GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No. 82

Dynamic Analysis of Location Choice byMultinational Firms

Tadashi Morita

September 2009; revised June 2012GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanDynamic Analysis of Location Choice byMultinational Firms ∗Tadashi Morita †Abstract
This paper constructs a North-South endogenous growth model to investigate
how the organizational forms of ﬁnal goods ﬁrms evolve. Initially, the ﬁnal goods
ﬁrms in the North obtain intermediate goods from Northern ﬁrms and produce in
the North. When trade costs are suﬃciently low, as the economy develops, the ﬁnal
goods ﬁrms produce the ﬁnal goods in the North and obtain the intermediate goods
from Southern ﬁrms. As the economy develops further, they produce the ﬁnal goodsin the South and obtain intermediate goods from Southern ﬁrms.
Keywords: Location Choice, North-South, R&D, Economic Growth
JEL classiﬁcation: F43, O31, R30∗

I am very grateful to Koichi Futagami for his considerable help. In addition, I have beneﬁted fromcomments and suggestions by Ichiro Daito, David Flath, Takeo Hori, Shinsuke Ikeda, Tatsuro Iwaisako,
Kazuo Mino, Yasuyuki Miyahara, Yoshiyasu Ono, Kaoru Sano, Kazuhiro Yamamoto and anonymousreferee, and seminar participants at Matsuyama University and Osaka University. This article was previously entitled, ’Dynamic Analysis of Outsourcing and FDI’.
†
Faculty of Economics, Osaka Gakuin University, 2-36-1 Kishibeminami, Suita, Osaka 564-8511,
JAPAN e-mail: t-morita@ogu.ac.jp1

1

Introduction

Recently, trade costs and communication costs have decreased rapidly and the volume oftrade has increased. Thus, ﬁrms in developed countries can easily obtain intermediate
goods from foreign countries. Indeed, many researchers argue that the importing of intermediate goods has increased. 1 In addition, foreign direct investment (FDI) conductedby multinational ﬁrms in developed countries has been rising. UNCTAD (2008) reported
that FDI inﬂows increased from $ 1.4 trillion in 2000 to $1.8 trillion in 2007. From these
data, ﬁrms in developed countries use many strategies to produce ﬁnal goods which relate to where they produce the ﬁnal goods and from where they obtain their intermediate
goods. In this paper, I investigate the relationship between the strategies of multinationalﬁrms and the trade costs and I analyze how multinational ﬁrms change their strategies
to produce the ﬁnal goods as economy develops.
I show examples how multinational ﬁrms change their strategies, Japanese cotton spinning companies before World War II and the scanner of Canon Inc-. After the treaty of
Shimonoseki was signed in 1895, China opened some cities to Japan and Japanese ﬁrms
were permitted to conduct FDI in China. However, at that time, because the wage ratein Japan was cheaper than that in China, Japanese cotton spinning companies produced
the cotton yarn in Japan and they exported the cotton yarn to Chinese market. After
World War I, the wage rate in Japan rose sharply. Then, Japanese cotton spinning companies conducted FDI in China. 14 Japanese cotton spinning companies conducted FDI
and they managed 35 factories in 1924. On the other hand, when Canon Inc-. produced
scanners in 1997, they obtained intermediate goods to domestic ﬁrms. As Taiwanese companies entered the scanner market and the price competition became severe, Canon wasconcerned about the high wage rate in Japan. As a result, Canon obtained intermediate
goods to Taiwanese ﬁrms whose wage rates were lower. Furthermore, Canon has produced
multifunction printers which have a scanner in Suzhou in China and obtained almost of all
the intermediate goods from near the factory recently. From these examples, the multinational ﬁrms have changed their strategies how to produce the ﬁnal goods and where toobtain the intermediate goods. In the past, multinational ﬁrms obtain their intermediate
goods from the country which they locate to produce the ﬁnal goods. However, in thesedays, multinational ﬁrms choose not only place of production but also choose where to obtain their intermediate goods. How have the multinational ﬁrms changed their strategies
1See, for example, Campa and Goldberg (1997), Feenstra and Hanson (1996), Hummels, Ishii, and Yi(2001), Yeats (2001), and Hanson, Mataloni, and Slaughter (2001, 2005).

2and what are the diﬀerent strategies that the multinational ﬁrms take.
Some researchers have used static models to investigate these issues. Antràs andHelpman (2004) analyzed how ﬁnal goods producers with diﬀerent productivity levels
choose whether to outsource the production of intermediate goods. They showed that thestrategies of ﬁnal goods producers crucially depend on their productivity. Grossman andHelpman (2005) investigated the choice of location from which ﬁnal goods ﬁrms obtain
their intermediate goods. They showed that productivity improvement in the South
shifted outsourcing from North to South. However, although these studies oﬀer useful
insights into ﬁrms’ ﬁnal goods strategies, the researchers analyzed only static equilibria.2
In the literature on product cycles, many studies have constructed North-South growthmodels in which Northern ﬁrms can produce both in the North and the South. Helpman(1993), Lai (1998), Glass and Saggi (2002), Glass and Wu (2007), and Tanaka, Iwaisako,
and Futagami (2009) assumed that undertaking foreign direct investment (FDI) is costless.
They showed that there exists an equilibrium in which some ﬁrms produce in the Northand some ﬁrms produce in the South. In a separate strand of the literature, Tanaka,
Iwaisako, and Futagami (2007) assumed that ﬁnal goods producers incur the costs insuccessfully negotiating licenses with Southern ﬁrms. When the ﬁnal goods producers
have successfully negotiated licensing, they can produce in the South. Although thesestudies considered the location of production, they did not focus on the switching point
at which ﬁnal goods producers change their production locations from developed countries
to developing countries.
Morita (2010) constructed an endogenous growth model of the variety-expansion typein which ﬁnal goods ﬁrms choose the intermediate ﬁrms from which they obtain theirintermediate goods. In an earlier paper, I explored transition paths and the switching
point at which ﬁnal goods producers obtained goods from foreign suppliers rather thanfrom domestic suppliers. However, Morita (2010) focused on the one strategy of multinational ﬁrms where to obtain their intermediate goods and could not explain that themultinational ﬁrms perform FDI as the economy develops. Compared to Morita (2010),
I generalize that model to allow Northern ﬁrms not only to obtain intermediate goods to2

Some researchers have used dynamic models to investigate these issues. Naghavi and Ottaviano(2009) used a North-South endogenous growth model with oﬀshoring to investigate the extent to which
ﬁnal goods producers outsource the production of intermediate goods to the South. They analyzed two
types of equilibrium: one in which all producers outsources the production of intermediate goods, andanother in which none of them outsource. In addition, Gao (2007) developed an endogenous growthmodel and analyzed the relationship between trade costs and the location in which intermediate goods
are produced. He showed that, as trade costs fall, the number of intermediate goods produced in theSouth increases. However, these studies analyze only the steady state.

3either the North or South, but also to base their production of ﬁnal goods in either the
North or South. This enables me to explain the diﬀerent behaviors of Japanese cotton
spinning companies before World War II and Canon Inc-.
In this model, I assume that it is costless to trade intermediate goods within a country
but not between countries. In addition, I assume that when ﬁnal goods ﬁrms produce
their ﬁnal goods in the South, they incur a management cost because they have to monitor the Southern workers. The ﬁnal goods ﬁrms choose the location that maximizes
proﬁt. I show why ﬁrms change the location of ﬁnal goods production as the economy
develops. Suppose that, initially, a ﬁnal goods ﬁrm produces ﬁnal goods in the North andobtains intermediate goods from Northern ﬁrms. When transportation costs are suﬃciently low, as the economy develops, the ﬁnal goods ﬁrm obtains the intermediate goods
from Southern ﬁrms. Then, as the economy develops further, the ﬁnal goods ﬁrm obtains
the intermediate goods from the South and also produces the ﬁnal goods in the South.
However, when transportation costs are suﬃciently high, the ﬁnal goods ﬁrms does notchoose the strategy that they locate in the North and obtain the intermediate goods fromSouth like Japanese cotton spinning companies before World War II. Then, in this case, as
the economy develops, the ﬁnal goods ﬁrm obtains the intermediate goods from Southernﬁrms and also produces the ﬁnal goods in the South.
The remainder of the paper is structured as follows. In Section 2, I introduce themodel. In Section 3, I derive the equilibrium path of the model and prove that it isunique. Section 4 concludes this paper.

2

The Model

I develop a dynamic general equilibrium model where ﬁnal goods producers obtain theintermediate goods from intermediate producers either in their home country or in foreigncountries and they produce the ﬁnal goods in their home country or in foreign countries.
Our model has a similar structure to the model of Grossman and Helpman (1991, Ch. 3).
The world economy consists of two countries, the North and the South denoted byl ∈ {N, S}. The population size in the world is one. Each individual lives forever andis endowed with Ll units of labor services, which are supplied inelastically at each pointof time. There exist three types of goods: a homogeneous good, intermediate goods, andﬁnal goods. The homogeneous good is produced only in the South, 3 and the ﬁnal goods
3

In this paper, a homogeneous good is produced in both countries with one unit of labor per unit.
This paper focuses on the case that the wage rate of North is larger than the wage rate of South and4can be produced in the both countries. The intermediate goods can be produced in either
country. Individuals consume the homogeneous good and the ﬁnal goods. R&D activities
can be conducted only in the North. 4
The ﬁnal goods ﬁrms purchase intermediate goods either from the Northern ﬁrms orfrom the Southern ﬁrms, and produce the ﬁnal goods in the North or in the South in orderto maximize their proﬁts. Therefore, there are four possible strategies. First, the ﬁnal
goods ﬁrms obtain the intermediate goods from the Northern ﬁrms and produce the ﬁnal
goods in the North. Second, they obtain the intermediate goods from the Southern ﬁrms
and produce the ﬁnal goods in the North. Third, they obtain the intermediate goods fromthe Northern ﬁrms and produce the ﬁnal goods in the South. Finally, they obtain theintermediate goods from the Southern ﬁrms and produce the ﬁnal goods in the South. In
the ﬁrst case, there are three sectors in the North: the R&D sector, the ﬁnal goods sector,
and the intermediate sector. In the South, there is only one sector: the homogeneous
good sector. In the second case, there are two sectors in the North: the R&D sector andthe ﬁnal goods sector. In the South, there are two sectors: the homogeneous good sector
and the intermediate sector. In the third case, there are two sectors in the North: theR&D sector and the intermediate sector. In the South, there are also two sectors: thehomogeneous good sector and the ﬁnal goods sector. Finally, in the fourth case, there is
only one sector in the North: the R&D sector. In the South, there are now three sectors:
the homogeneous good sector, the intermediate sector, and the ﬁnal goods sector.

2.1

Individuals

Individuals in both countries have identical preferences:∫ ∞e−ρ(ν−t) U (ν)dν, 0 < ρ < 1,(1)twhere ρ is the constant subjective discount rate. U (ν) is the instantaneous utility perperson at time ν and is speciﬁed as follows:
U (ν) = y(ν) +1
X(ν)µ ,µ

0 < µ < 1,(2)

the market of the homogeneous good is perfect competitive. Therefore, a homogeneous good is neverproduced in the North in equilibrium.
4
Lai (1998), Glass and Saggi (2002), Grossman and Helpman (1991, ch. 11), Grossman and Helpman(2005), and Tanaka (2006) constructed a North-South model in which innovation sector is located onlyin the North and imitation sector is located only in the South. They assume that South does not havethe know-how needed to invent new goods. This paper omits the imitation sector because I focus on thestrategies of the ﬁnal goods ﬁrms.

5

where µ is a parameter, y(ν) stands for consumption of the homogeneous good at time ν,
and X(ν) is a composite good at time ν that is made up of diﬀerentiated ﬁnal goods. For
simplicity, I drop the time index ν from all the variables. A composite good X is given
by:

[∫ n
α

X=

] α1

x(i) di,0 < α < 1,

(3)0

where x(i) represents consumption of diﬀerent ﬁnal goods i and n denotes the number
1
of ﬁnal goods and n is endogenous variable. 1−αis the elasticity of substitution between

any two varieties of ﬁnal goods in a given sector. If α is close to one, the goods are nearly
perfect substitutes and the sector is highly competitive. If α is close to zero, the goods
are distinct products and the sector becomes monopolistic. I assume that α > µ to ensure
the concavity of preferences over the x(i).
The utility maximization problem of the individual can be solved in two steps. The
ﬁrst step is to solve the following static optimization problem:

1
y+ X µ ,x,x(i)µ∫ nsubject to y+
P (i)x(i)di = E,
max0where the homogeneous good is chosen to be the numeraire, P (i) stands for the price ofthe product i, and E is the total expenditure. From the ﬁrst-order condition, I can obtainthe following inverse demand function:
P (i) = X µ−α x(i)α−1 , i ∈ [0, n] .

(4)

The second step is to solve the intertemporal optimization problem. From (4), theindirect utility function is given by:(

1
U =E− 1−µ) [∫ n
P (i)

α
α−1

] µ(α−1)α(µ−1)
.
di

(5)

0

As is clear from the indirect utility function (5), the marginal utility of expenditures isconstant. The intertemporal budget constraint is given by∫ ∞∫ ∞−r̄(s) lle
E (s)ds = A (0) +e−r̄(s) wl (s)ds,
0

(6)

0

where r̄(s) is the average interest rate between times 0 and s and r̄(s) =∫s0r(t)dt/t.

Al (0) is the present value of asset owned by a worker in country l, and wl is the wage rate6

in country l. In this model, I focus on the equilibrium that individuals consume at eachperiod. Therefore, the market interest rate at time t, r(t), must be equal to the subjective
discount rate:r(t) = ρ2.2

Production

2.2.1

The homogeneous good sector

for all t .

(7)

Production of the homogeneous good uses labor only. The homogeneous good is produced
only in the South. The production of one unit of the homogeneous good requires one unitof Southern labor. I assume that the homogeneous good market is perfectly competitive.
Thus, the price of the homogeneous good becomes equal to the wage rate in the South. As
the homogeneous good is chosen to be the numeraire, the wage rate in the South becomes
unity, that is, wS = 1. I assume that the homogeneous good can be traded costlessly.
2.2.2

The intermediate goods sector

The intermediate goods can be produced in the both countries. Production of one unitof each intermediate good requires one unit of labor. I assume that perfect competition
prevails in the intermediate goods markets in both countries. Thus, the marginal costof intermediate goods produced in the North is the wage rate in the North, wN . In thispaper, I focus on the case that wN > 1. The marginal cost of intermediate goods producedin the South is the wage rate in the South, wS = 1. I assume that no transportation costis incurred in transferring the intermediate goods when the ﬁnal goods ﬁrms and theintermediate good ﬁrms exist in the same country, but the ﬁnal goods ﬁrms have to pay
an iceberg cost τ (τ > 1) to obtain intermediate goods from the other country. Therefore,
the price of intermediate goods facing the ﬁnal goods ﬁrms is as follows:pm
N,N = wN ,(8)pm
N,S = τ,(9)pm
S,N = τ wN ,

(10)

pm
S,S = 1,(11)

where pmj,k is the price of intermediate goods when ﬁnal goods ﬁrms are in country j,j ∈ {N, S} and they obtain the intermediate goods from the suppliers in country k,k ∈ {N, S}. Hereafter, if there are two subscripts, the ﬁrst subscript denotes where ﬁnal

7

goods ﬁrms are located, and the second subscript denotes the country from which theﬁnal goods ﬁrms obtain the intermediate goods.
2.2.3

The ﬁnal goods sector

Production of each ﬁnal goods requires labor and a variety-speciﬁc intermediate good.
When the ﬁnal goods ﬁrms exist in the North, they use Northern labor and Northernintermediate goods. I assume that their unit cost function is as follows:
1−β m βcN (wN , pm
N,k ) = wN (pN,k ) , k ∈ {N, S} ,(12)

where pm
N,k is the price of the intermediate goods produced in country k when ﬁnal goods
ﬁrms are in the North, β is constant, and 0 < β < 1. On the other hand, when the ﬁnal
goods ﬁrms produce the ﬁnal goods in the South, I assume that they have to hire not
only Southern workers but also Northern workers in order to manage and train Southern
workers. The ﬁnal goods ﬁrms have to incur the cost because Northern workers haveto cross international borders and live in the South, which has a diﬀerent culture andlanguage from the North, in order not only to produce but also to monitor the Southernlabor. Speciﬁcally, I assume that management costs take the iceberg form to simplify thisanalysis. Hence, the cost of Northern labor is ²wN , ² > 1. Then, I assume that the unit
cost function of the ﬁnal goods ﬁrms is as follows:(1−γ)(1−β)γ(1−β)cS (²wN , pm
wSS,k ) = A(²wN )β(pmS,k ) ,β= A(²wN )γ(1−β) (pmS,k ) , k ∈ {N, S} ,(13)

where pm
S,k is the price of the intermediate goods produced in country k when the ﬁnal
goods ﬁrms are in the South, γ is a constant parameter, and 0 < γ < 1. A is also a
constant parameter.
By using (4), I can obtain the revenue of each ﬁnal goods producer as follows:
Rj,k (i) =P (i)x(i),
µ−α
=Xj,k
xj,k (i)α ,j, k ∈ {N, S} , i ∈ [0, n].

(14)where Rj,k is the revenue of the ﬁnal goods producer, Xj,k is the composite good andxj,k (i) is the consumption of diﬀerent ﬁnal goods i. I consider the proﬁt maximization
problem of the ﬁnal goods producers in both countries. In the Appendix A.1, I derive the
proﬁt function when the ﬁnal goods ﬁrms exist in the North and in the South. Using (8),8

(9), (10), and (11), I can write the proﬁt functions as follows:
µ−α

−µ

µ

πN,N = n α(1−µ) α 1−µ wN1−µ (1 − α),µ−α−µ(1−β)
1−µ

µ

−βµ

πN,S = n α(1−µ) α 1−µ wN
τ 1−µ (1 − α),−µ( ) 1−µµ−α−µγ(1−β)−βµAα(1−µ)
πS,N = n(²wN ) 1−µ (τ wN ) 1−µ (1 − α),α−µ( ) 1−µµ−α−µγ(1−β)Aα(1−µ)
πS,S = n(²wN ) 1−µ (1 − α).
α

(15)
(16)
(17)
(18)

Comparing (17) to (18), I can show that the proﬁt of the ﬁnal goods ﬁrms in the Southwhen they obtain the intermediate goods from the Southern ﬁrms is larger than that when
they obtain the intermediate goods from the Northern ﬁrms, that is, πS,S > πS,N , because
τ wN > 1. Therefore, when they are in the South, the ﬁnal goods ﬁrms never purchasethe intermediate goods from the Northern ﬁrms. Then, I obtain the following lemma.
Lemma 1. When the ﬁnal goods ﬁrms are in the South, they obtain the intermediate
goods only from the Southern ﬁrms.
I label the economy Regime NN when the ﬁnal goods ﬁrms are in the North and obtain
the production of intermediate goods from the Northern ﬁrms; Regime NS when the ﬁnal
goods ﬁrms are in the North and obtain the intermediate goods from the Southern ﬁrms;
and Regime SS when the ﬁnal goods ﬁrms are in the South and obtain the intermediate
goods from the Southern ﬁrms.

2.3

The R&D sector

The R&D activities of the present model follow Grossman and Helpman (1991, Ch. 3).
The ﬁnal goods producers in the North enter into the R&D race and ﬁnance the cost of
R&D by issuing equity in the stock market. The equity is bought by individuals who livein either country. The stock value of the ﬁnal goods producers at time t is equal to thediscounted sum of its proﬁt stream from time t. Suppose that the ﬁnal goods ﬁrms is ﬁrst
in one regime and they next move to another regime at time s and further to anotherregime at time s0 . Then, the stock value of the ﬁnal goods producers at time t is givenby:∫ s
vj,k =−ρ(τ −t)e−ρ(s−t)∫ s0πj,k dτ + et0+ e−ρ(s −t)

∫ s∞
s0

e−ρ(τ −s) πj 0 ,k0 dτ0e−ρ(τ −s ) πj 00 ,k00 dτ, j, j 0 , j 00 , k, k 0 , k 00 ∈ {N, S} , (19)9

where vj,k is the stock value. Diﬀerentiating (19) with respect to time t yields the following
no-arbitrage conditions:
ρvN,N =πN,N + v̇N,N ,(20)

ρvN,S =πN,S + v̇N,S ,(21)

ρvS,S =πS,S + v̇S,S .

(22)

The ﬁnal goods producers hire labor to develop blueprints. I assume that knowledge
spillovers in the R&D activities exist; that is, the more innovations that have been createdpreviously, the lower is the cost of innovating. I assume that LA units of labor for R&D
activities for a time interval dt produce dn = (LA n/a)dt new products. The cost of
the R&D activities is wN LA dt because the R&D sector is located only in the North.
Producing the blueprints creates a value for the ﬁnal goods producers of vj,k dn because
each blueprint has a market value of vj,k . I assume that there is free entry into the R&D
race. Therefore, the following free-entry condition must hold:vj,k ≤

2.4

awN
,
n

with equality whenever ṅ ≡dn
> 0, j, k ∈ {N, S} .
dt

(23)

The labor market

Labor market equilibrium requires that labor supply equals to labor demand. Under
Regime NN, the demand for labor in the North comes from the intermediate sector, theﬁnal goods sector, and the R&D sector. In the South, the demand for labor comes only
from the homogeneous good sector. Therefore, the labor market equilibrium conditions
are as follows:F
LN = LA + LM
N,N + LN,N ,LS = yN,N ,where yj,k is the labor demand for production of the homogeneous good in the South,FLMj,k is the labor demand for production of the intermediate goods in the North, and Lj,kis the labor demand for the production of the ﬁnal goods when the ﬁnal goods ﬁrms
producing in j country obtain the intermediate goods from k country. Under Regime NS,
the demand for labor in the North comes from the R&D sector and the ﬁnal goods sector.
In the South, the demand for labor comes from the homogeneous good sector and theintermediate goods sector. Therefore, the labor market equilibrium conditions are as

10

follows:
LN = LA + LFN,S ,LS = yN,S + LM
N,S .
Finally, under Regime SS, the demand for labor in the North comes from the R&D sector
and the ﬁnal goods sector. In the South, the demand for labor comes from the homogeneous good sector, the intermediate goods sector and the ﬁnal goods sector. Therefore,
the labor market equilibrium conditions are as follows:
LN = LA + LFS,S ,F,SLS = yS,S + LM
S,S + LS,S ,where LFS,S is the labor demand for the production of the ﬁnal goods in the South.
In the Appendix A.2, I derive the labor demand in each sector and under each regime.
Under Regime NN, I can rewrite the labor market clearing conditions in both countries
as follows:−µµ(1−α)1aṅ+ n α(1−µ) α 1−µ wN1−µ ,
n

North

LN =

South

LS = E − n α(1−µ) α 1−µ wN1−µ .

µ(1−α)

µ

(24)

−µ

(25)

Using (9), (A.11), (A.12), and (A.13), I can rewrite the labor market clearing conditionsof Regime NS as follows:
βµ−1µ(1−α)−βµ1aṅ+ n α(1−µ) (1 − β)α 1−µ wN1−µ τ 1−µ ,
n

North

LN =

South

LS = E + n α(1−µ) (αβ − τ ) α 1−µ wN 1−µ τµ(1−α)µ−µ(1−β)µ(1−β)−1
1−µ

(26)
.

(27)

Finally, using (11), (A.14), (A.15), (A.16), and (A.17), I can rewrite the labor market
clearing conditions of Regime SS as follows:−µ( ) 1−µµ(1−α)−µγ(1−β)AaṅN+ n α(1−µ) αγ(1 − β)
North
L =
(²wN ) 1−µ −1 ,nα−µ( ) 1−µµ(1−α)−µγ(1−β)A(²wN ) 1−µ [1 − αβ − α(1 − γ)(1 − β)] .
South
LS = E − n α(1−µ)
α

3

(28)
(29)

The Equilibrium PathIn this section, I examine the dynamics of the economy. First, I consider the dynamic
behaviors of Regime NN, Regime NS, and Regime SS separately. Finally, I integrate these
dynamic behaviors.
11

3.1

Dynamic behavior under Regime SS

In Regime SS, the equilibrium conditions are (7), the no-arbitrage condition, (22), thefree-entry condition, (23), and the labor market clearing condition, (28). From the labormarket clearing condition, I can derive the diﬀerential equation for the number of ﬁrms,
n, as follows:−µ[( ) 1−µ]µ(1−α)−µγ(1−β)n NA−1α(1−µ)
L − αγ(1 − β)nṅ =(²wN ) 1−µ
.
a
α

(30)

In the Appendix A.3, I derive the diﬀerential equation for the wage rate in the North, wN
as follows:−µ()( ) 1−µ()µ(1−α)−µγ(1−β)LN
1 α(1−µ)
A
αγ(1
−
β)
w˙N = ρ +wN − n(²wN ) 1−µ
1−α+
.
a
a
α
²

(31)

These two equations, (30) and (31), constitute the dynamic system of Regime SS. Then,the equation for ṅ = 0 locus is represented by:() 1−µ ( ) −µ
A Φ µ(1−α)αγ(1 − β) Φ
−1n αΦ ,
wN = ²
LN
α

(32)

N
where Φ = 1 − µ + µγ(1 − β). The ṅ = 0 locus is concave and limn→∞ ∂w= 0. On the∂nother hand, the equation for w˙N = 0 locus is given by:) 1−µ ( ) −µ(
A Φ µ(1−α)²(1 − α) + αγ(1 − β) Φ
−1n αΦ .
wN = ²aρ + LNα(33)

The w˙N = 0 locus has a similar shape to ṅ = 0 locus. Thus, the origin is the onlyintersection point of the two curves, ṅ = 0 and w˙N = 0. Figure 1 depicts the phaseαaρ
diagram for this system on the (n, wN ) plane. When 1−α< LN holds, w˙N = 0 locus is
αaρabove ṅ = 0 locus for any wN > 0. Hereafter, I focus on 1−α< LN .5 On the saddle path,the number of ﬁrms and the wage rate in the North increases perpetually. In this model,
I assume that the knowledge spillovers in the R&D activities exist. Therefore, an increasein the number of the ﬁrms raises the demand of labor in the North. Then, the wage ratein the North increases steadily.
µ(1−α)−ΦI show the saddle path stability in Regime SS. Let us deﬁne zSS ≡ n α(1−µ) (²wN ) 1−µ−µ( A ) 1−µ. Then, diﬀerentiating zSS with respect to time t and using (30) and (31) yields
αthe following diﬀerential equation for zSS :
)
( )
(
zSS
˙
µ(1 − α) ṅ
Φ
w˙N
=
−
zSS
α(1 − µ) n
1 − µ wN[]
µ − α − αµγ(1 − β) LNΦΓSS zSS=−ρ +,α(1 − µ)a
1−µa(1 − µ)5αaρEven when 1−α> LN , the following discussion can be applied.

12

wN

(I)
w˙N = 0(33)ṅ = 0(32)nFigure 1: Phase diagram under Regime SS

whereΓSS ≡ (1 − µ) [²(1 − α) + αγ(1 − β)] + µγ(1 − β) [(² − 1)(1 − α) + αγ(1 − β)] > 0.
˙
= 0 schedule is a straight line through the negative intercept with a
From ² > 1, zzSSSS

positive slope in Figure 2. There is a unique steady state at E. The steady state value isas follows:[α(µ + Φ) − µ] L
∗
zSS
=
αΓSS

N

+ aαρΦ

.

(34)∗
Because the steady state E is unstable and zSS is a jump variable, zSS jumps to zSS
atthe initial time. In the steady state, the relationship between the number of ﬁnal goods,
n, and the wage rate in the North, wN becomes as follows:−1

wN = ²( ) −µA Φ ∗ −1+µ µ(1−α)(zSS ) Φ n αΦ .
α(35)This equation represents the stable saddle path. Therefore, the dynamic equilibrium
follows the stable saddle path shown by the solid locus with arrow (I) in Figure 1.

3.2

Dynamic behavior under Regime NN

In Regime NN, the equilibrium conditions are (7), the no-arbitrage condition, (20), thefree-entry condition, (23), and the labor market clearing condition, (24). From the labormarket clearing condition, I can derive the diﬀerential equation for the number of ﬁnal
goods, n, as follows:][−1µ(1−α)
1n N
1−µα(1−µ)
1−µṅ =α wN .
L −n
a
13

(36)

zSS
˙
zSS
zSS
˙
=0
zSS

E
∗
zSS

zSS

Figure 2: Phase diagram about zSS

Using (7), (20), (23), and (36), I can obtain the diﬀerential equation for the wage rate in
the North, wN , in the same way as (31). Then, the diﬀerential equation for the wage ratein the North is

(

)
−µ
LN
1 µ(1−α) µ
wN − n α(1−µ) α 1−µ wN1−µ .
w˙N = ρ +(37)aaThese two equations, (36) and (37), constitute the dynamic system of Regime NN. The
equation for ṅ = 0 locus is given by the following:()1−µµ(1−α)
1
α
wN =
αn
.
LN

(38)

N
The ṅ = 0 locus is concave and limn→∞ ∂w= 0. On the other hand, the equation for∂nw˙N = 0 locus is given by:()−(1−µ) µ µ(1−α)
wN = aρ + LN
α n α .

(39)

The w˙N = 0 locus has a similar shape to ṅ = 0 locus. Thus, the origin is the onlyintersection point of the two curves ṅ = 0 and w˙N = 0. Figure 3 depicts the phaseαaρ
diagram for this system on the (n, wN ) plane when the inequality 1−α< LN holds.

Similarly to Regime SS, I can obtain the stable saddle path (See Appendix A.4). In
the steady state, the relationship between the number of ﬁnal goods, n, and the wage ratein the North, wN becomes as follows:( )−µ ()−1+µµ(1−α)
A(α − µ + αµ)LN + aαρ
wN =
n α .
α
α {1 − µ(1 − α)}This equation represents the stable saddle path. Therefore, the dynamics in Regime NN
follows the stable saddle path shown by the arrow (II) in Figure 3.

3.3

Dynamic behavior under Regime NS

In RegimeNS , the equilibrium conditions are (7), the no-arbitrage condition, (21), thefree-entry condition, (23), and the labor market clearing condition, (26). In the same way
14

wN

(II)
w˙N = 0(39)ṅ = 0(38)nFigure 3: Phase diagram under Regime NN

as in the former section, I can derive ṅ = 0 locus and w˙N = 0 locus. From the labormarket clearing condition, I can derive the diﬀerential equation for the number of ﬁrms,
n, as follows:][−(1−βµ)µ(1−α)
−βµ
1
n N
1−µ
α(1−µ)wN
τ 1−µ .
ṅ =L − α 1−µ (1 − β)na(40)

In the same way as (31), I can obtain the diﬀerential equation for the wage rate in the
North, wN , as follows:
(
w˙N =

LN
ρ+
a

)
wN −

−µ(1−β)µ(1−α)µ−βµ1 − αβ α(1−µ)
nα 1−µ wN 1−µ τ 1−µ .
a(41)These two equations, (40) and (41), constitute the dynamic system of Regime NS. Then,the equation for ṅ = 0 locus is represented by:
(
wN =

1−β
LN

1−µ
) 1−βµ−βµ

1µ(1−α)

α 1−βµ τ 1−βµ n α(1−βµ) .

(42)

NThe ṅ = 0 locus is concave and limn→∞ ∂w= 0. On the other hand, the equation for∂nw˙N = 0 locus is given by:(wN =1 − αβaρ + LN1−µ) 1−βµµ−βµµ(1−α)

α 1−βµ τ 1−βµ n α(1−βµ) .

(43)

The w˙N = 0 locus has a similar shape to ṅ = 0 locus. Thus, the origin is the onlyintersection point of the two curves, ṅ = 0 and w˙N = 0. Figure 4 depicts the phaseαaρ
diagram for this system on the (n, wN ) plane. When 1−α< LN holds, w˙N = 0 locus isabove ṅ = 0 locus for any wN > 0.

15

wN
(III)
w˙N = 0(43)ṅ = 0(42)nFigure 4: Phase diagram under Regime NS

Similarly to the previous regimes, I can obtain the stable saddle path (See Appendix
A.4). In the steady state, the relationship between the number of ﬁnal goods, n, and thewage rate in the North, wN becomes as follows (See appendix A.4):[](α − µ + αµ(1 − β))LN + (1 − βµ)aαρ
∗zN S =.
αΓN SThis equation represents the stable saddle path. Therefore, the dynamics in Regime NS
follows the stable saddle path shown by the arrow (III) in Figure 4.

3.4

Boundaries between the three regimes

The ﬁnal goods ﬁrms have to decide where they produce the ﬁnal goods and from where
they purchase the intermediate goods. Then, the ﬁnal goods ﬁrms compare levels of
the proﬁt in each regime and choose a regime where they can earn the largest proﬁt. I
investigate the condition under which the proﬁt in Regime NN become the same as the
proﬁts in Regime NS; that is πN,N = πN,S . From (15) and (16), the boundary condition
between Regime NN and Regime NS is given by:
wN = τ.

(44)

Next, I show the condition under which the proﬁt in Regime NN becomes the same as the
proﬁts in Regime SS; that is πN,N = πS,S . From (15) and (18), the boundary condition
between Regime NN and Regime SS becomes as follows:11−Ψ

wN = A Ψ ² Ψ ,

16

(45)

where Ψ = 1 − γ(1 − β) and 0 < Ψ < 1. Finally, from (16) and (18), I investigate theboundary condition between Regime NS and Regime SS as follows:
1

−β

γ

wN = A Ψ−β τ Ψ−β ² 1−γ .

(46)

From the three boundary conditions, (44), (45), and (46), I can obtain the following
proposition (see Appendix A.5 for the proof).
Proposition 1. When transportation costs are small and management costs are large,11−Ψthat is, τ < A Ψ ² Ψ , Regime of the economy is chosen from three regimes, Regime NN,
Regime NS, and Regime SS. On the other hand, when transportation costs are large and11−Ψmanagement costs are small, that is, τ ≥ A Ψ ² Ψ , Regime of the economy is chosen fromtwo regimes, Regime NN and Regime SS.
11−Ψ

Thus, when the inequality, τ < A Ψ ² Ψ , holds, the economy has three regimes, that is,
Regime NN, Regime NS, and Regime SS. When the wage rate in the North is low, Figure
5 shows that the economy is in Regime NN. As the wage rate in the North increases,
the economy turns into Regime NS. Finally, a further increase in the wage rate in the
North get the economy turn into Regime SS. When trade costs are suﬃciently small,obtaining the intermediate goods from southern ﬁrms becomes easier for the ﬁnal goods
ﬁrms. When management costs and in the parameter of cost function, A are suﬃciently
large, the cost of FDI becomes large compared to purchasing the intermediate goods
from Southern ﬁrms. Then, the ﬁnal goods ﬁrms obtain the intermediate goods from theSouthern ﬁrms and there exists Regime NS . On the other hands, when the inequality,11−Ψτ ≥ A Ψ ² Ψ , holds, Figure 6 shows that the economy has only Regime NN and Regime SS.
When the wage rate in the North is high, the economy is in Regime SS. Intuitively, whentransportation costs are suﬃciently high, the cost of obtaining the intermediate goods
from abroad is high. Therefore, the ﬁnal goods ﬁrms do not purchase the intermediate
goods from abroad. When management costs is suﬃciently small, the cost of producing
the ﬁnal goods in the South becomes lower and the ﬁnal goods ﬁrms does not purchasethe intermediate goods from abroad.
11−ΨLemma 2. When τ < A Ψ ² Ψ

holds, a decrease in the transportation costs and anincrease in the management costs enlarge the area of Regime NS. On the other hand, when11−Ψτ ≥ A Ψ ² Ψ holds, an increase in the management costs enlarges the area of Regime NN
and an increase in trade costs does not aﬀect the sizes of those regimes.
Proof :

See the Appendix A.6.
17

wN

wN
Regime SS

(44)

(46)
Regime SS
Regime NS

(45)

(45)
(44)

(46)

Regime NN

Regime NS

1

1

n

n
1

1

1

1−Ψ

Figure 6: τ ≥ A Ψ ² Ψ

1−Ψ

Figure 5: τ < A Ψ ² Ψ
1−Ψ

When τ < A Ψ ² Ψ , Regime of the economy is chosen form three regimes, that is
NN Regime, NS Regime, and SS Regime from proposition 1. A decrease in trade costsmeans that the cost of purchasing the intermediate goods from abroad decreases and ﬁrms
purchase the intermediate goods from the foreign countries easier. An increase in the
management costs means that the cost of producing in the foreign countries is increasing.
Then, the producing the ﬁnal goods in the North to purchase the intermediate goods fromabroad is relatively more proﬁtable than producing in the South. Therefore, the area of11−ΨRegime NS is enlarged. When τ ≥ A Ψ ² Ψ , Regime of the economy is chosen from two
regimes, that is NN Regime, and SS Regime from proposition 1. Then, if the management
costs increase, the cost of producing in the South increases and ﬁnal goods ﬁrms prefer
Regime NN to Regime SS. On the other hand, an increase in the transportation costs
does not aﬀect the behavior of ﬁnal goods ﬁrms. Because in Regime NN and Regime SS,
ﬁnal goods ﬁrms do not freight the intermediate goods to cross the border, the ﬁnal goods
ﬁrms do not incur trade costs.

3.5

The equilibrium path from Regime NN to another regime

In the previous subsections, I explored the dynamic behaviors in Regime NN, Regime NS,
and Regime SS, and showed the boundary conditions among Regimes. Now, I integrate11−Ψ

the three phase diagrams of these regimes into one. When the inequality, τ < A Ψ ² Ψ ,holds, the economy will transit through three regimes, that is, Regime NN, Regime NS,
and Regime SS. I can depict the phase diagram for the system on the (n, wN ) plane.
Figures 1, 3, 4, and 5 are integrated into Figure 7. In Figure 7, the arrow ABCD shows
the traditional dynamics. These results are stated as the following proposition.

18

1

1−Ψ

Proposition 2. Suppose that the economy is initially in Regime NN. When τ < A Ψ ² Ψ ,αaρand 1−α< LN holds, there exists an equilibrium path along which the economy enters into

Regime SS, passing through Regime NS.
wN

D
(33)

Regime SS

C
(45)

(42)
(38) Regime NS

B

(43)

Regime NN

A
1

n

n()
1

1−Ψ

αaρ
< LN
Figure 7: Phase diagram when τ < A Ψ ² Ψ and 1−αSuppose that the initial number of ﬁrms is suﬃciently small. In this economy, theﬁrms have rational expectation. The ﬁrms expect that the economy become Regime SS
and the economy follows the stable saddle path in Regime SS in the future. To follow thestable saddle path in Regime SS, the economy follows the divergent path in Regime NS toreach the intersection point C of the stable saddle path with the boundary line between
Regime NS and Regime SS. Then, to follow the divergent path in Regime NS, the economy
follows the divergent path in Regime NN to reach the intersection point B of the divergent
path in Regime NS with the boundary line between Regime NN and Regime NS.
11−Ψ

In order to get some clear results, I show the equilibrium path when τ < A Ψ ² Ψnumerically. In this example, the labor supply in the North is LN = 2.5. The subjective
discount rate is ρ = 0.05 and the parameters of utility are α = 0.5 and µ = 0.49,
respectively. The parameters of production function are A = 1.1, β = 0.7, and γ = 0.4
respectively. The parameter of R&D sector is a = 0.2. I assume that the parameters of11−Ψ

management costs and trade costs are ² = 4 and τ = 1.2 to hold τ < A Ψ ² Ψ . Figure 8
represents the number of ﬁnal goods ﬁrms in the equilibrium path and Figure 9 represents
the wage rate in the North in the equilibrium path. The points A, B, and C in Figures 8
and 9 correspond to the points in Figure 7. The horizontal lines of both Figures represent
19time. In the Regime NN, both the number of ﬁnal goods ﬁrms and the wage rate in the
North raise moderately because at initial time, the number of ﬁnal good ﬁrms is small
and the innovative activities are not vigorous in this Regime. As the economy reaches tothe point B, the ﬁnal good ﬁrms obtain their intermediate goods from South. The labor
supply to R&D sector increases and innovative activities become vigorous. Therefore,both number of ﬁnal goods ﬁrms and the wage rate in the North increases sharply. In
Regime SS, the economy follows the stable path and both the number of ﬁnal good ﬁrms
and the wage rate in the North increases exponentially.
WN

n

3

6
2.5

C

5

C

2

4

A

1.5
3
2

A

B

n

B

w_N

1
0.5

1

Regime NN Regime NS

0

t

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36

0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
0.28
0.3
0.32
0.34
0.36

0

Regime NN
Regime NS

Regime SStRegime SS

Figure 8: The numerical example of the num- Figure 9: The numerical example of the wage
1 1−Ψ
1 1−Ψ
ber of ﬁnal goods ﬁrms when τ < A Ψ ² Ψ
rate in the North when τ < A Ψ ² Ψ11−ΨOn the other hand, when the inequality, τ ≥ A Ψ ² Ψ , the economy will transit throughtwo regimes, Regime NN and Regime SS. I can depict the phase diagram similarly and
Figures 1, 3, 4, and 6 are integrated into Figure 10. In Figure 10, the arrow EF G shows
the traditional dynamics. These results are stated as the following proposition.
1

1−Ψ

αaρ
Proposition 3. When τ ≥ A Ψ ² Ψ , and 1−α< LN holds, there exists an equilibrium

path along which the economy evolves from Regime NN to Regime SS.
Suppose that the initially number of ﬁrms is suﬃcient small. The ﬁnal goods ﬁrms
expect that the economy becomes Regime SS and the economy follows the stable saddle
path in Regime SS in the future. To follow the stable saddle path in Regime SS, theeconomy follows the divergent path in Regime NN to reach the intersection point F of
the stable saddle path with the boundary line between Regime NN and Regime SS.
11−Ψ

I show the numerical example when τ ≥ A Ψ ² Ψ . In this case, I assume that the trade
costs are larger than the former case, that is τ = 2. Figure 11 represents the number
20

wN

G
(33) Regime SS

F

(38)

(44)

Regime NN

E
1

n

n(0)
1

1−Ψ

αaρ
Figure 10: Phase diagram when τ ≥ A Ψ ² Ψ and 1−α< LN

of ﬁnal goods ﬁrms in the equilibrium path and Figure 12 represents the wage rate in
the North in the equilibrium path. The points E and F in Figures 11 and 12 correspondto the points in Figure 10. The horizontal lines of both Figures represent time. In theRegime NN, both the number of ﬁnal goods ﬁrms and the wage rate in the North risemoderately. As the economy reaches to the point F , the Regime changes from Regime NN
to Regime SS. The labor supply to R&D sector increases sharply and innovative activities
become vigorous. Then, the economy follows the stable path and both the number of ﬁnal
good ﬁrms and the wage rate in the North increases exponentially.
WN

n
0.14

8

0.12

7

0.1

6
5

0.08

n

0.06

4

w_N
3

0.04

F

1

t
Regime SS

Regime NN

0.3

0.28

0.26

0.24

0.2

0.22

0.18

0.16

0.14

0.1

0.12

0.08

0.06

0
0

0

Regime NN

F

E

2

0.04

E

0.02

0.02

t

Regime SS

Figure 11: The numerical example of the Figure 12: The numerical example of the
1 1−Ψ
number of ﬁnal goods ﬁrms when τ > wage rate in the North when τ > A Ψ ² Ψ
1 1−Ψ
AΨ ² Ψ

21

4

Conclusion

This paper constructs a North-South endogenous growth model to investigate how the
production location of ﬁnal goods ﬁrms change. The ﬁnal goods ﬁrms have four strategies,
based on the regions in which they produce the ﬁnal goods and from which obtain theintermediate goods. This paper shows why the location choice of the ﬁnal goods ﬁrms
change as the economy develops and why the ﬁnal goods ﬁrms did not adopt a strategy ofobtaining intermediate goods from abroad until a few decades ago. Suppose that, initially,
the ﬁnal goods ﬁrms obtain the intermediate goods from Northern ﬁrms and produce theﬁnal goods in the North. When trade costs are suﬃciently low and management costs
are suﬃciently high, as the economy develops, ﬁnal goods ﬁrms produce the ﬁnal goodsin the North and purchase the intermediate goods from the Southern ﬁrms. Then, as
the economy develops further, the ﬁnal goods ﬁrms produce the ﬁnal goods in the Southand obtain the intermediate goods from the Southern ﬁrms. However, when trade costs
are suﬃciently high, the ﬁnal goods ﬁrms do not choose the strategy that they produce
the ﬁnal goods in the North and obtain the intermediate goods from Southern ﬁrms like
Japanese cotton spinning companies before World War II. Then, as the economy develops,
the ﬁnal goods ﬁrms produce the ﬁnal goods in the developing countries obtaining theintermediate goods from ﬁrms in the developing countries.
In this paper, we assume that the R&D sector is located only in North. However, inthese days, innovative activities also are conducted in developing countries. In addition,
multinational ﬁrms in developed countries outsource R&D to the developing countries. I
can extend this model and innovative activities can be operated in both two countries ormultinational ﬁrms outsource innovation activities to the developing countries. This isfuture research problems.

A

Appendix

A.1

Derivation of the proﬁt functionsFirst, I focus on the case where the ﬁnal goods ﬁrms are in the North. Then, I consider
the proﬁt-maximization problem of the ﬁnal goods producers in the North. Thus, theﬁnal goods producer i maximizes proﬁts as follows:µ−α
1−β m βπN,k (i) = XN,k
xN,k (i)α − wN
(pN,k ) xN,k (i), k ∈ {N, S} , i ∈ [0, n],

22

(A.1)

where πN,k is the operating proﬁt of the ﬁnal goods ﬁrms in the North when they obtain
the intermediate goods from country k. The proﬁt-maximizing output is given by:
α−µ

1

1−β

β

α−1
α−1 , k ∈ {N, S} .
xN,k (i) = α 1−α XN,kwNα−1 (pm
N,k )

(A.2)

By substituting (A.2) into (3), I obtain:1−α

1−(1−β)−β1−µ , k ∈ {N, S} .
XN,k = n α(1−µ) α 1−µ wN 1−µ (pm
N,k )

(A.3)

Using (A.2) and (A.3), I can obtain the following output and the proﬁt functions when
the ﬁnal goods ﬁrms produce the ﬁnal goods in the North:µ−α

1−(1−β)−β1−µ ,xN,k (i) = n α(1−µ) α 1−µ wN 1−µ (pmN,k )

πN,k = nµ−αα(1−µ)αµ
1−µ−µ(1−β)
1−µ

wN

−βµ

1−µ (1 − α), k ∈ {N, S} , i ∈ [0, n].
(pm
N,k )

(A.4)
(A.5)

The proﬁts of the ﬁnal goods producers decrease with the wage rate in the North wN andthe number of the ﬁnal goods ﬁrms n.
Next, I turn to the case where the ﬁnal goods ﬁrms are in the South and derive
the proﬁt function of the ﬁnal goods ﬁrms similarly to the above case. The ﬁnal goods
producer i maximizes the following proﬁts:µ−αβπS,k (i) = XS,k
xS,k (i)α − A(²wN )γ(1−β) (pmS,k ) xS,k , k ∈ {N, S} , i ∈ [0, n].

(A.6)

The proﬁt-maximizing output is given by:
1( ) α−1α−µγ(1−β)βA
α−1
α−1 , k ∈ {N, S} .
(²wN ) α−1 (pmxS,k (i) =
XS,k
S,k )
α

(A.7)

By substituting (A.7) into (3), I obtain:
XS,k = n1−αα(1−µ)−1( ) 1−µ−γ(1−β)−βA
1−µ , k ∈ {N, S} .
(²wN ) 1−µ (pm
S,k )
α

(A.8)

Using (A.7) and (A.8), I can obtain the following output and the proﬁt functions when
the ﬁnal goods ﬁrms are in the North:
xS,k (i) = nµ−αα(1−µ)−1( ) 1−µ−γ(1−β)−βA
1−µ ,(²wN ) 1−µ (pm
S,k )
α

(A.9)

−µ
( ) 1−µ−µγ(1−β)−βµA
1−µ (1 − α), k ∈ {N, S} , i ∈ [0, n].
πS,k = n(²wN ) 1−µ (pm(A.10)S,k )α
The proﬁts of the ﬁnal goods producers decrease with the wage rate in the North wN andµ−αα(1−µ)

the number of the ﬁnal goods ﬁrms n.
23

A.2

Derivation of the labor market equilibrium condition

To derive the labor demand of the ﬁnal goods sector and the intermediate goods sector
when the ﬁnal goods ﬁrms are in the North, I apply Shepard’s lemma on (12) and use(A.4), which yields:∫ n
LFN,k =0∂cN (wN , pm
N,k )xN,k (i)di∂wNµ(1−α)

1

βµ−1

−βµ

1−µ ,
= n α(1−µ) (1 − β)α 1−µ wN1−µ (pm
N,k )

∫ n
LM
N,k =
0

(A.11)∂cN (wN , pm
N,k )xN,k (i)dim∂pN,kµ(1−α)

1−µ(1−β)= n α(1−µ) βα 1−µ wN 1−µ (pmN,k )µ(1−β)−11−µ, k ∈ {N, S} .

(A.12)

Then, the production of one unit of the homogeneous good requires one unit of labor.
Thus, production of the homogeneous good equals the labor demand. I can write theproduction of the homogeneous good by using the budget constraint, (4), (A.3), and(A.4).
[∫ n] µ(1−α)α(1−µ)
yN,k = E −P (i)x(i)di[∫0 n]α(µ−α)α−1
α=E−x(i) XN,k di0=E−nµ(1−α)
α(1−µ)µ−µ(1−β)−βµ1−µ , k ∈ {N, S} .
α 1−µ wN 1−µ (pmN,k )(A.13)On the other hand, to derive the labor demand of the ﬁnal goods sector and theintermediate goods sector when the ﬁnal goods ﬁrms are in the South, I apply Shepard’s
lemma on (13) and use (A.9), which yields∫ n∂cS (wN , pm
S,k )xS,k (i)di∂(²wN )0−µ( ) 1−µµ(1−α)−µγ(1−β)
−βµ
A
1−µ ,
= n α(1−µ) αγ(1 − β)(²wN ) 1−µ −1 (pmS,k )α

LFS,k =

(A.14)∫ n∂cS (wN , pm
S,k )xS,k (i)di∂(wS )0−µ( ) 1−µµ(1−α)−µγ(1−β)
−βµ
A
1−µ ,
= n α(1−µ) α(1 − γ)(1 − β)(²wN ) 1−µ (pm
S,k )
α

LF,S
S,k =

24

(A.15)

∫ n∂cS (wN , pm
S,k )xS,k (i)di
m
∂pS,k
0
−µ
( ) 1−µµ(1−α)−µ(1−β)
−βµ
A
−1
1−µ
= n α(1−µ) αβ(²wN ) 1−µ (pm, k ∈ {N, S} .
S,k )
α

LM
S,k =

(A.16)

Then, I can obtain the production of the homogeneous good by using the budget constraint, (4), (A.8), and (A.9).
[∫ nyS,k = E −µ(1−α)] α(1−µ)P (i)x(i)di
0

=E−n

A.3

µ(1−α)
α(1−µ)−µ( ) 1−µ−µ(1−β)−βµA
1−µ , k ∈ {N, S} .
(²wN ) 1−µ (pm)
S,kα(A.17)

Derivation of the diﬀerential equation (31)

I derive the diﬀerential equation for the wage rate in the North. I diﬀerentiate the Free
entry condition (23) by t as follows:vSS˙ =aw˙N n − awN ṅ
.
n2(A.18)

Substituting (7) and (A.18) into (22), I obtain:ρaw˙NawN ṅ
awN= πSS +
−
.
n
n
n n

(A.19)

Then, I substitute (18) and (30) into (A.19) and I can obtain the diﬀerential equation for
the wage rate in the North as follows:µ(1−α)−µγ(1−β)−µLN
1 − α A 1−µ)wN −( ) (²wN ) 1−µ n α(1−µ)
aaαµ(1−α)−µγ(1−β)−µαγ(1 − β) A 1−µ−( ) (²wN ) 1−µ n α(1−µ)
a²αµ(1−α)−µγ(1−β)LN
1 A −µαγ(1 − β)= (ρ +)wN − ( ) 1−µ (²wN ) 1−µ n α(1−µ) (1 − α +
).
a
a α
²

w˙N = (ρ +A.4

(A.20)

Proof of the stable saddle paths in Regime NN andRegime NS

First, I show the proof of the stable saddle path in Regime NN. Suppose that I deﬁne−1 ( ) −µµ(1−α)zN N ≡ n α(1−µ) wN1−µ Aα 1−µ . Then, diﬀerentiating zN N with respect to time t and using(36) and (37) yields the following diﬀerential equation for zN N :
)
( )
(
zN˙ N
µ(1 − α) ṅ
1
w˙N
=
−
zN N
α(1 − µ) n
1 − µ wN[() N]
ρ(µ − α) − αµ L
1 − µ(1 − α) ( zN N )−.
=+α(1 − µ)
a
1−µ
1−µ
a
25

From µ < α, the ﬁrst parentheses is negative. Then, zzNN˙ NN = 0 schedule is a straight lineαthrough the negative intercept with slope 1−µ> 0 in Figure 13. There is a unique steady0state at E . The steady state value is as follows:
∗
zN
N =

(α − µ + αµ)LN + aαρ.
α {1 − µ(1 − α)}(A.21)

0∗
Because the steady state E is unstable and zN N is jump variable, zN N jumps to zN
N at

the initial time. In the steady state, the relationship between the number of ﬁnal goods,
n, and the wage rate in the North, wN , becomes as follows:( )−µ ()−1+µµ(1−α)
A(α − µ + αµ)LN + aαρ
wN =
n α .
α
α {1 − µ(1 − α)}(A.22)

Therefore, the dynamic equilibrium follows the stable saddle path shown by the solid
locus with arrow (II) in Figure 3.
zN˙ N
zN N
zN˙ N
zN N

=0

0

E
∗
zN
N

zN N

Figure 13: Phase diagram about zN N

Next, I show the saddle path stability in Regime NS in the same way. Suppose I deﬁne−1+βµ ( ) −µµ(1−α)−βµzN S ≡ n α(1−µ) wN1−µ Aα 1−µ τ 1−µ . Then, diﬀerentiating zN S with respect to time t andusing (40) and (41) yields the following diﬀerential equation for zN S :
)
( )
(
zN˙ S
µ(1 − α) ṅ
1 − βµ w˙N=−
zN Sα(1 − µ) n
1 − µ wN[()](µ − α) − αµ(1 − β) LN(1 − βµ)ρ
ΓN S zN S=−,+α(1 − µ)a
1−µa(1 − µ)where
ΓN S ≡ (1 − βµ)(1 − αβ) − µ(1 − α)(1 − β)> (1 − α)(1 − βµ − µ(1 − β)) = (1 − α)(1 − µ) > 0.
From µ < α and 1 − α < 1 − αβ, zzNN˙ SS = 0 schedule is a straight line through the negative00

intercept with a positive slope in Figure 14. There is a unique steady state at E . The
26

steady state value is as follows:[](α − µ + αµ(1 − β))LN + (1 − βµ)aαρ
∗zN S =.
αΓN S(A.23)

00∗
Because the steady state E is unstable and zN S is jump variable, zN S jumps to zN
S atinitial time. In the steady state, the relationship between the number of ﬁnal goods, n,
and the wage rate in the North, wN , becomes as follows:−µ( ) 1−βµµ(1−α)βµ−1+µA∗
1−βµ τ 1−βµ n α(1−βµ) .
wN S =(zN)
S
α(A.24)

Therefore, the dynamic equilibrium follows the stable saddle path shown by the arrow(III) in Figure 4.
zN˙ S
zN S
zN˙ S
=0
zN S

00

E
∗
zN
S

zN S

Figure 14: Phase diagram about zN S

A.5

Proof of proposition 1

Suppose that the boundary condition between Regime NN and Regime SS is bigger thanthe condition between Regime NN and Regime NS in terms of the wage rate in the North,11−Ψ

wN . Using (44) and (45), this inequality holds: τ < A Ψ ² Ψ . Then, I compare theboundary condition between Regime NN and Regime SS, (45), to the condition between
Regime NS and Regime SS, (45), as follows:
1

−β

γ

1

1−Ψ

A Ψ−β τ Ψ−β ² 1−γ − A Ψ ² Ψ−β[ 1 1−Ψ ] Ψ−β
γ
1
1 1−Ψ
Ψ−β
1−γ
Ψ
Ψ
>A
A ²
− AΨ ² Ψ²][−Ψ{ 1 1−Ψ } Ψ−β
γ
1
1 1−Ψ
=A Ψ ² Ψ A Ψ−β ² 1−γ A Ψ ² Ψ
−1
=0,
1

1−Ψ

where I derive the above inequality to use τ < A Ψ ² Ψ . If this inequality holds, Regime ofthe economy is chosen from three regimes, that is Regime NN, Regime NS, and Regime SS.
27

On the other hand, if the inverse inequality holds, Regime of economy is chosen from two
regimes, that is, Regime NN and Regime SS.

A.6

Proof of Lemma 2
11−ΨWhen τ < A Ψ ² Ψ holds, the boundary line is (44) and (46). Diﬀerentiating (46) withrespect to τ and ² yields
)
−β
γ
γ
−Ψ
1
1
∂ ( Ψ−ββ
τ Ψ−β ² 1−γ =AA Ψ−β τ Ψ−β ² 1−γ < 0,
∂τ
Ψ−β
(
)
−β
γ
−β
γ
1
1
∂
γ
A Ψ−β τ Ψ−β ² 1−γ =A Ψ−β τ Ψ−β ² 1−γ −1 > 0.
∂²
1−γ
1

(A.25)(A.26)

1−ΨWhen τ ≥ A Ψ ² Ψ holds, the boundary line is (45). Diﬀerentiating (45) with respect
to ² yields∂ ( 1 1−Ψ ) 1 − Ψ 1 1−Ψ −1
AΨ ² Ψ =AΨ ² Ψ
> 0.
∂²
Ψ(A.27)

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