Multiple growth and urbanization patterns in an endogenous growth model with spatial agglomeration

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Journal of Development Economics 75 (2004) 167–199

Multiple growth and urbanization patterns

in an endogenous growth model with

spatial agglomeration

Hiroki Kondo*

Department of Economics, Shinshu University, Asahi 3-1-1, Matsumoto, Nagano 390-8621, Japan

Received 1 January 2002; accepted 1 July 2003

Available online

Abstract

This paper presents a model which integrates an economic geography model with an endogenous

growth model and examines how the specialization pattern and global economic growth rate are

determined in an economy with decreasing transaction costs. If transaction costs decrease slowly and

the agglomeration process has not yet been completed, economic activities tend to agglomerate in the

region with large land mass and a high growth rate will be attained. On the other hand, a sharp

decrease in transaction costs can lead to agglomeration in a region with a paucity of land. In this

case, the further decrease in transaction costs decreases the global economic growth rate.

D 2004 Elsevier B.V. All rights reserved.

JEL classification: O18; O41; R11; R12

Keywords: Endogenous growth; Spatial agglomeration; Migration; Self-fulfilling expectation; Transaction costs

1. Introduction

High economic growth has often accompanied rapid industrialization and urbanization.

Such prominent growth, in the form of a radical intersectoral and interregional reallocation

of resources, has been triggered by integrations in factor and goods markets. In particular,

rapid progression in information technologies, along with continued improvements in

shipping, aircrafts, and other aspects of transportation, has enabled the efficient, long-

distance supply of a wide range of goods, which in turn has contributed to integration in

0304-3878/$ - see front matter D 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.deveco.2003.07.008

* Tel.: +81-263-37-2326; fax: +81-263-37-2344.

E-mail address: [email protected] (H. Kondo).

H. Kondo / Journal of Development Economics 75 (2004) 167–199168

the goods markets. Under these conditions, technological external effects of modern

industries have led to the concentration of economic activities in a limited number of

regions and sustainable economic growth.

In this paper, we examine how the urbanization and specialization patterns as well as

the economy-wide growth rate are determined in an economy with an integrated labor

market and decreasing transaction costs. This paper presents a model of two regions with

differing amounts of immobile factors such as land. Although households desire both a

large variety and quantity of differentiated goods, as introduced by Dixit and Stiglitz

(1977), such goods cannot be traded interregionally without transaction costs. For this

reason, firms and households tend to agglomerate in one region. Since households are

interregionally mobile, where the agglomeration occurs does not matter from income

distribution viewpoints, but rather it matters from the viewpoints of resource allocation.

In the case that the agglomeration process is in the initial stage, the region where

economic activities concentrate depends not only on historical factors, but also on the

expectations about the future. On the other hand, in the case that the agglomeration process

has been nearly completed in one region, it is impossible for another region to leapfrog the

first with respect to the degree of agglomeration. Moreover, the specialization and

urbanization pattern and global economic growth rate are jointly determined depending

on the degree and speed of the improvements in transaction facilities in the goods market,

and on the stage of agglomeration under which the improvements proceed.

The region with a large land mass can become the core of the economy in the long run

even if the agglomeration in that region was initially insufficient. As agglomeration

proceeds, the economy-wide growth rate increases, and remains high in the long run. The

more slowly the transaction costs decrease, the more likely it is that this process will

emerge.

In contrast, a rapid improvement in transaction costs will allow economic activities to

be concentrated in a smaller region. With a sharp decrease in transaction costs, the

differentiated goods sector in the region with a larger land mass faces fiercer competition

from imports. Therefore, the development of new products will shift to the smaller region,

which will then show greater agglomeration in the long run. Under such an agglomeration

pattern, a further improvement in transaction costs will deteriorate the economy-wide

growth rate. Since households are interregionally mobile, the benefits and the costs of

agglomeration must be balanced in equilibrium. That is, the benefits of cheaper

differentiated goods should compensate the costs of a higher population density in the

core. However, in the case that a region with small land mass is the core, the population of

the region will not necessarily be larger. A decrease in transaction costs enables house-

holds in the peripheral region to consume differentiated goods at lower costs, diminishing

the necessity of residing in the core. The population will then move from the core to the

periphery, which will have the effect of increasing the wages, reducing the capital

accumulation in the core, and thereby reducing the economy-wide growth rate.

This paper integrates an economic geography model and an endogenous growth

model—two systems that have previously been treated as discrete. The new economic

geography models such as Fujita (1988), Abdel-Rahman (1988), Abdel-Rahman and

Fujita (1990), and Krugman (1991a) introduce Dixit and Stiglitz (1977) types of

differentiated goods and transaction costs, and analyze the agglomeration process of

H. Kondo / Journal of Development Economics 75 (2004) 167–199 169

economic activities. It has been shown that households are willing to reside in the region

with more differentiated goods firms to benefit from the lower transaction costs. On the

other hand, the fiercer competition in such a region makes profits per firm lower. However,

the market scale in this region is larger since more households want to reside there. If the

latter market expansion effects dominate the former fiercer competition effects, firms and

households agglomerate in one region enhancing their profits and utility levels with one

another. The factors that cause such an agglomeration process are called ‘‘pecuniary

external effects.’’ On the other hand, Krugman (1991b) introduces a factor of production

with ‘‘technological external effects’’ and analyzes in which region such a factor

agglomerates, focusing on the role of expectation as well as historical inertia. He shows

that if the agglomeration process is still in its initial stage, expectations play a crucial role

in the choice of the long-run agglomeration pattern. In addition, the more localized the

technological external effects, the larger the role of the expectations. Matsuyama (1991)

introduces the sector with technological externalities and analyzes the role of expectation

employing the more generalized framework with high-powered mathematics, but geo-

graphical aspects are absent. Ottaviano (2001) applies Matsuyama’s method to the

economic geography models where pecuniary external effects cause agglomeration. This

shows that the more prominent the pecuniary externalities, the larger the role of the

expectations.

In these economic geography models, capital accumulation or the increase in the total

number of differentiated goods is not explicitly introduced. In these models, the dynamics

are the reallocation process of differentiated goods firms and population between the

regions. In reality, however, the number of the differentiated goods is increased by R&D

investments motivated by the profits from monopolies. The R&D investments, in turn, are

financed by savings of households who maximize their intertemporal utilities. Conse-

quently, the knowledge capital that is necessary for producing differentiated goods is

accumulated, and it is shared by households as assets in the shape of patents or the stocks

of differentiated goods firms. Moreover, in the R&D activities, dynamic technological

external effects operate such that the previous R&D activities enhance the efficiency of the

present R&D activities. Romer’s (1990) endogenous growth model analyzes sustainable

growth in the number of differentiated goods and thus in the knowledge capital,

incorporating the intertemporal optimization behaviors of firms and households, and the

dynamic technological external effects. The dynamic external effects are geographically

local in nature as stressed by Lucas (1988). However, little attention has been given to the

geographical aspects in the field of economic growth theory.1

Grossman and Helpman (1990, 1991) applied the framework of Romer (1990) to a two-

country model and showed that the differentiated goods sector will be concentrated in a

single country if the dynamic technological external effects are quite local in scope. The

specialization pattern is determined only by historical inertia, which is called ‘‘hysteresis.’’

Baldwin et al. (2001) introduced transaction costs on differentiated goods into the

Grossman and Helpman (1991) model and investigated the relationships between the

1 Faini (1984, 1996), Henderson (1987), and Ioannides (1994) have introduced geographical aspects into

traditional economic growth models such as the Solow model, an optimal growth model, and an overlapping

generations model.


transaction costs, agglomeration and trade patterns, and growth rate. The differentiated

goods sector tends to agglomerate in one region in the case that transaction costs are low

and dynamic technological external effects are quite local. The specialization pattern is

determined only by historical inertia, as is in the work of Grossman and Helpman (1991).

However, in the works of Grossman and Helpman (1990, 1991) and Baldwin et al.

(2001), labor mobility was not introduced. Therefore, we can observe that differentiated

goods firms concentrate in one region, but we cannot observe that it complementally

causes the agglomeration of population. That is, among the pecuniary external effects and

the technological ones, only the latter can be observed. If labor mobility is absent, it is

relatively less plausible that economic activities agglomerate geographically, as shown in

the works of Krugman and Venables (1995) and Ioannides (1999). In reality, however, the

geographical scope of the technological externalities is smaller than the scope that

households can move. Therefore, it is necessary to introduce interregional migration

and then take into account the pecuniary external effects, as well as the technological

external effects, to consider the growth and urbanization patterns in an economy. However,

in the model with both of the two external effects, agglomeration is far more plausible than

in new economic geography models with pecuniary externalities, or in the works of

Krugman (1991b) and Baldwin et al. (2001) with technological externalities. Hence, to

make a fairer analysis, we introduce the congestion effects that makes population disperse

among regions.

Walz (1996) is one of a few researches that allow labor mobility in an endogenous

growth model with spatial agglomeration. However, the role of the transaction facilities

was analyzed with a focus on the steady state equilibrium under which the agglomeration

process was completed. In reality, however, the growth rate and specialization patterns in

the long run can differ depending on to what degree the improvements in transaction

facilities proceed, and at what stage of agglomeration they occur. Therefore, it is crucial to

consider the transitional dynamics as well as steady states. If the agglomeration process

has not yet been completed, a sharp decrease in transaction costs can lead to agglomeration

in a region with a paucity of land, which results in a less efficient outcome. In this case, a

further decrease in transaction costs decreases the economic growth rate. In addition, Walz

(1996) focused mainly on cases in which there was no difference in wages. In general

settings, however, wages can be unequal. Since households are interregionally mobile, it

does not matter from income distribution viewpoints, but it may reflect the inefficient

allocation of resources. Indeed, the phenomenon that a decrease in transaction costs

reduces the growth rate can be observed in the case that unequal wages emerge.

2. The basic model

Consider an economy that consists of two regions, A and B. The economy is endowed

with one unit of households that are interregionally mobile, and each region is endowed

with a land mass of Ni(i =A, B). The land mass in region B is larger than that in region A.

In the economy, two sets of final goods are produced: differentiated goods (X) and

traditional goods (Y). In addition to these goods, households’ utility levels depend on the


available land mass per capita in their residential region too. In the region with higher

population density, the utility from land is lower.

A household residing in region i faces the following optimization problem:

MAX

Z l

t

falnCiXt þ ð1� aÞlnCi

Yt þ blnðNi=LiÞge�qðs�tÞds; ð1Þ

CiXt ¼

Z nAt þnBt

0

ðCijtÞ

cdj

" #1c

; ð2Þ

s:t: Ai ¼ rAi þ wi � Ei: ð3Þ

In Eq. (1), CXi and CY

i denote the consumption of final goods per capita and Li denotes the

population in region i. In Eq. (2), Cji denotes the consumption of differentiated goods j, and

nti denotes the measure of differentiated goods in region i. The elasticity of substitution

between any two differentiated goods is e= 1/(1� c)z 1. In the flow budget constraint

(Eq. (3)), Ai denotes the financial assets per capita in region i. The financial assets are

interregionally mobile. Therefore, an identical return prevails, which is denoted by r. Also,

wi represents the wages, and Ei denotes the consumption expenditure per capita in region i:

Ei ¼Z nAþnB

0

pijCijdjþ pYC

iY;

where pji( ja(0, nA + nB) and pY denote the prices of differentiated goods and traditional

goods in region i, respectively. We assume that both differentiated goods and traditional

goods are tradable. However, differentiated goods cannot be interregionally traded without

some transaction costs as is common in the new economic geography models. The

transaction costs reflect various kinds of costs of doing business. These include not only

transport costs but also information costs, such as search costs and the costs of commuting

with separated customers. Thus, the prices of differentiated goods will differ between

regions.

As the solution to the dynamic optimization problem, we obtain the following

conditions2:

Cij ¼

ðpijÞ�e

Z nAþnB

0

ðpijVÞ1�e

djV

aEi; i ¼ A;B; ð4Þ

2 First, we derive Eqs. (4) and (5) that maximize the instantaneous utility under a value of Ei, and describe the

instantaneous utility as a function of Ei. Then, we derive Eq. (6) that maximize the intertemporal utility. In the first

step, we obtain the maximization conditions ofPA;B

i piYCiY ¼ ð1� aÞ=f and pijx

ij mn

AþnB

0 ðxijÞcdji¼ ða=fÞðxijÞ

ch

,

where f is the Lagrange multiplier. Integrating the second equation yields mnAþnB

0 pijxijdj ¼ a=f. Then we can see

that the consumption expenditures for differentiated goods and traditional goods are aEi and (1� a)Ei,

respectively. The second equation can be rearranged as pijxij ¼ ðpijÞ

1�e mnAþnB

0 ðxijÞdji�e

ða=fÞeh

. Integrating it yields

aEi ¼ mnAþnB

0 ðpijÞ1�e

djimn

AþnB

0 ðxijÞcdji�e

ða=fÞ�ehh

With these two equations, we can see that Eq. (5) holds.


CiY ¼ ð1� aÞEi

py; i ¼ A;B; ð5Þ

Ei

Ei¼ r � q; i ¼ A;B: ð6Þ

Each differentiated good is monopolistically supplied by the firm that invented that

good. One unit of each differentiated good is produced using one unit of labor as input.

For dealing with differentiated goods, we introduce the iceberg form of transaction costs.

That is, to sell one unit to the other region, sz 1 units must be exported. Then, the

production function in which differentiated goods are measured by the amount actually

sold can be written as follows:

xij ¼ lij; i ¼ A;B; ð7Þ

xij* ¼ lij*=s; i ¼ A;B; ð7ÞV

where xji and xj

i* denote the actually sold differentiated good j in region i domestically and

in another region, and lji and lj

i* are the devoted labor force, respectively. Then, the cost to

sell one unit domestically is wi and that to sell in the other region is swi. From Eqs. (4), (7)

and (7)V, we can describe the profits per firm as follows:

pij ¼

ðpijÞ1�e � wiðpijÞ

�e

Z nAþnB

0

ðpijÞ1�e

djV

aLEi þðp�i

j Þ1�e � swiðp�ij Þ�e

Z nAþnB

0

ðp�ij Þ1�e

djV

að1� LÞE�i; i ¼ A;B:

ð8Þ

L denotes the population in region A, and � i denotes the region other than i. The choice

of the price that maximizes Eq. (8) yields

pi ¼ wi

c; ð9Þ

pi* ¼ swi

c: ð9ÞV

where pi is the price of the differentiated good which is produced and sold in region i

domestically, while pi* is the price of the good which is produced in region i but

transported to another region. By aggregating the demand for differentiated good j (Eq.

(4)) in each region and inserting Eqs. (9) and (9)V into this formula, we derive the

consumption of differentiated good j in each region. Taking this consumption level

together with Eqs. (7) and (7)V, we can derive the labor force devoted to the production of

differentiated good j. Aggregating this labor force for all differentiated goods located in


each region yields the total labor force employed in the differentiated goods sector in each

region:

lAX ¼ cahwA

� �LEA

h þ ð1þ hÞ/xþ ð1� LÞEB/

h/ þ ð1� hÞx

; ð10aÞ

lBX ¼ cað1� hÞwB

� �LEA/x

h þ ð1� hÞ/xþ ð1� LÞEBx

h/ þ ð1� hÞx

; ð10bÞ

with x=(wB/wA)1� e, /=(s)1� e, (0 </V 1) and h = nA/(nA + nB), (0V hV 1). A larger

value of / indicates lower transaction costs. Inserting Eqs. (9) and (9)Vinto Eq. (8), we canget the profits per firm:

pA ¼ ð1� cÞanA þ nB

� �LEA

h þ ð1� hÞ/xþ ð1� LÞEB/

h/ þ ð1� hÞx

; ð11aÞ

pB ¼ ð1� cÞanA þ nB

� �LEA/x


h/ þ ð1� hÞx

: ð11bÞ

We assume that differentiated goods are produced only in the region where they were

invented. In many cases, the knowledge capitals to produce the differentiated goods

include not only the technological ones, but also the tacit ones like cultural and social

factors, which are hard to adopt in the other region. Hence, Eqs. (11a) and (11b) are not

necessarily equal.

It takes one unit of labor to produce one unit of traditional goods. Traditional goods are

supplied in a perfect competitive market, which implies marginal cost pricing:

pY ¼ w; with w ¼ minfwA;wBg ð12Þ

Aggregating the demand for traditional goods (Eq. (5)), we can derive the labor force

devoted to the traditional goods sector in region i. When there exists a difference in wage

between regions, these labor forces are

liY ¼ ð1� aÞfLEA þ ð1� LÞEBgwi

; l�iY ¼ 0; wi

tVw�it ð13Þ

On the other hand, in the case of equal wages, they are

lAY ¼ ð1� aÞsfLEA þ ð1� LÞEBgwA

; ð14aÞ

lBY ¼ ð1� aÞð1� sÞfLEA þ ð1� LÞEBgwB

; ð14bÞ


where s(0V sV 1) denotes the market share of region A in the production of traditional

goods.

Households hold the share of the firm producing differentiated goods as financial

assets, which brings dividend pi. Equilibrium in the financial market requires:

At ¼ nAvA þ nBvB; ð15Þ

r ¼ v i þ pi

vi; i ¼ A;B; ð16Þ

where vi denotes the market prices of share of differentiated goods firms in region i.

The number of differentiated goods is increased by R&D activities motivated by the

profits from monopolies. The number of newly developed differentiated goods is

expressed by the following equation:

dni ¼ ðni þ kn�iÞliR&D; i ¼ A;B; ð17Þ

where lR&Di represents the labor devoted to the R&D activities in region i. In Eq. (17),

we assume dynamic external effects. That is, the more R&D activities have been

accumulated, the more efficient the present R&D activities are. Also, k(0V kV 1)

denotes the geographical scope of such external effects. The extreme cases of k = 1and k= 0 indicate perfect and nonexistent interregional external effects, respectively. We

assume partially localized external effects.3 We use gi = dni/ni to express the growth rate

of the number of differentiated goods in region i. With gi, Eq. (17) can be rewritten as

follows:

gihi=ðhi þ kh�iÞ ¼ liR&D: ð17ÞV

R&D activities create a value of vi profits, the present value of the profits the new good

brings. In the case that it exceeds the R&D cost of wi/(ni + kn� i), unbounded labor will

engage in the R&D activities. Such a case cannot arise in general equilibrium. Then, the

profits must be equal to or less than the R&D costs:

MAX vA � wA

nA þ knB; vB � wB

knA þ nB

� �V0: ð18Þ

For R&D to be active in region i, Eq. (18) must be bind for that region. This condition is

called a free entry condition. When Eq. (18) binds only in region A, we can obtain the

3 Geographically, these effects of increasing returns to scale tend to be local, as discussed by Lucas (1988).

The empirical analysis such as that of Jaffe et al. (1993), Henderson et al. (1995), and Ciccone and Hall (1996)

supports rather regional in scope spillover.


growth rate of the measure of differentiated goods using Eqs. (11a) (11b) and (16) as

follows:

hgA

h þ kð1� hÞ ¼ð1� cÞa

wA

� �fh þ kð1� hÞg LEA


h/ þ ð1� hÞx

� r þ wA

wA

� �: ð19aÞ

When Eq. (18) binds only in region B, we get

ð1� hÞgBkh þ ð1� hÞ ¼

ð1� cÞawB

� �fkh þ ð1� hÞg

� LEA/xh þ ð1� hÞ/x

þ ð1� LÞEBxh/ þ ð1� hÞx

� r þ wB

wB

� �ð19bÞ

In the region where Eq. (18) does not bind, it follows that

gi ¼ 0: ð20Þ

In the case of Eq. (18) being bind for both regions, R&D activities take place in both

regions. From Eqs. (11a) Eqs (11b) and (16), gA and gB are determined at the levels that

satisfy the following two equations:

hgA þ kð1� hÞgBh þ kð1� hÞ ¼ ð1� cÞa

wA

� �fh þ kð1� hÞg

� LEA


h/ þ ð1� hÞx

� r

þ wA

wA

� �: ð21aÞ

khgA þ ð1� hÞgBkh þ ð1� hÞ ¼ ð1� cÞa

wB

� �fkhð1� hÞg

� LEA/xh þ ð1� hÞ/x

þ ð1� LÞEBxh/ þ ð1� hÞx

� r

þ wB

wB

� �: ð21bÞ


In which region Eq. (18) binds depends on the sizes of the differentiated goods firms’

stock prices and the costs of the R&D activities in two regions. The latter depends on hwhich has been historically determined. On the other hand, the former depends on the

expectation about in which region the differentiated goods firms will concentrate in the

future, as well as the historically determined h. In the case that the profits of the

differentiated goods firms can be higher in the region with relatively larger number of

differentiated goods, the stock prices can be higher in the region where the firms are

expected to concentrate in the future. Consequently, the differentiated goods sector can

concentrate in the region where the number of differentiated goods is now smaller and then

the R&D costs are higher, if such an expectation is shared. We will discuss these points in

Section 3.

In this model, households get utilities from land. In reality, the land is owned by

households, rented in housing market, and traded in financial market. Hence, the

determination of land rents and land prices should be taken into consideration. However,

the discussions are not essentially changed even if these factors are taken into account.

Since households get utilities from exclusive use of land, rents can be charged for it.

That is, the payments and receipts of land rents between households emerge. The land

rents are higher in the region where more households are willing to reside. The equilibrium

rents are determined at the level where the demand for the land per capita just equals to the

supply of the land per capita, Ni/Li. Consequently, the objective function can be described

as Eq. (1), in the case that the land is owned privately too. On the other hand, the return

from the land is the same irrespective of which region’s land households own due to the

arbitration in financial market.4 Then, introducing the ownership of land does not

essentially change households’ budget constraint.

3. Equilibrium

3.1. Dynamic equilibrium path

In this section, we consider the dynamic equilibrium path of the model. Henceforth, we

take the wages in region A as numeraire.

In the case of equal wages (i.e., wB = 1 and x = 1 hold), we can derive the labor market

clearing conditions, with Eqs. (10a), (10b), (14a), (14b) and (17)V, as follows:

hh þ kð1� hÞ

gA þ hca

LEA

h þ ð1þ hÞ/ þ ð1� LÞEB/h/ þ ð1� hÞ

þ sð1� aÞfLEA þ ð1� LÞEBg ¼ L ð22aÞ

4 In financial market, the land prices are determined as the present value of the future land rents. Households

can earn the higher land rents from the land in the region with higher population density, but the purchase of such

a land costs more due to the higher land prices. The returns of the land in region A, the one region B and the

stocks of differentiated goods firms are all equalized.


1� hkh þ ð1� hÞ

gB þ ð1� hÞca LEA/

h þ ð1þ hÞ/ þ ð1� LÞEB

h/ þ ð1� hÞ

þ ð1� sÞð1� aÞfLEA þ ð1� LÞEBg ¼ 1� L ð22bÞ

On the other hand, in the case that region A has the higher wages, we can derive the labor

market clearing conditions, with Eqs. (10a), (10b), (14a), (14b) and (17), as follows:

hh þ kð1� hÞ

gA þ hca

LEA


h/ þ ð1� hÞx

¼ L ð23aÞ

1� hkh þ ð1� hÞ

gB þ ð1� hÞca

wB

� �LEA/x


h/ þ ð1� hÞx

þ ð1� aÞfLEA þ ð1� LÞEBgwB

¼ 1� L ð23bÞ

In the opposite case that region B has the higher wages, the labor market clearing

condition in region A can be derived by adding the term (1� a) {LEA+(1� L)EB} to the

left-hand side of Eq. (23a) and the one in region B can be derived by subtracting its third

term from the left-hand side of Eq. (23b).

In these conditions, the first term including gi in the left-hand side denotes the labor

force devoted to R&D activities. Differentiated goods firms can operate only in the region

where they exploited the goods. Therefore, R&D is active only in the region where the

expected present value of the future profits the newly exploited good brings can be equal

to the today’s R&D costs. That is, only in the region with Eq. (18) binding, gi can be

positive as Eqs. (19a), (19b). On the contrary, in the region where Eq. (18) does not bind,

gi is zero as Eq. (20). In the case with Eq. (18) binding for both regions, gi must satisfy Eq.

(21a) and (21b). The second and third terms in Eqs. (22a), (22b), (23a), and (23b)

represent the labor forces employed in the productions of differentiated goods and

traditional goods, respectively. In the case of unequal wages, the third term is zero in

the region with higher wages.

Since households are interregionally mobile, they can acquire an equal level of utility

regardless of the region where they live. The condition is derived as follows:

EA

EB

� �¼ NB

NA

� �L

1� L

� �h/ þ ð1� hÞxh þ ð1� hÞ/x

� � abðe�1Þ

" #ð24Þ

(See Appendix A for the derivation.) The utility level in each region depends on the

following three elements: the wages, the land mass per capita, and the price level of

differentiated goods. In the region where the wages are higher, the consumption

expenditure is higher too. Hence, the higher the relative wages in region A, the larger

the left-hand side of Eq. (24). In this sense, the left-hand side of Eq. (24) refers to the

relative benefits of residing in region A. On the other hand, the larger the size of the


population in region A, the smaller the land mass per capita and then the utility level

there. In addition, the smaller the relative number of differentiated goods in region A,

the higher the relative price of goods for residents and then the smaller the utility level

there. These two factors are summarized in the right-hand side of Eq. (24), which can be

said to represent the relative costs of residing in region A. Consequently, Eq. (24) means

that these three factors must be balanced in an equilibrium. The wages obtained by

solving the equilibrium conditions including Eq. (24) are not necessarily equal between

regions, since wages are one of the three factors which determine utility level. In the

following analyses, whether or not we are in the case under which wages are equal must

always be checked.

The dynamic equilibrium path in this economy is described by the simultaneous

differential equations which consist of Eq. (6) and

h ¼ hð1� hÞðgA � gBÞ: ð25Þ

Under given EA, EB and h, the valuables r, gA and gB in Eqs. (6) and (25), together with s

and L, are determined as the solution of the system consisting of the labor market clearing

conditions (22a) and (22b), the equal utility conditions (Eq. (24)), and the Eqs. (19a),

(19b), (20), (21a), and (21b). However, wages differ between regions, if the obtained s is

out of the range of sa[0, 1]. In such a case, r, gA and gB in Eqs. (6) and (25), together with

x and L, are determined as the solution of the system consisting of the labor market

clearing conditions (23a) and (23b), the equal utility conditions (Eq. (24)), and the Eqs.

(19a)–(21b). The obtained r, gA, gB, L and s or x, in turn, drive EA, EB and h according to

the simultaneous differential Eqs. (6) and (25).

We have described the dynamics of the model, but solving it directly is formidably

difficult. Thus, to investigate the characteristics of the dynamics, we employ the following

procedure. First, we investigate the steady states of the system. If there exist multiple

steady states, it is important to analyze the differences in wages, consumption expendi-

tures, and growth rate between the steady states. Then, we investigate the stability of each

steady state and the transitional dynamics.

3.2. Steady state equilibria

From the simultaneous differentiated Eqs. (6) and (25), we can see that if interest rate is

equal to subjective discount rate: i.e.,

r ¼ q ð26Þ

and if one of the equations h = 1, h = 0 and gA= gB holds, the economy is in a steady state

equilibrium where consumption expenditures, wages and the allocation of population

between regions remain fixed, and the number of differentiated goods grow at a constant

rate. First, we will investigate the consumption expenditures, the wages, the population

and the growth rate on a steady state equilibrium in which differentiated goods firms and

R&D activities concentrate in only one region: i.e., h = 1, gA>0 and gB = 0 hold, or h = 0,gA= 0 and gB>0 hold. Next, we will analyze an steady state equilibrium where gA= gB


holds and then h remains constant in the range of ha(0, 1). Whether these steady state

equilibria are supported as stable equilibria will be examined in the next subsection.

Henceforth, we call such steady state equilibria where differentiated goods firms are

concentrated in only one region core–periphery steady state equilibria, and the region

where agglomeration emerges the core.

As we have discussed, wages are not necessarily equal between regions. In the case that

wages are equal in a core–periphery steady state equilibrium, from household’s budget

constraint (Eq. (3)), EA=EB holds. We introduce E to represent these equal levels of

consumption expenditure. By inserting h= 0 or h = 1 into the equilibrium conditions, we

can obtain the consumption expenditure, the economic growth rate and the population size

in the core as follows:

E ¼ 1þ q ð27Þ

g ¼ ð1� cÞað1þ qÞ � q: ð28Þ

Lcore ¼ 1

1þ /a=bðe�1ÞðN periphery=N coreÞð29Þ

On the other hand, in the case that wages are unequal in a core–periphery steady state

equilibrium, traditional goods sector operates only in the region with lower wages. Since

differentiated goods firms concentrate in the core region, the traditional goods sector

operates in the peripheral region. Then, wages are lower in the peripheral region

(wperiphery <wcore). In this case, the consumption expenditure, the wages and the population

in each region, and the growth rate can be obtained by solving the following simultaneous

equations:

gcore ¼ ð1� cÞafLcoreEcore þ LperipheryEperipherygwcore

� q: ð30Þ

gcore þ cafLcoreEcore þ LperipheryEperipherygwcore

¼ Lcore; ð31aÞ

ð1� aÞfLcoreEcore þ LperipheryEperipherygwperiphery

¼ Lperiphery; ð31bÞ

Ecore

Eperiphery¼ N periphery

N core

� �Lcore

Lperiphery

� �/

abðe�1Þ

� �b

: ð32Þ


These equations are obtained by inserting h = 0 or h = 1 and Eq. (26) into Eqs. (19a),

(23a), (23b), and (24).5

It is crucial to elucidate the factors which determine whether wages are equal or not in

steady state equilibria.

We begin with the case that wages are unequal in a core–periphery equilibrium with

region A being the core. In this case, the left- and right-hand side of Eq. (32) refer to the

relative benefits and costs of residing in core region A, respectively. In an equilibrium,

these must be balanced. In Fig. 1, the curves BB and CC indicate the left-hand side and the

right-hand side as the functions of L, respectively.

From Eqs. (31a) and (31b) we can see that the larger the labor supply in region A, the

smaller the region A’s relative wages, permanent income, and hence relative consumption

expenditure. Then, the curve BB decreases on the right-hand side.

Obviously, the larger L is the larger the amount of land per capita in region B. That is,

the larger L is the larger the amount of land per capita a household in region A must

abandon. Thus, the curve CC, which indicates the relative costs of residing in region A,

increases on the right-hand side. In addition, the smaller the transaction costs, the higher

the curve locates, as CVCV. Owing to transaction costs, households in region B must pay

more in consuming differentiated goods. However, the smaller the transaction costs, the

less they have to pay for differentiated goods and hence the larger the relative opportunity

costs of residing in region A.

The equilibrium population in region A is determined at the intersect of two curves,

indicated by L. However, we must note that, in order for wages to be unequal in a core–

periphery equilibrium with region A being the core, the curves BB and CC must intersect

at a point higher than 1. This can be guaranteed if and only if the structural parameters

satisfy the following:

NB

NA

� �a � ð1� aÞqð1þ qÞð1� aÞ

� �/

abðe � 1Þz1 ð33Þ

See Appendix B for the derivation. Unless Eq. (33) holds, wages are equal in a steady state

equilibrium with region A being the core. R&D activities and differentiated goods firms

concentrate in region A, while traditional goods sector operates in both regions. In this

case, E, gA and L in equilibrium are Eqs. (27)–(29), respectively.

Next, we consider the case that wages are unequal in a core–periphery equilibrium with

region B being the core. By a similar treatment, we can see that such equilibrium exists if

and only if the following condition holds:

NA

NB

� �a � ð1� aÞqð1þ qÞð1� aÞ

� �/

abðe � 1Þz1: ð34Þ

5 gcore, wB, EA, EB and L in equilibrium can be obtained by solving Eqs. (30), (31a), (31b), and (32) and the

equations EA= 1 + q and EB =wB + q, which are obtained from household’s budget constraint (Eq. (3)). One of

Eqs. (30), (31a), and (31b), EA= 1 + q and EB =wB +U is redundant. Then, the number of the equations coincides

the number of valuables. However, obtaining them in closed form explicitly is difficult.

Fig. 1. Core–periphery steady state equilibrium with differences in wages.


Unless Eq. (34) is satisfied, wages are equal in a steady state equilibrium with region B

being the core. The equilibrium E, gB and 1� L are Eqs. (27)–(29), respectively.

From Eqs. (33) and (34), we can see that wages tend to be unequal in a core–periphery

equilibrium in the case that the relative land mass in the core region (Ncore/Nperiphery) is

small, the share of the expenditure for traditional goods in total consumption expenditure

(1� a) is small, and transaction costs are low (/ is large). To understand it intuitively,

suppose the case that wages are equal, though these factors are satisfied. Traditional goods

are produced in both regions. However, there are sufficient labor forces in the peripheral

region to satisfy the total demand for the traditional goods. Due to the low transaction

costs, the difference in the price level of differentiated goods is small, and so the relative

benefits in residing in the core are small. In addition, households can enjoy the large land

mass in the periphery. Hence, many households are willing to reside in the periphery.

However, the traditional goods sector is not so large as to absorb the labor supply in the

peripheral region, even if all traditional goods are produced there. Consequently, excess

supply occurs in the labor market in the periphery, which reduces the wages there.

The land mass of region B is larger than that of region A. Therefore, unequal wages in a

core–periphery equilibrium with region B being the core are more unlikely to occur even

in the case of very low transaction costs. Without loss of generality, we consider the

parameter set under which wages never differ between regions in a core–periphery

equilibrium with region B being the core irrespective of the levels of transaction costs, but

wages can differ in a core–periphery equilibrium with region A being the core in the case

of sufficiently small transaction costs. That is, we focus on the parameter set under which

Eq. (34) does not hold for all, but there exists /˜a(0, 1) that satisfies Eq. (33) in the range

/V/.


Fig. 2 indicates the relationships between transaction costs / and the growth rate, the

differences in wages, and the shares of the core in population and traditional goods, in

core–periphery equilibria. The thin line indicates these relationships for the case that

region B is the core, while the broken line shows them when region A is the core.

In the case that the region A is the core, the households in region B import differentiated

goods from region A. The higher the transaction costs, the more households in region B pay

for differentiated goods. Therefore, there must be a sufficient land mass per capita to

compensate for this disadvantage in region B. That is, the population density in region A is

higher than those in region B. However, the population in region A is not necessarily larger

than that in region B since the land mass in region A is smaller than that in region B. A

decrease in transaction costs enables the households in region B to consume differentiated

goods at lower costs. The relative benefits of residing in region A fall. Accordingly, the

population moves from region A to region B, diminishing the differences in population

Fig. 2. Transaction costs and key valuables.


density. As long as / < /holds, such a change in population is absorbed in the change of the

share in traditional goods s. That is, labor supply and demand move in the same direction in

region A. Thus, no gap in wages occurs between the two regions and the rate of growth does

not change. In contrast, if transaction costs decrease into the range /V/, the relative wagesin region A increase (wB decreases) and the rate of growth deteriorates.6 Such a decrease in

transaction costs renders the labor force in region A too small to fulfill a huge demand for

labor at an equal wage rate even if all the traditional goods are produced in region B. The

relative wage in region A begins to rise. Hence, R&D activities are constrained. Because of

the differences in wages, the differences in land mass per capita remain even in the case

without transaction costs (/ = 1).

On the other hand, in the case that larger region B is the core, the core will contain a

much larger population. A labor force sufficient to maintain R&D activities at a constant

level will remain even after sharp decreases in transaction costs.

Next, we investigate the steady state equilibrium where gA= gB holds and then hremains constant in the range of ha(0, 1). Henceforth, we call such steady state equilibria

in which the differentiated goods sector disperses for the two regions symmetric steady

state equilibria.

To begin with, we investigate the consumption expenditures, the economic growth rate

and the allocation of population in the case that wages are equal in a symmetric

equilibrium. If gA= gB holds, the right-hand sides of Eqs. (21a) and (21b) must be equal.

In such a case, from the right-hand sides of Eqs. (21a) and (21b), we can see that the ratio

of the profits of differentiated goods firms must be equal to the ratio of the R&D costs:

pA

pB¼ L

h þ ð1� hÞ/ þ ð1� LÞ//h þ ð1� hÞ

L/

h þ ð1� hÞ/ þ ð1� LÞ/h þ ð1� hÞ

¼ kh þ ð1� hÞh þ kð1� hÞ : ð35Þ

From Eqs. (24) and (35), we can derive the combination of (h, L) in the symmetric steady

state equilibrium. We denote it as (h, L). With this (h, L), and Eqs. (22a) Eqs. (22b) and

(26), we can derive the consumption expenditure and the growth rate as follows:7

E ¼ 1þ qh

h þ kð1� hÞþ 1� h

kh þ ð1� hÞ

( )ð36Þ

g ¼ ð1� cÞa h

h þ kð1� hÞþ 1� h

kh þ ð1� hÞ

( )�1

þq

24

35� q: ð37Þ

Then, we consider whether wages can differ in symmetric steady state equilibria.

In the case that h is in the range of 0 < h < 1, considering whether wages are equal or notis difficult. As we have discussed, we can examine whether wages differ or not in core–

6 These changes can be captured by the shifts of CC (from CC to CVCV) in Fig. 1.7 Consumption expenditure in steady state equilibria can be also derived from household’s budget constraint

(Eq. (3)), equilibrium condition in financial market (Eq. (15)), free entry condition (Eq. (18)), and Eq. (26).


periphery equilibria by checking whether Eq. (33) or Eq. (34)is satisfied or not. On the

other hand, in the cases other than the core–periphery equilibria, examining whether

wages differ or not is difficult, since it depends on h in the range of 0 < h < 1, as well asNcore/Nperiphery, a, and /. However, in the case that h is in the range of 0 < h < 1, wages tendto be equalized for the following reason.

Suppose the case that Eq. (33) is satisfied, but Eq. (34) is not. This is the case under

which wages are equal in core–periphery equilibria with region B being the core (h = 0),but wages differ in the equilibria with region A being the core (h = 1). The benefits of

residing in region A are smaller in the case of h < 1 than in the case of h= 1, then L, the

population in region A, is smaller. However, households get utilities from per capita land

mass as well as differentiated goods. The smaller the population, the larger the land mass

per capita and then the higher the utility level. Hence, L does not move so dramatically as

h does, though they move in the same direction. Therefore, smaller h means smaller total

labor supply L and much smaller labor demand by the differentiated goods sector in region

A. If the wages in region A remain higher than those in region B, there emerges excess

supply in the labor market in region A. Consequently, wages in region A decrease. With

much smaller h, wages are equalized between regions and then the traditional goods sector

operates in region A too.

To sum up, in the case that transaction costs are in the levels of / < / and neither Eq.

(33) nor Eq. (34) is satisfied, wages are equalized not only in core–periphery equilibria

(h= 0, 1), but also in all the other h including symmetric equilibria. On the other hand, in

the case that transaction costs are small so as to satisfy /V/ and hence Eq. (33) is fulfilled

but Eq. (34) is not, wages will differ with h near unity.

3.3. Transitional dynamics

In this subsection, we investigate and the stability of each steady state equilibrium and

the transitional dynamics. We focus on the case that neither Eq. (33) nor Eq. (34) holds and

hence wages are equal for all h. After that, we intuitively discuss the transitional dynamics

in the case that transaction costs satisfy /V/ and Eq. (33) holds but Eq. (34) does not, and

hence, wages differ with h near unity.

In analyzing the equilibrium dynamics, it is important to examine in which region R&D

is active. The R&D activities can be observed only in the region where the benefits of the

activities are balanced with their costs and then Eq. (18) binds. The benefits of the R&D

activities are determined as the present value of the sequence of the differentiated goods

firms’ profits, and they can be changed depending on the expectation about in which

region the differentiated goods firms will concentrate. On the other hand, the R&D costs

depends on h which has been historically determined. First, we consider the relationship

between the differentiated goods firms’ profits and the R&D costs. Then, we consider the

relationship between the stock prices of the differentiated goods firms and the R&D costs,

and where R&D is active and the differentiated goods firms agglomerate. It will be shown

that the firms can concentrate even in the region where the number of differentiated goods

is now smaller and then the R&D costs are higher, if such an expectation is shared.

The curve LL in Fig. 3 indicates the combinations of L and h which satisfy the

equal utility condition (24). The larger the h is, the less the households in region A

Fig. 3. Conditions of equal utility, equal profits and equal R&D costs.


have to pay for differentiated goods than the households in region B. Therefore, to

equalize the utilities between the two regions, population in region A should be larger

and thus the land mass per capita there should be smaller. Hence, the curve LL

increases on the right hand side. If a combination of L and h locates in the area upper

(lower) than the curve LL, the utility levels of households in region B (A) are higher. L

and h must be on the curve LL because households are freely mobile between regions.

On the other hand, the curve CC indicates the combinations of L and h which make

the difference in profits of differentiated goods firms between two regions be zero. An

increase in h makes the competition in region A fiercer, decreasing the profits. To

compensate this disadvantage, L must increase to expand the markets there. Hence, the

curve CC increases on the right hand side. If a combination of L and h locates in the

area upper (lower) than the curve CC, the profits of differentiated goods firms in

region A (B) are higher. Also, the curve QQ shows the combinations of L and h on

which Eq. (35) is satisfied and then the ratio of the profits is equal to the ratio of the

R&D costs. The slope of the curve QQ is less steeper than that of CC. In the case of

global technological external effects (k= 1), QQ corresponds to CC. On the other hand,

technological external effects are limited within the region (k= 0), the slope of QQ is

negative.

The intersect of LL and QQ corresponds to the symmetric equilibrium (h, L) that wehave discussed in the previous subsection. On the other hand, the points on LL of h = 0 andh = 1 corresponds to the core–periphery equilibrium where region B is the core and the

one where region A is the core, respectively.


Suppose that the economy is now at (h, L), the intersect of the curves LL and QQ, and

all share the expectation that the ratio of the profits of producing differentiated goods will

remain equal to the ratio of the R&D costs as Eq. (35) in the future forever. Under such an

expectation, the ratio of the stock prices of differentiated goods firms is also equal to the

ratio of the R&D costs, and then Eq. (18) binds for both regions. R&D activities will

operate in both regions. h does not change from h, and hence, the initial expectation is self-fulfilled. L, E and g remain at the levels of L, Eqs. (36) and (37), respectively. Note that

h>0.5 and the intersect locates in the area lower than CC.8 At h = 0.5, R&D costs are equal

between the two regions. Population density is also equal, but it means that the population

in region A is smaller than in region B. Then, the benefits of R&D activities are smaller in

region A, which contradicts the characteristics of symmetric steady state equilibria. At a hlarger than 0.5, R&D costs in region A are smaller. Though the profits from producing

differentiated goods in region A are little a bit smaller than in region B, the costs and

benefits of R&D activities are balanced at h>0.5.As shown in Fig. 3, the slope of LL is steeper than that of CC, and consequently

steeper than that of QQ, the core–periphery equilibria which we have discussed in the

previous subsection do really exist. To understand this intuitively, we consider the case

that differentiated goods industries concentrate in region A (h = 1). At h = 1, LL is above

CC. Under these conditions, differentiated goods firms in region B earn larger profits than

those in region A. If all share the expectation that h = 1 will be kept in the future forever,

the stock prices as well as the profits of the firms in region A are higher. In addition, LL is

above QQ at h = 1. That is, region A has a greater advantage with respect to the R&D costs

than it has with respect to the profits and stock prices. Consequently, Eq. (18) binds and

the R&D activities can be observed only in region A. h remains at h= 1, and then the

initial expectation is self-fulfilled.

Next, we use the following procedure to analyze the existence and characteristics of the

dynamic equilibrium path on which the differentiated goods sector is concentrated in one

region. First, we derive a dynamic path in which Eq. (18) binds for region A, which means

that gAwill be positive as in Eq. (19a) but gB will be zero as in Eq. (20). We then check to be

sure that Eq. (18) does not bind for region B, which warrants that such a dynamic path can be

supported as an equilibrium one. A similar procedure can be used to check the existence of a

dynamic equilibrium path on which the economic activities are concentrated in region B.

The phase diagram in Fig. 4 shows the dynamics of h and E under a given /. SeeAppendix C for the details about Fig. 4. Which prevails, Fig. 4a or b, depends on in which

region (18) binds. For instance, in describing Fig. 4a, all the equilibrium conditions are taken

into consideration under the assumption that Eq. (18) binds only in region A. Note that to be

sure that the dynamic path in the Fig. 4a can be supported as an equilibrium one, we must

check whether the relative price of the differentiated good firm’s stock in region A to that in

region B is higher than the relative cost of the R&D activities in region A to that in region B,

and hence, Eq. (18) do really bind only in region A. To do so, we check the sign of

f ðhÞ ¼ vAðhÞvBðhÞ �

kh þ ð1� hÞh þ kð1� hÞ ; ð38Þ

8 In the case of NA=NB, the curves LL, CC and QQ all intersect at (0.5, 0.5).

Fig. 4. (a) Transitional dynamics leading region A to be the core. (b) Transitional dynamics leading region B to be

the core.


where vi(h) is the stock price on a dynamic path that causes economic activities to be

oncentrated in region A, starting from a state value of h. A positive sign of Eq. (38)

means that Eq. (38) binds only in region A.9 That is, in region B, the profits of R&D

9 Since Eq. (18) binds in region A, positive Eq. (38) means that Eq. (18) does not bind in region B.


activities are smaller than the R&D costs. In this case, such a dynamic path can be

supported as an equilibrium one. In contrast, a negative sign of Eq. (38) means that

the profits of R&D activities in region B exceed the costs, which contradicts the

condition (18). In this case, a dynamic path from such a level of h to h = 1 cannot be

supported as an equilibrium one. Using a similar treatment, we can confirm the

existence of the dynamic equilibrium path on which economic activities are concen-

trated in region B.10

Both dynamic paths from h = h to h = 1 in Fig. 4a and from h = h to h = 0 in Fig. 4b can

be the candidates of equilibrium paths. The former path can be supported if it is expected

that producing differentiated goods in region A will be so profitable for vA(h)/vB(h) toexceed {kh+(1� h)}/{h + k(1� h)} and then f(h) is positive. Under such an expectation,

R&D activities operate only in region A, rendering h larger. Therefore, L and h move

northeast on the curve LL in Fig. 3. These move the area upper than the curve QQ. pA/pB

equals {kh+(1� h)}/{h + k(1� h)} initially, but will exceed it for the indefinite future.

Consequently, the relative stock prices vA(h)/vB(h) do exceed {kh+(1� h)}/{h+ k(1� h)}and Eq. (38) will be strongly positive at h = h. As a result, the initial expectation is self-

fulfilled. Similarly, we can see that the equilibrium path from h = h to h = 0 can be attained

if such an expectation prevails at the time when h = h holds.11 On the transitional

dynamics, we can see that the interest rate is initially lower than the discount rate q,but it soon exceeds—and then finally converges to—this discount rate. Then, consumption

expenditure decreases in the short run, but eventually increases to the level of Eqs. (31a)

and (31b). The growth rate is initially higher, but it eventually approaches the level of Eq.

(26). In the resulting urbanization process, the population and differentiated goods sector

are concentrated in one region.

We should notice that from a h near h, either of the two dynamic paths, leading to

the different two core–periphery steady states, can be supported as the equilibrium

path. Which of these two paths is chosen depends on expectations about the future as

well as historical factors. In the case that a dynamic path leading region A to be the

core starts from a value below h, the profits of producing differentiated goods in region

A will be smaller in the short term. Then, vA(h)/vB(h) decreases as h decreases. In

addition, the smaller the h, the higher the relative R&D costs in region A. Therefore,

f(h) is an increasing function of h. We have seen that, under the expectation that region

A will be the core, f(h)>0 holds. Then, the h which makes f(h) zero (we set such a h as

h) is smaller than h . Similarly, when region B is expected to be the core, the h for

which f(h) is zero (we set such a h as h¯) is larger than h.12 Consequently, from a h in

the range of ha(h, h), two dynamic equilibrium paths can emerge, leading to different

core–periphery steady state equilibria.13 In this range, the choice of which dynamic

10 In this case, however, we must check whether f < 0 holds.11 Under the expectation that region B will be the core, dynamic equilibrium path from an initial state value

of h to h= 0 exists if f(h) < 0 holds. By a similar procedure, we can see that it is supported at h= h.12 This is because f(h) is an increasing function of h and because f(h) < 0 holds under the expectation that

region B will be the core.13 In Appendix D, we prove the non-emptiness of (h, h), though in doing so, we restrict ourselves to the case

where k is nearly equal to 1.


equilibrium path occurs will depend on both historical factors and expectations about

the future. The differentiated goods sector can concentrate even in the region where the

number of differentiated goods is now smaller and then the R&D costs are higher, if

such an expectation is shared.

The existence of a range ha(h, h) from which either of the multiple equilibria can be

attained is due to the fact that LL is steeper than CC, as shown in Fig. 3, and so the profits

of differentiated goods firms are larger in the region with a relatively larger number of

differentiated goods. Such a situation can emerge if the parameters satisfy the following

relationship:

a > bðe � 1Þ: ð39Þ

An increase in the relative number of the differentiated goods in a region causes

immigration to this region. The larger the preference of the differentiated goods (the

larger a is and the smaller e is), and the less the utility is lowered by a decrease in land

mass per capita (the smaller b is), the larger the scale of the immigration. In the case

that a, b and e satisfy Eq. (39), the increase in the population is large enough for the

increase in profits by the accompanied market expansion to dominate the decrease in

profits by the fiercer competition. That is, the pecuniary external effects emerge, which

are focused on in new economic geography models like Krugman (1991a). The

pecuniary external effects can be so large as to compensate the disadvantage in the

R&D costs, and then the differentiated goods sector can concentrate even in the region

where the number of differentiated goods is historically smaller, as long as h is in the

range of (h, h). In the case that h is out of the range (h, h), the pecuniary external effects

are active but not so large as to compensate the gap in the R&D costs. Consequently, in

the case that h is near unity or zero, the agglomeration pattern in the future is

determined only by the historical inertia.

In the work of Grossman and Helpman (1991), there is neither interregional migration

nor transaction costs. In such settings, there cannot be a difference in the profits of

differentiated goods firms between regions. Thus, the region in which R&D activities take

place will depend only on the difference in R&D costs. In the case that technological

external effects are localized, the R&D costs are smaller in the region where the more

differentiated goods firms have located. Therefore, which region becomes the core depends

only on historical factors. In the work of Baldwin et al. (2001), transaction costs are

incorporated, but interregional migration is not. Hence, an increase in the relative number of

differentiated goods makes competition fiercer but never makes the market expand,

rendering the profits per firm lower. As a result, R&D activities and then differentiated

goods firms tend to disperse. However, the lower the transaction costs, the less dramatically

the profits per firm are reduced by an increase in rivals. In addition, the more localized the

technological external effects, the more dramatically the costs of R&D activities are

reduced by an increase in rivals. To sum up, the lower the transaction costs and the more

localized the technological external effects, the more likely the differentiated goods sector

tend to agglomerate in the region that historically has seen a greater accumulation of R&D

activities.


This paper takes into account both interregional migration and transaction costs.

Differently from Grossman and Helpman (1991) and Baldwin et al. (2001), pecuniary

external effects emerges, and then the core–periphery steady state equilibria are more likely

to arise. The size of transaction costs / does not matter with respect to whether

agglomeration emerges or not.14 Instead of it, households’ preference of land b is one of

the important factors which cause agglomeration.

Moreover, the choice of which core–periphery equilibrium occurs depends on

expectations about the future as well as historical factors, due to the interregional

migration. Population being equal, an increase in h will lead to tougher competition

and lower profits in region A. In this paper, however, an increase in h causes the

migration from region B to region A too. In this case, as we have seen, the profits are

larger in the region with relatively larger number of differentiated goods, if a, b and esatisfies Eq. (25). Therefore, when the expectation prevails that h, which is not

particularly large now, will grow in the future and hence the profits in region A will

increase, the present disadvantage in R&D costs can be compensated by the future

advantage in the profits. If the stock prices vA, the present value of the future profits,

arise so much as to compensate the gap in R&D costs between regions, then R&D

will take place in region A and the expectation will be self-fulfilled. Moreover, it

should be noted that the more global the technological external effects are (the larger

k is), the less serious the initial gap in R&D costs between regions is, and hence the

more likely the expectation about the future can reverse the historical agglomeration

pattern.

Krugman (1991b) shows that the more localized the technological external effects, the

more likely the historical agglomeration pattern is reversed by expectation about the

future. In this paper, we introduce the pecuniary external effects as well as the

technological external effects focused on in Krugman (1991b). The pecuniary external

effects, rather than the technological ones, are essential for the expectation about the future

to reverse the historical agglomeration pattern. Moreover, in contrast to Krugman (1991b),

the more global the technological external effects, the larger the role of the expectation in

determining the choice of equilibrium paths.

Analyzing the stability of the symmetric steady state equilibrium is difficult, since h is

in the range of (h, u) from which multiple equilibrium paths can emerge depending on the

expectation about the future. Therefore, we consider the restricted case such that the

reverse in the order of the sizes in the ratio of the profits and the ratio of the R&D costs is

never expected. That is, the equilibrium path never crosses the curve QQ in Fig. 3. Under

such an expectation formation, h = 1 is attained in the case that an initial h is in ha(h, 1),and h= 0 is attained in the case that an initial h is in ha(0, h). In this sense, the symmetric

equilibrium h = h is unstable.

In this subsection, we focused on the case that there was no gap in wages. However, as

we have seen in the last part of Section 3.2, wages differ in the case that transaction costs

satisfy /V/, and hence, Eq. (33) is satisfied but Eq. (34) is not, and that h is sufficiently

14 Though the slopes of LL and CC in Fig. 3 are changed by a change in transaction costs, the relative

steepness remains unchanged. Then, Eq. (39) does not contain /.


near unity. Investigating the transitional dynamics in the case of unequal wages is

formidably difficult. However, the characteristics of the transitional dynamics are not

changed for the following reason.

In Fig. 3, the broken lines are LL, CC, and QQ in the case that h is sufficiently

near unity, and hence, the wages in region A are higher than those in region B.

Therefore, the utility level in region A is higher than that in the case of equal wages,

even if population L is slightly larger. Thus, the broken LL locates upper than the thin

LL. However, the higher wages in region A mean higher costs and therefore lower

profits for producing the differentiated goods there. For the ratio of the profits

between regions to be equal to the one in the case without the difference in wages,

the market size in region A must be larger. Hence, the broken CC and QQ are above

the thin ones. Consequently, the broken LL and CC locate upper than the thin ones

and then their relative location will not be changed unless the parameters take very

particular values. Therefore, the characteristics of transitional dynamics under a / that

satisfies /V/ are not essentially changed.

4. Specialization pattern, economic growth and welfare implications

We will now consider the economic welfare of each steady state equilibrium in the

model. Then, we will analyze which steady state equilibrium tends to be attained focusing

on changes in transaction costs.

To begin with, we compare symmetric steady state equilibria with core–periphery

steady state equilibria.

As we have discussed in Section 3.2, in core–periphery steady state equilibria with

equal wages, the consumption expenditure and the growth rate are determined as Eqs.

(27) and (28) and are independent of k and /. On the other hand, in symmetric

equilibria, the consumption expenditure and the growth rate are determined as Eqs. (36)

and (37), and depend on k and /.15 Moreover, the comparison between Eqs. (27) and

(36) elucidates that the more localized the technological external effects (the smaller the

k), the smaller the growth rate and the larger the consumption expenditure. Due to

external effects that are partially localized, the greater the volume of differentiated goods

that are concentrated in one region, the lower will be the costs of R&D. Consequently,

the smaller the geographical scope of externalities, the more plausible it is that the

symmetric steady state equilibrium is inferior and that history plays a crucial role in the

determination of the ultimate steady state core–periphery equilibrium. Hence, it is

important to consider which of the two core–periphery equilibria is preferable, and to

which equilibrium the economy tends to evolve.16

15 That is because h in these equations depends on /, as will be discussed soon.16 In the case that labor does not move and the differentiated goods market is less integrated (transaction

costs are particularly high), the results will be near the ones in Baldwin et al. (2001) as we have discussed in

Section 3.3. The symmetric steady state equilibria where the differentiated goods sector tends to disperse for the

two regions will emerge. Hence, in the case of autarky, the growth rate is smaller than in the case with labor

mobility and with low transaction costs.


In core–periphery steady state equilibria with equal wages, we can calculate the

intertemporal economic welfare per capita as follows:

lnE þ bln N core þ /a

bðe�1ÞNperiphery� �

qþ

ae � 1

� �g

q2 : ð40Þ

See Appendix A for the details. In the case that region B is the core, wages are always

equalized, and hence, Eq. (40) holds for all /. In contrast, in the case that region A is

the core, Eq. (40) holds only for the transaction costs in the range of /V /. Then, aslong as / stays in the range of /V /, E and g in Eq. (40) are constant as Eqs. (27)

and (28), respectively, and the same irrespective of which region is the core. However,

we can see that the case that region B is the core is preferable for the following two

reasons, even though the transaction costs are in the range of /V /.First, the amount of land mass per capita is larger in the case that region B is the core.

From Eqs. (24) or (29), we can see that the ratio of population density in the core to that in

the periphery, i.e., (Ncore/Lcore)/(Nperiphery/Lperiphery), is the same irrespective of which

region is the core. However, in the case that region B is the core, both Ncore/Lcore and

Nperiphery/Lperiphery are larger than when region A is the core. The growth rate g and the

consumption expenditure E being equal, an equilibrium with a larger amount of land per

capita is preferable. Indeed, Eq. (40) is larger in the case that region B is the core

(Ncore =NB and Nperiphery =NA), than in the case that region A is the core (Ncore =NA and

Nperiphery =NB).

Second, from the discussion in Section 3.2 and Fig. 2, we have seen that a sharp

decrease in transaction costs causes a steep decline in the rate of growth in the case

that region A is the core. The higher population density in region A does not

necessarily mean that the population is larger, since the land mass in this region is

smaller. A decrease in transaction costs enables the households in peripheral region B

to import and consume differentiated goods at lower costs. The relative benefits of

residing in region A fall. Hence, the labor supply in region A decreases, lifting relative

wages in region A and hence the costs of R&D. Consequently, economic growth rate

and then welfare deteriorate due to the decrease in transaction costs. On the other

hand, in the case that region B has been the core, the core contains a much larger

population. Then, a decrease in transaction costs reduces a labor force in the core not

so dramatically as to lift relative wages there and deteriorate the growth rate. Then, the

growth rate and the consumption expenditure in the case region B being the core are

constant at levels of Eqs. (27) and (28), respectively. In this case, from Eq. (40), we

can see that a decrease in transaction costs always improves economic welfare.

It is crucial to consider how the preferable agglomeration pattern with region B being

the core can be attained.

The curve MM in Fig. 5 depicts the relationship between transaction costs and the

symmetric solution h. The core–periphery solutions are also depicted (h = 0 and h = 1).The evolution of h under a fixed / is analyzed in Section 3.3 and Fig. 4. The

movements of h are captured in Fig. 5 as vertical ones from MM to h = 0 or h = 1. Fig.5 shows that lower transaction costs (larger /) results in smaller symmetric solution h.

Fig. 5. Core–periphery and symmetric steady state equilibria, and transitional dynamics.


That is, though the shares in differentiated goods are equal between S0 and S1, the S0,

the case of large transaction costs locates below MM, and the S1, the case of small

transaction costs locates above MM. From S0 (S1), the dynamic path to B0 (A1),

rendering region B (A) the core emerges, unless strong opposite side of expectation is

formalized. The intuitive explanation of the reason is as follows.

Region B has a smaller number of differentiated goods. For utility levels to be equalized

between regions, population density should be lower in region B. However, the population

in region B is not necessarily smaller since there is larger land mass. At S0, in addition, the

market in region B is protected from the import of differentiated goods from region A due to

the larger transaction costs. Hence, the profits of the differentiated goods sector in region B

can be higher. R&D activities can be observed only in region B, rendering region B to be

the core. On the other hand, at S1, the differentiated goods sector in region B faces fiercer

competition due to the imports from region A. The profits of the differentiated goods firms

in region B (A) are smaller (larger) than in the case of S0. If the profits are higher in region

A, the R&D is active only in the region A, which causes a core–periphery equilibrium with

region A being the core.

Based on the discussion above, we investigate the effects of a decrease in transaction

costs from /0 to /1 or /2.

If the transaction costs decrease slowly, we can observe the dynamic path that starts from

S0, passes near B0 and B1, and approaches B2 in Fig. 5. Region B will leapfrog region A in

terms of the market share in differentiated goods, and will become the core in the long run.

In contrast, if the transaction costs decrease sharply, we can observe the dynamic path that

starts from S0, passes near A1, and approaches A2, or that passes near S1 and S2 and

approaches A2. The latter one can be observed in the case of a particularly sharp decrease in

the transaction costs. As we have seen in Section 3.3, the growth rate increases in the

process of the agglomeration. However, after the agglomeration process has been

completed rendering region A to be the core, a further decrease in transaction costs


depresses the growth rate as we have seen in Section 3.2 and Fig. 2. Consequently,

transaction costs sharply decrease when the agglomeration in region B with a large land

mass is still in its initial stage, and a less efficient agglomeration pattern tends to emerge in

which region A with a paucity of land becomes the core.

5. Conclusion

We have presented a model that integrates an endogenous growth model with a new

economic geography model. Our model has one symmetric steady state equilibrium and

two core–periphery steady state equilibria. Which of the two core–periphery equilibria

will be attained depends on the expectations about the future as well as historical factors

when the agglomeration process is in its initial stage. However, the smaller the

geographical scope of externalities, the more plausible it is that history plays a crucial

role in the determination of the ultimate steady state. In addition, the size of transaction

costs plays a crucial role in the choice of the equilibrium. Hence, it is important to

consider which of the two core–periphery equilibria is preferable and tends to emerge,

focusing on the timing of the improvements in transportation facilities and the stage of

agglomeration.

The core–periphery equilibrium in which a larger region is the core, there is no

difference in wages and growth rate is high, and the equilibrium in which a smaller region

is the core, a difference in wages emerges, and the rate of growth is low. Then, the former

is preferable. A slow decrease in transaction costs can render a region with large land mass

the core region even if the agglomeration there was previously insufficient. However, our

results show that a rapid decrease in transaction costs leads to a core–periphery

equilibrium with the smaller region being the core. It must be noted that after such an

equilibrium is firmly established, further improvements in transaction costs will only

decrease the global growth rate.

Acknowledgements

The author would like to thank Akihiko Kaneko and two anonymous referees for their

valuable comments and suggestions. Of course, the responsibility for errors remains mine

alone.

Appendix A. Equal utility condition

There should be no difference in utility levels regardless of the region a household

resides. Since an identical r prevails, consumption expenditure grows at a same rate in the

two regions. Therefore, instantaneous utilities must be equalized. The instantaneous

utilities are

i i i
alnCX þ ð1� aÞlnCY þ blnCZ; i ¼ A;B: ðA1Þ


With Eqs. (2), (4), (9) and (9)V, the first terms of both sides can be rewritten as

alnCAX ¼ aln nA

ðwAÞ�e

nAðwAÞ1�e þ nBðswBÞ1�e

!e � 1

e

� �aEA

( )1�1e

24

þ nBðswBÞ�e

nAðwAÞ1�e þ nBðswBÞ1�e

!e � 1

e

� �aEA

( )1�1e

35

1

1�1e

ðA2aÞ

¼ ae � 1

� �ln½nAðwAÞ1�e þ nBðswBÞ1�e þ alnEA þ alna

e � 1

e

� �

¼ ae � 1

� �ln½ðh þ ð1� hÞ/xÞðnA þ nBÞðwAÞ1�e þ alnEA þ alna

e � 1

e

� �

alnCBX ¼ aln nA

ðswAÞ�e

nAðswAÞ1�e þ nBðwBÞ1�e

!e � 1

e

� �aEB

( )1�1e

24

þ nBðwBÞe

nAðswAÞ1�e þ nBðwBÞ1�e

!e � 1

e

� �aEB

( )1�1e

35

1

1�1e

¼ ae � 1

� �ln½nAðswAÞ1�e þ nBðwBÞ1�e þ alnEB þ alna

e � 1

e

� �

¼ ae � 1

� �ln½ðh/ þ ð1� hÞxÞðnA þ nBÞðwAÞ1�e þ alnEB

þ alnae � 1

e

� �: ðA2bÞ

Inserting Eqs. (A2a) and Eqs. (A2b) into Eq. (A1), and using Eq. (5) to substitute CYi in

Eq. (A1) yield

a1

e � 1

� �ln½h þ ð1� hÞ/x þ 1

e � 1

� �lnðnA þ nBÞ þ lnEA þ bln

NA

L

� �;

ðA3aÞ

a1

e � 1

� �ln½h/ þ ð1� hÞx þ 1

e � 1

� �lnðnA þ nBÞ þ lnEB þ bln

NB

1� L

� �:

ðA3bÞ


The terms that are less important or identical between regions are abbreviated. Since these

must be equalized, Eq. (24) holds.

We can calculate the intertemporal utility in each region in the core–periphery

equilibrium with equal wages by inserting Eq. (A3a) Eq. (A3b), and h = 1 or h = 0 into

Eq. (1):

Core :lnE þ bln N core

Lcore

� �q

þa

e�1

� �g

q2; ðA4aÞ

Periphery :a 1

e�1

� �ln/ þ lnE þ bln Nperiphery

Lperiphery

� �q

þa

e�1

� �g

q2: ðA4bÞ

In the case that wages are equal in the core–periphery equilibrium, the population in the

core can be obtained as Eq. (29), by solving Eq. (24) with x = 1, EA=EB =E and h= 1 or

h = 0. Inserting Eq. (29) into Eqs. (A4a) and Eqs. (A4b) yield the same result as follows:

lnE þ bln N core þ /a

bðe�1ÞNperiphery� �

qþ

ae�1

� �g

q2: ðA5Þ

Inserting Eqs. (27) and (28) into Eq. (A5) yields Eq. (40).

Appendix B. Derivation of condition (33)

With Eqs. (30) and (31a), we get

LEA þ ð1� LÞEB ¼ Lþ qa

: ðA6Þ

Inserting Eq. (A6) into Eq. (31b) yields

wB ¼ ð1� aÞðLþ qÞað1� LÞ : ðA7Þ

Inserting Eq. (A7) into EB =wB + q (or inserting EA= 1 + q into Eq. (A6)) yields

EB ¼ ð1� aÞðLþ qÞað1� LÞ þ q: ðA8Þ

We can see that the larger L is, the larger EB�EA is. Then the curve BB is right side down.

For an equilibrium with the wages in region A being larger than in region B to exist, the

curves BB and CC in Fig. 1 must intersect at a point higher than 1. It is guaranteed if the


right-hand side of Eq. (32) is larger than 1 at L where the left-hand side of Eq. (32) is just

one. Namely,

NB

NA

� �L

1� L

� �/

abðe�1Þz1: ðA9Þ

From Eq. (A7), such a L can be derived as follows:

L ¼ a � q þ aq: ðA10ÞInserting Eq. (A10) into Eq. (A9) yields the condition (33).

Appendix C. Transitional dynamics in Fig. 4

Since Fig. 4a is described under the assumption that Eq. (18) binds only in region A,

gA>gB = 0 holds and then h˙z 0 holds. On the other hand, whether or not E increases

depends on whether or not (E, h) locates higher than the curve PAPA, which indicates (E,

h) for which the evolution of E stops. Summing the labor market clearing conditions (22a)

and (22b) yields

hh þ kð1� hÞ

gA þ ðac þ 1� aÞE ¼ 1: ðA11Þ

r and gA in Eqs. (6) and (25), together with L, are determined as the solution of the system

consisting of Eqs. (19a), (24), and (A11) under given E and h. From Eq. (24), L can be

described as the function of h. We set this function as L(h). From Eq. (6), we can see that E

grows if the r is larger than q. Hence, the combinations of (E, h) on the curve PAPA are the

ones which satisfy Eq. (A11) and the following equation:

hh þ kð1� hÞ

gA ¼ ð1� cÞafh þ kð1� hÞg

� LðhÞh þ ð1� hÞ/ þ ð1� LðhÞÞ/

h/ þ ð1� hÞ

E � q: ðA12Þ

By a similar treatment, the curve PBPB on which the evolution of E stops in Fig. 4b can be

described. PAPA and PBPB must intersect at h˜. In deriving these curves, resource

constraints are also taken into consideration, and then they intersect at Eqs. (36).

Appendix D. Non-emptiness of (h, h )

We have seen that in Fig. 4a, a dynamic path which starts from a h larger than h and

leads to the core–periphery steady state where region A is the core can be supported as an

equilibrium path. On the candidate of dynamic equilibrium paths in Fig. 4a, there exists

the point where E is equal to its long-run level (Eqs. (27)) and h is smaller than h. We set

such a h as h*. This point can exist if k is equal to or sufficiently near 1. Here, we consider

whether or not the dynamic equilibrium path from h = h* to h = 1 exists.


From household’s budget constraint (Eq. (3)), consumption expenditure at time zero

when h = h* holds can be written as follows:

E0 ¼ q½H0 þ A0; ðA13Þwhere H0 is the present value of wage income: H0 ¼ ml0 expð� mt0 rðsÞdsÞdt. Because weconsider the case that Eq. (18) binds in region A, it follows that

A0 ¼ nAvA þ nBvB ¼ h*=fh*þ kð1� h*Þg þ nBvB: ðA14ÞIf vB < 1/{knA + nB}(nBvB < (1� h*)/{kh*+(1� h*)}) is satisfied, we can say that a

dynamic equilibrium path from h = h* to h = 1 can be supported as an equilibrium one.

We have seen that the interest rate is initially lower than the discount rate q, but it soonexceeds—and then finally converges to—this discount rate. Therefore, it holds thatZ l

0

rðsÞds ¼Z l

0

qds and

Z t

0

rðsÞds <

Z t

0

qds ¼ qt;

and thus

H0 ¼Z l

0

expð�Z t

0

rðsÞdsÞdt >Z l

0

expð�qtÞdt ¼ 1

q: ðA15Þ

Since Eq. (A13) is equal to Eqs. (27),

E0 � El ¼ q½H0 þ A0 � ½1þ q ðA16Þ

¼ q H0 �1

q

� �þ q½A0 � 1:

¼ 0

Inserting Eqs. (A14) and (A15) into Eq. (A16) elucidates that

nBvB þ h*

h* þ kð1� h*Þ� 1 < 0:

It can be rearranged as follows:

nBvB <kð1� h*Þ

h*þ kð1� h*Þ

¼ ð1� h*Þk�1h*þ ð1� h*Þ

<ð1� h*Þ

kh*þ ð1� h*Þ


Consequently, if all share the expectation that region A will be the core in the future at

the time when h is in h*, the expectation can be self-fulfilled. Then, [h*, h] is non-empty

and (h, h), which contains the [h*, h] as a partial set, is non-empty too.

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