(s, h) ≤ 0 ⇐⇒ φ ≤ ραM τM ⇐⇒≤.
∂s(1 + ρ)αM κ̄τAConsequently, the following condition must hold so that the potential function has a nonpositive gradient in the neighborhood of the origin:κ̄ − αM
τM≥.
τA(1 + ρ)αM κ̄
16

(41)

The meaning of this condition is the same as that in the case of Fujita and Krugman (see(38)). However, it is noteworthy that condition (38) is more strict compared with thecondition (41). In fact, because αA > (κ̄−αM )/κ̄ holds, (38) implies (41). When k converges
to σ − 1, (38) and (41) are equivalent because RHS of (41) converges to RHS of (38).
Therefore, we conclude that condition (38) implies that lims→0+ ∂Ω/∂s < 0.
Under conditions (39) and (41) (or (38)), two types of lock-in eﬀect of the city in thepresent model contribute to sustainability of the monocentric spatial conﬁguration. To
show the ﬁrst type of eﬀect, consider that the condition (41) does not hold. It implies thatφ > ραM τM > 0. If φ is positive, then the wage rate is decreasing with distance from the city.
Therefore, That (41) is not true means that the wage rate is decreasing suﬃciently fast when
we move a short distance away from the origin, and a ﬁrm ﬁnds it more proﬁtable to moveaway from the center.19 If (41) is strictly satisﬁed or if (38) holds, then the potential function
has negative gradient in the neighborhood of the origin. Therefore, in the neighborhood of
the origin, no ﬁrm has an incentive to move away from the origin. This is the ﬁrst type oflock-in eﬀect of the city.
To elucidate the second type of the lock-in eﬀect of the city, we assume that the condition(39) does not hold. If (39) is not true, then there will be some locations in the neighborhoodof the origin where the market size (and thus the number of supplied varieties) is greater than
the area of the city. Consequently, because of the love of variety, some locations will exist in
the neighborhood of the origin, and at these locations consumers can have a higher utility
level than that of the city. As explained earlier, the equilibrium wage rate is determined as areservation wage that ensures an identical utility level for workers in any location. Therefore,
if (39) does not hold, there will be some locations in the neighborhood of the origin where themarket size is larger and the wage rate is lower than those of the city. It is readily apparent
that these locations are more suitable for the production of M -goods than the city. For
that reason, (39) must hold to maintain the monocentric spatial conﬁguration. If (39) holds,
then the value of the potential function is less than unity in the neighborhood of the origin.
Therefore, in the neighborhood of the origin, no ﬁrm has an incentive to enter the market atany other location because production of M -goods is unproﬁtable at those locations. Thisis the second type of the lock-in eﬀect of the city, and it is not considered in Fujita andKrugman (1995) or any other literature.20
Because the fringe distance h increases with respect to the population size L, the second
type of the lock-in eﬀect becomes strong when L is larger. The gap of the market size between
the city and any other location increases with respect to the population size. Therefore, thegap of the number of consumable varieties between the city and any other location also
increases when L increases. This is the reason that the second type of the lock-in eﬀect
becomes large when L becomes strong. In the existence of both types of the lock-in eﬀect of
the city, new cities never emerge in the neighborhood of the origin as long as both conditions(39) and (41) hold.
Finally, let us consider the case in which population size L diverges to inﬁnity. In this19

For this reason, (38) must hold to ensure the sustainability of the monocentric spatial conﬁguration
when k → σ − 1.
20e.g., see Behrens (2007), Fujita and Hamaguchi (2001), Fujita and Mori (1996), and Fujita et al. (1999a).

17

case, the fringe distance also diverges to inﬁnity. Therefore, we can rewrite the potentialfunction asΩ(s; ∞) ≡ lim Ω(s, h)h→∞[](2kτM −τA )s
σ
α
α
−
τ
M +1
A 2kτM e
A
φ−τ
ks
(
)
M
= D(∞)e σ−1+,222kτM − τA

where(D(∞) ≡ lim D(h) =
h→∞

αA τA
αM 2

(42)

) αρM k+1−σ
σ−1.

The term outside the brackets in (42) is smaller than unity and decreases with respect tos when (40) and (41) (or (38)) are satisﬁed. Consequently, this term works as a damping
force for the potential function when we move away from the center. The term inside thebrackets in (42) works as a booster of the potential function because it increases with s.
4.4. The conditions for the stability of the monocentric spatial conﬁguration
We now are ready to derive the condition for the monocentric spatial conﬁguration to
be sustainable. In the circumstances under which (40) and (41) (or (38)) are satisﬁed,Ω(s; ∞) has a non-positive gradient in the neighborhood of the origin and the number of its
local minimum point is only one at most on (0, h]. Therefore, when the population size is
suﬃciently large, the monocentric spatial conﬁguration is sustainable if and only ifσlim Ω(s; ∞) < 1 ⇐⇒ lim e[2kτM −τA +( σ−1 φ−τM )k]s ≤ 1,

s→∞

s→∞

is satisﬁed. It is readily apparent that
(
[
]
)
σ
k−ρ
kτA τM
2kτM − τA +φ − τM k = (ρ − αM )
+
,
σ−1
ρ
τA
ρk
and therefore we can eventually conclude that the monocentric spatial conﬁguration is sustainable for suﬃciently large L if and only if αM ≥ ρ is satisﬁed.21 Given this condition, theassociated curves of ΩF K are always below 1 everywhere under large L (see Table 1). To
rule out this case, the no-black-hole condition αM < ρ must hold.22 We can summarize thestability conditions under which (41) or (38) is satisﬁed as in Table 2.
Insert Table 2
When αA τA < 2αM and αM < ρ hold, the monocentric spatial conﬁguration is unstable
for a suﬃciently small or large population size. However, the reasons diﬀer greatly according
to the cases. When the population size is too small, the monocentric spatial structure
become unstable because the supply of the A-good to the city is too small to maintain it.
Particularly, if the transport costs of A-good are too high, then the supply of the A-good tothe city is always insuﬃcient and the monocentric spatial conﬁguration is never sustainable21
22This condition is equivalent with the condition so that Ω(s; ∞) has a non-positive gradient on (0, h].
For details about the no-black-hole condition, see Fujita et al. (1999b).

18

for any population size.23 When the population size is suﬃciently large, the monocentric
spatial structure become unstable because the periphery market is suﬃciently large to ensurethat a ﬁrm ﬁnds it more proﬁtable to enter the market at the remote area.
An interesting case arises, when αA τA < 2αM and αM < ρ hold. The monocentric
spatial conﬁguration is unsustainable with suﬃciently small population size which in tern,
becomes sustainable with medium population size, and un sustainable when the population
size suﬃciently large.
Insert Figure 1
Fig.1 illustrates this point. In this numerical example, we assign αA = αM = 0.5,
τA = 0.9, τM = 1.0, σ = 4.0, k = 4.5 and aA = 0.5. We can see that when the population
size L increases from 0.2e, the potential curve ﬁrst exceeds 1 in the neighborhood of theorigin; then the potential curve becomes below 1 everywhere; and ﬁnally it exceeds 1 in the
periphery.
5. Conclusion and Discussion
In this paper, we presented a continuous location-space NEG model based on Fujita
and Krugman (1995) with heterogeneous ﬁrms. We examined conditions under which all
manufacturing productions will concentrate in a single city.
Introducing Melitz type ﬁrm heterogeneity, we can consider diﬀerences of two types:
one is the diﬀerence of productivity among ﬁrms; another is the diﬀerence in the numberof consumable goods among locations. Furthermore, we have described the existence of thediﬀerence of the number of consumable goods among locations makes the city more attractive
and strengthens the lock-in eﬀect which leads to the monocentric spatial conﬁguration toa sustainable state. In addition, we have shown how the sustainability of the monocentric
spatial structure depends on the transport technology and population size in the economy.
When the population size in the economy is too small, the monocetric spatial structure will
be unstable because the supply of the A- good to the city is insuﬃcient to maintain the city.
Particularly when the transport costs of A-good are too high, the supply of the A-good tothe city will always be insuﬃcient and the monocentric spatial conﬁguration will never besustainable for any population size.
These results lend historical insight into economic geography. When the population size
is small, economic activities would disperse in the economy and ratio of urban population
to total population in the economy would be low. Even if population grew gradually, as
long as transport technology level for agricultural goods was low, there would be no largecity and the ratio of urban population to total population would still be low. When the

23

In the original FK model, the monocentric spatial conﬁguration also becomes unstable when the transport
costs of A-good are suﬃciently high relatively to that of M -goods. However the reason diﬀers from thatof the present model. High transport costs lead the high A-good price relative to M -goods in the city.
Therefore, workers have an incentive to leave the city. Because the equilibrium wage is determined as areservation wage which ensures identical utility level for workers in any locations, the wage rate decreases
when we move a short distance away from the center. Therefore, the monocentric spatial structure becomes
unstable because the potential function has the positive gradient in the neighborhood of the center.

19

population size became suﬃciently large and the transportation cost of agricultural goods
became suﬃciently low, then a large city would be expected to form.
Our analysis of the model presented in this paper, however, is preliminary. Some important works remain to be conducted in the future. We have examined the monocentric
spatial conﬁguration and have shown that it might become unsustainable according to the
population size. In the analysis of the present model, however, we have not described what
happened when the monocentric spatial structure became unstable. Because, perhaps, theeconomy will have more than one city when the monocentric spatial structure becomes unstable, one avenue of the future studies is to investigate spatial conﬁgurations including
multiple cities, as did Fujita and Mori (1997) and Fujita et al. (1999a).
Venables (1996) shows the vertical linkage between the manufacturing and intermediategood sectors becomes one important force to concentrate economic activities. We haveignored this force in the present model. It seems that our model can be modiﬁed to include
the vertical linkage between the manufacturing and intermediate-good sectors, introducing
the ﬁnal goods sector, which uses a continuum of intermediate goods as input (e.g., Fujita
and Hamaguchi (2001)). Therefore, a second direction of future studies is to develop a modelthat has a vertical linkage between the manufacturing and intermediate-goods sectors.
One other directions of the future studies is the development of a model that can yieldan answer to the question of why the economy tends to move toward agglomeration. In theanalysis of the present model, we have assumed that all production of manufacturing goodsin the economy initially concentrates in a single city. However, we have not questioned why
this structure emerges initially. It is merely an assumption. Initially, some cities might
emerge simultaneously or there might be no city. Consequently, it is necessary to develop amodel that can analyze conditions under which a symmetric spatial structure is unstable andthe economy moves to agglomeration, under the same framework with the present model.
That point is left as a subject for future research.
Acknowledgments
The author thanks to Nobuaki Hamaguchi, Noritsugu Nakanishi and Kazuhiro Yamamotofor valuable comments and suggestions. The author also thanks to Keisuke Kondo, Yang Xi,
Yuji Matsuoka and Chihiro Inaba.
Appendix A.
Diﬀerentiating ãM (x) with respect to a∗M (x), we can obtain the following equation[()σ−1 ]a∗M (x) ∂ãM (x)a∗M (x)µ(x, a∗M (x))
ãM (x)=1−, x ∈ X.
(A.1)
ãM (x) ∂a∗M (x)σ−1a∗M (x)Since, by deﬁnition, ãM (x) is smaller than a∗M (x), ∂ãM (x)/∂a∗M (x) is positive.

20

Appendix B.
Substitution of (16) into (14) yields that[]
( ∗)σ−1 α f + f G(a∗ (c)) + ∫ h f G(a∗ (z))dz
M
E
M
M
−h
aM (x)Y (x)=,∗ãM (x)f G (aM (x))
Y (c)x ∈ X.

(B.1)

It implicitly determines the cutoﬀ level a∗M (x).24 The above equation also implies thata∗M (x) and ãM (x) are functions of relative market size Y (x)/Y (c), and a∗M (c) and ãM (c) do
not depend on Y (c) and Y (x).
Using the implicit function theorem and (A.1), the following equation holds:
[
]−1
(
)σ−1
Y (x)/Y (c)∂a∗M (x)ã(x)
M= a∗M (x)µ(x, a∗M (x)) ∗+σ−1a∗M (x) ∂(Y (x)/Y (c))
aM (x)x ∈ X, x ̸= c.

> 0,

(B.2)

Thus, we obtain (17).
Using from (8) to (11) and n(x) = nE G(a∗M (x)), the price index is given by
1
1−σ

P (x) =Thus,

τM |x|

nE w(c)eρ[∫ ∗aM (x)

1
] 1−σ
′

a1−σ
M G (aM )daM,x ∈ X.

0dP (x)/da∗M (x)a∗M (x)G(a∗M (x))
−1
=
< 0.
∫ a∗M (x) 1−σ ′
P (x)
σ−1
a G (aM )daM
0

(B.3)

M

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Note that ãM (x) is also a function of a∗M (x). See (11).

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23

Table 1. Stability conditions for the monocentric equilibrium (k → σ − 1)
(1 + ρ)αM τM ≥ αA τA
αM ≥ ρ
αM < ρ
Always stable
Stable for small L(1 + ρ)αM τM < αA τA
Always unstable

Source: Table 1 of Fujita and Krugman

Table 2. Stability conditions for the monocentric equilibrium in general case

For small L
UnstableαA τA ≥ 2αM

αA τA < 2αM
For large L
αM ≥ ρ
αM < ρ
StableUnstable

Always Unstable

1.5
1.4
1.3
1.2
L=e (h=0.97)Ω(s,h)

1.1
1.0
0.9

L=0.8e (h=0.83)

0.8
L=0.6e (h=0.66)

0.7
0.6
0.5
0.0

L=0.4e (h=0.48)
L=0.2e (h=0.26)
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

s

Figure 1: Potential Curve
Note: αA = αM = 0.5, τA = 0.9, τM = 1.0, σ = 4.0, k = 4.5, and aA = 0.5.

24