GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.196

Trade costs, wage difference, and endogenous growth

Akinori Tanaka and Kazuhiro Yamamoto

May 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Trade costs, wage di¤erence, and endogenous
growthAkinori Tanakay Kazuhiro Yamamotoz
May 11, 2011

Abstract
In this paper, we develop an endogenous growth model with two countries in which the international trade of di¤erentiated goods requires different trade costs and equilibrium wages in the two countries. If the laborproductivity in one country’s agricultural sector is higher than that of theother country, the wages will also be higher.
In this model, there is a case in which the small country has a higher
share of manufacturing …rms than the larger country, and the innovation
sector locates in the small country, since the cost for production of the
manufacturing sector and innovation sector is higher in the large country
than in the small country.
First, when trade costs are high, the share of manufacturing …rms in alarge country increases with a decline in trade costs. However, the sharethen decreases with a decline in trade costs when trade costs are low.
Finally, all …rms agglomerate in the small country with lower production
costs. If trade costs are very high, the innovation sector will locate in thesmall country. If trade costs take an intermediate value, it will locate in
the large country. If trade costs become very low, it will re-locate in thesmall country. The growth rate moves non-monotonously in a W-shaped
curve when there is a reduction in trade costs. This happens because
the growth rate is a¤ected by the number of manufacturing …rms and thelocation of the innovation sector.

JEL Classi…cation: F0, O31.
Key words: market size, wage di¤erential, growth rates, innovation sector.
We are grateful to Takanori Ago, Colin Davis, Koichi Futagami, Kyoko Hirose, Tatsuro
Iwaisako, Yoshitsugu Kanemoto, Fu-Chuan Lai, Yuji Matsuoka, Tomoya Mori, Se-il Mun, Jun
Ohshiro, Mitsuru Ota, Dan Sasaki, Takatoshi Tabuchi, Hajime Takatsuka, Yuki Takayama,
and Dao-Zhi Zeng for their helpful comments and suggestions. Of course, we are responsible
for any remaining errors. The second author acknowledges the …nancial support from theJapan Society for the Promotion of Science through a Grant-in-Aid for Scienti…c Research((A)2124302), a Grant-in-Aid for Scienti…c Research ((B)21330055) and a Grant-in-Aid forYoung Scientists ((B)197001760).
y Address: Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka 560-0043, Japan; E-mail: gge006ta@mail2.econ.osaka-u.ac.jp
z Graduate School of Economics, Osaka University: E-mail: yamamoto@econ.osaka-u.ac.jp1

1

Introduction

In this paper, a Grossman-Helpman (1991) [5] and Romer (1990) [11]-type endogenous growth model that has two countries is developed. The activity of theinnovation sector results in an increased variety of di¤erentiated goods. In ourmodel, there is some degree of knowledge spillover between the two countries.
Given the knowledge spillover, the innovation costs in a particular country decrease as the number of …rms locating there increases. We assume that laborproductivity in the agricultural sector are di¤erent between two countries, whichleads to di¤erent equilibrium wages in the two countries. In our model, there
are three sectors, agricultural, manufacturing, and innovation, and there are
two countries. In the agricultural sector, homogenous goods are produced only
by labor with constant return production functions. These homogenous goods
are traded internationally with no trade costs. Therefore, the price of thesegoods is the same in both countries. We assume that labor productivity in the
agricultural sectors di¤ers in the two countries. As a result, the equilibrium
wages in the two countries di¤er.
If the equilibrium wage in two countries is the same, the country with a large
market always absorbs more …rms than the other one. Therefore, when the equilibrium wage in two countries is the same, the innovation sector always locatesin the large market country, and the growth rates increase with agglomerationof manufacturing …rms in that country. In this case, the share of manufacturing…rms in the large market country increases with a decline in trade costs.
In the standard international trade theory, there are comparative advantages
between two countries. However, in the standard new economic geography literature, such as that by Fujita, Krugman, and Venables (1999), there are notcomparative advantages between two countries. Some studies have examined theendogenous growth-new economic geography models, such as those by Baldwin,
Martin, and Ottaviano (2001), Martin and Ottaviano (1999), (2001), Yamamoto(2003), and Minniti and Parello (2011). However, in each of these studies, it isassumed that there are no comparative advantages between two countries andthat the production costs of the agricultural and manufacturing sectors are thesame. No study has investigated the e¤ects of the production cost di¤erential
in the context of the endogenous growth - new economic geography. It is important to study the endogenous growth - new economic geography model insuch a context. In this paper, we present a model of endogenous growth andnew economic geography in which production cost di¤erentials are introduced.
If wages in the large market country are higher than those in the othercountry, the higher wages will lower the share of manufacturing …rms in the large
market country. Let us assume that trade costs are initially high and gradually
decrease. In this case, the share of manufacturing …rms in the large country
increases with a decline in trade costs. In a range of trade costs, we observe full
agglomeration in the large country. However, when trade costs become lower,
full agglomeration in the large country is broken, and some manufacturing …rms
switch their location from the large to the small market country. Finally, weobserve that all manufacturing …rms agglomerate in the small market country2

when trade costs are very low. In this process, the innovation sector changes its
location twice when critical numbers of manufacturing …rms are reached. When
trade costs are very high, the innovation sector locates in the small market
country. When the trade costs have an intermediate value, the innovation sector
locates in the larger market country. When trade costs are very low, theinnovation sector locates once again in the small market country. The growthrate moves non-monotonously in this process. The growth rate is raised by the
number of …rms of the same country because of the spillover of local knowledge.
Our model shows that not only the market size but also the production costs ofthe manufacturing sector determine the location of the innovation sector, which
derives economic growth.
Our model is closely related to that of Martin and Ottaviano (1999) [9].
When shipping of di¤erentiated goods incurs trade costs, more …rms locate in
the large market country than in the small market country to economize on trade
costs. In addition, if the equilibrium wages in two countries are the same, theinnovation costs are always lower in the country that has the large market. Thisoccurs because the higher the number of …rms that locate in a particular country,
the lower the innovation costs in that country due to the partial spillover of local
knowledge. Thus, when the equilibrium wage is the same in the two countries,
the growth rate increases monotonically with agglomeration in the large market
country. Our paper expands on the ideas of Martin and Ottaviano (1999) [9]; inour model, wages in the two countries di¤er from each other. In this paper, weshow that wage di¤erences between two countries have a similar e¤ect to market
size di¤erences on the locational share of manufacturing …rms. In other words,
it is more pro…table for …rms to locate in a country with lower production costs.
If wages in a small market country are su¢ ciently low, the small market country
will have a higher share of manufacturing …rms than the large market country.
Furthermore, the innovation sector will locate in the small market country, since
the production costs for the manufacturing sector and innovation sector will behigher in the large market country than in the small market country. In thiscase, agglomeration in the small market country fosters the economic growthrate. The growth rate reaches its maximum value when all of the …rms locate
in the small (lower wage) country. This occurs when the trade costs are very
low. Thus, our paper has richer implications and presents the mechanism of
the history of economic development and agglomeration. Although it has beenreported in many studies that trade liberalization results in increased growth
rates, there are cases in the process of economic development in which theregulation of trade has resulted in increased growth rates. For example, Komiya
et al. (1984) argued that the Japanese government regulated trade for theprotection and expansion of domestic industry between 1950 and 1970. Dueto this regulation, Japan achieved rapid growth. Therefore, there are cases inwhich trade liberalization increases the growth rate, and there are cases in which
trade liberalization lowers the growth rate. By introducing a wage di¤erential,
we can explain why both situations occur in the real world.
This paper is organized as follows. In the next section, we present thebasic model. In Section 3, we analyze the model and present the steady state3equilibrium. In Section 4, we report the e¤ects of trade costs, wage di¤erence,and market size on the location of manufacturing …rms, the innovation sector,
and the growth rate. Section 5 is the conclusion.

2

The Model

In this section, we introduce the model. There are two countries, 1 and 2.
Variables referring to 1 have the subscript 1, and those referring to 2, the subscript 2. Each country is endowed with a …xed amount of labor, L1 and L2(L1 > L2 ). Thus, country 1 has a larger amount of labor than country 2. This
also means that country 1 has a larger market than country 2. Labor can beused to produce homogenous agricultural goods, di¤erentiated manufactured
goods, and blueprints. While labor can be mobile between sectors in the same
country, it cannot be mobile between di¤erent countries. For variety to beachievable, a blueprint has to be invented. The blueprint is then protected bya patent that cannot expire that initially belongs to the country in which innovation took place. Once the blueprints are invented, the patent can be sold to
any …rm located in either country. The innovation and the production process
are, therefore, conducted by di¤erent economic agents and possibly in di¤erent
countries.
The intertemporal utility function of the consumer in country s (s = 1; 2) is
as follows:
Z 1
Us =e t (Yst + log Mst ) dt;
(1)
0

where

Mst =

Z n1t
0

m1st (i)

1

di +

Z n2

m2st (i)

1

1

di

;

> 1:

(2)

0

Here, Yst is the consumption of agriculture goods at time t, Mst is the consumption of the composite of manufactured goods at time t, is the subjective
discount rate, and is the constant parameter. mrst (i) denotes the consumption of the variety i th manufactured goods produced by a …rm in country r(r = 1; 2). nrt is the number of varieties produced by a …rm in country r at timet. nrt is also the number of operating …rms in country r at time t. Nt n1t +n2t
denotes the total number of varieties at time t. is the constant parameter that
represents the elasticity of substitution among di¤erentiated goods. Following
Grossman and Helpman (1991), the market has been characterized by free …nancial movements between two countries. Thus, the interest rate of both countries
is the same at all times (r1t = r2t = rt ). The intertemporal optimization behavior of the consumer brings about the next equation
rt = :

4(3)

We can derive the following instantaneous demand functions (we take homogeneous goods as the numeraire, such that pA = 1),
Mst =

Pst =

Z n1t

1

p1st (i)

Pst

di +

;

(4)

Z n2t

1

1

p2st (i)

1

di

;

(5)

0

0

Yst = Est
mrst (i) =

;

(6)

Pst 1
;
prst (i)(7)

where Pst is called the ‘price index’in country s at time t. prst (i) is the consumer
price of variety i, which is produced in r and consumed in s, and Est represents
the instantaneous expenditure of a consumer in country s at time t.
Here, we describe the production structure of the agricultural sector. A
homogeneous agriculture goods market is perfectly competitive. We assume
that the productivities of labor in the agricultural sector di¤er between the twocountries. In country 1, a1 units of agriculture goods are produced with 1 unitof labor. In country 2, 1 unit of labor produces a2 units of agricultural goods.
We assume that the international trade of homogenous goods incurs no trade
costs. Therefore, since we assume that agricultural goods are produced in bothcountries at the equilibrium1 , the equilibrium wages in the two countries become
w1 = a1 , w2 = a2 .
In the manufacturing goods sector, manufacturing …rms operate under DixitStiglitz [2]-type monopolistic competition. Each …rm produces di¤erentiatedgoods, and each variety is produced by one …rm. To start a production activity,
a …rm in country r is required to buy one unit of a patent produced by theinnovation sector at market price vrt , which plays the role of …xed costs for the…rms. Moreover, a …rm locating in a country uses c units of labor in its country
as the marginal input to produce one unit of manufactured goods. Potential…rms can freely enter into a production activity as long as the operating pro…ts
are positive and can choose to locate in a country where pro…ts are higher.
Under this production structure, each manufacturing …rm sets the following
constant markup price:
pr =

1

cwr =

1

car ; r = 1; 2:(8)

cwr = car represents the marginal costs for the manufacturing …rms. Theinternational trade of manufactured goods incurs ‘iceberg’-type trade costs. If
a …rm sends x units of goods to a foreign country, it must dispatch x units of
1 We assume thatis su¢ ciently small so that the total demand for agricultural goods is suf…ciently large, that is, (E1t
) L1 + (E2t
) L2 a1 L1 and (E1t
) L1 + (E2t
) L2
a2 L2 . In this model, E1t
a1 and E2t
a2 . Therefore, we assume that (a1) L1 +(a2) L2 a1 L1 and (a1) L1 + (a2) L2 a2 L2 .

5

goods.
> 1 represents the trade costs. Thus, consumer prices are prs = prif r = s; and prs = pr if r 6= s: With constant markup pricing, the operating
pro…t of each …rm and the price index are written as,
rt =

Pit =

c

nit a1i

car
cwr
qrt =
qrt ;
1
1

(9)

1

+ njt a1j

1

; i = 1; 2: j = 1; 2: i 6= j;
(10)
1
1
where
2 (0; 1) represents the freeness of trade.
We assume that the capital market is perfectly competitive. We assume
that there are risk-free assets and their interest rate is rt : The value of the …rm(which is the market price of the patent) is equalized to the present value of thesum of discounted pro…t over time. From (9), it represents
Z 1
car
qr d :(11)vrt =e r( t)1
tDi¤erentiating (11) with respect to t, we obtain the no- arbitrage condition for
capital investment vrt :car
qrt + v_ rt = rvrt :(12)1
In the innovation sector, we assume that innovation …rms produce 1 unit ofa patent by using Is units of labor. For innovators in country s, the innovation
costs for a patent are written as ws Ist (s = 1; 2). Innovators choose theirown location with no relocation cost. Then, innovators choose s, where they
can minimize innovation costs ws Ist . If w1 I1t < (>)w2 I2t , then the innovator
locates in country 1(2). We assume that Is depends on the number of home andforeign varieties of manufacturing …rms, as follows:
Iit =nit + njt

; i = 1; 2: j = 1; 2: i 6= j;(13)where 2 (0; 1) represents the degree of international knowledge spillover2 .
In this innovation technology, the agglomeration of manufacturing …rms in acountry lowers the innovation costs in this country, since we assume that 2
1
as the share of manufacturing …rms in country 1.
(0; 1). Let us de…ne s n1n+n
2@w1 I1t@w2 I2t
We can check @s < 0 and @s > 0 as N n1 +n2 …xed. A cost-minimizing
innovator chooses its location according to the share of manufacturing …rms incountry 1, s
n1 =N , and the relative wage in country 1 to country 2, aa21 .
Therefore, we can describe the location behavior of innovators as follows:8no< country 2 ifs; aa21 2 s; aa21 j0 s sb; 0 < aa21no ;The location of the innovator is: country 1 ifs; aa21 2 s; aa21 jb
s s 1; 0 < aa21
(14)
2 In Hirose and Yamamoto (2007) [6], knowledge spillover from a foreign country is asymmetric in two countries. We can extend our model with the assumption of asymmetric knowledge spillover.

6where sb

a1 a2(1 )(a1 +a2 )

=

a1
a2

(1

)

a1
a2 +1

:

@b
s

1+

=

a1
a2

@

a1
a2 +1

a1
a2

> 0: As the

productivity for agricultural goods of country 1 relative to that of country 2
becomes larger, wages in country 1 become higher relative to those in country
2. Thus, if a1 =a2 is su¢ ciently large, innovation …rms locate in country 2.
On the other hand, if the share of manufacturing …rms in country 1 is large,
positive technological externality lowers the innovation costs in country 1. Then,if country 1 absorbs many manufacturing …rms relative to country 2 (s is large),
the innovation sector locates in country 1.

3

Equilibrium conditions and steady states

3.1

Market clearing conditions

In this section, we characterize and solve the equilibrium conditions of the model.
The total production quantities of a …rm of each variety in countries 1 and 2
must satisfy q1 = L1 m11 + L2 m12 and q2 = L2 m22 + L1 m21 , respectively.
Introducing (7), (8), and (5), the market clearing conditions for each variety ofgoods areq1t (s)q2t (s)

= a1
= a2

(
(

1)
cNt

st a11

1)
cNt

(1

L1
+ (1 st ) a12
L2
1st )a2 + st a11
a1
a2

For the following analysis, we de…ne A
this de…nition, we can see that

dA
d

+
+

(1
st a11L2(15);+ st a11
L1(16):+ (1 st ) a12st )a121

and= L1 =(L1 + L2 ). With

< 0; alim A = 1; a lim A = 0 and

a1
a2

1
a2 !0

1
a2 !1

1=2 < < 1.
We must clarify the parameter conditions in which the …rm’s location equilibrium becomes an interior solution or a corner solution. The di¤erence of
operating pro…ts across two countries is
(s)

1

2 =

(A
)
(L1 + L2 )N
sA + (1 s)+( A 1) (1):(1 s) + s A(17)(s) is a decreasing function of s: A manufacturing …rm determines its location
taking s as given. When (s) > (<)0, a …rm locates in country 1 (2). When(s) = 0, it is indi¤erent for a …rm wheter it locates in country 1 or 2. Since
enterance and exit of manufacturing …rms are free, …rms enter into one country
as long as its pro…ts are positive and higher than the other country. We de…ne0, then s = 0. Second, if
s as an equilibrium value of s. First, if (0)(1) 0, then s = 1. Third, if (0) > 0 and (1) < 0, then s is an interior
solution (0 < s < 1) and (s ) = 0. In order to determine which equilibrium
is realized, we analyze the model by identifying three cases: (a)0 < A,(b) 1 A < 1 and (c) < A < 1 .
7

(a) The case of 0 < A
leads to both A
0 and A 1 < 0 in the
numerators of (17). Thus, (s) < 0for all s 2 [0; 1] and for all 2 21 ; 1 . Then,,(0) < 0 for all 2 12 ; 1 . This means that s = 0. Then, when 0 < A
all manufacturing …rms agglomerate in country 2.
(b) The case of 1 A < 1 leads to both A> 0 and A 1 0 in the
numerator of (17). Thus, (s) > 0 for all s 2 [0; 1] and for all 2 12 ; 1 :Then,(1) > 0 for all 2 12 ; 1 . This means s = 1. Therefore, in the case of1
A < 1, all manufacturing …rms agglomerate in country 1.
> 0 and A 1 < 0. The(c) The case of < A < 1 leads to both A
sign of (17) is not clear. However, since (17) is a decreasing function of s, it has
interior solutions in some parameter conditions: (0) > 0 and (1) < 0. The(1
A)
conditions of (0) > 0 and (1) < 0 are equivalent to (0 <) A(1+
<
)(1
) <
1
A
(1+ )(1) (< 1).

Under this condition, s satis…es 0 < s < 1 andOtherwise, it has corner solutions. In other words, if(s ) = 0.

(1
A)
A(1+ )(1) , then,1
A
s = 0, and, if(1+ )(1) , then, s = 1.
From the above discussion ((a), (b), and (c)), the equilibrium share of …rms
in country 1 iss ( ; A; ) =

8
>
<
>
:

0
A(1+ )(1
)
(1
A)(A

if
(1
)

A)

if
if

1

( ; A; ) 2nf ; A; j0( ; A; ) 2; A; j 0 <( ; A; ) 2 f ; A; j 1 <(1 A )
1 A
where 0A(1+ )(1); 1(1+ )(1).
By di¤erentiating an interior solution with respect tothe following lemma.

0 ; 0 < A < 1go
1
1; < A <1; 0 < A < 1g
(18)

and A, we obtained

Lemma 1 (1) The interior solution of s ( ; A; ) is an increasing function of: That is, @s> 0. (2)The interior solution of s ( ; A; ) is an increasing@
function of A. That is, @s@A > 0.
A(1+ )(1@s
Proof. (1) @ = (1 A)(A )) 0 (Note 1
A > 0 and A> 0): (2) See
the Appendix.
Lemma 1 says that the share of …rms in country 1 rises if (1) the relative
market size of country 1 becomes larger or if (2) the relative wage in country 1
becomes lower. We can depict the equilibrium share of manufacturing …rms as
in Figure 1.

3.2

Equilibrium locations of the innovation sector

The innovation …rm determines its location according to s and aa12 as (14). Some
calculations lead us to the next lemma.

8Lemma 2 (1) Let us assume that aa12 < 1: More than half of the manufacturing…rms locate in a large market country, and the innovation sector locates in thiscountry. That is, s > 12 ; and sb < 21 . (2) Let us assume that aa12 > 1: There
1exists s^^ is raised as aa12 becomes larger (A becomes smaller). In
2 , and s
1

addition, when A becomes smaller than 1, the innovation sector always
locates in country 2, even if s = 1.
1
Proof. (1) (18) leads to s
^ 21 < 0. Thus, we see
2 > 0. (14) leads to sthat s^ < s . This means that the innovation sector locates in country 1. (2)From sb =

a1
a2

(1

)

a1
a2 +1

1 1

addition, if A

, s^

1
2

0. Moreover,, then sb =

@b
s
@

a1
a2

(1

)

a1
a2 +1

a1
a2

1+

=

a1
a2 +1

> 0. In

a1
a2

> 1.
1

1
For sb to be smaller than 1 for A = ; we assume that; that is,
the parameter of the extent of knowledge spillover is small. The location of
the innovator at equilibrium is derived by combining (14) with (18). In Figure
1, we identify the class of switch points of the innovator’s location choice as the
sb curve. The sb curve is the points; aa12 ; , which satis…es the next equation:
a1
a2

(1

)

a1
a2 + 1=This relation is satis…ed at points

A(1 + )(1
)
(1
A)(A

(1
)

A)

:

(19)

= 21 ; A = 1 , and ( = 1; A = ) for any
1. From Lemma 2 sb curve locates above s = 21 line. In the case of 1
< ,for any ; the sb line lies between the s = 1 line and the s = 12 line. By Lemma
1, the iso-s curves are downward-sloping in Figure 1. In addition, sb is increasing
as A is decreasing. Therefore, we can see that the sb curve is downward-sloping.
In Figure 1, the innovation sector locates in country 1 at the right and upperside from the sb curve (when country 1 has a larger market and lower wages).
Conversely, the innovation sector locates in country 2 at the left and lower sideof the sb curve (when country 2 has a larger market and lower wages).
In the following analysis, we restrict the values of parameters a1 a2 , and
1 1< . This means that the wage in country 1 is higher than that incountry 2 and the threshold s^ lies between 21 and 1.

3.3

Growth rates at steady states

As in the study by Grossman and Helpman (1991) [5], this model has a uniquesteady state in which the mass of variety is expanding at a constant rate overtime. Since the innovation sector is assumed to be under free entry, at equilibrium, the innovation cost must be equalized to the value of the patent. Then,1
vt = Nt (s + a(1s )) when the innovation sector locates in country 1. s is constant over time from (18), and v must decrease at the same rate as Nt increases;
N_ tthus, vv_ tt = N: At equilibrium, the total sales of a manufacturing …rm aret9

1 +L2 ). Introducing these equalities into (12), we derive
a1 q1 = a1 q2 = ( 1)(Lc Nt
the growth rate at steady states (we de…ne g as the rate of variety expanding
N_ t
at steady states, that is, g N
).
t

a1
a2

g s;

where

; a1 + a2
a1
a2
a1
a2 +1

=

(

(L1 +L2 )(1 s)+ s(1)(a1 +a2 )(L1 +L2 )s+ (1 s)(a1 +a2 )if

the innovator locates in country 2.

if

the innovator locates in country 1.
(20)1= a1a+a.
2

In the next section, we analyze the e¤ect of parameters on the share ofmanufacturing …rms and steady state growth rates.

4E¤ects of parameters on the location of …rms
and the growth rates

4.1

E¤ects of the change of aa21 and a1 + a2 on s and g

In this subsection, we analyze the e¤ect of change aa12 and a1 + a2 on the geographical con…guration of …rms and growth rates.
First, di¤erentiating (20) with respect to s
( (L +L )
(1 )
1
2
@g
(1
)(a1 +a2 ) < 0 if the innovator locates in country 2.
=(1 )(L1 +L2 )@s
if the innovator locates in country 1.
(a1 +a2 ) > 0(21)
The sign of @g
dependsonthelocationoftheinnovation
sector.
Next,
the
@s
lemma represents this property.
Lemma 3 The agglomeration of manufacturing …rms in the country where theinnovation sector locates enhances the economic growth rate.
Let us assume that the innovation sector locates in country 2. When the
agglomeration of a manufacturing …rm in country 1 progresses, the unit requirement for producing a patent in country 2 increases, since international knowledge spillover is imperfect. The growth rates then become lower. Conversely,
when the innovation sector locates in country 1, agglomeration in country 1
makes the innovation sector more e¢ cient. Then, the growth rates become
higher.

10

Second, di¤erentiating (20) with respect to aa12 , we obtain:3
@g
@

a1
a2

@g @
@ @ a1a2
8(L1 +L2 )
>
<

=

=

(L1 +L2 )>:(1(1 s)+ s)2 (a1 +a2 )(s+ (1 s))
2 (a +a )1
2@> 0 ifthe innovator locates in country 2.

<0

the innovator locates in country 1.

a

@ a1
2
@
@

a1
a2

if

Raising aa21 means that the relative wages in country 1 become higher. Thus,
the innovation cost in country 1 becomes larger relative to that in country 2.
Therefore, @g
is negative when the innovation sector locates in country 1.
a1@

Conversely,

a2

@g
@

is positive when the innovation sector locates in country 2.

a1
a2

Third, di¤erentiating (20) with respect to a1 + a2 ;( (L1 +L2 )[(1 s)+ s]< 0 if the innovator locates in country 2.
@g(1)(a1 +a2 )2=(s+ (1 s))(L1 +L2 )@ (a1 + a2 )<0
if the innovator locates in country 1.
(a +a )21

4.2

2E¤ects of change of , aa12 , and a1 + a2 on g

We substitute s = s
g

;

;A

a1
; a1 + a2 ;
a2

a1
a2

;

characterized by (18) into (20),
g s

;A

a1
a2

;

;

a1
; a1 + a2 :
a2

(22)

We di¤erentiate (22) with respect to parameters , aa21 , and a1 + a2 :@s
From @g= @g
and @s> 0, we obtain @g< (>)0 when the innovation
@
@s @
@
@
sector locates in country 2(1). This means that, if the market size of the country
in which the innovation sector locates becomes larger, manufacturing …rms become more agglomerated in that country. Lemma 3 means that the innovation
activity becomes more e¢ cient, and growth rates become higher, depending on
the market size of the country where the innovation sector locates.
Di¤erentiating (22) with respect to aa21 ; we derive the following equations:

8 @g @s@A@g>+> 0 if
>
>
>
@s
@A
@
(a
=a
)
@
(a
1
2
1 =a2 )
>
|{z}
|{z}
>
| {z } | {z }
<
+

@g
+
=
@g @s
@A
@g
@ (a1 =a2 ) >>+< 0 if>>@s @A @ (a1 =a2 ) @ (a1 =a2 )>>: |{z}|{z}| {z} | {z }+3 Notice that

@
a

@ a1
2

+

> 0.

11

the innovator locates in country 2.
:
the innovator locates in country 1.

The change of relative wages has two e¤ects on the growth rate. One is the
direct e¤ect, in which innovation costs change. The other is the indirect e¤ect,in which the share of manufacturing …rms changes. Because these two e¤ects: Let us assume that thehave the same direction, we can see the sign of @(a@g1 =a2 )
innovator locates in country 1 and the relative wages in country 1 increase. The
innovation costs become higher by the direct e¤ect, and innovation technology
becomes ine¢ cient through the indirect e¤ect of the share of …rms in country
1. Therefore, the growth rates become lower when the innovation sector locatesin country 1 and the relative wages in country 1 increase.
= @(a1@g+a2 ) . In the following table, we summarize the e¤ects
Here, @(a@g1 +a2 )
of each variable on the growth rate g .
country 2 country 1+A+
a1 + a2
In this table, country 2 and country 1 represent the country in which theinnovation sector locates. It is noteworthy that A is a decreasing function of@ga1= 0; from @s=0a2 , and then the sign is reversed. At corner solutions, @@and @(a@g= @(a@g; from @s@A = 0. In this case, the e¤ects on the growth
1 =a2 )
1 =a2 )
rate g are as follows:country 2
0A
a1 + a2

country 1
0+

We depict Figure 1, in which the e¤ects of and A aa12 on g are shown:
The arrow in Figure 1 indicates the direction in which the growth rates arehigher.

4.3

E¤ects of the change ofon s and g

To study the e¤ects of trade costs on the location of a manufacturing …rm, withand A …xed, we check the sign of @s@ . It is di¢ cult to derive the di¤erentialof (18) directly. However, we obtain the following lemma by calculations shownin the Appendix.
1
Proposition 1 (1) Let us consider that the parameters satisfy
1+A2 . When
trade costs are high, s becomes larger by the decline in trade costs. When trade
costs are su¢ ciently low, s becomes smaller than the decline in trade costs. In
other words, @s
0 for is small and @s
0 for is large. Finally, when@@is very large, all manufacturing …rms agglomerate in country 2. (2) Moreover,1
1+2(1 A2 ) 2
1
in the case in which parameters satisfy the relation2
1+A2 ,
when trade costs take an intermediate value, full agglomeration in country 1 isrealized.

12

Proof. See Appendix.
Proposition 1 says that, when trade costs are high, declining the trade costs
facilitates agglomeration to the large market (and higher wage) country. On
the other hand, when trade costs are small, declining the trade costs fosters
agglomeration in the small market (and lower wage) country. Reduction of
trade costs has two e¤ects. One is that locating in the larger market country
becomes more pro…table because …rms can transport manufactured goods withlower trade costs. Firms, then, need not locate in the small market country.
This e¤ect is reported in many studies of new economic geography, such as
those by Martin and Ottaviano (1999) [9] and Martin and Ottaviano (2001)[10]. Another e¤ect is that, when trade costs become lower, the market size
become less important, and the wage di¤erence becomes more important fora …rm’s location choice 4 . Indeed, in the = 1 case, i.e., perfect free trade,all manufacturing …rms agglomerate in country 2, where wages are lower, i.e.,s = 0 for every 2 21 ; 1 and A 2 (0; 1). When trade costs are high, the formere¤ect dominates over the latter e¤ect, and then s increases. With a decline
in trade costs, the di¤erence in the two e¤ects becomes small; then, the lattere¤ect dominates over the former e¤ect. As a result, s decreases. Finally, atvery low trade costs, all manufacturing …rms agglomerate in country 2 (Figure
2).
Next, we study the e¤ects of trade costs on the location of the innovation
sector. From (14), with aa21 ( 1) …xed, the innovator determines its locationaccording to s only. Then, we can draw s^ line horizontally between 12 and 1
1
in Figure 2. By taking proper ; for any and A such that
1+A2 ; the relation ofs^ is satis…ed. This relation means that, when trade costs are very
high because the share of …rms in country 1 is smaller than the threshold, theinnovation sector locates in country 2. With a decline in trade costs, because s1
1+2(1 A2 ) 2
rises and reaches s = 1 under the parameter condition of, the2
innovation sector changes its location from country 2 to country 1 at some(denote ^ 1 ). Moreover, with a decline in trade costs, because full agglomeration in country 1 is broken and manufacturing …rms change their location fromcountry 1 to country 2, the innovation sector moves again from country 1 tocountry 2 at some (denote ^ 2 ).
We study the economic growth rate at a steady state; Lemma 3 means that
agglomeration in the country where the innovation sector locates fosters thegrowth rate. Thus, when innovation sector locates in country 1, the growth rate
moves in the same direction as s moves. When the innovation sector locatesin country 2, the growth rate moves in the opposite direction against s . In
4 In Gao (2007) [4], because globalization makes the manufacturing/agriculture wage ratio
in country 2 higher and manufacturing labor supply becomes larger, the manufacturing …rms
in country 2 expand, with a decline in trade costs.

13

addition, at the full agglomeration case, the growth rate is as follows:
g 1;
g 0;

a1
a2
a1
a2

; a1 + a2=; a1 + a2=(L1 + L2 )(L1 + L2 )1;
(a1 + a2 )1
(1
) (a1 + a2 ):< 1, and then 1. This implies
From a1
a2 , we can see that 12g (1; ) g (0; ). Thus, we can draw the relation between trade costs and thegrowth rate as shown in Figure 3. After summarizing this discussion, we obtainthe next proposition:1

1+2(1 A2 ) 2
andProposition 2 Consider that the parameter conditions of2s^ are satis…ed. (1) As trade costs decrease, the growth rates at the steadystate follow a W -shaped curve; that is, they have two low growth-rate valleys.
(2) When trade costs are very low, the growth rates reach their maximum value.

5

Conclusion

In this paper, we have constructed a model in which equilibrium wages in two
countries are di¤erent from each other. Di¤erences in wage rates and market
sizes generate particular patterns of growth and agglomeration in the economy.
If the equilibrium wages in two countries are the same, the country which has
a large market always absorbs more …rms than the other country. Therefore,
when the equilibrium wages in two countries are the same, the innovation sector
always locates in the large market country, and growth rates are raised withagglomeration of manufacturing …rms in that country. In this case, the shareof manufacturing …rms in the large market country increases with a decline intrade costs.
However, if wages in the large market country are higher than those in
the other country, higher wages lower the share of manufacturing …rms in thelarge market country. We show the relationship between the proportion of
manufacturing …rms and the reduction in trade costs. When trade costs are
high, the share of manufacturing …rms in the large market country increases
with a decline in trade costs, and we observe full agglomeration in the large
market country. However, when trade costs become low, full agglomerationin the large market country is broken, and some manufacturing …rms locate
in the small market country. We show that, …nally, all manufacturing …rms
agglomerate in the small market country when trade costs are very low. In
this process, growth rates …rst decrease with a decline in trade costs. Afterthese movements, the growth rates take W -shaped movements. As we saw
above, we have studied the e¤ects of reduction in trade costs on manufacturing…rms’ location and economic growth rates. By introducing the di¤erence of14

wages in two countries, the results become richer, and the equilibrium in which
manufacturing …rms agglomerate in a small market country and the innovation
sector locates in that country can be observed.

Appendix
The proof that the …rm share s = s ( ; A; ) is an increasing
function of A
We show that s ( ; A) is an increasing function of A: Di¤erentiating (18) withrespect to A; we obtain
@s
@A

= s
=

"

"

1+ 2 2 A+A)(1 + A) (A
)
#
) + 2
1+ 2 2 A s:(1 + A) (A)(1 + ) (1) + 2
A (1 + ) (1)(1

(1 + ) (1#(23)

From an interior solution condition, the denominator of (23) satis…es (1 + A) (A
0: Therefore, we check the sign of the numerator of this equation. (18) can betransformed as follows:(1 + ) (1

2

) +

=

1

) +

2

(A

A

) s +

A

Then, the numerator of (23) is(1 + ) (1
=

1

(A

A

=

A

=

1
(1
A

s +

A

1+

) s +

A

2

2 A s
1+

2

2 A s

A

s ) + s A > 0:In other words, @s@A > 0 has been showed.

The proof that the …rm share s = s ( ; A; ) is an increasing
function of for small and a decreasing function of forlarge :We analyze the sign of @s@ : Totally di¤erentiating (18), we can derive
ds =

@s
@s
@s
d +
dA +
d :
@
@A
@
15

)>

Setting ds = 0; dA = 0;
@s
@s
d +
d
@
@

0=

@s
=
@
d
d

@s d
@ d
plane. Since @s@is the slope of the iso-s curve in the
d
d

@s
@

0 from Lemmadd0; on the other hand, if
0, then @s
0. We
@
A
1. We will draw the iso-s curves on the

1, if
0, thenconsider the case of 0plane (Figure 4).
(1) s = 0 curve

A (1 + ) (1
)
(1
(1
A) (A
)
A (1 + ) (1

1d
d

A

=

1

A

(1
1

=

+

A)

=

)

1

0

=

(1

1
1+

A)

1
1

2

) + 2 (1 A)
2
(1
A)

0

(2) s = 1curve
A (1 + ) (1
)
(1
(1
A) (A
)
(1 + ) (1

1d
d

=
=

When

= 0, 1 dd

=

A)
)

=

1

=

(1

A)

A

1
1
1
A 1+
1
2
A A 2
(1 + ) (1) (1A)

A < 0. When

= A, 1 dd

3= (1A AA2 )2 > 0. The

numerator of 1 dd is negative for small and positive for large , while thedenominator of 1 dd is always positive. Then, the s = 1 line is U-shaped.
(3) general s 2 (0; 1) curve
1d
1 s
=
d
As + (1

1
1+

s )
16

+

1
1

A
1

A

(24)

1
1d
=
!0 d

s

lim
Since s =ddat

= 0, ifj =0 > 0: This means that, if

1 + A2
As

1
d
0: If
1+A2 , then d j =01
@s0.
1+A2 , then @ j =0

1 s
A
1d
=
+
d
A
(1 + A) (1(24) can be transformed as follows;lim!A

1 s (1 + A)
1d=+d
[As + (1 s )] (1 + ) (1

A)

1
< 1+A
2 , then

>0

1 A
) (1

A)

(25)

1
The …rst term of the right-hand side of (25) is positive if s
1+A but negative
1
if s
1+A . The second term of the right-hand side of (25) is positive. First,
1 d
1
if s
1+A , thend is su¢ ciently positive for all . Thus, the iso-s curve is1
upward-sloping. Second, if s > 1+A; then 1 dd is an increasing function of .

(Since the denominator of the …rst term of (25) is an increasing function, thenthe …rst term of (25) is an increasing function of (Note that 1 s (1 + A) < 0).
The second term of (25) is an increasing function of .) Thus, the iso-s curveis convex.
From this discussion, we can draw the iso-s curves onplane (Figure
1114). If s
;
then
the
iso-s
curve
is
U-shaped.
If
s
< 1+A
2
2 , then
1+A
1+A
1
the iso-s curve is convex and an increasing function of . If s < 1+A, thenthe iso-s curve is an increasing function of .
Therefore, if@s1
1+A2 , then @and @s@0 for small

0 for large .

Parameter conditions for full agglomeration in country 1
When all …rms agglomerate in country 1 at equilibrium, equation s ( ; A; ) = 1
for has two real number solutions. (18) can be transformed as(1 + ) (1
)
2
A+1

=
=

(1
0:

A) ;

For this equation to have two real number solutions, the following must be
satis…ed:
D

A2

4 (1

)

0:

That is,1

1 + 2 1 A2 2:217

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[4] Gao, T. (2007) Trade costs, international production shifting, and growth,
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18

A
1
d

s D = 12sD = 1s D = s!

1d
sD = 0

0
1
2

V

1

Equilibrium share of manufacturing …rms

sD
1

s!
V
1
2

0
0

d! 1

d! 2

A

1

d

The relation of trade costs and share of manufacturing …rms

19

gD
gÝ0, FÞ
gÝ1, FÞ

gÝs!, FÞ

0
d! 1

0

d! 2

A

1

d

The relation of trade costs and steady state growth rate

V
1

s!

sD = 1

V1
1
1+A

1
2

sD = 0

0
0

d! 1

d! 2

A

1

d

The relation of trade costs and market size and share of …rms

20