CSIS Discussion Paper No. 87

Asymmetric Transport Costs and Economic Geography

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

October 2007

Abstract
This paper has explored the impacts of the asymmetry of transport costs with regard to
the directions of shipments upon economic geography. In doing so, we have focused on theasymmetry arising from the optimizing behavior of transport ﬁrms: they charge diﬀerent
prices for the transport services involving shipments in diﬀerent directions in response to thediﬀerence in the marginal costs of shipment and the price elasticities of the demands. Twocases are studied: the case where the economic activities are distributed equally between
regions and the case where they are concentrated in one region. It has been shown thatwhether each of those two patterns is supported as a stable long-run equilibrium patterndepends on the values of several parameters in a complicated way.
Keywords: marginal cost of shipment, joint production, oligopolistic transport sector,
substitution between imports and domestic products, import price eﬀect, export price eﬀect
JEL Classiﬁcation Numbers: F12 (Models of Trade with Imperfect Competition and
Scale Economies), R13 (General Equilibrium and Welfare Economic Analysis of Regional
Economies), R49 (Other: Transportation Systems)
E-mail address: takaaki-t@csis.u-tokyo.ac.jp
I would like to thank Masahisa Fujita for stimulating discussions and comments. I am greatly beneﬁted
from the comments by Tomoya Mori and Se-il Mun. I also appreciate comments from an anonymous refereeand seminar participants at various institutions. This research is partly supported by the Grant in Aid for
Research (No. 18600004) by Ministry of Education, Science and Culture in Japan.

1Introduction

We often encounter the situation in which the levels of transport costs diﬀer from each otherif the directions of shipments are diﬀerent, that is to say, the transport cost necessary toship goods from one region to another is not equal to the cost necessary to ship them inthe opposite direction. Casual observation suggests that the levels depend on the sizes ofthe demands for transport services or shipments: if the demand for the shipment from one
region to another is larger than the counterpart in the opposite direction, the former costs
more than the latter. This asymmetry of the transport costs is not a thing which is so
trivial that one can ignore when explaining the determination of the geographical patternof economic activities.
Consider, for instance, an economy with two regions, West and East, where more economic activities are located in West than in East but the diﬀerence is quite small. Because
the amount of the shipment from West to East is larger than that from East to West, we
expect that the transport cost to ship goods in the former direction is higher than the costto ship them in the latter direction. It implies that the import is less expensive in theWest than in East up to the diﬀerence in the transport cost, other things being equal. Thisaﬀects the relative sizes of the force that promotes geographical agglomeration of economic
activities (a ‘centripetal force’) and the force that promotes their geographical dispersion (a‘centrifugal force’), which determine the geographical pattern. On the one hand, consumers
ﬁnd themselves more aﬀordable in West, because of the less expensive imports there, thanin East, ceteris paribus. This gives them an incentive to locate in West, which works as acentripetal force. On the other hand, ﬁrms attempt to exploit the opportunity of the lesscompetition pressure from the imports by locating in East rather than in West; a centrifugal
force. Unfortunately, which force, the centripetal or centrifugal force, dominates the otheris ambigous. In any case, nonetheless, we can assert that the asymmetry of the transport
costs gives a substantial impact on the relative sizes of the two counteracting forces, andthus, greatly aﬀects the spatial distribution of economic activities.
The purpose of this paper is to study how the asymmetry of transport costs aﬀects
the economic geography. For that purpose, a two region model with a transport sector isconstructed. As in the standard models in the literature, I focus on two distribution patterns,
the symmetric distribution pattern, in which mobile factors are distributed equally in thetwo regions, and the agglomeration pattern, in which they are concentrated in one region;and examine the conditions for each of them to be supported as a stable equilibrium pattern
taking into account the possibility that the transport costs become asymmetric.
Here, it is important to emphasize that in this piece of research, the asymmetry is not
something to be assumed but something to be generated as a result of the behaviors of1

transport ﬁrms. Although there would be many reasons for the asymmetry, the following
two are particularly considered in this paper.
The ﬁrst reason is related to a technology. Notice that the shipments in one direction andin the opposite direction are usually joint products of a transport ﬁrm. One consequence isthat the marginal cost of shipping in a certain direction depends on the amounts of not only
the shipment in that direction but also on the shipment in the opposite direction. This point
would be better understood by a simple example. Let us consider a carrier based in Chicago,say, which owns some cargo trucks. It probably keeps them at its sites within or around
Chicago when they are idle. This means that the trucks delivering goods to, say, Seattle,
will eventually have to come back to Chicago. Suppose that the carrier gets the order toship 100 units of goods from Chicago to Seattle. To fulﬁll the order, 10 trucks need to bedispatched. Then, they can carry goods from Seattle to Chicago on their returning trip with
virtually 0 marginal cost unless the amount of the return cargo exceeds 100 units, because
the 10 trucks must return to Chicago anyway. In general, if there is a physical requirementthat transport equipments eventually return to the place of their departure, the marginal
cost to ship goods in one direction diﬀers in size from that in the opposite direction: it ispositive in the direction with a greater demand but 0 in the opposite direction.
The second reason for the asymmetry is a monopolistic industry structure of the transport
sector. As long as the price elasticities of the demand for the transport services in the two
directions are not equal to each other, carriers with a certain size of monopoly power attempt
to price discriminate. If larger demand is associated with lower price elasticities, they charge
a higher price for the transport service in the direction with a greater demand.
So as to pay full attention to those two factors, I consider the situation in which transport ﬁrms produce the transport services (shipments) in the two directions jointly, and themarginal costs to produce them are inter-dependent as has been explained in the above illustrative example. Furthermore, it is assumed that they compete with each other à la Cournot
in an oligopolistic market. Then, I show that whether each of the two patterns is supported
as a stable long-run equilibrium depends on several key parameters. For one thing, higher
elasticity of substitution in consumers’ preference makes the symmetric distribution patternmore likely to occur while the agglomeration pattern less likely to occur. Furthermore, thehigher the marginal cost of shipment associated with the binding capacity, the less likely
it is that the symmetric distribution pattern and agglomeration pattern are supported as astable long-run equilibrium patterns.
For all its importance, the asymmetry of the transport costs has seldom attracted attentions of researchers. Even in the ﬁeld of new economic geography, which underscores
the role of the transport costs in the determination of the geographical pattern of economic2

activities, the possibility of the asymmetry has been ignored (see Fujita et al. (1999); Fujita
and Thisse (2000); and Fujita and Thisse (2002)). This disregard is, for the most part, owning to the fact that to give proper regard to the two reasons of the asymmetry mentioned
earlier involves embodying the transport sector explicitly in the model and analyzing howeach transport ﬁrm behaves. It obviously makes the analysis much more complex, whichmost of the studies have been avoided.1 There are a few exceptions. Takahashi (2006b) discusses the behaviors of individual transport ﬁrms to obtain welfare implications in a generalequilibrium setting. Behrens et al. (2006), on the other hand, examine the impacts of theregulation of a transport sector upon the welfare levels paying attention to the incentives oftransport ﬁrms, though in a very simple way.
The paper is organized as follows. In the next section, I present a basic model. Section 3
formulates the transport sector and solves the problem faced by each transport ﬁrm. In thesubsequent two sections, the symmetric distribution pattern and the agglomeration pattern
are examined in order. We derive the conditions for each pattern to be supported as a stableequilibrium pattern. Finally, Section 6 concludes.

2

Model

As a basic framework, I use the model used in Takahashi (2006a), which is a modiﬁed versionof the analytically solvable model by Forslid and Ottaviano (2003).

2.1

Basic Framework

There are two regions, denoted by 1 and 2; and two goods, an intermediate and a ﬁnal
good. Total labor force consists of workers and entrepreneurs. The workers are not endowedwith the skill necessary to produce the ﬁnal good and thus work in the intermediate sector.
There are 2 units of workers in the economy, who cannot migrate across the regions. They
are distributed equally in the two regions: 1 unit lives in each region. The entrepreneurs
are, on the other hand, endowed with the skill so that they can engage in the production of
the ﬁnal good. Furthermore, they can freely move across the two regions. Their number is
ﬁxed at n, of which λ1 n live in region 1 and λ2 n live in region 2 (λ1 + λ2 = 1). Instead of
λ1 and λ2 , a parameter λ ∈ [0, 1] with λ ≡ λ1 (λ2 = 1 − λ) will be often used when doing
so is more convenient.
1 Some works discuss the endogenous determination of the transport costs;

but do not deal with thetransport sector explicitly. Mori and Nishikimi (2002), for instance, examine the eﬀect of the economy ofdensity on transport costs. Bougheas et al. (1999), and Mun and Nakagawa (2005) study the impacts ofinfrastructure investment. Finally, Takahashi (2006a) examines the selection of the transport technology,
which aﬀects the levels of transport costs.

3

The intermediate is a homogeneous product and produced in a competitive sector. Eachworker produces 1 unit of the intermediate, whose price, therefore, equals his wage rate ineach region. The cost to ship the intermediate from one region to the other is assumed to
be 0 so that the prices of the intermediate and, consequently, the wage rates of workers areequalized in the two regions. The wage rate is taken as a numeraire.
On the contrary, the ﬁnal good is a diﬀerentiated product and produced by a monopolistically competitive sector. Each entrepreneur owns a ﬁrm which produces a variety in theregion of her residence. Thus, there are n varieties in the economy of which λi n are produced
in region i (i = 1, 2). Taking a unit appropriately, we can consider that each ﬁrm produces
1 unit of a variety from 1 unit of the intermediate using the skill of an entrepreneur. All therevenue left after the payment for the intermediate is taken by the entrepreneur. In otherwords, the proﬁt of a ﬁrm is equal to 0:
pq − 1 · q − w = 0,(1)

where p and q denote a price and an amount of a variety produced, and w a wage ratereceived by an entrepreneur.
The workers and entrepreneurs have the same preference represented by a utility function,∫n
U = [ 0 x(k)ρ dk]1/ρ , where ρ ∈ (0, 1), and x(k) denotes the amount of the kth variety.
The entrepreneurs sell their varieties at the price that maximizes their own wages. Since
not only they are subject to the same technological condition but also all the varieties enter
the utility function in a symmetric manner, they charge the same price as long as they arelocated in the same region. Thus, I denote the prices of a variant produced in respective
regions by p1 and p2 .
Next, I introduce transport costs. It takes no cost to carry the ﬁnal good within the same
region. However, in order to carry it across the regions, consumers need to buy transport
services, which are provided by transport ﬁrms or ‘carriers’. The prices of the services are
assumed to be proportional to the values of the good to be shipped. Let τi ∈ (0, 1) be
the proportional coeﬃcient for the transport service that carriers the good from region i toregion j (i = 1, 2 and j 6= i). Here, remember that the subscript of τ refers to the region of
the origin of the shipment. Then, in order to have 1 unit of a variety shipped from regioni to region j, consumers pay τi pi , which is a price of the transport service, or, in a moreordinary expression, a transport cost, in that direction. The delivered price in region j of
the k-th variety produced in region i is, therefore, equal to (1 + τi )pi . Here, we are taking
into account the possibility that the transport costs may diﬀer from each other depending on
the direction of shipment, that is, τ1 is not necessarily equal to τ2 . This point distinguishes
the model from the conventional ones. In what follows, it is more convenient to use notationti ≡ 1 + τi as a measure of the prices of the transport services or transport costs rather than4τi (i = 1, 2).
Let Xii and Xij be the total demands for a variety produced in region i, by the region i
consumers and region j consumers, respectively. Applying the standard procedure to derive
consumers’ demand functions yields
σ−1
Xii = p−σYii Pi(i = 1, 2)(2)(i = 1, 2; j 6= i)

(3)

and
−σ σ−1
Xij = t−σYj .
i pi Pj

Here, σ ≡ 1/(1 − ρ) > 1 is an elasticity of substitution, Yi is an aggregate income and1[ ()] 1−σ1−σ 1−σPi = n λi p1−σ
+
λ
t
p
j
i
j
j

(i = 1, 2; j 6= i)(4)is a price index in region i. Since the amount of each variety produced in region i, qi , must
be equal to its demand, we have(i = 1, 2; j 6= i)qi = Xii + Xij .

(5)

An entrepreneur in region i maximizes her wage given by wi = pi qi − 1 · qi (i = 1, 2) (see(1)), setting the price at
p1 = p2 = p ≡

σ
.
σ−1

(6)

The wage received by an entrepreneur is reduced to
wi =

1
qi .
σ−1

(i = 1, 2)(7)Moreover, because transport ﬁrms earn no proﬁt, as will be explained later, the aggregate
income consists of workers’ and entrepreneurs’ earnings:
Yi = 1 + λi nwi .

(i = 1, 2)(8)This completes a description of a basic framework.

2.2

Short-Run Equilibrium

Before deriving the short-run equilibrium, it is useful to introduce one additional parameter,θi ≡ λi + λj t1−σ(i = 1, 2; j 6= i). Notice that θi−1 = pXii /(Yi /n), where the numeratorjrepresents the actual level of the spending on each home variety by region i consumers while
the denominator represents the hypothetical level that would prevail if they consume thehome varieties and the imports equally. Because the imports are in fact more expensive
than the home products up to the transport cost, they spend more money on each homevariety than each foreign variety. Thus, θi−1 > 1 as long as the transport cost from region5

j to region i is positive (tj > 1). The inverse of θi measures this home product bias eﬀect.
With this interpretation, it is straightforward to explain how θi depends on tj and λi . First,dθi /dtj < 0: as the transport cost of the import declines, θi−1 decreases in correspondencewith the abatement of the home product bias eﬀect. When the transport cost becomes0 (tj = 1) at one extreme, θi−1 reaches 1: consumers spend their income equally among
the home products and the imports. Second, dθi /dλi > 0: as more varieties come to be
produced in the home region, the competition among the home varieties becomes severer,
which relatively reduces the spending on each home variety.
A system of equations (2) to (8) determines a short-run equilibrium, in which entrepreneurs’ distribution, λ, is given. Substituting (2) to (7) into the two equations in(8) and solving these equations simultaneously, we can obtain the equilibrium values of Y1
and Y2 :()σθi σθj − λj + λi t−σiYi =−σ . (i = 1, 2; j 6= i)(σθ1 − λ1 )(σθ2 − λ2 ) − λ1 λ2 t−σ
1 t2

(9)

Y1 and Y2 , in turn, give the equilibrium values of the demand variables, Xii ’s and Xij ’s,by (2) and (3):
Xii
Xij

=
=

(σ − 1)Yi(i = 1, 2)σnθi(σ − 1)t−σi Yj.
(i = 1, 2; j 6= i)
σnθj

(10)

This would be a good place to note that the transport costs aﬀect the wage rates throughsubstitution eﬀect between imports and domestic products. Suppose that ti rises, which
implies that the region i products become more expensive in region j (j 6= i). Then, theentrepreneurs in region i will be disappointed to ﬁnd that the demand for their export
decline, which causes the fall in their wage rate (an export price eﬀect), ceteris paribus.
At the same time, those in region j will be happy because the domestic demand for theirproducts expands due to the rise in the import price and therefore their wage rate rise (animport price eﬀect). To sum up, the rise in ti is followed by a fall in wi by the export priceeﬀect and a rise in wj by the import price eﬀect, other things being constant.
However, this is not the end of the story: in addition to these direct eﬀects, there areindirect eﬀects through the changes in the regional incomes. The fall in wi and the rise in wj
bring about a decrease in Yi and an increase in Yj , respectively. For wi , the decrease in Yi
works adversely but the increase in Yj works favorably. The overall direction of the change
depends on the sizes of the changes in respective regional incomes and the sensitivities of
the wage rate to these changes. When the two regions are suﬃciently ‘alike’, the decreasein Yi gives a greater impact on wi than the increase in Yj because of the home productbias eﬀect. In that case, wi tends to fall: the indirect regional income eﬀect reinforces the
direct eﬀects. However, if region j is much ‘bigger’ than region i, that is, if it has a quite6

large population and/or a big advantage in the transport costs (the transport cost to ship
the goods from that region to the other is much less expensive than the counterpart in the
opposite direction), the eﬀect of the increase in Yj dominates that of the decrease in Yi ; andtherefore the changes in the regional incomes result in the rise in wi .

2.3

Long-Run Equilibrium

In the long run, entrepreneurs move freely across the regions. The long-run equilibriumis the pair of prices and distribution of entrepreneurs for which they have no incentive tochange their locations. Let vi (λ) be the indirect utility for an entrepreneur living in region
i:
[
(
)]
1
(σ − 1)(nθi ) σ−1 σθj − λj + t−σσθi + λj t−σ
i
j
wi
[
] .
vi (λ) ==Piσn (σθ − λ )(σθ − λ ) − λ λ t1−σ t1−σ
1

1

2

2

1 2 1

(i = 1, 2; j 6= i)

(11)

2

Then, an interior distribution pattern λ ∈ (0, 1), for which at least some entrepreneurs arelocated in each region, is a long-run equilibrium pattern if v1 (λ) = v2 (λ). The agglomeration
pattern with λi = 1 is so if vi (1) ≥ vj (1) (i = 1, 2; j 6= i).
In addition, suppose that the equilibrium distribution pattern is perturbed by the relocation of inﬁnitesimally small numbers of entrepreneurs. If they cannot become better oﬀ,
the equilibrium pattern is considered to be stable. Formally, the equilibrium pattern λ isstable if and only ifv (λ) ≥ v2 (λ − ε) and v2 (λ) ≥ v1 (λ + ε)
 1
v1 (1) ≥ v2 (1 − ε) v (0) ≥ v (ε)
21when λ ∈ (0, 1)
when λ = 1
when λ = 0,

where ε > 0 is a number arbitrarily close to 0.
This condition can be expressed in a more convenient manner. First, recall that v1 (λ) =v2 (λ) for any equilibrium pattern with λ ∈ (0, 1). Therefore, v1 (λ) ≥ v2 (λ−ε) in the ﬁrst line
is equivalent to v2 (λ) ≥ v2 (λ − ε) at the equilibrium pattern. The similar argument applies
to the second inequality. Second, for the equilibrium pattern with λ = 1 (an agglomeration
pattern), we can distinguish two cases, the case with v1 (1) = v2 (1) and the case withv1 (1) > v2 (1). In the ﬁrst case, the same logic applies: v1 (1) ≥ v2 (1 − ε) is equivalent tov2 (1) ≥ v2 (1 − ε). In the second case, the requirement for the stability is automaticallysatisﬁed as long as v2 (λ) is continuous at λ = 1, which, it will be shown, holds true. The
similar argument can be used for the equilibrium pattern with λ = 0 (another agglomeration
pattern). Putting these results altogether, the stability condition for the equilibrium pattern7

λ can be rewritten as follows:
dv1 (λ)dv2 (λ)≤ 0 and≥ 0 holds when λ ∈ (0, 1), both


dλ
dλ


dv2 (1)at least either≥ 0 or v1 (1) > v2 (1) holds when λ = 1, and


dλ





 at least either dv1 (0) ≤ 0 or v1 (0) < v2 (0) holds when λ = 0.
dλ

3

(12)

Transport Sector

There are m ≥ 1 transport ﬁrms or carriers, which produce transport services. The unit isnormalized so that 1 unit of the services is necessary to ship 1 unit of the ﬁnal good fromone region to the other. This implies that the demand for the transport service from regioni to region j 6= i is equal to that for the import of the ﬁnal good in region j from region i,that is, λi nXij , which is expressed as a function of t1 , t2 and λ:()−σ
λi t−σi (σ − 1) σθi − λi + λj tj
Zi (t1 , t2 : λ) ≡ λi nXij =−σ ,(σθ1 − λ1 )(σθ2 − λ2 ) − λ1 λ2 t−σ
1 t2(i = 1, 2, j 6= i)(13)where (10) is used. On the other hand, the total supply of the service from region i to region∑m
j 6= i is equal to k=1 zik , where zik is the amount of the service in that direction provided
by the k-th carrier. The prices of the transport services, t1 and t2 , are determined so thatthe demand for each service becomes equal to its supply. In other words, they are given as
solutions to the following system: Z1 (t1 , t2 : λ)


=





 Z2 (t1 , t2 : λ) =

m
∑

z1k

k=1

m
∑

(14)
z2k .

k=1

Furthermore, the transport services are produced from the intermediate. It is convenient
to distinguish two components. One is the intermediate used as a ﬁxed input: each carrierneeds to use F units of the intermediate no matter what the production levels are. Thus,
it bears the ﬁxed cost whose amount is equal to F . The other is the intermediate used as avariable input. The number or the scale of the equipments necessary to ship the ﬁnal good,such as cargo ships, freight cars and cargo trucks, evidently depends on the amount of theshipment. The payment for them comprises a variable cost.
For this variable cost, furthermore, I impose an important assumption: there is a physicalrequirement that transport equipments eventually return to the place of their departure.
Then, as has been explained in the introduction by a simple illustrative example, the variable
cost of carriers depends on the amount of shipment in one of the two directions, the amountthat is larger than the other, namely, max[z1k , z2k ]. In addition, it is assumed that the variable8

cost is proportional to zik when zik ≥ zjk (j 6= i) with the proportionality constant being equal
to c.
Recalling that carriers receive pτi ≡ p(ti − 1) for 1 unit of the transport service fromregion i to region j 6= i, we can write down the k-th carrier’s proﬁt as
π k ≡ p(t1 − 1)z1k + p(t2 − 1)z2k − c max[z1k , z2k ] − F.

(15)

Each carrier chooses its level of production given the other carriers’ levels à la Cournot, thatis, it maximizes its proﬁt subject tozik + Zi−k = Zi (t1 , t2 : λ)(i = 1, 2),(16)

where Zi−k is the amount of the transport services from region i to region j 6= i produced∑by the other ﬁrms, i.e., Zi−k ≡ l6=k zil .
In its decision making, each carrier anticipates how its production levels aﬀect the transport prices. Totally diﬀerentiating each of the two equations in (14) and solving the derived
system simultaneously, we obtain



∂t1
∂t2


Z22
1 
 ∂z1k ∂z1k = ∂t1 ∂t2  Z00 −Z12∂z2k ∂z2k

−Z21 

,

Z11

(17)

where Zij ≡ ∂Zi (t1 , t2 : λ)/∂tj and Z00 ≡ Z11 Z22 − Z12 Z21 (i = 1, 2; j 6= i). Here, take alook at Zij more closely. Since Zi (t1 , t2 : λ) = λi nXij , changes in the transport prices aretransmitted to Zi (t1 , t2 : λ) only through Xij . As is clear from (3), t1 and t2 aﬀect Xij via
three channels: ﬁrst, directly; second, through Yj ; and ﬁnally, through Pj . It is not plausible,
however, that the carriers take into account such far-ﬂung eﬀects as those of the changes
in regional incomes and price indices. More appropriate description would be that they
consider only the direct eﬀects, or, in other words, that they are ‘regional income takers’
and ‘price index takers’. Under this assumption, they evaluate Zii at −σZi (t1 , t2 : λ)/ti
while Zij at 0 (i = 1, 2; j 6= i). If this is the case, we have



∂ti
ti


= −

k
σZi (t1 , t2 : λ)∂zi(i = 1, 2, j 6= i)

 ∂ti


 ∂z k = 0

(18)

j

by (17). The regional income and price index taker assumption is not only appropriate fora description of the real world but also helpful in keeping the model simple.
In the rest of the paper, furthermore, we will focus on the symmetric equilibria in which
all the carriers produce the same amounts of the transport services. Let us use the asterisk
to denote the variables at the symmetric equilibrium. By deﬁnition, we havem∗ zi∗ = Zi (t∗1 , t∗2 : λ)9(i = 1, 2)(19)

for given λ.
Finally, the number of carriers is determined by free entry and exit. To keep the modeltractable, I ignore the integer constraint. Then, the 0-proﬁt condition prescribes
(
)
(
)
[
]
p t∗1 −1 Z1 (t∗1 , t∗2 : λ)+p t∗2 −1 Z2 (t∗1 , t∗2 : λ)−c max Z1 (t∗1 , t∗2 : λ), Z2 (t∗1 , t∗2 : λ) −m∗ F = 0.
(20)

4

Symmetric Distribution Pattern

In this section, we examine the equilibrium at the symmetric distribution with λ = 1/2 andthe condition for a symmetry breaking.

4.1

Short-Run Equilibrium

First of all, let us derive the short-run equilibrium realized when the distribution of entrepreneurs is symmetric or almost symmetric. Without loss of generality, suppose thatλ ≥ 1/2. The analysis is made up of two steps. First, we consider the situation in which all
the competitors of one carrier, the k-th carrier, are producing the same amounts of the transport services in the following two senses; the same amounts among carriers (zil = zih , h 6= l)
and the same amounts with respect to the two directions of shipment (zil = zjl , j 6= i). It
will be shown that the best strategy of the k-th carrier is to do the same as the competitors
do. Second, because this result gives a justiﬁcation for us to limit our attention to the casein which the amounts of the services produced by each ﬁrm are equal with respect to thedirections of shipment, we solve the k-th carrier’s proﬁt maximization problem with theconstraint of zik = zjk . The solutions are then evaluated under the supposition that all the
carriers are symmetric.
Let us begin with the ﬁrst step. All the carriers except the k-th produce z 0 units of z1
and z 0 units of z2 . Suppose that the shipment of the k-th carrier is not balanced: without
loss of generality, we assume that z1k < z2k . Then, can it raise the proﬁt by increasing z1k ?
Using (18), we obtain
∂π k
∂z1k
∂π k
∂z2k

]
[ {
}
z1k
0
−1
= p t1 1 −
σZ1 (t01 , t02 : λ)
[ {
]
}
k
z
2
= p t02 1 −− 1 − c,σZ2 (t01 , t02 : λ)(21)

where t01 and t02 represent particular levels of transport prices corresponding to this unbalanced situation. Because z1k < z2k , z2k becomes a capacity constraint and thus increasing z2k
involves an additional marginal cost, c. This is why ∂π k /∂z2k contains the negative term of−c while ∂π k /∂z1k does not.
10

The market clearing prescribes
Zi (t01 , t02 : λ) = zik + (m − 1)z 0(22)

because each of m − 1 carriers produces z 0 units (see (16)). Using this, we can show that
∂π k /∂z1k always exceeds ∂π k /∂z2k as long as λ is suﬃciently close to 1/2. Thus, the carrier
can raise its proﬁt by either enhancing the production of z1k or reducing that of z2k . Thisimplies that it chooses the production levels with z1k = z2k .
Lemma 1. Suppose that all the carriers except the k-th produce z1 = z2 = z 0 units of thetransport services. Then, there is ε > 0 such that the k-th carrier chooses the productionlevels with z1k = z2k for any λ ∈ [1/2, 1/2 + ε].

The proof is relegated to the appendix.
Because the k-th carrier chooses z1k and z2k with z1k = z2k , its problem is to maximize the
proﬁt given by (15) with the constraint that z1k = z2k . It is straightforward to derive the ﬁrst
order condition. Furthermore, recall that we focus on the symmetric equilibrium in which
all the carriers produce the same amounts of the transport services, that is, z1∗ units of z1
and z2∗ units of z2 . Applying (19) and z1∗ = z2∗ to the ﬁrst condition yields[]
2σ + c(σ − 1) mSSS
t1 + t2 =,σmS − 1(23)

where the superscript S refers to the equilibrium at the symmetric or almost symmetric()()
distribution pattern. What is more, because Z1 tS1 , tS2 : λ = Z2 tS1 , tS2 : λ (see (19)), (13)implies that[( )σ( )σ ]()(σ − 1) (1 − λ)2 tS1 − λ2 tS2+ σλ(1 − λ) tS1 − tS2 = 0.

(24)

Finally, the 0-proﬁt condition, (20), is reduced to[] ()
2σ + c(σ − 1) Z1 tS1 , tS2 : λ − F mS (σ − 1)(σmS − 1) = 0,

(25)

(
)
(
)
where Z1 tS1 , tS2 : λ = Z2 tS1 , tS2 : λ is used again. Those three equations give the equilibrium prices and the equilibrium number of the carriers. Thus, we have established thefollowing proposition.
Proposition 1. There is ε > 0 such that there exists an equilibrium with z1k = z2k = z S for
all k and for any λ ∈ [1/2, 1/2 + ε]. The equilibrium prices and the equilibrium number ofthe carriers are given as the solutions of the system consisting of (23) to (25).

11Next, let us examine the special case of the symmetric distribution pattern, λ = 1/2.
First of all, it immediately follows from Proposition 1 and (23) that there exists a unique[]
2σ + c(σ − 1) m̄S
(
)
t̄ =
> 1,
2 σ m̄S − 1equilibrium with

S

(26)

where t̄S and m̄S represent the values of tS and mS when λ = 1/2, respectively. It is
straightforward to see that t̄S decreases with m̄S : when the sector is more competitive, theequilibrium price becomes lower. This is a standard result of the Cournot oligopoly model.
Furthermore, (25) gives m̄S as a solution to[]2 2σ + c(σ − 1)
− F m̄S = 0,
2 + σ m̄S (σ − 1)(2 + c) + 2α(σ − 1)(σ m̄S − 1)

(27)

[

{
} ]σ
m̄S 2σ + c(σ − 1)α≡.
2(σ m̄S − 1)

whereThus, we have established the following result:Corollary 1. When λ = 1/2, there is a unique equilibrium with t1 = t2 = t̄S and m = m̄S .
The two variables are implicitly determined by (26) and (27).

It is easily veriﬁed that
dm̄S
< 0,
dc

dm̄S
< 0 and
dF

dm̄S
< 0.
dσ

(28)

Notice that t̄S decreases with m̄S , that ∂ t̄S /∂c > 0 and that ∂ t̄S /∂F = 0. Therefore, it
immediately follows from (28) thatdt̄S
> 0 and
dc

dt̄S
> 0.
dF

(29)

The direction of the change in t̄S with respect to σ is ambiguous. Numerical simulation,
however, suggests the following. When c is relatively large, t̄S increases with σ (see Fig.
1 (a), which describes the case with c = 1 and F = 0.01). When c is relatively small, t̄S
ﬁrst increases and then decreases (see Fig. 1 (b), which depicts the case with c = 0.1 and
F = 0.01).

4.2

Long-Run equilibrium

As has been mentioned, a distribution pattern is supported by a long-run equilibrium if
v1 (λ) = v2 (λ), provided that λ ∈ (0, 1). It is straightforward to see that this is satisﬁed
at the symmetric distribution pattern, λ = 1/2, and therefore, it is a long-run equilibrium
pattern. Thus, the rest of this section is devoted to the examination of the stability of the12symmetric distribution pattern. Henceforth, the argument λ of vi ’s is often suppressed forshortness’ sake as long as it yields no confusion, and, in addition, all the derivatives areevaluated at λ = 1/2 unless otherwise mentioned.
Before proceeding to the analysis of stability, we need to introduce one additional assumption. Recall that the stability is determined by whether an entrepreneur (or more
preciously, an inﬁnitesimally small mass of entrepreneurs) becomes better oﬀ when she relocates. Theoretically, her relocation (change in λ) aﬀects the equilibrium number of thecarriers operating in the transport sector, mS (through (25)), which in turn inﬂuences theprices of the transport services, tS1 and tS2 (through (23) and (24)). Notwithstanding, I treat
mS as a constant as far as the stability of the equilibrium distribution pattern is concerned.
There are three justiﬁcations. First, the entry to and exit from the transport sector often
take a long period of time: they will take place after all the adjustments of consumers’
locations have ﬁnished. Therefore, to judge the stability of the equilibrium pattern, we canfocus on the period of time that is not too long for the carriers’ entry and exit to occur
but is long enough for the entrepreneurs’ location changes to be completed. Second, it is
not likely that an entrepreneur, when making the decision about the relocation, takes into
account the fact that the number of the active carriers changes if she relocates. Rather,
she would take it for granted that her action gives no impact on it. Third, the assumption
makes the analysis much more tractable.
Applying this assumption, we can easily determine the directions of changes in the transport prices at or near the symmetric distribution. Totally diﬀerentiating (23) and (24)provided that mS is ﬁxed yields( )σ4(σ − 1) tS[( )σ−1 ] > 0
σ 1 + (σ − 1) tS

dtS1
dλ

=

dtS2
dλ

dtS
= − 1 <0
dλ

and
(30)

at λ = 1/2. The result is because as a region gets bigger, the demand for the shipment fromthat region to the other swells while that in the opposite direction shrinks.
Now, we can show that the condition for the stability represented in the ﬁrst line of (12)is rewritten in terms of the relative levels of indirect utilities, v̂(λ) ≡ v1 (λ)/v2 (λ) (in thispaper, the circumﬂex always refers to the ratios of the variables with respect to region 1 tothose with respect to region 2):
Lemma 2. The symmetric distribution pattern is a stable equilibrium pattern if and only ifdv̂(1/2)/dλ ≤ 0.

The proof is relegated to the appendix.
13

Now, we are ready to examine the stability condition, dv̂(1/2)/dλ ≤ 0. Let ŵ ≡ w1 /w2and P̂ ≡ P1 /P2 . Then, (11) implies thatdv̂dŵ dP̂
=
−
,
dλ
dλ
dλ

(31)

because v̂ = ŵ = P̂ = 1 at λ = 1/2. Since our main concern is on the question of
how the possibility of the asymmetric transport costs aﬀects the stability, it would be worth
separating the eﬀects of the induced changes in the transport costs. To do so, it is convenient
to look at the change in the relative wage rate and the relative price index one by one.
Let us begin with the change in the wage rates. Equations (2) - (7), along with the facts
that Ŷ ≡ Y1 /Y2 = 1, that θ̂ ≡ θ1 /θ2 = 1 and that dtS2 /dλ = −dtS1 /dλ (see (30)) at λ = 1/2,
imply that[
( S )σ {}]()( S )σ
¯
¯
S
S
2σ
t
+
t
σ
+
(σ
−
1)t
t
− 1 dŶ ¯¯dŵdθ̂ ¯¯
dtS1( )σ, (32)=−+−( ) [( )σ ]dλdλ
dλ ¯tS const dλ ¯tS const
1 + tSβ tS tS 1 + tS
1
1

with

¯
dŶ ¯¯dλ ¯tS const=¯dθ̂ ¯¯dλ ¯tS const

=

1

1

( )σ
8 tS (1 + tS )( )[( )σ ] > 0,β tS tS + tS[( )]σ
4 tS − 1( )σ > 0,tS + tS

where all the derivatives and tS are evaluated at λ = 1/2 and( )( )σβ tS ≡ 1 + σtS + (σ − 1) tS > 0.
Equation (32) indicates that the impact of the change in the distribution of entrepreneurson the relative wage rate is decomposed into three parts.
First, it aﬀects the relative wage rate through the transport costs. Suppose that λ rises.
As we have seen in (30), t1 rises and t2 falls, which provokes the substitution between imports
and domestic products. The rise in t1 , on the one hand, induces the region 2 consumers
to substitute the domestic products for the imports from region 1, which results in the fallin w1 (the export price eﬀect in terms of region 1) and the rise in w2 (the import priceeﬀect in terms of region 2). The fall in t2 , on the other hand, promotes the substitutionof the imports for the domestic products in region 1, which again lowers w1 (the importprice eﬀect in terms of region 1) and enhances w2 (the export price eﬀect in terms of region2). Consequently, the relative wage rate declines. As we have explained before, however,there is also an indirect eﬀect through the changes in the regional incomes (regional income
eﬀect). The two regions are suﬃciently ‘alike’ in the sense explained in Section 2.2, theregional income eﬀect reinforces the direct eﬀects just mentioned. Hence, the total eﬀect on
the relative wage rate is negative, which is captured by the ﬁrst term in (32).
14

Second, the distribution of entrepreneurs aﬀects the regional incomes even if the transport
cost remained the same. The relative regional income rises as a result of the increase in λ.
This is because of the demand linkage eﬀect: when a region has more population, the demand
for the varieties produced in that region increases. This eﬀect is represented by the term¯
dŶ /dλ¯tS const .
1

Third and ﬁnally, the change in the distribution of entrepreneurs is transmitted to thewage rates through the change in the size of the home product bias eﬀect even if the transport
costs remain unchanged. As more varieties come to be produced in a region, competition
among them becomes intensiﬁed. This counteracts the home product bias eﬀect and raises θin that region. Therefore, the increase in λ results in the rise in θ̂. This eﬀect is represented¯
by dθ̂/dλ¯tS const .
1

Next, we turn our eyes to the change in the price index. It is straightforward to see
]
[
}
{( )
dP̂
2
dtS1
σ
S
S
[
< 0,
(33)
=−
− t + (σ − 1)( )σ ] 2 tdλdλ(σ − 1) tS + tS
where again all the derivatives are evaluated at λ = 1/2. Two factors aﬀect the ratio of theprice index. The ﬁrst is a direct eﬀect: as more entrepreneurs choose to locate in region 1,
the mass of varieties produced there increases, which lowers the relative price index. The
second is the eﬀect through the change in the transport costs. When λ increases, t1 rises
and t2 falls. As a result, the region 1’s import from region 2 relatively falls compared tothat of the region 2’s import from region 1; and, consequently, the ratio of the price indicesfalls. Both eﬀects are negative and therefore the total eﬀect is also negative.
Substituting the relevant expressions to (31) and arranging it yields][( S)( S ) dtS1
dv̂
2
[
= ( )
( )σ ] 2γ t + (σ − 1)δ t dλdλβ tS (σ − 1)(1 + tS ) t + tS

(34)

where all the derivatives are evaluated at λ = 1/2, and
][
]
[( )
( )σ ]( )[( )σ
( )
( )σ
σ
,
γ tS
≡ 2 tS (σ − 1)(1 + tS ) tS − 1 + β tS tS − tS σ + (2 − σ) tS( S)( S )σ [
]
(
)
[
]
2σ−1
δ t
≡ 1−σ tσ + (σ − 2)tS − 2 tS
σ 2 + (σ − 1)2 tS .
If dtS1 /dλ were 0, the equation dv̂/dλ would give a conventional break point for the( )symmetric distribution pattern. In that case, the pattern would be stable if γ tS ≤ 0( )and unstable otherwise. Notice that γ tS > 0 for tS that is suﬃciently close to 1. Thisis because γ(1) = 0 and dγ(1)/dtS = 8σ(σ − 1) > 0. On the other hand, we note that( )( )γ tS < 0 for suﬃciently high tS , because γ tS approaches negative inﬁnity as tS goesto inﬁnity. Thus, the symmetric pattern is, for a given value of σ, stable if tS given by(26) becomes suﬃciently high while it is unstable if tS becomes suﬃciently small. The
15

conclusion that the high (low, resp.) transport cost tends to make the symmetric patternstable (unstable, resp.) is along the line of the conventional literature of the new economic
geography.
What is missing in the literature is, however, the term containing dt1 /dλ at the square
brackets in (34). This term represents the eﬀect through the change in the transport cost due
to the proﬁt maximizing behaviors of the carriers. Since dt1 /dλ > 0, this eﬀect works toward( )the direction to stabilize the symmetric pattern if δ tS < 0 and the direction to destabilize
( )
( )
( )
it if δ tS > 0. Several observations follow with respect to δ tS . First, δ tS = (tS −1)2 > 0( )for any tS at σ = 1. By continuity, it implies that δ tS > 0 for σ suﬃciently close to 1.
Therefore, the transport cost eﬀect works against the stabilization when σ is suﬃciently( )small. Second, δ tS < 0 for any σ > 2.2 Thus, the eﬀect stabilizes the symmetric pattern( )whenever σ > 2. Third, notice that dδ tS /dtS > 0. We have already seen that as c and/or
F rises, the equilibrium t̄S rises (see (29)) because fewer carriers come to operate in the( )sector. Consequently, δ tS increases for given σ: as a result of the rise in c and/or F , it
becomes more likely that the transport eﬀect works to destabilize the symmetric pattern.
We have established the following.
Proposition 2. i) The transport cost eﬀect works to stabilize, rather than to destabilize,
the symmetric pattern if σ > 2; and furthermore, it works to destabilize the pattern if σ is
suﬃciently small.
ii) As c and/or F rises, it becomes more likely that the transport cost eﬀect works to destabilize the symmetric pattern.

Finally, what can we say about the overall stability? To examine the stability in termsof the three parameters, c, F and σ, is not straightforward because the key equations arehighly nonlinear and some of them give the solution only implicitly. Thus, the best strategy
would be to rely on some numerical analysis. Fig. 2 shows how c aﬀects dv̂/dλ for a given
value of F = 0.01 and σ. The three panels describe the cases with diﬀerent values of σ,
namely, σ = 1.5, σ = 2 and σ = 3. When σ is low (σ = 1.5), the symmetric pattern isstable for small c but unstable for large c, that is, it ‘breaks’ when c is greater than a criticalvalue. When σ is high (σ = 2 or σ = 3), the symmetric pattern is stable even for suﬃciently
small c: it never breaks. Thus, concerning the stability of the symmetric pattern, the size ofparameter c matters only when σ is low. The numerical simulation suggests that the sameconclusion applies to parameter F : its size matters only when σ is low; and in that case,the higher F is, the more likely it is that the symmetric pattern becomes unstable.

2 This is because 1 < σ(tS )σ [σ + (σ − 2)tS ].

16

5

Agglomeration

In this section, let us focus on the case in which almost all the entrepreneurs are located inregion 1, that is, λ is suﬃciently close to 1. The other polar case with λ being suﬃciently
close to 0 is similarly analyzed. The short-run equilibrium is characterized ﬁrst; and thenits sustainability is discussed.

5.1

Short-Run Equilibrium

When λ is suﬃciently close to 1, we expect that carriers produce the transport service fromregion 1 to region 2 more than that from region 2 to region 1 since most of the varieties areproduced in region 1, which implies that almost all the demand for transport services comesfrom the consumers in region 2: most of the variants are produced in region 1. Indeed, itturns out that there is an equilibrium in which all the carriers produce z1∗ and z2∗ units withz1∗ > z2∗ for given λ.
This is shown through several steps. First, it is shown that no carrier has an incentive
to choose z1 and z2 with z1 < z2 when all the other carriers produce z10 and z20 units withz10 > z20 . Second, the proﬁt maximization problem of a representative carrier, which is
subject to the constraint z1 > z2 , is solved. Here, I focus on the case of symmetric carriersin the sense of (19): any of them produces z1∗ and z2∗ units. Third and ﬁnally, it is veriﬁedthat the solution to the maximization problem indeed satisﬁes z1∗ > z2∗ .
Let us begin with the ﬁrst step by examining the proﬁt maximization of a single carrier,say the k-th carrier, when every other carrier produces z10 and z20 units, given λ. Supposethat the k-th carrier produces z1k and z2k units with z1k ≤ z2k . Its marginal proﬁts are
given by (21), because z2k becomes the capacity constraint. Furthermore, Zi (t01 , t02 : λ) =zik + (m − 1)zi0 , where t01 and t02 are the prices realized in this speciﬁc situation. In addition,
let t01 and t02 be the prices when all the carriers including the k-th produce z10 and z20 units.
If ∂π k /∂z1k > ∂π k /∂z2k for any z1k and z2k with z1k ≤ z2k , then the k-th carrier can raise its
proﬁt by relatively increasing the production of z1 compared to that of z2 . In this case,therefore, the carrier is inclined to choose z1k and z2k with z1k > z2k , which I prove in thefollowing lemma.
Lemma 3. Suppose that all the carriers except the k-th produces z10 and z20 units of thetransport services with z10 > z20 . Then, as long as t01 > t02 , the k-th carrier chooses the
production levels with z1k > z2k .

The proof is tedious and relegated to the appendix.

17

Now, let us proceed to the second step. Because Lemma ?? guarantees, as long as t01 > t02 ,that the k-th carrier has no incentive to choose z1k and z2k with z1k ≤ z2k , it suﬃces to solve
the proﬁt maximization problem with the constraint z1k > z2k . The necessary condition is
given as
∂π k
∂z1k
∂π k
∂z2k

[ {
]
}
z1k
= p t1 1 −−1 −c = 0
σZ1 (t1 , t2 : λ)
]
[ {
}
z2k
−1
= 0.
=
p t2 1 −σZ2 (t1 , t2 : λ)Assuming the symmetry among carriers, we can obtain the following equations:[]σ + c(σ − 1) mAt∗1 = tA≡1
σmA − 1σmAt∗2 = tA≡.
2
σmA − 1

(35)

(36)

Therefore, we have indeed t∗1 > t∗2 , which is assumed in Lemma ?? as t01 > t02 . Here, it isworth noting that parameter c, the marginal cost of the binding transport service, plays acritical role in the determination of the transport prices. At a limit case with c = 0, the twotransport prices become equal. As it rises, the divergence between them increases. Thus, ccorresponds to the degree of the asymmetry in the transport prices.
The last step is to verify that the solution is consistent with the supposition that z1∗ > z2∗ .
Take a look at the limit case with λ = 1. The total amounts of the transport services are
given asσ−1,Z2 (t∗1 , t∗2 : 1) = 0.
(37)
σt∗1
Since Z1 (t∗1 , t∗2 : 1)/m∗ > Z2 (t∗1 , t∗2 : 1)/m∗ , we have z1∗ > z2∗ for λ = 1. Because no regime
Z1 (t∗1 , t∗2 : 1) =change occurs, as has been discussed in Lemma ??, the solution to the maximization problem
is continuous. Consequently, for λ suﬃciently close to 1, it must still hold that z1∗ > z2∗ .
These steps establish the following result:
Proposition 3. There is ε > 0 such that for any λ ∈ [1 − ε, 1], there exists an equilibriumwith z1k = z1A and z2k = z2A for all k with z1A > z2A . The equilibrium prices satisfy (36).
Finally, let us examine the special case of the agglomeration, namely, the case with λ = 1.
√
Substituting (36) and (37) into (20) yields m = m̄A ≡ 1/µ, where µ ≡ F σ is a parameter
which measures the size of the ﬁxed cost. The transport prices are, consequently, given by
σ + c(σ − 1)
σ−µ
(38)
σ
A
t2 = t̄2 ≡.
σ−µCorollary 2. When λ = 1, the equilibrium prices and the number of carriers are given by
t1

= t̄A
1 ≡

A
A
t̄A
1 and t̄2 , and m̄ , respectively.

18

5.2

Long-Run Equilibrium

As has been mentioned earlier, the agglomeration pattern with λ = 1 is supported by thelong-run equilibrium if v1 (1) ≥ v2 (1), and if so, it is stable if at least either dv2 (1)/dλ ≥ 0
or v1 (1) > v2 (1) holds. Recall that the condition dv2 (1)/dλ ≥ 0 matters only when v1
and v2 happen to become equal to each other at the agglomeration pattern. Because there
is no reason to believe that such is an important and often observed situation in the realworld, I rather focus on the case with v1 (1) > v2 (1). In short, the case with v1 (1) =v2 (1) is disregarded as a singular situation. According to the terminology in the literature,
furthermore, I say that the agglomeration pattern is sustainable when it is supported as astable equilibrium pattern, that is, when v1 (1) > v2 (1), and unsustainable when v1 (1) <v2 (1). To put it another way, it is sustainable when v̂(1) > 1 and unsustainable whenv̂(1) < 1, where v̂(1) turns out to be equal to( A )σ(1 + t̄A )
σ t̄A
1 t̄2
( A )σ ( A )σ1v̂(1) =.
A
1 + σ t̄1 + t̄1t̄2 (σ − 1)

(39)

Our goal is to ﬁnd how this sustainability is aﬀected by parameters, c, σ and µ. To
begin with, the next result immediately follows from (39) (the proof is again relegated tothe appendix):
Proposition 4. i) When σ ≤ 2, the agglomeration pattern is sustainable for any c and µ.
ii) When σ > 2, it is unsustainable for suﬃciently large c.

The reason why suﬃciently large c makes the agglomeration pattern unsustainable is
not obvious. Indeed, c is related to the sustainability in a quite complicated way. The ﬁrst
Athing to note is that it aﬀects the sustainability only through t̄A
1 (as c rises, t̄1 rises as (38)indicates): (39) shows that v̂(1) does not depend on c directly. Therefore, in order to answer
the question of how c aﬀects the sustainability, it suﬃces to study the impact of the changeAin t̄A
1 on v̂(1), given that t̄2 is ﬁxed.

The impact is decomposed to that on the wage rates and that on the price indices, as
before. On the one hand, the relative wage rate, ŵ ≡ w1 /w2 , decreases with t̄A
1 . Too see
this, note that
w1

w2

=

=

)
(
1
1 + t̄A
Y2
1
Y1 + A = Anσt̄1
nt̄1 (σ − 1)][[
] (t̄A )σ−1 σ − 1 + (t̄A )−σ (t̄A )−σ (1 + σ t̄A )112
1
1 ( A )σ−1Y1
Y2 + A σ =t̄1.
nσnσ(σ − 1)(t̄2 )(40)

Now, we suppose that t̄A
1 rises. First, this directly aﬀects the wage rates through thesubstitution between the import and the domestic products, namely, through the export
19

price eﬀects and the import price eﬀects: w1 falls by the export price eﬀect while w2 risesby the import price eﬀect. Second, these changes alter the regional incomes: Y1 declines
and Y2 increases. These eﬀects are shown by the terms between the two equality signs ineach equation of (40): the direct eﬀects (the export price eﬀect for w1 and the import priceeﬀect for w2 ) come through the change in t̄A
1 and the indirect eﬀects (the regional income
eﬀects) come through the changes in Y1 and Y2 . For w1 , the sum of the export price eﬀect
and the eﬀect of the decline in Y1 dominates the eﬀect of the rise in Y2 and therefore thetotal eﬀect is negative: w1 decreases with t̄A
1 . For w2 , however, the direction of the overallchange is ambiguous. When σ > 2, the substitution is so large that the sum of the importprice eﬀect and the eﬀect of the rise in Y2 dominates the eﬀect of the decline in Y1 , and as
a result, w2 increases. When σ is suﬃciently close to 1, to the contrary, the decline in Y1
gives a severe impact and consequently, w2 declines. Yet even if w2 declines, the scale ofthe decline is relatively small compared to that of the decline in w1 . Therefore, ŵ ≡ w1 /w2decreases.
A
On the other hand, the relative price index, P̂ ≡ P1 /P2 , also decreases with t̄A
1 . As t̄1

rises, varieties, all of which are imported from region 1, become more expensive in region1

σ−1 /(σ − 1), rises. Because t̄A does
2. Consequently, the price index in region 2, P2 = σ t̄A1n
11

not aﬀect the price index in region 1, P1 = σn σ−1 /(σ − 1), P̂ falls.
To sum up, as a result of the rise in t̄A
1 , both the relative wage rate and the relative
price index fall. Because the two forces work on the relative indirect utility, v̂(1), in the
opposite directions, therefore, this information is not suﬃcient to determine whether v̂(1)increase or not. However, it can be veriﬁed that when t̄A
1 is large, the relative wage rate
declines more rapidly than the relative price index, which results in the decline of v̂(1).
Then, for suﬃciently large t̄A
1 , that is, for suﬃciently large c, v̂(1) becomes lower than 1 andthe agglomeration pattern becomes unsustainable.
Some numerical simulation analyses may be helpful for us to obtain a clearer image ofthe impacts of c on the sustainability. Fig. 3 shows how v̂(1) changes as c increases, for
given values of σ and F = 0.1. The four panels describe the cases with diﬀerent values of
σ. First, when σ < 2 (the ﬁrst panel shows the case with σ = 1.8), v̂(1) increases withc and approaches positive inﬁnity. Because v̂(1) > 1 at c = 0, the agglomeration patternis sustainable for any c ≥ 0. Second, when σ is of an intermediate size (the second panel
describes the case with σ = 2.2), v̂(1) > 1 at c = 0 and v̂(1) ﬁrst increases and then decreases.
><
Because it eventually goes to 0, there is a critical value c∗ such that v̂(1) = 1 if c = c∗ .
<>Consequently, the agglomeration pattern is sustainable for c < c∗ while unsustainable forc > c∗ . Finally, when σ is large, v̂(1) decreases with c. If σ is moderately large (the casewith σ = 3 is described by the third panel), v̂(1) > 1 at c = 0. Therefore, there is a critical

20

value c∗ explained above. If σ is too large (the case with σ = 7 is depicted by the last panel),
however, v̂(1) < 1 at c = 0. Thus, v̂(1) < 1 for any c ≥ 0: the agglomeration pattern isunsustainable for any c ≥ 0.
We can obtain similar results for the change in F . Fig. 4 shows that, given c = 5, theagglomeration pattern is sustainable for any F ≥ 0 when σ < 2 (see the ﬁrst panel withσ = 1.8), that it is sustainable for F < F ∗ but unsustainable for F > F ∗ when σ is of anintermediate size (see the second panel with σ = 2.3), and that it is unsustainable for any
F when σ is large (see the last panel with σ = 3).
Finally, Fig. 5 describes the loci of (F, c) that makes v̂(1) equal to 1 for given values of
σ. Because ∂v̂(1)/∂c < 1 at v̂(1) = 1, v̂(1) < 1 at the area above each locus while v̂(1) > 1
at the area below it. Therefore, the agglomeration pattern is more likely to be sustainable
when c is lower. This result is consistent with our earlier ﬁnding. Furthermore, the ﬁgure
also suggests that lower σ is more likely to be associated with the sustainability. Lastly,
the eﬀect of F is not necessarily monotone. To see this, consider the case with σ = 3 andc = 0.65, for instance. For small F (F < 0.0211) and large F (F > 0.171), we have v̂(1) > 1:
the agglomeration pattern is unsustainable. For the intermediate values of F , however,v̂(1) < 1 and therefore, it is sustainable. Nonetheless, this non-monotonicity occurs only
when c is suﬃciently small: when c is large enough, v̂(1) < 1 for any F .
The next observation summarizes these simulation results.
Observation 1. As the varieties become more substitutable (σ becomes higher) and/or thetransport costs become more asymmetric (c becomes higher), one ﬁnd sustaining the agglomeration pattern become more diﬃcult.

6

Concluding Remarks

This paper has explored the impacts of the asymmetry of transport costs with regard to
the directions of shipments upon economic geography. In doing so, we have focused on theasymmetry arising from the optimizing behavior of transport ﬁrms: they charge diﬀerent
prices for the transport services involving shipments in diﬀerent directions in response to thediﬀerence in the marginal costs of shipment and the price elasticities of the demands. Twocases are studied: the case where the economic activities are distributed equally between
regions and the case where they are concentrated in one region. It has been shown that higher
elasticity of substitution in consumers’ preference makes the symmetric distribution patternmore likely to occur while the agglomeration pattern less likely to occur. Furthermore, thehigher the marginal cost of shipment associated with the binding capacity, the less likely21

it is that the symmetric distribution pattern and agglomeration pattern are supported as astable long-run equilibrium patterns.

22References[1] Behrens, K. (2004) “On the location and ‘lock-in’ of cities: geography vs. transportation
technology,” mimeo.
[2] Behrens, K., C. Gainé and J.-F. Thisse. (2006) “Is the regulation of the transport sector
always detrimental to consumers?” mimeo.
[3] Bougheas, S., P. O. Demetriades and E. L. W. Morgenroth. (1999) “Infrastructure,transport costs and trade,” Journal of International Economics, 47, 169-189.
[4] Forslid R. and G. I. P. Ottaviano. (2003) “An analytically solvable core-periphery model,”Journal of Economic Geography, 3, 229-40.
[5] Fujita, M., P. Krugman and A. J. Venables. (1999) The Spatial Economy: Cities, Regions, and International Trade, The MIT Press, Cambridge.
[6] Fujita, M. and J.-F. Thisse. (2000) “The formation of economic agglomerations: oldproblems and new perspectives,” in J.-M. Huriot, J.-F. Thisse (Eds.), Economics ofCities, Cambridge University Press, Cambridge, pp. 3-73.
[7] Fujita, M. and J.-F. Thisse. (2002) Economics of Agglomeration, Cambridge University
Press, Cambridge.
[8] Mori T. and K. Nishikimi. (2002) “Economies of transport density and industrial agglomeration,” Regional Science and Urban Economics, 32, 167-200.
[9] Mun S. and S. Nakagawa. (2005) “Cross-border transport infrastructure and aid policies,” mimeo.
[10] Takahashi, T. (2006a) “Economic geography and endogenous determination of transportation technology,” Journal of Urban Economics, 60, 498-518.
[11] Takahashi, T. (2006b) “Is the transport sector too large? Welfare analysis of the trademodel with a transport sector,” mimeo.

23

Appendix
Proof of Lemma 1.
Suppose that z1k < z2k . This immediately implies that Z1 (t01 , t02 : λ) < Z2 (t01 , t02 : λ) (see(22)). However, (13) implies that
1 + (σ − 1)(t02 )σ + σt02
Z1 (t01 , t02 : 1/2)=.
00Z2 (t1 , t2 : 1/2)
1 + (σ − 1)(t01 )σ + σt01
Therefore, t01 > t02 in a suﬃciently small neighborhood of λ = 1/2, that is to say, there exists
ε for which t01 > t02 for any λ ∈ [1/2, 1/2 + ε]. On the other hand, z1k < z2k together with(22) implies that 1 − z1k /σZ1 (t01 , t02 : λ) > 1 − z2k /σZ2 (t01 , t02 : λ). These ﬁndings establishthat ∂π k /∂z1k > ∂π k /∂z2k for any λ ∈ [1/2, 1/2 + ε]. Therefore, the carrier can always raise
its proﬁt either by augmenting z1k or by reducing z2k . Consequently, the proﬁt is maximizedat (z1k , z2k ) with z1k ≥ z2k . By a similar argument, however, we can prove that the proﬁt
maximization prescribes z1k ≤ z2k . Hence, z1k = z2k at the equilibrium. QED

Derivation of (28).
Notice that totally diﬀerentiating (27) yields
km dm̄S + kc dc + kF dF + kσ dσ = 0,
where
km

≡

kc

≡

]
[
][
−F − 2ζ1 σ(m̄S )−1 (σ − 1) 2σ + c(σ − 1) 2α(m̄S − 1) + m̄S (2 + c) < 0,[]
−4ζ1 (σ − 1)(σ m̄S − 1) 1 + α(σ − 1)2 < 0,kF≡−m̄S < 0,kσ≡−ζ1 (ζ2 + ζ3 + ζ4 ) < 0,

and

with
ζ1
ζ2
ζ3
ζ4

]−2
>0
2 − 2α(σ − 1) + σ m̄S (σ − 1)(2 + c + 2α)[]≡ 2(2 + c) 2(m̄S σ 2 − 1) + cm̄S (σ − 1)2 > 0,[{}]
≡ 4α 2(σ − 1) + 2σ 2 (m̄S − 1) + c(σ − 1) m̄S (σ − 1) + σ(m̄S − 1) > 0,[][
] m̄S 2σ + c(σ − 1)
≡ 4α(σ − 1)(σ m̄S − 1) 2σ + c(σ − 1) ln> 0.
2(σ m̄S − 1)
≡

[

and

Here, the inequalities hold because σ > 1 and m̄S ≥ 1.
Proof of Lemma 2.
What we have to show is that dv1 /dλ ≤ 0 and dv2 /dλ ≥ 0 if and only if dv̂/dλ ≤ 0. The‘only if’ part is self-evident. To prove the ‘if’ part, I show that dv1 /dλ = −dv2 /dλ. For
that purpose, it is convenient to treat vi (λ) represented by (11) as functions of t1 , t2 andλ. First, the short-run equilibrium condition (23) implies that dtS1 /dλ = −dtS2 /dλ. Second,v1 and v2 are symmetric with respect to t1 and t2 in the sense that ∂v1 /∂t1 = ∂v2 /∂t2 and24∂v1 /∂t2 = ∂v2 /∂t1 . Third and ﬁnally, v1 and v2 are also symmetric with respect to λ1 and
λ2 , that is, ∂v1 /∂λ = ∂v2 /∂(1 − λ) = −∂v2 /∂λ. Consequently, we have
)
(
dv1
dv2
dt1 ∂v1dt2 ∂v1∂v1
dt1 ∂v2∂v2
dt2 ∂v2
=−
=
+
+
=−
+
+
.
dλ
dλ ∂t1dλ ∂t2∂λdλ ∂t2dλ ∂t1∂λdλThen, it is straightforward to see that dv1 /dλ has the same sign as dv̂/dλ. QEDProof of Lemma 3.
Suppose that z1k ≤ z2k . The equations (21) imply that
[ {
}
{
}]
∂π k
z2k
∂π k
z1k
0
0
− t2 1 −+ c.
− k = p t1 1 −
σZ1 (t01 , t02 : λ)σZ2 (t01 , t02 : λ)∂z1k∂z2
On the one hand, the term in the ﬁrst braces is greater than that in the second. This isbecause the two inequalities z10 > z20 and z1k ≤ z2k imply thatz0
z1k
< 10 ,
k
z2
z2

(41)

which is a necessary and suﬃcient condition for z1k /σZ1 (t01 , t02 : λ) < z2k /σZ2 (t01 , t02 : λ). To
prove that t01 > t02 , on the other hand, notice that (41) implies that
Z1 (t01 , t02 : λ)(m − 1)z10 + z1k
Z1 (t01 , t02 : λ)
mz10
=
.
=
<
0
0
0
0
k
Z2 (t1 , t2 : λ)mz2Z2 (t01 , t02 : λ)(m − 1)z2 + z2

(42)

Now, suppose that t01 ≤ t02 . If t01 > t02 , it turns out that
]
[
0
0 σ
λ (σ − 1)λ(t2 ) + (1 − λ)(1 + σt2 )Z1 (t01 , t02 : λ)][=Z2 (t01 , t02 : λ)(1 − λ) (σ − 1)(1 − λ)(t01 )σ + λ(1 + σt01 )][λ (σ − 1)λ(t02 )σ + (1 − λ)(1 + σt02 )Z (t0 , t0 : λ)] = 1 10 20[≥,
Z2 (t1 , t2 : λ)(1 − λ) (σ − 1)(1 − λ)(t01 )σ + λ(1 + σt01 )which contradicts (42). Hence, t01 > t02 . QEDProof of Proposition 4.
i) If σ ≤ 2, we have][( ) ( )][( )( A )σ [( A )2 ( A )σ ]A σ A σ
A σ
+
1
,
−
t̄
>
0
>
−
t̄
t̄
σ t̄A
−
1
+
σ
t̄
t̄
t̄
1
1
2
1
2
1
2
( A )2 ( A )σ
≥ 0. The inequality implies that v̂(1) > 1.
− t̄1
because t̄A
2 > 1 and t̄1Aii) As c expands unboundedly with the other parameters being kept constant,[ t̄1 approaches( )σ−2 ]A
positive inﬁnity whereas t̄2 remains the same. Then, v̂(1) approaches σ/ (σ − 1) t̄A,
1
which goes to 0 if σ > 2. By continuity, therefore, v̂(1) < 1 for suﬃciently large c. QED

25

tS
1.6

1.4

1.2

2

3

4

5

6

(a) The Case with c = 1 and F = 0.01

tS

1.08

1.06

2

3

4

5

(b) The Case with c = 0.1 and F = 0.01

Fig. 1. Effect of σ on tS

6

σ

σ

dˆv / dλ
5
4
3
2
1

c

10

20

-1

(a) The Case with σ = 1.5dˆv / dλ
10

20

c

-0.5

-1

-1.5

(b) The Case with σ = 2dˆv / dλ
2

4

-5

-10

-15

(c) The Case with σ = 3

Fig. 2. Effect of c on dˆv / dλ when F = 0.01cvˆ (1)
4

3

2

10

20

c

30

(a) The Case with σ = 1.8vˆ (1)

1.4

1.2

10

20

c

c*

(b) The Case with σ = 2.2vˆ (1)1c*vˆ (1)

5c

10(c) The Case with σ = 3

1

0.5

1

(d) The Case with σ = 7
Fig. 3. Effect of c on vˆ (1) when F = 0.12cvˆ (1)
4

3

2

0.2

0.4

0.6

0.8

F

(a) The Case with σ = 1.8vˆ (1)
1.2

1.1

0.2

0.4

0.6

F*

F

(b) The Case with σ = 2.3vˆ (1)
0.4

0.3

0.2

0.4

0.6

0.8

(c) The Case with σ = 3
Fig. 4. Effect of F on vˆ (1) when c = 5

F

0.8

c
σ=3

0.65
0.6

0.4

0.2

σ=5

σ=7
0.05

0.1

0.15

Fig. 5. The Loci of ˆv (1) = 1

F0.2