CSIS Discussion Paper No. 100

Directional imbalance of transport prices and economic
geography

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

September 2009

Abstract
This paper examines how and why transport prices become imbalanced with respect todirections of shipments and how it aﬀects economic geography. It is shown that the equilibrium transport price of the shipment in a particular direction is a nondecreasing functionof a relative size of the embarkation region. Furthermore, we show that the directional
imbalance of transport prices makes the symmetric pattern more likely to be stable while
the core-periphery patterns less likely to be sustainable. In short, the imbalance acts as acentripetal force.
Keywords: behavior of transport ﬁrms, break point, export price eﬀect, import price eﬀect,
joint production, marginal cost of shipment, return constraint, substitution between imports
and domestic products, sustain point
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 anonymous referees
and seminar participants at various institutions. This research is partly supported by the Grant in Aid for
Research (No. 18600004 and No. 20330045) by Ministry of Education, Science and Culture in Japan.

1Introduction

It is often observed that transport costs diﬀer depending on directions of shipments. Manyof us, for instance, know that a ﬂight from the East coast to the West coast can costconsiderably more or considerably less than the counterpart in the reverse direction. To
give another example, the report of the freight rates on three major liner trade routes by
UNCTAD (2008) indicates their marked diﬀerence between pairwise directions of shipments(see Table 1). In addition, Cariou and Wolﬀ (2006) compute the freight rates of maritime
transport between Europe and the Far East adjusting the BAF (bunker adjustment factor),and show that they exhibit a signiﬁcant directional imbalance during the period of 20002004: the rate for westbound cargo is always, and twice to third times for most periods,
higher than that for eastbound cargo. Finally, Jonkeren et al. (2008) report that the freightrates for trips originating from Rotterdam are 32 percent higher than those arriving there
during the period of 2003-2007.

Table 1. Freight rates (market averages) per TEU on the three major liner trade routes.
($ per TEU.)
Transpaciﬁc

Europe-Asia

Transatlantic

Asia-USA

USA-Asia

Europe-Asia

Asia -Europe

US-Europe

Europe-USA

First quarter

1,836

815

793

1,454

995

1,829

Second quarter

1,753

828

804

1,408

1,010

1,829

Thrid quarter

1,715

839

806

1,494

1,041

1,854

Fourth quarter

1,671

777

792

1,545

1,066

1,762

First quarter

1,643

737

755

1,549

1,032

1,692

Second quarter

1,675

765

744

1,658

1,067

1,653

Thrid quarter

1,707

780

777

1,952

1,115

1,725

Fourth quarter

1,707

794

905

2,054

1,147

1,766

First quarter

1,725

861

968

2,021

1,193

1,700

Second quarter

1,837

999

1,061

1,899

1,326

1,652

2006

2007

2008

Source: UNCTAD (2008).

One of the most important reasons for this directional imbalance is a diﬀerence in volumesof trade in each direction. Jonkeren et al. (2008), using the data on inland waterways
in northwest Europe, ﬁnd out that the freight rates of shipments in a particular direction
positively depend on the total amount of shipments in that direction. In a standard textbook,this is explained as follows (see, for example, Boyer (1997)). Consider the situation in1

which the demand for transport services in one direction is larger than that in the opposite
direction. The shipment in the former direction is sometimes called a ‘front-haul’ and that in
the latter direction a ‘back-haul’. Thus, the demand curve for the front-haul lies above thatfor the back-haul. An equilibrium is realized when the sum of the demand price (marginal
evaluation) of the front-haul and that of the back-haul at a particular number of round trips
equals the cost per round trip. In other words, the equilibrium is reached at the point where
the curve derived by adding the two demand curves vertically cuts the curve representing thecost per round trip. The total cost of a round trip is then divided between the front-haul andthe back-haul according to their demand prices. By construction, therefore, the front-haul
becomes more expensive than the back-haul at the equilibrium. Now, note that it is thecost per round trip but not the costs of each one-way trip that matters in this explanation.
This is because transport ﬁrms or ‘carriers’ produce transport services (shipments) in two
directions jointly. Main reason is that they usually face a physical constraint that transport
equipment such as cargo ships, freight cars and cargo trucks eventually return to a home
port, namely, a ‘return constraint’.
This intuitive explanation, though quite helpful to grasp the nature of the matter, has
limitations. For one thing, it lacks an analysis of underlying behaviors of individual carriers.
What decisions they make is not obvious, however, because their problem involves joint
production with a return constraint. What is more important, it implicitly assumes thatthe demand for the transport services in a particular direction is independent of the price ofthe services in the opposite direction. This is not true in the reality. A change in a price oftransport services in one direction aﬀects the demand behaviors of consumers in a destination
through a change in delivered prices of goods, which in turn gives an impact on wage rates
not only in the destination but also in the region of embarkation. This further changes
regional incomes as well as price indices in both regions. To take into account these eﬀects
altogether, we need to conduct a general equilibrium analysis, in which prices and quantitiesof transport services in each direction are all determined simultaneously. In contrary, there
have been surprisingly few attempts to discuss the determination of transport prices within ageneral equilibrium framework. One reason might be that they have long lacked a tractable
general equilibrium framework that can deal with the relationship between regional trade
and transport costs. Nevertheless, we are now provided with a variety of so-called ‘neweconomic geography’ models, which has been developed by a number of researchers since
Krugman (1991). The ﬁrst purpose of this paper is, thus, to explain from the behaviors ofindividual carriers what levels transport prices are determined at and why and how they are
directionally imbalanced, based on a general equilibrium model of new economic geography.

2

The study of the determination of transport prices, especially when the possibility of
their directional imbalance is taken into consideration, is a matter of great signiﬁcance inits own right. However, there is another important reason to study it. It is that transport
prices aﬀect locations of economic activities, which is a fundamental message of new economic geography. In particular, this suggests a possibility that the directional imbalance oftransport prices, especially such a long-term and steady imbalance as the one observable in
Table 1, gives an inﬂuence on economic geography. Suppose, for example, that the demand
for the transport services from a bigger region to a smaller one is larger than that in the
opposite direction. Then, the price of the services in the former direction is higher thanthat in the latter direction, which implies that exports of the bigger region become more
expensive than those of the smaller region at the places of consumption, ceteris paribus. It
is not surprising that such a systematic diﬀerence aﬀects allocation of economic activities
between two regions. The second purpose of this paper is, thus, to examine such eﬀects ofa directional imbalance of transport prices upon economic geography.
For all its importance, the directional imbalance of transport costs has seldom attracted
attentions of researchers in the ﬁeld of new economic geography (see Fujita et al. (1999);
Fujita and Thisse (2000); and Fujita and Thisse (2002)). A notable exception is a work by
Baldwin et al. (2003), who explicitly examine the imbalance. In their study, however, theimbalance is given and no transport sector is considered. Moreover, not only the directional
imbalance but even the determination of a transport price has rarely been treated in theliterature: in most studies, the price is regarded as an exogenous parameter and a decision
making process of transport ﬁrms is abstracted away. There is one exception: Behrens et al.
(2006) examine the impacts of regulation of a transport sector upon welfare levels, paying
attention to incentives of transport ﬁrms, though in a very simple way.1
In order to explain the determination of transport prices and examine the impact of
their directional imbalance upon economic geography, we construct a two region modelwith a transport sector. Carriers produce transport services in two directions jointly under
the return constraint mentioned earlier. Their decision making process is formulated as asimple game à la Bertrand. As in the conventional models in the literature, furthermore,
we characterize two types of distribution patterns, a symmetric pattern, in which a mobile
factor is distributed equally in the two regions, and core-periphery patterns, in which it isconcentrated in one region; and examine how the possibility of a directional imbalance of
1 Some works discuss an endogenous determination of transport costs but do not deal with a transport

sector explicitly. Mori and Nishikimi (2002), for instance, examine the eﬀect of economy of density ontransport costs. Bougheas et al. (1999), and Mun and Nakagawa (2008) study impacts of infrastructure
investment. Finally, Takahashi (2006) examines a selection of a transport technology, which determines
transport costs.

3

transport prices aﬀects the conditions for each type of distribution pattern to be supported
by a stable equilibrium.
Main ﬁndings are summarized as follows. First, the equilibrium transport price of theservices in a particular direction is a nondecreasing step-shaped function of the relative sizeof an embarkation region. More speciﬁcally, as the relative size of a region gradually increases
from a very low level, the price of the services from that region to the other ﬁrst rises, thenbecomes constant and then rises again. A corollary of this ﬁnding is that the transport pricein a ‘binding direction’, the direction with a larger amount of shipment, is higher than thatin the opposite direction, a ‘slack direction’. Second, when two regions are suﬃciently alikein size, the amounts of inter-regional shipments in respective directions are equal. When one
region is much bigger than the other, instead, volumes of trade are imbalanced. Third, thedirectional imbalance of transport prices works to stabilize the symmetric pattern. Lastly
but not the least, as we allow for the directional imbalance, the core-periphery patterns
become less likely to be sustainable. The last two results demonstrate that the directional
imbalance acts as a centripetal force.
The paper is organized as follows. In the next section, a basic model is presented. Section
3 formulates a transport sector and derives a Nash equilibrium of a game played by carriers.
In the subsequent section, we explore the eﬀects of a directional imbalance of transport costs
upon economic geography. Finally, Section 5 concludes.

2

Model

As a basic framework, we use a modiﬁed version of an analytically solvable model by Forslid
and Ottaviano (2003), which is presented as a ‘footloose entrepreneur model’ in Baldwin etal. (2003).

2.1

Basic Framework

There are two regions, denoted by 1 and 2; two sectors, agriculture and manufacturingsectors; and two factors, workers and entrepreneurs.
Workers, whose number is denoted by l, are distributed equally in two regions and cannot
migrate across regions. There are, on the other hand, n entrepreneurs, of which λ1 n live
in region 1 and λ2 n live in region 2 (λ1 ∈ [0, 1], λ2 ≡ 1 − λ1 ). Instead of λ1 , a parameterλ ∈ [0, 1] with λ ≡ λ1 will be used when it is more convenient. Entrepreneurs can freely
move across regions.
The agricultural sector is competitive and produces a homogeneous product through aconstant returns to scale technology. More speciﬁcally, aA workers are used to produce 1

4

unit of agricultural goods. This implies that their price in region i is equal to aA wiA , wherewiA is a wage rate of workers in that region (i = 1, 2). The agricultural goods can be shippedfrom one region to the other with no cost so that their prices and, consequently, wage rates
of workers are equalized in two regions. The wage rate is taken as a numeraire.
To the contrary, the manufacturing sector is monopolistically competitive and produces adiﬀerentiated product through an increasing returns to scale technology. Each entrepreneurowns a ﬁrm which produces a variety in the region of her residence. Thus, there are nvarieties in the economy, of which λi n are produced in region i (i = 1, 2). Each ﬁrm
produces 1 unit of a variety using aM workers and skill of an entrepreneur. All the revenue
left after the payment for workers is taken by an entrepreneur. In other words, the proﬁt of
a ﬁrm is equal to 0:
pi qi − aM qi − wi = 0,(i = 1, 2)(1)where pi and qi denote a price and an amount of a variety produced in region i, and wi a
wage rate received by an entrepreneur in that region.
The workers and entrepreneurs have the same preference represented by a utility funcµ1−µtion, U = (CM ) (CA ). Here, CA is a consumption of agricultural goods and CM =∫n
ρ1/ρ
[ 0 x(k) dk]
is an aggregate of a manufacturing consumption with x(k) denoting theamount of the kth variety (ρ ∈ (0, 1)).
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, we denote the prices of a variety produced in each region
by p1 and p2 .
Next, let us introduce transport costs. It takes no cost to ship a manufacturing productwithin the same region. In order to ship it across regions, however, consumers need touse transport services, which are provided by transport ﬁrms or ‘carriers’. Carriers take a
portion of manufacturing products by way of compensation for the transport services they
provide. The portion is proportional to the amount of shipment and depends on its direction.
Let τi ∈ (0, 1) be the amount taken by a carrier when it carries 1 unit of a manufacturing
product from region i to region j (i = 1, 2 and j 6= i). (Bear in mind that the subscript ofτ refers to the region of the origin of shipment.) That is to say, transport cost or a price oftransport services from region i to region j is equal to τi pi .
Three comments are in order. First, in our setting, transport costs become proportional
to the prices of the products to be shipped, that is, they are ad valorem.2 Second, what
2 The assumption of the ad valorem transport costs is common among all the economic geography models

based on a Dixit-Stiglitz framework. It is primarily for the sake of analytical convenience. In addition,5

distinguishes our model from conventional ones is that we are taking into account the possibility that transport costs may diﬀer depending on directions of shipment, that is, τ1 is
not necessarily equal to τ2 . Third and lastly, the delivered price in region j of a variety
produced in region i is equal to (1 + τi )pi . In what follows, ti ≡ 1 + τi may be used as analternative measure of transport costs rather than τi whenever it is useful (i = 1, 2). Notethat ti ∈ (1, 2) by deﬁnition.
Let Xii and Xij be the total demands for a variety produced in region i by region i
consumers and region j consumers, respectively. Applying a standard procedure to derive
consumers’ demand functions yields
σ−1
Xii = µp−σ
Yi ,
i Pi

−σ σ−1
Xij = µt−σYj .
i pi Pj(i = 1, 2; j 6= i)(2)Here, σ ≡ 1/(1 − ρ) > 1 is an elasticity of substitution, Yi is an aggregate income and1[ ()] 1−σ+ λj t1−σp1−σ
Pi = n λi p1−σjji(i = 1, 2; j 6= i)(3)is a price index of the manufacturing products consumed in region i. As will be shown later,
all the manufacturing products taken by carriers for transport services are used up as inputs
for their production. Therefore, the amount of each variety produced in region i, qi , must
be equal to the sum of demands by consumers in respective regions:(i = 1, 2; j 6= i)qi = Xii + ti Xij .

(4)

An entrepreneur in each region maximizes her wage, wi , given by (1) (i = 1, 2), setting
prices at
p1 = p2 = p ≡

σ
aM .
σ−1

(5)

She therefore receives a wage equal towi =1
aM qi .
σ−1

(i = 1, 2)(6)

Moreover, because no carriers earn a positive proﬁt as will be shown in the next section, theaggregate income consists of workers’ and entrepreneurs’ earnings:
Yi =l+ λi nwi .
2(i = 1, 2)(7)

Finally, by choosing appropriate units, the analysis becomes simpler. First, for the
agricultural goods, we take a unit so that the labor input coeﬃcient aA becomes equal to 1.
a number of empirical studies have found a positive relationship between shipping prices and values of
the goods shipped. A 1 % increase in the value per kilo, for instance, results in a 0.55% increase in thetransport cost (import charges) for the US imports (Blonigen and Wilson (2008)) and a 0.34% increasein the transport cost in 16 Latin-American countries (Wilmsmeier et al. (2006)). The main reason for the
positive relationship is that high valued products tend to demand more extensive insurance coverage, stricter
security, and an additional care in the process of handling.

6

Second, for the manufacturing goods, we do the same so that aM = (σ − 1)/σ, which implies
that p = 1. This implies that the measure of transport costs, ti ≡ 1 + τi , actually gives adelivered price of the manufacturing products that are produced in region i and consumed
in region j (i = 1, 2; j 6= i). Third, the number of workers, l, is normalized to 2. Lastly,
the number of entrepreneurs, n, is normalized to 1. This completes a description of a basic
framework.

2.2

Short-run equilibrium

A system of equations (2) to (7) determines a short-run equilibrium, for which entrepreneurs’
distribution, λ, is given. Substituting (2) to (6) into the two equations in (7) and solving
them simultaneously, we can obtain the equilibrium values of Y1 and Y2 :[]Yi = σΓ σθ1 θ2 + µθi (λi φi − λj ) ,(i = 1, 2; j 6= i)

(8)

[
]−1
where Γ ≡ (σθ1 − µλ1 )(σθ2 − µλ2 ) − σ 2 λ1 λ2 θ1 θ2and θi ≡ Pi1−σ . In addition, φi ≡ ti1−σ()
represents ‘freeness of trade’ (i = 1, 2). By deﬁnition, φi ∈ Φ ≡ 21−σ , 1 . Equation (8), in
turn, gives the equilibrium values of demand variables, Xii ’s and Xij ’s, by (2).
Furthermore, a volume of trade from region i to region j is equal to Zi ≡ λi nti Xij(i = 1, 2, j 6= i). It is convenient to deﬁne their ratio:][b (σ − µ) + φ2 (σ + µ)b 1 λλφb ≡ Z1 =[] ,ZZ2b 1 (σ + µ)
φ2 σ − µ + λφ

(9)b ≡ λ1 /λ2 . (In this paper, circumﬂexes are used to indicate ratios of the variables
where λpertaining to two regions or two directions.) It is straightforward to see that
b
Z

{ > }
=
1
<

if

b
λ

( ) σ−1{ > } ( φ ) 12
2
t1
2
=
=
.
<
φ1
t2

(10)

Thus, whether Z1 exceeds Z2 or not depends on transport costs in two directions and a
distribution of entrepreneurs. More speciﬁcally, if a region is relatively bigger and/or thetransport cost of an outbound trip is relatively lower compared to that of an inbound trip,
then the outbound trade is more likely to outweigh the inbound trade.

2.3

Long-run equilibrium

In a long run, entrepreneurs move freely across regions seeking a higher level of indirect
utility. Let vi (λ) be the indirect utility for an entrepreneur living in region i:vi (λ) = K

µ [
]
wi
σ−1
σ(θj + θi φi ) + µλj (φ1 φ2 − 1) ,µ = KσΓθiPiwhere K ≡ µµ (1 − µ)1−µ > 0.
7(i = 1, 2; j 6= i)(11)This would be a good place to note that transport costs aﬀect a relative level of indirect
utility, vb (λ) ≡ v1 (λ) /v2 (λ), through three channels.
¯
d ln vb(λ)
d ln Yb ∂ ln w
∂ ln wb ¯¯bd ln Pb+=,− µ¯d ln ti∂ ln ti Yb const.
d ln ti ∂ ln Yb
d ln ti(i = 1, 2)(12)

where wb ≡ w1 /w2 , Yb ≡ Y1 /Y2 and Pb ≡ P1 /P2 .

First, as a result of a change in transport costs, consumers substitute between imports anddomestic products, which alters a relative wage rate. Suppose that ti rises. Then, regioni products become more expensive in region j (j 6= i), which induces consumers in that
region to shift part of their consumption from now relatively more expensive imports tonow relatively less expensive domestic products. This undermines the demand for region i
products by region j consumers and thus gives an adverse eﬀect on wi , other things being
equal. This negative eﬀect is referred to as an export price eﬀect. At the same time, it
stimulates the demand for domestic products in region j and therefore raises w2 . This
import price eﬀect gives a negative impact on the relative wage rate. These two negativeeﬀects are all together referred to as a terms of trade eﬀect, which is represented by the ﬁrst
term in the right hand side of (12).
Second, the second term describes an eﬀect through regional incomes. The fall in wi andthe rise in wj due to the terms of trade eﬀect bring about a decrease in Yi and an increasein Yj , respectively. For wi , the decrease in Yi works adversely but the increase in Yj works
favorably. The overall direction of a change of wi depends on the sizes of the changes in
Yi and Yj and the sensitivities of wi to these changes. Unless region j is much ‘bigger’
than region i, that is, unless the former region has a quite large population and/or a big
advantage in transport costs (tj is much lower than ti ), however, the eﬀect of the decreasein Yi dominates that of the increase in Yj .3 Then, wi declines. Otherwise, we have theopposite: changes in regional incomes result in a rise in wi .
Third and ﬁnally, we have a channel through price indices, represented by the last termof (12). As ti goes up, the delivered price of region i products and, therefore, a price index
increase in region j, which raises a relative price index. This gives an adverse impact on therelative level of indirect utility.
Now, migration behaviors of entrepreneurs follow a simple equation, which is common
in the literature:λ̇ = ζ [v1 (λ) − v2 (λ)] ,(13)

where ζ > 0 denotes speed of adjustment.
A distribution of entrepreneurs is at a long-run equilibrium if it is a stationary pointof (13). That is to say, an interior distribution (λ ∈ (0, 1)) is supported by a long-run
3 Note that this applies to the case where two regions are of equal size. It is because a wage rate responds

more to the income of a home region than to that of a foreign region, ceteris paribus.

8equilibrium if v1 (λ) = v2 (λ). The core-periphery pattern with λi = 1 is so if vi (1) ≥ vj (1)(i = 1, 2; j 6= i).
In addition, equation (13) implies that an equilibrium distribution, denoted by λ∗ , isstable ifdbv (λ∗ )

≤ 0 holds when λ∗ ∈ (0, 1),



dλ


db
v (1)
at least either≤ 0 or vb (1) > 1 holds when λ∗ = 1, and


dλ




v (0)

 at least either db≤ 0 or vb (0) < 1 holds when λ∗ = 0.
dλ3Determination of transport prices

In this section, we examine determination of transport prices paying a special attention to
how they become divergent depending on directions of shipments.
To begin with, let us normalize the unit of transport services so that 1 unit of them isnecessary to ship 1 unit of a manufacturing product across regions. This implies that the
total demand for the transport services from region i to region j 6= i is equal to the volumeof trade in that direction, i.e., Zi . On the other hand, the total supply of the services is∑mgiven by h=1 zih . Here, zih is the amount of services from region i to region j 6= i provided
by an h-th carrier; and m is the number of carriers, which is given. Therefore, the markets∑m∑mof transport services clear if Z1 = h=1 z1h and Z2 = h=1 z2h .
Each carrier must incur two kinds of costs to produce transport services. One is thecosts that depend on the total amounts of shipment in a round trip. They include the costs
to stow and discharge a cargo and a rental cost of containers, for instance. For simplicity,
I assume that a ﬁxed amount, c > 0, of manufacturing products are necessary to handle 1)(
unit of shipment. Thus, carrier h incurs this type of costs equal to c z1h + z2h (rememberthat p = 1). The other is the costs related to a ‘capacity’ of shipment. Suppose that carriers
face the return constraint mentioned earlier, which requires that transport equipment suchas cargo ships, freight cars and cargo trucks eventually return to a home port region. In thiscase, part of costs depend on the scale of the equipment dispatched for a round trip, or, we
may call it, a capacity of shipment, which is determined by the size of a front-haul, namely,
max[z1h , z2h ]. Here again we limit our attention to the simplest case: carriers need to use
manufacturing products whose amount is proportional to max[z1h , z2h ]. Thus, this ‘capacity
cost’ is given by k max[z1h , z2h ] (k ≥ 0). In sum, the total costs to produce transport services()
is equal to c z1h + z2h + k max[z1h , z2h ].4
4 Alternatively, we might assume that carriers use labor instead of manufacturing products in the produc-tion of transport services. Under this assumption, however, the manufacturing products taken by carriers by
way of compensation for transport costs are to be placed in a market and eventually consumed by someone.

9

It is important to emphasize that when the amounts of shipments diverges between thetwo directions, marginal (and average) costs of shipment diﬀer depending on its direction.
To see this, suppose that z1h > z2h , without loss of generality. In order to increase a fronthaul, namely, a shipment from region 1 to region 2, it is necessary for carrier h to rent
an additional cargo ship, a freight car or a cargo truck. Such an expansion of transport
equipment or capacity induces the carrier to pay a marginal cost of k as well as c. However,it does not have to incur such a cost as long as the addition is made to a back-haul, namely,a shipment from region 2 to region 1. Therefore, the marginal cost of a front-haul is c + k
while that of a back-haul is c.
Moreover, it is assumed that when there is an imbalance, the marginal costs do not
exceed the price of manufacturing products, i.e., 1,. In other words, we assume that c < 1
and c + k < 1. Since k ≥ 0, only the latter inequality matters:
Assumption 1. c + k < 1.
This assumption, as will become clear, guarantees that transport prices are lower than theprice of manufacturing products, 1.
Now, recalling that each carrier receives τi when it carries 1 unit of manufacturing
products from region i to region j 6= i, we can write down an h-th carrier’s proﬁt as
π h ≡ (τ1 − c) z1h + (τ2 − c) z2h − k max[z1h , z2h ].

(14)

Carriers choose the prices and the quantities of transport services that maximize the
proﬁt given by (14), playing a two stage game, given total demands Z1 and Z2 . In the ﬁrst
stage, they simultaneously quote prices of transport services, τ1h ’s and τ2h ’s (h = 1, · · ·, m).
The carriers who quote the lowest price of τi are entitled to sell in the next stage any amountof transport services up to Zi /mi where mi is the number of such carriers. Those who failto quote the lowest price cannot sell any services. Carriers are thus put in a competitiveenvironment in the sense that each of them faces an inﬁnitely high elasticity of demand.5
In the second stage, they decide the amounts of services to produce, z1h and z2h .
This modiﬁcation involves two problems. First, because the structure of a model changes, we can no longer
consider a conventional model as a special case of our model. As a result, the comparison of results becomesdiﬃcult. Second, technically speaking, it means that ti is to be removed from (4). It turns out that thischange, though it seems minor, makes the analysis much more complicated and prevents us from deriving
clear-cut results.
5 It is often maintained that linear shipping, which is one of the two distinct markets of ocean transport
for non-oil cargo, has been characterized by collusive agreements, or shipping conference (see Sjostrom(2004) and the studies cited there). However, their eﬀectiveness is still an open question (see Hummels(2007)). Other industries such as tramp shipping, trucking and charter plane industries are allegedly morecompetitive. In this paper, we concentrate on a benchmark case of a competitive market.

10

We focus on the Nash equilibrium of this transport game that meets the following threerequirements. First, each carrier earns a nonnegative proﬁt. Second, the markets of transport services clear. Third and lastly, behaviors of carriers are symmetric, that is, all of themquote the same prices and produce the same amounts. In what follows, the term ‘equilibrium’ will refer to the Nash equilibrium satisfying all these requirements. To characterizeb 6= 1 and that withthe equilibrium, it is convenient to distinguish two cases: the case with Z
b = 1.
Zb 6= 1. Let us designate the direction in which total demand for transFirst, suppose that Z
port services is larger as a ‘binding direction’ and the opposite direction as a ‘slack direction’.
Furthermore, we denote the embarkation region of the shipment in a binding direction by b
and that in a slack direction by s. Thus, Zb > Zs (b = 1, 2; s 6= b). The market clearing andthe symmetric behaviors of carriers demand that each of them produce Z1 /m and Z2 /m of∑respective services. Each carrier then earns a proﬁt equal to l=1,2 (τl − c) Zl /m − kZb /m.
b 6= 1, price markups of transport services in each
The ﬁrst point to make is that as long as Z
direction must be 0 at the equilibrium: τb − c − k = 0 and τs − c = 0. If the price markups
were positive, a carrier could get a higher proﬁt by charging a slightly lower price than its
competitors. For instance, if τb − c − k > 0, a carrier could lower its quote a little fromτb and increase the production from Zb /m to Zb , which gives a higher proﬁt. If the pricemarkups were negative, on the other hand, it could enhance its proﬁt by slightly reducing
its supply. If τb − c − k < 0, for example, a carrier could contract the production of transport
services in a binding direction so that its amount gets smaller than Zb /m (but still larger
than Zs /m).
This implies that the equilibrium price equals c + k in a binding direction while c in aslack direction. Several comments are in order. First, because Zb /m > Zs /m by deﬁnition,
the shipment in a binding direction is a front-haul and that in a slack direction is a back-haulfor each carrier. Therefore, we can say that each carrier sets the prices equal to the marginal(and average) costs of shipment in each direction. This indicates that our transport gameis a variant of the Bertrand price setting game. Second, each carrier earns 0 proﬁt at theequilibrium, which is a typical result of the Bertrand price setting game. Third and lastly,
the next result immediately follows:

Proposition 1. The transport price in a binding direction is no lower than that in a slack
direction.
This result is consistent with an empirical ﬁnding by Jonkeren et al. (2008).
Now, suppose that b = 1, that is, the direction from region 1 to region 2 is binding.
[b>λbU ≡ (c + k +b > 1 and (τ1 , τ2 ) = (c + k, c) in this case, (10) implies that λBecause Z

11

/
] σ−1
[
/
] σ−1
b<λ
bL ≡ (c+1) (c+k+1) 2 .
b < 1) occurs if and only if λ1) (c+1) 2 . Similarly, b = 2 (Zb 6= 1, then the equilibrium
Consequently, we conclude that if there is an equilibrium with Z
pair of transport prices is given by (c + k, c)(τ1∗ , τ2∗ ) ≡ (c, c + k)

if

b>λ
bU
λ

if

b<λ
bL .
λ

(15)

Furthermore, we can easily show that (τ1∗ , τ2∗ ) is indeed supported by a Nash equilibriumof the transport game.

Lemma 1. The strategy proﬁle in which all the carriers quote the same pair of prices,(τ1∗ , τ2∗ ), is a Nash equilibrium in the transport game.
The proof is tedious and relegated to the Appendix.
b = 1. First of all, (10) requires that a pair ofNext, consider the remaining case with Z
transport prices satisfy(

τ1 + 1τ2 + 1

) σ−1
2

b
= λ.

(16)

Second, notice that the proﬁt of each carrier at the equilibrium is now equal to (τ1 + τ2 −2c−k)Z1 /m. The nonnegative proﬁt requires that τ1 +τ2 −2c−k ≥ 0. If τ1 +τ2 −2c−k > 0,
however, a carrier would make an attempt to undercut either or both prices to get a higher
proﬁt. Therefore, we must have
τ1 + τ2 − 2c − k = 0.

(17)This implies that the proﬁt of each carrier becomes 0 at the equilibrium.
Solving (16) and (17) for τ1 and τ2 , we conclude that if there is an equilibrium withb = 1, then the equilibrium pair of transport prices is given by
Z
( 2
)
2
b σ−1 (2c + k + 1) − 1 2c + k + 1 − λ
b σ−1
λ
o o
(τ1 , τ2 ) ≡
.
,
2
2
b σ−1
b σ−1
1+λ
1+λ

(18)

b (and therefore, in λ) while τ o is decreasing in
It is readily veriﬁed that τ1o is increasing in λ2b (and therefore, in λ). Transport prices coincide with each other at the symmetric pattern
λb = 1) and diverge more and more as the distribution goes farther away from it.
(λWe need a set of conditions for this pair of transport prices to be supported by a Nashequilibrium. Suppose that a price of transport services in a certain direction is lower thanc. Then, each carrier has an incentive to cut the service in that direction and earn a higher
proﬁt. Therefore, a Nash equilibrium requires that τ1o − c ≥ 0 and τ2o − c ≥ 0. Formally,
this is shown as follows.

12Lemma 2. The strategy proﬁle in which all the carriers quote the same pair of prices,(τ1o , τ2o ), is a Nash equilibrium in the transport game, if and only if τ1o −c ≥ 0 and τ2o −c ≥ 0.

The proof of this lemma, which is relegated to the Appendix, is similar to that of Lemma
1 with some minor changes. By (18), the two conditions τ1o − c ≥ 0 and τ2o − c ≥ 0 can be
rewritten as follows:bL ≤ λ
b≤λ
bU .
λ

(19)

Because these two cases with Z 6= 1 and Z = 1 are exhaustive, we have established thefollowing result.
Proposition 2. The equilibrium transport prices are given by (τ1∗ , τ2∗ ) in (15) when (19)does not hold, and (τ1o , τ2o ) in (18) when (19) holds. At the equilibrium, all the carriers earn
0 proﬁt.
The ﬁrst panel of Fig. 1 shows transport prices as functions of λ for σ = 3, c = 0.2 andk = 0.3. The solid line represents τ1 and the dotted line τ2 . Because τ1o increases and τ2odecreases with λ, the transport prices exhibit step-like shapes. As σ rises, furthermore, anincreasing part of τ1 becomes ﬂatter and the range of λ associated with it widens.
— Insert Fig. 1 around here. —Three remarks are worth adding. First, the 0 proﬁt result implies that the manufacturing
products taken by carriers by way of compensation for the transport services are just used
up as inputs for their production. This enables us to analyze the transport sector andthe manufacturing sector separately, which greatly simpliﬁes the analysis. Second, as has
been postulated, the derived transport prices do not exceed the price of manufacturing
products, namely, 1. When they are given by (τ1∗ , τ2∗ ), it is self-evident. When they are
given by (τ1o , τ2o ), on the other hand, it may need some explanation. Note that τ1o reachesb=λbU because it increases with λ.
b However, we can verify by a tediousits maximum at λcomputation that τ1o < 1 at that point as long as Assumption 1 holds. Hence, τ1o < 1 forb ∈ [λbL , λbU ]. Third andb ∈ [λbL , λbU ]. A parallel reasoning proves that τ o < 1 for any λany λ2ﬁnally, the equilibrium prices are always positive.
b > 0 (for constant φ1 and φ2 ).
b λTo close this section, notice that (9) implies that ∂ Z/∂
b increases with λ. This establishes the
Therefore, whenever (τ1 , τ2 ) is given by (τ1∗ , τ2∗ ), Z
following result.
b < λbL , the volume of trade from region 1 to region 2 is smaller
Proposition 3. When λb > λbU , to the contrary, it is larger. In boththan that in the opposite direction. When λ13bL ≤ λb≤λbU , the volumes of tradecases, it increases with a size of region 1. Finally, when λare equal for two directions and they are independent of sizes of two regions.
b as a function of λ (again σ = 3, c = 0.2 and k = 0.3). It
Diagram (b) of Fig. 1 shows Z
b is continuous but that its derivative is not.
indicates that Z

4Impacts of the directional imbalance of transport prices
upon economic geography

In this section, we study how the directional imbalance of transport prices aﬀects economic
geography. Two types of distribution patterns, core-periphery patterns and a symmetricpattern are discussed in order. Then, the eﬀects of changes in parameters, in particular, thetwo parameters related to the costs to produce transport services, upon economic geography
are brieﬂy discussed.

4.1

Core-periphery pattern

To begin with, we consider a core-periphery pattern in which entrepreneurs are concentrated
in region 1, that is, λ = 1. Thus, region 1 is a ‘core’ region while region 2 is a ‘periphery’region. The other core-periphery patten with λ = 0 can be similarly analyzed.
b which becomes inﬁnitely large at λ = 1, exceeds λbU , (τ1 , τ2 ) is given
First of all, since λ,by (τ1∗ , τ2∗ ) = (c + k, c).
Proposition 4. When entrepreneurs are concentrated in region 1, transport prices are givenby (τ1 , τ2 ) = (c + k, c).
As has been mentioned earlier, the core-periphery pattern with λ = 1 is supported bya long-run equilibrium if vb(1) ≥ 1; and if so, it is stable when at least either db
v (1)/dλ ≤ 0
or vb1 (1) > 1 holds. Recall that the condition db
v (1)/dλ ≤ 0 matters only when v1 and v2
happen to become equal to each other at λ = 1. Because there is no reason to believe thatsuch a measure-0 occurrence has a particular importance in the real world, one would rather
dismiss it as a singular case. Thus, we focus on the other condition, vb(1) > 1, namely,[]µφ2 (σ + µ) σ − µΥ ≡ φ1σ−1+− σ < 0,(20)22φ1which is a suﬃcient condition for the core-periphery pattern to be supported by a stablelong-run equilibrium, or a condition for it to be sustainable according to a terminology inliterature.

14

The ﬁrst point to make is that (20) is always satisﬁed if
σ < 1 + µ.6(21)

When (21) holds, therefore, the core-periphery pattern becomes sustainable whatever transport prices are. (21) is known as a black hole condition in the literature. When σ is suﬃciently high, to the contrary, Υ becomes positive. Indeed, we can verify that Υ approaches(1 + φ1 φ2 )/2φ1 − 1 > 0 as σ goes to the positive inﬁnity.7
Next, to explore the sustainability further, let us draw the locus of Υ = 0 in a φ1 − φ2plane. Because it corresponds to a sustain point of freeness of trade in a conventional modelwith no directional imbalance of transport prices, we call it a sustain curve. Fig. 2 depicts
such curves for four diﬀerent values of σ, namely, σ = 1.5, σ = 1.9, σ = 3 and σ = 4, with
µ being ﬁxed at 0.8. Now, (20) indicates that Υ increases with φ2 . At the area below eachcurve, therefore, Υ is negative, that is, the core-periphery pattern is sustainable. At thearea above it, in contrast, the pattern is unsustainable.
— Insert Fig. 2 around here. —The slope of a sustain curve is equal to¯
dφ2 ¯¯∆,= 2
dφ1 ¯Υ=0
φ1 (σ − 1)(σ + µ)where ∆ ≡ (σ − µ)(σ − µ − 1) − φ1 φ2 µ(σ + µ). When the black hole condition, (21), issatisﬁed, ∆ apparently becomes negative; and therefore, the sustain curve has a downward
slope at any (φ1 , φ2 ) ∈ Φ2 (remember that Φ ≡ (21−σ , 1)). In Fig. 2, the curve for σ = 1.5
represents this case. When the black hole condition is not satisﬁed, to the contrary, it is notstraightforward to determine the sign of ∆. From the ﬁgure, however, one would conjecturethat the sustain curve is upward sloping for any relevant pair of (φ1 , φ2 ), that is, for any(φ1 , φ2 ) ∈ Φ2 . The following lemma, whose proof is relegated to the Appendix, conﬁrms
that the conjecture is true.
Lemma 3. When the black hole condition is not satisﬁed, the sustain curve is upward sloping for any (φ1 , φ2 ) ∈ Φ2 .
We are now ready to discuss the relationship between a directional imbalance of transport
prices and the sustainability of the core-periphery pattern. Notice that the size of theimbalance is measured by parameter k since (τ1 , τ2 ) = (c + k, c).
6 Note that Υ = σ(Υ0 Υ00 − 1) where Υ0 ≡ φ−(σ−µ−1)σ−11and Υ00 ≡ [φ1 φ2 (σ +µ)+ σ −µ]/(2σ). For one thing,Υ0 < 1 if (21) holds, because φ1 < 1. Moreover, Υ00 < 1 since φ1 φ2 (σ + µ) + σ − µ < (σ + µ) + σ − µ = 2σ
for φ1 < 1 and φ2 < 1. Hence, Υ < 0.
7 Because φ ≥ φ and φ > 2−1/φ , we have φ > 2−1/φ , which is equivalent to (1+φ φ )/2φ −1 > 0.
2
1
1
1
2
1
1 2
1

15

When σ is suﬃciently low that the black hole condition is satisﬁed, on the one hand, achange in k has no impact on the sustainability. This is because the core-periphery patternis sustainable no matter what the levels of transport prices are.
When σ is so high that the black hole condition does not hold, on the other hand,parameter k does aﬀect the sustainability. Since it does so only through φ1 , we have¯dΥ ¯¯dφ1 ∂Υ
t1σ−µ−2==∆,dk ¯Υ=0dk ∂φ12σwith all variables being evaluated at Υ = 0. Now, because Lemma 3 implies that ∆ > 0 on
the sustain curve, dΥ/dk|Υ=0 is positive: Υ decreases as k falls. Consequently, a fall in k
makes the core-periphery pattern more likely to be sustainable. Thus, we have establishedthe following proposition.
Proposition 5. i) When σ is suﬃciently low that the black hole condition is satisﬁed, thecore-periphery pattern is sustainable irrespective of the size of directional imbalance.
ii) When σ is so high that the black hole condition does not hold, less divergent transport
prices are associated with higher sustainability of the core-periphery pattern.
To understand ii) of the proposition intuitively, let us return to the identity (12) that
decomposes the impact of a change in t1 on vb(1), given that t2 is ﬁxed, into the terms of tradeeﬀect, regional income eﬀect and price index eﬀect. Suppose that k falls. Then, t1 declines
but t2 remains unchanged, which scales down the directional imbalance of transport prices.
For one thing, as a result of the decline in t1 , consumers in region 2 would try to substituteimports for domestic products, if they had been consuming domestic products. This importprice eﬀect reduces the wage rate of a potential entrepreneur in region 2, which raises therelative wage rate, w.
b On the other hand, there is no export price eﬀect. This is because no
consumers in region 2 actually make the substitution: in fact, no manufacturing production
is conducted in that region. Consequently, the terms of trade eﬀect solely consists of theimport price eﬀect, and is negative. What is more, as long as the core-periphery pattern isconcerned, relative regional income is independent of transport prices: there is no regional
income eﬀect. Finally, as t1 declines, varieties become less expensive in region 2. Because t1
does not aﬀect the price index in region 1, Pb rises, and thus a positive price index eﬀect.8
It turns out that the import price eﬀect dominates the relative price index eﬀect for any(φ1 , φ2 ) ∈ Φ2 . As a result, the relative indirect utility increases and the core-periphery
pattern becomes more likely to be sustainable as k falls.
Another way to see the relationship between the directional imbalance of transport prices
and the sustainability is examining directly the eﬀects of changes in parameters on the
8 The terms of trade eﬀect and the price index eﬀect are given by −(σ − 1)(1 − b)/ [1 − b + φ φ (1 + b)]
1 2

and µ, respectively.

16sustainability. Fig. 3 describes the loci of Υ = 0 in a k − c plane for four diﬀerent valuesof σ with µ again being ﬁxed at 0.8. It is easily veriﬁed that Υ = 0 when k = c = 0 or
φ1 = φ2 = 0. Therefore, each locus passes through the origin. If the black hole condition
holds, furthermore, we can prove that a locus has a negative slope.9 This case is represented
by the locus for σ = 1.5. Here, Υ becomes negative when (k, c) lies above the locus andpositive when it lies below it. For any admissible pair of (k, c), therefore, Υ < 0 and thecore-periphery pattern is sustainable. If the black hole condition is not met, instead, thesign of the slope of a curve depends on parameters. For relatively low σ, it has a downward
slope for any k ∈ [0, 1). In the ﬁgure, the locus for σ = 1.9 describes this case. The sign of Υ
at a particular point in the plane can be determined in the same manner as in the previous
case. As σ rises, the locus rotates counterclockwise around the origin. For a high level ofσ, the locus has a shape of a reverse image of character “C”. That is to say, it is downward
sloping for high values of c and upward sloping for low values of it (see the locus for σ = 4
in Fig. 3). In this case, Υ is negative “inside” the character “C”, and positive “outside” it.
— Insert Fig. 3 around here. —From the ﬁgure, several observations can be obtained. In the ﬁrst place, it is apparent
that for a given value of c, a fall in k makes the core-periphery pattern more likely to besustainable, which reﬂects the result obtained earlier (see ii) of Proposition 5). In the second
place, let us examine the eﬀect of a fall in c when k is given. To begin with, we consider
a special case with k = 0, where our model is reduced to a conventional model with nodirectional imbalance of transport prices. In this case, there is a critical point, a ‘sustain
point’, of c: the core-periphery pattern is sustainable if c is lower than it, whereas it isunsustainable if c is higher than it. It is because lower c implies lower transport costs, whichenable entrepreneurs in the core to deliver their products to the workers in the peripheryat less expense. Now, let us turn our attention to the case with k > 0, the case with thedirectional imbalance. For suﬃciently high c, the core-periphery pattern is unsustainable.
As c falls, it eventually reaches a critical point corresponding to the sustain point mentioned
above; and thereafter, the pattern becomes sustainable. What is happening here is virtually
the same as in the case of no directional imbalance. As c falls further, however, it hits anothercritical point. For c lower than it, the core-periphery pattern again becomes unsustainable.
In sum, as k gradually falls, the pattern becomes sustainable, unsustainable, and thenagain sustainable. Such a non-monotonic behavior is not observed in a conventional model.
Indeed, the second critical point approaches 0 as k goes to 0; and therefore, the lower
˛

9 Note that dc/dk ˛= ∆(c + 1)
Υ=0

.h

i
φ1 φ2 (σ − 1)(σ + µ)(c + k + 1) − ∆(c + 1) . Because ∆ < 0 when theblack hole condition holds, the derivative is negative.

17

‘unsustainability range’ of c disappears at k = 0. The reason for this novel result is that as
c falls, a relative size of the directional imbalance of transport prices expands although its
absolute size, which is equal to k, remains unchanged. Consequently, when c becomes too
low, the negative eﬀect of imbalance on the sustainability grows to the extent that it morethan oﬀsets the positive eﬀect of low transport costs brought about by low c. Finally, we
have observed that as σ rises, the locus rotates counterclockwise. This implies that the area
for which Υ is negative shrinks. In this sense, the core-periphery pattern becomes less likely
to be sustainable as σ rises. This result is the same as the one derived in a conventionalmodel with no directional imbalance of transport prices.

4.2

Symmetric patternb = 1).
Next, we consider a symmetric pattern (λ = 1/2 or λbL < 1 < λbU , transport prices are given by (τ o , τ o ) = (c + k/2, c + k/2) at
Because λ1 2that pattern. Furthermore, the inequality implies that a distribution suﬃciently close to thesymmetric pattern also satisﬁes (19). More formally, there exists ε > 0 such that conditionbL < λb < λbU is satisﬁed for any λb ∈ (1 − ε, 1 + ε). These ﬁndings are summarized as
λfollows.

Proposition 6. When entrepreneurs are distributed equally or nearly equally in two regions,transport prices are given by (τ1 , τ2 ) = (c + k/2, c + k/2).
What is more, the symmetric pattern is supported by a long-run equilibrium becausevb(1/2) = 1. In addition, the equilibrium is stable if dbv (1/2)/dλ ≤ 0, which is equivalent tob ≤ 0. Notice that λb aﬀects vb both directly and indirectly through a change
d ln vb(1/2)/d ln λin transport prices:d ln vb(1/2)
bd ln λ==¯∂ ln vb(1/2) ¯¯d ln to1 ∂ ln vb(1/2) d ln to2 ∂ ln vb(1/2)++¯
bb ∂ ln t1b ∂ ln t2∂ ln λd ln λd ln λ(t1 ,t2 ) const.
¯∂ ln vb(1/2) ¯¯d ln to1 ∂ ln vb(1/2)+2,¯bb ∂ ln t1∂ ln λd ln λ(t1 ,t2 ) const.

(22)

b=
where toi ≡ τio + 1 (i = 1, 2). Here, the last equality follows from the fact that d ln to1 /d ln λ//b and ∂ ln vb(1/2) ∂ ln t1 = −∂ ln vb(1/2) ∂ ln t2 . If to were to be equal to to for−d ln to2 /d ln λ21b and the second term in theb would become equal to d ln to /d ln λ;b then d ln to /d ln λany λ,
2
1
second line of (22) would disappear. This means that the term represents the eﬀect due toa directional imbalance of transport prices.
Since t1 = t2 and φ1 = φ2 at the symmetric equilibrium, their equilibrium values aredenoted simply by t and φ, respectively, in the rest of this section. With evaluating all the18b = 1, we obtain from (11) and (18)derivatives at λ¯(1 − φ)Ψ0∂ ln vb(1/2) ¯¯
[
]
= Ψ≡
¯
b
(σ − 1)(1 + φ) σ − µ + φ(σ + µ)
∂ ln λ(t1 ,t2 ) const.

and2t−σ Ω0[],(1 + φ) σ − µ + φ(σ + µ)[]where Ψ0 ≡ φ(σ + µ)(σ + µ − 1) − (σ − µ)(σ − µ − 1) and Ω0 ≡ µ σ − µ + φ(σ + µ) − 2σ(σ − 1).
2

d ln to1 ∂ ln vb(1/2)
b ∂ ln t1
d ln λ=Ω≡Here, Ψ represents the direct eﬀect that arises provided that transport prices remain
constant. It is what one can ﬁnd in a conventional model with exogenous and non-divergent
transport prices (see Baldwin et al. (2003), among others). Because Ψ0 is increasing in φ,/>there is a critical value, φB ≡ (σ − µ)(σ − 1 − µ) (σ + µ)(σ − 1 + µ) such that Ψ = 0 if<>
φ = φB . This is a break point in a conventional model: if freeness of trade is higher than<φB , the symmetric equilibrium ‘breaks’ and becomes unstable. Notice that when the blackhole condition given by (21) is satisﬁed, φB becomes negative; and therefore, Ψ > 0 for any
φ ∈ Φ: the symmetric pattern becomes always unstable in this case.
In our model, however, there is an additional term, Ω, which represents the indirect eﬀect
through a change in transport prices, or more speciﬁcally, the eﬀect due to a directional
imbalance of transport prices. Notice that Ω0 increases with φ and becomes negative at
φ = 1 as long as the black hole condition does not hold. Hence, in that case, for any
admissible value of transport price, φ ∈ Φ, we have Ω0 < 0 and therefore Ω < 0. The
negativity of Ω means that the symmetric equilibrium becomes more likely to be stable if
we allow for the endogeneity and the imbalance of transport prices. Formally, we have thefollowing proposition:

Proposition 7. As long as the black hole condition is not satisﬁed, the range of φ for which
the symmetric equilibrium becomes stable is wider when transport prices are endogenously
determined and allowed to be imbalanced with respect to directions than when they are not.
To understand why Ω < 0, recall that the eﬀect of a change in transport prices on thebrelative level of indirect utility is decomposed into three parts (see (12)). Suppose that λincreases a little from 1 so that region 1 becomes slightly bigger than region 2. Then, thetransport price in the binding direction, i.e., the direction from region 1 to region 2, risesb results in a (σ − 1)−1 %while that in the slack condition falls. Indeed, a 1 % increase in λincrease in to1 and a (σ − 1)−1 % decrease in to2 . Let us focus on the increase in to1 . First,
the relative wage rate decreases by the terms of trade eﬀect, which contains the export priceeﬀect and the import price eﬀect. Second, the regional income eﬀect works in the same
direction as the terms of trade eﬀect, because we are now dealing with the case where thetwo regions are suﬃciently alike in size. Third and lastly, the relative price index declines. It

19turns out that the sum of the terms of trade eﬀect and the regional income eﬀect dominates
the price index eﬀect; and consequently, the relative level of indirect utility declines.10 This
works as a centripetal force and the symmetric equilibrium becomes more likely to be stable.
The eﬀect of the decrease in to2 is similarly explained.
The question which is not yet answered is how the ‘break point’ behaves in our modelwith the additional term of Ω. Since Ψ + Ω is nonlinear, it is diﬃcult to analytically obtainfurther consequences: we need to rely on numerical simulations. Now, ﬁxing µ at 0.8, we candraw the value of Ψ + Ω as a function of φ. Diagram (a) of Fig. 4 describes such curves forfour diﬀerent levels of σ. Now, suppose that σ gradually falls. For suﬃciently high σ (σ = 3
and σ = 4, for example), Ψ + Ω is negative and the symmetric equilibrium is stable for any
φ ∈ Φ. As σ rises, the curve rotates counterclockwise or shifts upward. When σ reaches acritical value (σ = 2.221), its top touches the horizontal axis. Subsequently, the curve cuts
the horizontal axis twice. Then, for σ = 1 + µ (= 1.8), the curve cuts the horizontal axis
at φ = 0 and φ = 1. Thereafter, the black hole condition becomes satisﬁed. The curve lies
above the axis for an entire range of φ ∈ Φ.
— Insert Fig. 4 around here. —One could understand the fact that the ‘Ψ + Ω curve’ is bell-shaped with the help ofDiagrams (b) and (c), which describe Ψ and Ω, respectively, as functions of φ for given levelsof σ. As long as φ is not too high, Ω is negative; and therefore, the Ψ + Ω curve lies below
the ‘Ψ curve’. At the same time, Diagram (c) shows that an absolute value of Ω increases
with φ unless φ is too high. Consequently, the Ψ + Ω curve lies farther away from the Ψ
curve for higher values of φ.
Moreover, it is a novel outcome that φ has a non-monotonic impact on Ψ+Ω when σ is of
medium level. Suppose that σ is greater than 1 + µ but suﬃciently close to it, as represented
by σ = 1.9. For such σ, Ψ + Ω becomes negative at low values and high values of φ andpositive at intermediate values of it (see Diagram (a) of Fig. 4). Therefore, the symmetricequilibrium becomes stable when φ is either low or high and becomes unstable when it
is in-between. A conventional model does not exhibit such a non-monotonic property: itprescribes that the symmetric equilibrium become stable when φ is lower than a breakpoint and become unstable when φ exceeds it. To give an intuitive explanation for this nonmonotonicity, we again consider a situation where region 1 happens to become slightly bigger
b inﬁnitesimally increases from 1. As to the result that the symmetricthan region 2, or, λequilibrium is stable for suﬃciently low freeness of trade, nothing new enters the picture:
10 The terms of trade eﬀect, the regional income eﬀect, and the price index eﬀect are given by −φ(σ −‹‹ˆ˜1) (1 + φ), −2µφ(σ − 1)(1 − φ) (1 + φ)2 σ − µ + φ(σ + µ) , and µφ/(1 + φ), respectively.

20

relocating to the smaller region is proﬁtable for entrepreneurs, because by doing so, they
can acquire the domestic demand in the smaller region, whose amount is considerably large
due to a high delivered price of imports. To the contrary, the result that the equilibrium isstable for suﬃciently high freeness of trade is a novel feature of our model. If there were no
directional imbalance of transport prices, entrepreneurs would have an incentive to relocate
to the larger region (Ψ is positive); but only a weak incentive (Ψ is small) for suﬃciently
high freeness of trade. (Indeed, when manufacturing products are traded freely (φ = 1), thisb brings about a directional imbalance
incentive disappears (Ψ = 0).) Now, the increase in λby raising a transport price in a binding direction and reducing that in a slack direction.
As has been explained, this weakens the incentive of entrepreneurs to relocate to the larger
region through the terms of trade eﬀect, regional income eﬀect and the price index eﬀect.
For high freeness of trade, the intrinsic incentive to relocate to the larger region is so weakthat it is overturned by the eﬀect of the directional imbalance. That is, they actually have an
incentive to relocate to the smaller region. Therefore, the symmetric equilibrium becomesstable when freeness of trade is very high.
Finally, we can draw the locus of the break point as a function of σ. In Fig. 5, which isdrawn for µ = 0.8 as before, the solid line represents the break point in a conventional modelwith no endogenous determination of transport prices nor the possibility of their directional
imbalance, that is, the locus of Ψ = 0. The symmetric equilibrium is stable when a pair ofparameters lies below the line, that is to say, when freeness of trade is lower than the breakpoint value. Instead, it is unstable for a pair above the line, namely, a pair with freeness of
trade exceeding the break point value. Furthermore, the line is upward sloping: the higher σis, the more likely the symmetric equilibrium is to be stable. On the other hand, the dottedline represents the break point in our model with the endogenous determination of transport
prices and the possibility of their directional imbalance, namely, the locus of Ψ + Ω = 0.
The symmetric equilibrium is stable for a pair of parameters at the right of the line, andunstable for that at the left of it. Notice that both lines cut the horizontal axis at σ = 1 + µ,
the critical value of σ for the black hole condition. Furthermore, the maximum value of σon the dotted line (σ = 2.221) is the value for which the top of the Ψ + Ω curve touches thehorizontal axis in Diagram (a) of Fig. 4.
— Insert Fig. 5 around here. —4.3

Overall impacts of changes in parameters

One of the greatest achievements of new economic geography is that it has revealed theimpacts of changes in transport costs upon economic geography. As is asserted in the21

literature, transport costs have been declining signiﬁcantly over the centuries. Researchers
in the ﬁeld have proved through a rigorous analysis that this is expected to strengthen acentripetal force and reduce a centrifugal force, and, as a result, tend to make economic
activities concentrate in a fewer regions, which has indeed happened in the real world. In
this paper, however, transport costs are not exogenous but determined through each carrier’s
decision makings. What is exogenous is the cost parameters, k and c. In this section, thus,
we compete the analysis by examining in a spirit of the literature of new economic geography
what happens to economic geography if those parameters gradually decline over time.
In the ﬁrst place, let us consider a decline in k for given c. For suﬃciently high σ,
the core-periphery patterns are unsustainable while the symmetric equilibrium is stable forany k ∈ [0, 1). Instead, for suﬃciently low σ that the black hole condition is satisﬁed, thecore-periphery patterns are sustainable while the symmetric equilibrium is unstable for anyk ∈ [0, 1). Thus, an interesting case occurs for intermediate levels of σ. Fig. 6 shows
so-called ‘pitchfork diagrams’ for two values of σ when µ and c are ﬁxed at 0.8 and 0.2,
respectively. The horizontal axis measures k while the vertical axis λ.
— Insert Fig. 6 around here. —For σ = 3 (Diagram (a)), the core-periphery patterns are sustainable only for low valuesof k (k < 0.553). This is because a decline in k results in a fall in the transport pricein a binding direction, which enhances the relative indirect utility in the core through apositive and dominating terms of trade eﬀect and a negative but dominated price index
eﬀect. A ‘sustainability range’ of k, namely, the range of k for which the core-periphery
patterns become sustainable, furthermore, appears at a lowermost part of k for σ lower thana critical value (5.048), and expands as σ falls. On the other hand, the symmetric equilibrium
is stable for any k ∈ [0, 1). Therefore, when k is high, it is expected that economic activities
are dispersed between two regions. As k declines, the dispersed pattern probably continues
but there emerges a possibility that economic geography switches to the agglomeration by
accident.
For σ = 1.9 (Diagram (b)), the core-periphery patterns are sustainable for any k ∈ [0, 1).
The symmetric equilibrium is, on the other hand, stable only for low values of k (k ≤ 0.466).
We have explained this by comparing levels of Ψ and Ω as follows (see Fig. 4). When k
is low, freeness of trade is high at the symmetric equilibrium. For high freeness of trade,
entrepreneurs would have a small incentive to relocate to the larger region if there were no
directional imbalance of transport prices. The directional imbalance, however, reduces the
incentive. It turns out that the latter eﬀect of the imbalance more than oﬀset the former
so that they actually have an incentive to relocate to the smaller region. In addition, a22‘stability range’ of k, that is, the range of k for which the symmetric equilibrium becomesstable, contracts as σ decreases and disappears for σ lower than a critical value (1.837).
Finally, one would conceive a following scenario for this conﬁguration of sustainability andstability. When k is high, manufacturing activities are concentrated in one region. As
k becomes suﬃciently low, however, the possibility of dispersion arises in addition to theagglomeration.
Furthermore, notice that in each diagram of Fig. 6, two dotted arcs are drawn. They
represent equilibrium distributions which are interior but not symmetric. The upper arc
represents the distribution supported by an equilibrium with (τ1 , τ2 ) = (c + k, c). We canbU , which is not shown in the diagrams
verify that it actually lies above the line depicting λbL , represents the(see (15)). Similarly, the lower arc, which lies below the line depicting λdistribution supported by an equilibrium with (τ1 , τ2 ) = (c, c + k) (see (15)). It follows fromnumerical computation that these equilibria with an asymmetric interior distribution are
unstable. Thus, we can disregard them.11
Lastly, let us take a look at a change in c, ﬁxing the levels of µ and k at 0.8 and 0.3,
respectively. First, when σ is suﬃciently high or suﬃciently low, the result is similar tothe one obtained for a change in k. Thus, let us focus on intermediate levels of σ. For
one thing, Diagram (a) of Fig. 7 depicts a pitchfork diagram for σ = 3. The symmetricequilibrium is stable for any k ∈ [0, 1). The sustainability range of c is, on the one hand,an interval, (0.019, 0.797), which we can also observe in Fig. 3. As has been explained, thecore-periphery patterns are not sustainable for too low c, because the relative size of thedirectional imbalance of transport prices, which works against the sustainability, becomes
too large. The sustainability range emerges for σ lower than a critical value (3.332) andexpands toward both sides as σ declines. What is more, when σ = 1.9 (see Diagram (b)),
the core-periphery patterns are sustainable for any k ∈ [0, 1). The symmetric equilibrium
is stable for low values of c (c ≤ 0.283) and unstable for high values of it. The stability
range of c contracts as σ decreases and disappears for σ smaller than a critical value (1.825).
Finally, two dotted arcs in each diagram again represent unstable equilibrium distributions,associated with the transport prices given by (τ1 , τ2 ) = (c + k, c) and (τ1 , τ2 ) = (c, c + k),
respectively.
— Insert Fig. 7 around here. —11 As long as σ 6= 1+µ, in addition, there is no asymmetric interior distribution supported by an equilibrium

b2 for these prices, we have v
with (τ1 , τ2 ) = (τ1o , τ2o ). This is explained as follows. Because φ2 = φ1 λb(λ) =

b−
λ

σ−µ−1
σ−1

. Unless σ = 1 + µ, therefore, vb(·) is an increasing or a decreasing function of λ. Consequently,vb(λ) = 1 has a unique solution, λ = 1/2, as long as σ 6= 1 + µ.

235Concluding Remarks

In this paper, we have explained from the behaviors of individual transport ﬁrms how
and why transport prices become imbalanced with respect to directions of shipments, andexamined how it aﬀects economic geography in a general equilibrium framework of new
economic geography. It is found that the equilibrium transport price of the shipments
in a particular direction is a nondecreasing step-shaped function of a relative size of theembarkation region, which implies that the transport price in a binding direction is higher
than that in the slack direction. Furthermore, it is shown that volumes of trade are equal
between two directions when regions are suﬃciently alike in size, whereas they are diﬀerent
when one region is much bigger than the other. In addition, the directional imbalanceof transport prices makes the symmetric pattern more likely to be stable and the coreperiphery patterns less likely to be sustainable. To put it in other words, the imbalance acts
as a centripetal force.

24

References[1] Baldwin, R., R. Forslid, P. Martin, G. Ottaviano and F. R.-Nicoud. (2003) Economic
Geography and Public Policy, Princeton University Press, Princeton.
[2] Behrens, K., C. Gainé and J.-F. Thisse. (2006) “Is the regulation of the transport sector
always detrimental to consumers?” mimeo.
[3] Blonigen, B. A. and W. W. Wilson. (2008) “Port eﬃciency and trade ﬂows,” Review ofInternational Economics, 16, 21-36.
[4] Bougheas, S., P. O. Demetriades and E. L. W. Morgenroth. (1999) “Infrastructure,transport costs and trade,” Journal of International Economics, 47, 169-189.
[5] Boyer, K.. (1997) Principles of Transportation Economics, Addison-Wesley, Reading.
[6] Cariou, P. and F.-C. Wolﬀ. (2006) “An analysis of Bunker Adjustment Factors and freightrates in the Europe/Far East market (200-2004),” Maritime Economics and Logistics, 8,
187-201.
[7] Forslid R. and G. I. P. Ottaviano. (2003) “An analytically solvable core-periphery model,”Journal of Economic Geography, 3, 229-40.
[8] Fujita, M., P. Krugman and A. J. Venables. (1999) The Spatial Economy: Cities, Regions, and International Trade, The MIT Press, Cambridge.
[9] Fujita, M. and J.-F. Thisse. (2000) “The formation of economic agglomerations: oldproblems and new perspectives,” in J.-M. Huriot and J.-F. Thisse (eds.), Economics ofCities, Cambridge University Press, Cambridge, pp. 3-73.
[10] Fujita, M. and J.-F. Thisse. (2002) Economics of Agglomeration, Cambridge University
Press, Cambridge.
[11] Hummels, D. (2007) “Transport costs and international trade in the second era ofglobalization,” Journal of Economic Perspectives, 21, 131-154.
[12] Jonkeren, O., E. Demirel, J. van Ommeren, and P. Rietveld. (2008) “Endogenous transport prices and trade imbalances,” Tinbergen Institute Discussion Paper, 2008-088/3.
[13] Mori T. and K. Nishikimi. (2002) “Economies of transport density and industrial agglomeration,” Regional Science and Urban Economics, 32, 167-200.
[14] Mun S. and S. Nakagawa. (2008) “Cross-border transport infrastructure and aid policies,” Annals of Regional Science, 42, 465-486.

25

[15] Sjostrom, W. (2004) “Ocean shipping cartels: A survey,” Review of Network Economics,
3, 107-134.
[16] Stopford, M. (2009) Maritime Economics, Routledge, Oxon.
[17] Takahashi, T. (2006) “Economic geography and endogenous determination of transportation technology,” Journal of Urban Economics, 60, 498-518.
[18] UNCTAD. (2008) Review of Maritime Transport 2008, United Nations Publications,
New York.
[19] Wilmsmeier, G., J. Hoﬀmann, and R. J. Sanchez. (2006) “The impact of port characteristics on international maritime transport costs,” in K. Cullinane and W. K. Talley(eds.), Port Economics, Elsevier, Amsterdam, 16, 117-140.

26

Appendix
Proof of Lemma 1.
In this proof, we examine whether carrier h can obtain a positive proﬁt by deviating fromthe Nash strategy, given that every other carrier sticks to it. Suppose that zih ≥ zjh (i, j =1, 2; i 6= j). Then, the proﬁt of the deviator is written as
[
]
∑(
) h
( ∗
)( h
) ∑( ∗) hhhhhτl − c zl − kzi = − τj − c zi − zj +τl − τl zl ,
l=1,2

l=1,2

where the equality follows from the fact that τ1∗ + τ2∗ − 2c − k = 0. Because τj∗ − c ≥ 0 (see))(((15)), τj∗ − c zih − zjh ≥ 0. Furthermore, if the deviator charges a price higher than τl∗ ,()
it will get zlh = 0 and, therefore, τl∗ − τlh zlh = 0. On the other hand, if it undercuts τl∗ , it
(
)
(
)
∑
may receive a positive demand and τl∗ − τlh zlh ≥ 0. Therefore, l=1,2 τl∗ − τlh zlh ≥ 0.
Hence, the deviating carrier can never earn a positive proﬁt.
Proof of Lemma 2.
The proﬁt of a deviator, carrier h, is now represented by
[
]
∑(
)
(
)(
)
) ∑( oτl − τlh zlh
τlh − c zlh − kzih = − τjo − c zih − zjh +
l=1,2

l=1,2

with zih ≥ zjh (i = 1, 2; j 6= i), where the equality follows from (17). By the same reason()∑
explained in the proof of Lemma 1, l=1,2 τlo − τlh zlh ≥ 0. First, let us prove the suﬃcient))((condition. If τ1o − c ≥ 0 and τ2o − c ≥ 0, then τjo − c ≥ 0. Therefore, τjo − c zih − zjh ≥ 0.
Hence, the deviating carrier cannot earn a positive proﬁt. Second, in order to prove thenecessary condition, suppose that τ2o −c < 0. Then, carrier h can choose z1h and z2h arbitrarily()
so that z1h > z2h while keeping the prices at τ1h , τ2h = (τ1o , τ2o ). By doing so, it earns a)(∑positive proﬁt because (τ2o − c) z1h − z2h + l=1,2 (τlo − τlo ) zlh < 0. This is a contradiction.
Hence, it must be the case that τ2o − c ≥ 0 at the equilibrium. We can similarly prove thatthe condition τ1o − c ≥ 0 is also necessary.
Proof of Lemma 3.
Because Υ = 0 at (φ1 , φ2 ) = (1, 1), the sustain curve passes through (1, 1). Note that at that√>>point, ∆ = (σ −µ)(σ −µ−1)−µ(σ +µ) = 0 if σ = σ 0 ≡ (1+3µ+ 1 + 2µ + 9µ2 )/2. Three<<cases are distinguished. First, if σ > σ 0 , then ∆ > 0 for any (φ1 , φ2 ) ∈ Φ2 because φ1 < 1and φ2 < 1. (The sustain curve for σ = 4 in Fig. 2 represents this case since σ 0 = 3.146
for µ = 0.8.) Second, suppose that σ < σ 0 . Then, at (1, 1), the sustain curve is downward
sloping. Note that d2 φ2 /dφ21 < 0 when evaluated at ∆ = 0. That is, the sustain curve is¯
concave at its stationary points. This, together with the continuity of dφ2 /dφ1 ¯Υ=0 , implies
that it has at most one stationary point. Consequently, there are two possibilities. One is
the possibility that the curve is downward sloping at any φ1 ∈ Φ. (The curve for σ = 1.9
in Fig. 2 represents this case.) In this case, it passes through φ2 > 1, that is, φ2 ∈/ Φ, forany φ1 ∈ Φ. Hence, we do not have to consider this case. The other is the possibility that()()
the curve is upward sloping for any φ1 ∈ 21−σ , φ1 and downward sloping for φ1 ∈ φ1 , 1
where φ1 ∈ Φ gives a stationary point. (The curve for σ = 3 in Fig. 2 represents this case.)
By construction, the curve takes φ2 > 1 at interval φ1 ∈ [φ1 , 1). Therefore, if the curve27

passes through (φ1 , φ2 ) ∈ Φ2 , we must have φ1 < φ1 . Hence, the curve is upward slopingat such a point. Third and ﬁnally, suppose that σ = σ 0 . In this case, the sustain curve
becomes stationary at (1, 1). Because it is concave at that point, the curve is upward sloping
for any φ1 ∈ Φ.

28

τ1 , τ2
τ1
0.4

τ2

0.2

0.5

λ

1

(a) Transport prices.

Z
2

1

0.5

1

λ

(b) Relative volume of trade.
Fig. 1. Effect of entrepreneurs’ distribution on equilibrium variables
when σ = 3, μ = 0.8, c = 0.2 and k = 0.3.

φ2
σ=4
σ=3

1

σ = 1.9σ = 1.5
0.5

0.5

1

Fig. 2. Sustain curves when μ = 0.8.

φ1

c

0.75

σ=3

0.5

σ=4

0.25

0.25

σ = 1.5

0.5

σ = 1.9

Fig. 3. Loci of ϒ = 0 when μ = 0.8.

k

0.5

σ = 1.5
0.5

1

φ

σ = 1.9

0.5

1

σ=3
1.5

1

0.5

σ=4
(a) Ψ + Ω.
σ = 1.5

σ = 1.9
σ=3
0.5

1

φ

σ=4
0.5

(b) Ψ.
0.5

σ = 1.5
0.5

1

φ

σ = 1.9

0.5

1

σ=4

σ=3
(c) Ω.

Fig. 4. Ψ + Ω, Ψ and Ω as functions of φ when μ = 0.8.

φ
1

0.5

1.8 2 2.221

3σFig. 5. The loci of break point when μ = 0.8.

λ
1

0.5

0.25

0.553

0.75

1

0.75

1

k

(a) σ = 3.

λ
1

0.5

0.25

0.466

k

(b) σ = 1.9.

Fig. 6. Effect of k on economic geography
when μ = 0.8 and c = 0.2.

λ
1

0.5

0.019

0.25

0.5

0.797

1

c

(a) σ = 3.

λ
1

0.5

0.283

0.5

0.75

1

c

(b) σ = 1.9.
Fig. 7. Effect of c on economic geography
when μ = 0.8 and k = 0.3.