GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.24

Dynamic Analysis of Outsourcing

Tadashi Morita

December 2008

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, JapanDynamic Analysis of Outsourcing ∗Tadashi Morita †December 22, 2008

Abstract
This paper constructs a North-South endogenous growth model where
final good producers in the North determine whether they outsource theproduction of intermediate goods to the South or not. When the final good
producers outsource the production of intermediate goods to the Northernfirms, the price of intermediate goods is high, whereas the cost of outsourcing
is low. On the other hand, when they outsource to the Southern firms, theprice of intermediate goods is low, whereas the cost of outsourcing is high.
Using this model, this paper shows that, as the economy develops, the wage
inequality between the North and the South widens and that the outsourcing
location for the Northern final good producers switches from the North to
the South.
Keywords: Outsourcing, North-South, R&D, Economic Growth
JEL classification: F43, O31∗

I would like to express sincere gratitude to Koichi Futagami for his helpful comments. I also
acknowledge the helpful comments made by Cristina Echevarria, Takeo Hori, Kazuo Mino, Ryoji
Ohdoi, Yoshiyasu Ono, Kazuhiro Yamamoto, and seminar participants at Kobe University andOsaka University. Any remaining errors are mine.
†
Address: Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka,
Osaka, 560-0043, Japan. Email address: gge013mt@mail2.econ.osaka-u.ac.jp1

1

Introduction

When firms produce final goods, they use many varieties of intermediate goods.
The firms can buy the intermediate goods either from their home country or fromforeign countries. Recently, many firms have outsourced the production of neededintermediate goods: they buy them from all over the world and use them to produce
the final goods. For example, according to the 1998 World Trade Organization
annual report, in the production of an ”American” car, 30 percent of the car’s
value originates in Korea, 17.5 percent in Japan, 7.5 percent in Germany, 4 percent
in Taiwan and Singapore, 2.5 percent in the United Kingdom, and 1.5 percent in
Ireland and Barbados. Thus, only 37 percent of the production value is generated
in the United States. Thus, there are rarely all-U.S. products in the U.S..
Feenstra and Hanson (1996) found that the import of intermediate goods rapidly
increased during the period 1972 to 1990. Since there are no figures directly representing outsourcing, they measured it by calculating the share of the imported
intermediate goods within the total purchase of non-energy materials.1 The increase in international trade of intermediate goods could be because of the decisionof final good firms in the developed countries to switch from intermediate goods
suppliers in the home country to those in a country where the wage rate is lower,
in order to reduce costs.2
To investigate the outsourcing issues, researchers have used static models. Oneof these models is developed by Antràs and Helpman (2004). It is a two-country
Ricardian model of international trade. In their model, final good producers withdifferent productivity levels decide to whether to integrate the suppliers or to outsource the production of intermediate goods. Furthermore, they have to choose acountry to which to outsource the production of intermediate goods. Antràs andHelpman (2004) show that the final good producers’ strategy crucially depends on
1

Many researchers also analyze outsourcing empirically, for example, Campa and Goldberg(1997), Hummels, Ishii, and Yi (2001), Yeats (2001), and Hanson, Mataloni and Slaughter (2001,
2005).
2
When Canon Inc-. produced scanners in 1997, they outsourced the production of intermediate
goods to domestic firms. As Taiwanese companies entered the scanner market and the pricecompetition became severe, Canon was concerned about the high wage rate in Japan. As a result,Canon outsourced the production of intermediate goods to Taiwanese firms whose wage rates
were lower. Recently, because the wage rate in Taiwan became higher, Canon outsourced the
production of intermediate goods to China, where the wage rate is much lower.

2their productivity. Although this study has suggested some important results about
the final good firms’ strategies, it only analyzes the static equilibria.
In contrast to this static analysis, Naghavi and Ottaviano (2008) present a
North-South endogenous growth model with offshoring in order to investigate how
many final good producers outsource the production of intermediate goods to theSouth. They analyze the two types of equilibrium: first, all of the producers outsource the production of intermediate goods; second, only some of the producers
outsource the production of intermediate goods. Gao (2007) also proposes an endogenous growth model and analyzes the relationship between trade cost and theplace where the intermediate goods are produced. He shows that, as the trade cost
falls, the number of intermediate goods produced in the South increases. Althoughthese studies are based on dynamic models, they focus only on the steady state.
In this paper, we construct an endogenous growth model of the variety-expansion
type to investigate the dynamic choice problem of final good producers: whenshould they outsource the production of their inputs to the South? We exploretransition paths and the point at which final good producers outsource the production of intermediate goods from the domestic suppliers to the foreign suppliers. On
the one hand, when the final good producers outsource the production of intermediate goods to Northern firms, the price of intermediate goods is high, whereas the
management cost paid by the final good producers to obtain the intermediate goods
is low. On the other hand, when the final good producers outsource the productionof intermediate goods to Southern firms, the price of intermediate goods is low,
whereas the management cost is high. Therefore, the final good producers face theproblem of where to outsource the production of intermediate goods.
This paper shows that, as the economy develops, the wage inequality between
the North and the South widens, and the place to which the final good producers
outsource the production of their inputs is switched from the North to the South.
Moreover, in the steady state, a decrease in the management cost paid by the final
good producers to obtain the intermediate goods from Southern firms causes anincrease in the North of both the number of firms and the wage rate in the North.
Indeed, an increase in the productivity of R&D raises the wage rate in the North.
An increase in the labor endowment in the North also increases the number of firms.
An increase in the labor force in the South affects neither the number of firms nor
3the wage rate in the North.
The remainder of the paper is structured as follows. Section 2 introduces themodel. In Section 3, we derive the equilibrium path of the model and prove that
there exists a unique equilibrium path converging to the steady state. In Section 4,
we conduct comparative statics with respect to the management cost, population
size, and productivity of R&D. In Section 5, we deal with the case where theequilibrium path is indeterminate. Section 6 concludes.

2

The Model

We develop a dynamic general equilibrium model where final good producers outsource production of the intermediate goods to intermediate producers either intheir 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 indexed
by l ∈ {N, S}. The population size in the world is 1. Each individual lives foreverand is endowed with Ll units of labor services, which is inelastically supplied at eachpoint of time. There exist three types of goods: a homogeneous good, intermediate
goods, and final goods. The homogeneous good can be produced only in the South,
and the final goods can be produced only in the North. The intermediate goods canbe produced in either country. Individuals consume the homogeneous good and thefinal goods. Figure 1 shows the production structure of the world economy. In the
North, there can exist at most three sectors: a R&D sector, a final goods sector,
and an intermediate goods sector. In the South, there can exist at most two sectors:
an intermediate goods sector, and a homogeneous good sector. R&D activities canbe conducted only in the North. The final good firms outsource the productionof intermediate goods either to the Northern firms or to the Southern firms tomaximize their profits. When the final good producers outsource the production ofintermediate goods to the Northern suppliers, there are three sectors in the North:
the R&D sector, the final good sector, and the intermediate goods sector. In theSouth, there is only one sector: the homogeneous good sector. On the other hand,when the final good producers outsource the production of intermediate goods to
the Southern firms, there exist only two sectors in the North: the R&D sector and4the final good sector. In the South, there are now two sectors: the intermediate
goods sector and the homogeneous good sector.

North

South

Final Goods(X)Homogeneous
Good(x)Intermediate
Goods(m)R&DIntermediate
Goods(m)

Labor

Labor

Figure 1: Production structure of the world economy

2.1

Consumers

Individuals in both countries have identical preferences:Z ∞e−ρ(τ −t) U (τ )dτ, 0 < ρ < 1,(1)twhere ρ is the constant subjective discount rate. U (τ ) is is the instantaneous utility
per person at time τ . It is specified as follows:
U (τ ) = x(τ ) +1
X(τ )µ ,µ

0 < µ < 1,(2)

where µ is a parameter, x(τ ) stands for consumption of the homogeneous good attime τ , and X(τ ) is a composite good at time τ that is made up of differentiated
final goods. For simplicity, we drop the time index τ from all the variables. A
composite good X is given by:
·Z n
X=

α

x(i) di¸ α1,0 < α < 1,

(3)0

where x(i) represents consumption of different final goods i and n stands for the1number of final goods. 1−αis the elasticity of substitution between any two varieties5

in a given sector. If α is close to one, the goods are nearly perfect substitutes and thesector is highly competitive. If α is close to zero, the goods are distinct products
and the sector becomes monopolistic. We assume that α > µ in the following
analysis.
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
x+ X µ ,x,x(i)µZ nsubject to x+
P (i)x(i)di = E,
max0where the homogeneous good is chosen to be the numeraire, P (i) stands for the
price of the product i, and E is the total expenditure.
From the first-order condition, we can obtain the 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),the indirect utility function is given by:µ1
U =E− 1−µ¶ ·Z 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
is constant. The market interest rate at time t, r(t), must be equal to the subjective
discount rate, as follows:r(t) = ρ

2.2

for all t .

(6)

Production

We denote the wage rate in the North by wN , and the wage rate in the South by
wS . This economy has no transportation costs and tariffs.
Production of the homogeneous good uses labor only. The homogeneous goodis supposed to be produced only in the South. The production of one unit of thehomogeneous good requires one unit of Southern labor. We assume that the homogeneous good market is perfectly competitive. Thus, the price of the homogenous
good becomes equal to the wage rate in the South. Since the homogeneous good6

is chosen to be the numeraire, the wage rate in the South becomes unity, that is,
wS = 1.
Production of each final good requires a variety-specific intermediate good. We
assume that all final goods are produced only in the North. The production functionof firm i is given by:
x(i) = θm(i),(7)

where m(i) stands for an intermediate good used for the production of final good i,and θ is a productivity parameter. The intermediate input m(i) can be produced
in both countries. Production of one unit of each intermediate good requires one
unit of labor. We assume that perfect competition prevails in the intermediate
goods markets in both countries. Thus, when it is produced in the North, the priceof each intermediate good is equal to the marginal cost, that is, the wage rate in
the North. Similarly, the price of each intermediate good produced in the South
becomes the wage rate in the South.
We assume that the final good producers have to pay the management costs
when they outsource the production of intermediate goods. The labor input is required for management activities like supervision, quality control, communications
costs, and contracting with the intermediate suppliers. When the final good producers buy the intermediate goods from Northern firms, they need the Northernlabor input, f N to manage these problems, and when they buy the intermediate
goods from a Southern firm, they similarly need the Southern labor input f S .
Furthermore, the final good producers in the North face problems of differencesin language, laws, and customs between the countries when they outsource the production of intermediate goods to the South. Therefore, we assume that when theintermediate producers are located in the South, the labor input for the management tasks is higher than when they are located in the North, that is in their homecountry:f N < f S.

(8)

By using (4), we can obtain the revenue of each final good producer as follows:
R(i) =P (i)x(i)=X µ−α θα m(i)α . l = N or S, i ∈ [0, n].
7(9)

We next consider the profit-maximization problem of the final good producers.
Because the perfect competition prevails in the intermediate goods market, theprice of intermediate goods is given by wl (l = N or S). Thus the final good
producer i maximizes profits as follows:
π l = R(i) − wl m(i) − wN f l ,l = N or S, i ∈ [0, n].

(10)

The profit-maximizing input of the intermediate good is given by:
µ−α

1

−1

α

m(i) = X 1−α α 1−α θ 1−α wl1−α , l = N or S.

(11)

By substituting (7) and (11) into (3), we obtain:
1−α

1

−1

1

X = n α(1−µ) α 1−µ θ 1−µ wl1−µ ,i ∈ [0, n].

(12)

From (12), we see that X increases with the number of final good firms n and the
productivity θ. On the other hand, X decreases with the wage rate wl . Using (11)and (12), we can obtain the profit functions as follows:µµ ¶ 1−µµ−ααθl
π = n α(1−µ)(1 − α) − wN f l ,l = N or S, i ∈ [0, n].
wl

(13)

The profits of the final good producers decrease with the wage rate in the NorthwN and the number of the final good firms n. On the other hand, the profit of the
final good producer increases with the productivity θ.

2.3

R&D sector

The R&D activities of the present model follow the model of Grossman and Helpman (1991, Ch. 3). We assume that Southern firms cannot innovate and imitate
a new variety of final good. The final good producers enter into the R&D raceand finance the cost of R&D by issuing equity in the stock market. The equity
is bought by individuals who live in either countries. The stock value of the final
good producers at time t is equal to the present discounted value of its profit stream
subsequent to t. Suppose that the final good producers change the intermediatefirms from the North to the South at time s. Then, the stock value of the final
good producers at time t is given by:
Z s
Z ∞
N
−r(τ )(τ −t) N−r(s)(s−t)
v =eπ dτ + ee−r(τ )(τ −s) π S dτ.
t

s

8

(14)

Then, the stock value of the final good producers at time t, when they outsource
the intermediate goods to the Southern firms, is given by:
Z ∞
S
v =
e−r(τ )(τ −t) π S dτ.

(15)

t

Differentiation of (14) and (15) with respect to time t yields the following noarbitrage conditions:v˙N = −π N + r(t)µZ s−r(τ )(τ −t) Ne−r(s)(S−t)π dτ + e

t

¶

Z ∞
e

−r(τ )(τ −s) S

π dτ

,

s

= −π N + r(t)v N ;(16)
v˙S = −π S + r(t)v S .

(17)

The final good producers hire labor to develop blueprints. We presume that there
exist knowledge spillovers in R&D activities: the more innovations have been created previously, the lower the cost of innovating. We assume that LA units of laborfor R&D activity for a time interval dt produce a new variety of final good according
to:
dn =

LA n
dt.
a

(18)

The cost of the R&D activities is wN LA dt because R&D sector is located onlyin the North. To produce innovative the blueprints creates value for the final good
producers of v l dn since each blueprint has a market value of v l . We assume that
there is free entry into the R&D race. Therefore, the following free-entry condition
must hold:
vl ≤

2.4

awN
,
n

with equality whenever ṅ ≡

dn
> 0.
dt

(19)

Labor market

Labor market equilibrium requires labor supply to be equal to labor demand. Welabel the economy Regime N when the final good producers outsource the production of intermediate goods to suppliers in the North, and Regime S when theyoutsource to suppliers in the South. In Regime N, the demand for labor in the
North comes from the intermediate sector, the R&D sector, and the managementtasks. In the South, the demand for labor comes only from the homogeneous good9

sector. Therefore, the labor market equilibrium conditions become as follows:
N
LN =LA + LN
M + f n,LS =xN ,where xN is the labor demand for production of the homogeneous good in theSouth, LN
M is the labor demand for production of the intermediate goods in the
North, and f N n is the management tasks of the final good sector. On the other
hand, in Regime S, the demand for labor in the North comes from the R&D sector
and the management tasks. In the South, the demand for labor comes from thehomogeneous good sector and the intermediate goods sector. Therefore, the labormarket equilibrium conditions become as follows:
LN =LA + f S n,LS =xS + LSM,S ,where xS is the labor demand for production of the homogeneous good, LSM is thelabor demand for production of the intermediate goods in the South, and f S n is
the management input of the final good sector.
The production of one unit of the homogeneous good and of the intermediate
goods requires one unit of labor each. Thus, production of the homogeneous good
and the intermediate goods equals the labor demand. We can write the productionof the homogeneous good by using the budget constraint, (4), (7), (11), and (12).
Z nl
P (i)x(i)di
x =E−
0
¸
·Z n
α
x(i) di
=E−X
0

=E−n

µ(1−α)
α(1−µ)−µ³ w ´ 1−µ

l

αθ

.

(20)

We can derive the production of the intermediate goods by using (11) and (12) as
follows:
LlM =Z nm(i) di
0
µ(1−α

1

µ

−1

= n α(1−µ) α 1−µ θ 1−µ wl1−µ ,10l ∈ {N, S} .

(21)

In Regime N, we can rewrite the labor market clearing conditions in both countries
using (18), (21), and (20) as follows:−1µ(1−α)µ1aṅ
+ n α(1−µ) α 1−µ θ 1−µ wN1−µ + f N n,n−µ
³ w ´ 1−µµ(1−α)NS
α(1−µ)
L =E−n
.
αθ

LN =

North
South

(22)
(23)

Using (18), (21), and (20), we can rewrite the labor market clearing conditions inRegime S as follows:3aṅ+ f S n,
n

North

LN =

South

LS = E − n α(1−µ) (αθ) 1−µ + n α(1−µ) α 1−µ θ 1−µ .

µ(1−α)(24)µµ(1−α)1µ(25)

The Equilibrium PathIn this section, we examine the dynamics of the economy. We first consider the
dynamic behaviors of Regime N and Regime S separately. Finally, we integratethese dynamic behaviors.

3.1

The dynamic behavior in Regime N

The equilibrium conditions are (6), the no-arbitrage condition, (16), the free-entry
condition, (19), and the labor market clearing condition, (22). From the labormarket clearing condition, we can derive the differential equation for the numberof final goods, n, as follows:·¸−1µ(1−α)
µ
1
n N
1−µ
N
ṅ =
L − f n − n α(1−µ) α 1−µ θ 1−µ wN .
a

(26)

Using (6), (16), and (19), we can obtain the differential equation for the wage ratein the North, wN , as follows:
µ
¶
−µ
µ
LN
1 µ(1−α)w˙N = ρ +wN − n α(1−µ) (αθ) 1−µ wN1−µ .
a
a

(27)

These two equations, (26) and (27), constitute the dynamic system of Regime N.
Figure 2 depicts the phase diagram for this system on the (n, wN ) plane. Theintersection point of the two curves ṅ = 0 and w˙N = 0 at point E is the steadystate of this system. The equation for the ṅ = 0 locus is given by:¡¢−(1−µ) µ µ(1−α)
wN = L N − f N n
αθ n α .
11

(28)

N

The ṅ = 0 locus is upward sloping and wn approaches infinity as n tends to Lf N .
On the other hand, the equation for the w˙N = 0 locus is given by:¡¢−(1−µ)µ(1−α)(αθ)µ n α .
wN = aρ + LN

(29)

The w˙N = 0 locus is also upward sloping and concave. Furthermore, the slope of
this locus diminishes monotonically toward zero as n approaches infinity. Thus,
there exists t a unique steady state. The steady state value of n is given by:n∗ =(1 − α)LN − αaρ
.
fN

(30)

We can show that the steady state E becomes a saddle point (see Appendix).
ṅ = 0 locus
wN

w˙N = 0 locus
E

n
N
N
L
/f
n
Figure 2: Phase diagram of Regime N∗

3.2

The dynamic behavior in Regime S

The equilibrium conditions are (6), the no-arbitrage condition, (17), the free-entry
condition, (19), and the labor market clearing condition, (24). From the labormarket clearing condition, we can derive the differential equation for the numberof firms, n, as follows:

·
¸
n N
S
ṅ =
L −f n .
a

12

(31)

Using (6), (17), and (19), we can obtain the differential equation for the wage ratein the North, wN , as follows:µ¶µ
LN
1 µ(1−α)w˙N = ρ +wN − n α(1−µ) (αθ) 1−µ (1 − α)
a
a

(32)

In the same way as in the former section, these two equations, (31) and (32),constitute the dynamic system of Regime S.
Figure 3 depicts the phase diagram for this system on the (n, wN ) plane. The
0intersection point of the two curves ṅ = 0 and w˙N = 0 at point E is the steadystate of this system. The equation for the ṅ = 0 locus is represented by:
n=

LN
.
fS

(33)

N

The ṅ = 0 locus is a vertical line at n = Lf S . On the other hand, the equation for
the w˙N = 0 locus is given by:µ(1−α)µ(αθ) 1−µ (1 − α)n α(1−µ)
wN =.
aρ + LN(34)

The w˙N = 0 locus is also upward sloping and concave. Furthermore, the slope of
this locus diminishes monotonically toward zero as n approaches to infinity. Thus,
there exists a unique steady state. The steady state value of n is given by:
n∗∗ =

LN
.
fS

(35)

From (34) and (35), we can obtain the steady state value of wN :
(αθ)
∗∗
wN
=

µ
1−µ

(1 − α)aρ + LNµ

LN
fN¶ µ(1−α)α(1−µ).

(36)

0

We can show that the steady state E becomes a saddle point (see Appendix).

3.3

A boundary between Regime N and Regime S

When the final good firms produce, they have to outsource the intermediate goods
to either the North or the South. The final good firms decide to buy the intermediate goods from the suppliers in the North (South) when they can obtain higher
profits by buying them from the Northern (Southern) suppliers. We investigate
the condition under which the profits of final good producers in Regime N become13ṅ = 0 locus

wN

w˙N = 0 locus

∗∗
wN

E

0

LN /f S
Figure 3: Phase diagram of Regime Snthe same as the profits in Regime S, that is π N = π S . From (13), the boundary
condition between Regime N and Regime S is given by:
µ
n=

1−α
S
f − fN¶ α(1−µ)α−µαµ(αθ) α−µ −µ
1−µ α(1−µ)α−µ1 − wN 
wN

.

(37)

Figure 4 depicts this boundary condition. The boundary line has the inverse-Cµ−1

max
shape where n takes a maximum value at wN= (1 − µ) µ . As wN approaches

infinity, n approaches zero. The inequality π N < π S holds in the shaded area of
Figure 4, thus Regime S prevails in the shaded area and the economy follows theequilibrium condition of Regime S. The other area corresponds to π N > π S . Thismeans that the economy is in Regime N. Therefore, in this area, the economy
follows the equilibrium condition of Regime N. In the next section, these threefigures are integrated into one figure.
We can give the intuition of Figure 4 as follows. Suppose that the number offirms n takes a constant value. When the wage rate in the North is small, thefinal good producers obtain the intermediate goods from the North. As the wage
rate in the North increases, the intermediate goods made in the North become
expensive compared with the intermediate goods made in the South. Therefore, thefinal good producers outsource the production of intermediate goods to the South.
However, the greater the North-South wage rate difference, the more expensive the14management cost of outsourcing production to the South. Thus, the final good
producers cannot pay the management cost, give up outsourcing the productionof intermediate inputs to the South, and obtain them instead from the Northernfirms.
wN
Boundary
πN = πS
Regime N
πN > πS
Regime S
max
wN

πN < πS1nnmax
Figure 4: Boundary condition between Regime N and Regime S

3.4

Equilibrium path from Regime N to Regime S

In the previous subsections, we explored the dynamic behaviors in Regime N andRegime S and showed the boundary condition between Regime N and Regime S.
We now consider the relative positions of the ṅ = 0 and w˙N = 0 loci in Regime N
and Regime S in order to integrate the two phase diagrams of into one.
We first compare the curve ṅ = 0 in Regime N with that in Regime S. From
LN
fN

N

> Lf S , there exists an intersection between the curve ṅ = 0 in Regime N andthat in Regime S. Next, we compare the locus of w˙N = 0 in Regime N with that in
Regime S. The intersection of the curve w˙N = 0 in Regime N with that in Regime S,ñ, is:−α(1−µ)¢ α(1−µ)¡−αñ = aρ + LN µ(1−α) (αθ) 1−α (1 − α) µ2 (1−α) .

15

(38)

When the inequality n < ñ holds, the w˙N = 0 locus in Regime N is above that in
Regime S. On the other hand, when the inequality n > ñ holds, the w˙N = 0 locus
of Regime S is above the w˙N = 0 locus of Regime N.
We next focus on the equilibrium path that the economy evolves from Regime N
to Regime S. There are three sufficient conditions that guarantee the existence ofthe equilibrium path that the economy evolves. The first condition is that the steadystate in Regime S exists in the shaded area in Figure 4 and that the equilibrium
path doesn’t converge to the steady state in Regime N. In the steady state in∗∗Regime S, the wage rate in the North is wN. When the wage rate in the North∗∗, let n1 be the profits of final good producers in Regime N, which becomeis wN

the same as the profits in Regime S. Thus, the condition that the steady state inRegime S exists in the shaded area in Figure 4 is n∗∗ < n̂1 . This inequality can be
expressed as follows:µ

1−αSf − fN¶ α(1−µ)α−µ α(1−µ)−µ∗∗ 1−µ

 α−µ
αµ
1−w

(αθ) α−µ



N
∗∗
wN



>

LN
.
fS

(39)

The second condition should satisfy the following two preconditions: there exists
an intersection of the w˙N = 0 locus in Regime N with the boundary line; and theequilibrium path does not converge to the steady state in Regime N. First, we show
the former precondition. If an intersection exists, we can find the real numbers thatsatisfy the following equation from (29) and (37):
µ
n̂2 =

1−α
S
f − fN¶ α(1−µ)α−µα(1−µ)
−µ  α−µ·¸ 1−µµ(1−α)
N −(1−µ)(αθ)µ n̂2 α
 1 − (aρ + L )αµ(αθ) α−µµ(1−α)(aρ + LN )−(1−µ) (αθ)µ n̂2 α






,

(40)
where n̂2 is defined as the smallest real numbers satisfying this equation. If n̂2
exists, this implies the existence of an intersection between the w˙N = 0 locus inRegime N and the boundary line. The second precondition is that n̂2 is smaller
than the number of final good firms in the steady state in Regime N, (30). That is:n̂2 <(1 − α)LN − aαρ
.
fN

(41)

Note that if n̂2 is larger than the steady state in Regime N, then the equilibrium
path will remain in Regime N.
16

Finally, the third condition is that the intersection point of the w˙N = 0 locus inRegime S with (37) is below the intersection point of the two w˙N = 0 loci in bothregimes, that is, n̂2 > ñ. Therefore, the condition is represented by:−α(1−µ)¡¢−αn̂2 > aρ + LN (αθ) 1−α (1 − α) µ2 (1−α) .

(42)

When the three necessary conditions, (39), (41), and (42), are satisfied, we candepict the phase diagram for that system on the (n, wN ) plane. Figures 2, 3, and 4
are integrated into Figure 5. In Figure 5, the arrow shows the transitional dynamics.
These results are stated as the following proposition.
Proposition 1. Suppose that parameters satisfy conditions (39), (41), and (42).
Then there exists the equilibrium path along which the economy evolves from Regime N
to Regime S. Moreover, the steady state is a saddle point.
N

Suppose that the initial number of firms satisfies n(0) < Lf S . At the initial point,
the economy is in Regime N. Then n and wN increases and eventually the trajectory
crosses the boundary line, (37). At the intersection point of the trajectory with theboundary line, the final good firms can outsource the intermediate inputs to either
the Northern firms or the Southern firms. After the crossing, the economy turns
from Regime N into Regime S and follows the stable path towards the steady state
E0 . Therefore, the final good producers change the region to which they outsource
the production of intermediate inputs from the North to the South as the numberof firms increases and the wage gap between the North and the South increases.

4Comparative Statics

In this section, we conduct comparative statics analysis of the steady state.

4.1

A Lower the management cost in Regime S

A decrease in the management cost in Regime S is expressed by a decrease in f S .
For example, if the number of people in the South who can speak the language of
the North increases, the number of lawyers in the South increases, and the amountof bureaucratic corruption decreases, then the management cost can decrease.
17Regime S : ṅ = 0 locus

wN

Boundary line
Regime N : ṅ = 0 locus

E

∗∗
wN

Regime S : w˙N = 0 locus

0E

Regime N : w˙N = 0 locus1nNn∗∗ = fL S n̂1
Figure 5: The phase diagram from Regime N to Regime S
n(0) ñn̂2

In the steady state, a decrease in f S affects the number of firms and the wage
rate in the North as follows:
∂n∗∗
LN
=
−
< 0,
∂f S
(f S )2(43)∗∗µ(1−α) 1µ∂wNµ(1 − α)L α(1−µ)N
1−µ (1 − α)(
=
−
(aρ
+
L
)(αθ)
)
< 0.
∂f S
α(1 − µ)
fS
fS

(44)

In the steady state, a decrease in f S increases the number of firms and the wage
rate in the North. Intuitively, when the management cost in Regime S decreases inthe steady state, the labor force used in the management sector decreases and thatin the R&D sector increases. Therefore, the number of final good firms increases.
When the number of final good firms increases, R&D activities become much easier,from (18), and the labor demand for the R&D sector increases. Therefore, the wage
rate in the North increases. Thus, we can summarize these results as the following
proposition.
Proposition 2. A decrease in the management cost in Regime S promotes obtaining the intermediate goods from the South and the number of firms and relative
wage rate in the North.
18

4.2

A higher productivity of R&D

An increase in the productivity of R&D is expressed by a decrease in a. The effectof an increase in the productivity of R&D is given by:
∂n∗∗
=0
∂a
µ
∗∗
∂wN
ρ(αθ) 1−µ (1 − α)n∗∗=−<0∂a(aρ + LN )2

(45)
(46)

Therefore, an increase in the productivity of R&D positively affects the wage ratein the North but does not affect the number of the firms. When an increase in
the productivity of R&D occurs, from (18), R&D activity is easier, and the labordemand for the R&D sector increases. Therefore, the wage rate in the North
increases. In the steady state in Regime S, the number of final good firms increases
until all the labor in the North are hired in the management sector. Hence, thenumber of final good firms does not change when the productivity of R&D increases.
We can summarize the above analysis as the following proposition.
Proposition 3. An improvement in the productivity of R&D promotes the wage
rate in the North and does not affect the number of final good firms.

4.3

A larger labor endowment

An increase in the labor services of both regions is expressed by increases in LN
and LS . In the steady state, an increase in the labor services of the South does
not affect the number of the final good firms, n∗∗ , nor the wage rate in the North,∗∗wN. The labor demand of the intermediate sector does not change when the labor

services in the South increase in the steady state. People who are not hired in theintermediate goods sector are hired in the homogeneous sector. However, since thehomogeneous good is chosen to be the numeraire, both the wage rate in the Northand the number of final good firms do not change. On the other hand, in the steadystate, the effects of an increase in the labor services of the North are as follows:∂n∗∗
1= S >0
N
∂L
f
µ
·
¸
∗∗
µ−α
(αθ) 1−µ (1 − α) ∗∗ α(1−µ)∂wNN
N µ(1 − α)=−nL − (aρ + L ).
∂LN(aρ + LN )2α(1 − µ)
19

(47)
(48)

An increase in the labor services of the North increases the number of the final goodfirms whereas, the effect of the wage rate in the North is ambiguous. The change∗∗
in LN affects wN
through two channels: a direct effect and a change in n∗∗ . ”A

direct effect” means that the expansion of the labor services of the North reduces
the wage rate in the North. The first term in the parentheses in (48) represents thisdirect effect. On the other hand, when the labor supply increases, the number offirms increases. Since the expansion of the number of the final good firms increases
the Northern labor demand in the management sector, the wage rate in the North
rises. The second term in the parentheses in (48) represents this effect. Therefore,
the two effects work in the opposite direction on the wage rate in the North. Whenthe size of the labor endowment of the North is larger than µ(1−α)aρ, a direct effectα−µis larger than a change in n∗∗ and the wage rate in the North decreases. On theother hand, when the size of the labor endowment of the North is smaller thanµ(1−α)aρ, a direct effect is smaller than a change in n∗∗ , and the wage rate in the
α−µNorth increases. This can be summarized as follows.
Proposition 4. The increase in the size of the labor force in the North promotesa higher number of final good firms and a higher wage rate in the North if the). The increase in the population
population in the North is smaller (LN < µ(1−α)aρ
α−µin the South does not affect the number of the final good firms nor the wage rate in
the North.

5

Indeterminacy of the Equilibrium PathIn this section, we indicate that the equilibrium path can be indeterminate. There
are two sufficient conditions for the equilibrium path to be indeterminate. The
first condition is that the steady state in Regime S exists in the shaded area of
Figure 4, where Regime S prevails and the equilibrium path can converges to thesteady state in Regime S. The other condition is that the steady state in Regime N
exists in the Regime N area in Figure 4, and the equilibrium path can also converge
to the steady state in Regime N. The former condition is given by (39), and thelatter condition is that equation (40) does not have real solution. When these two
sufficient conditions are satisfied, the equilibrium path is indeterminate and we candepict the phase diagram for that system on the (n, wN ) plane in Figure 6. At the20

initial point, if individuals expect that the equilibrium path will converge to thesteady state in Regime N, the economy converges to the steady state in Regime N
and the wage rate in the North is determined. On the other hand, at the initialpoint, if individuals expect that the equilibrium path will converge to the steadystate in Regime S, the economy converges to the steady state in Regime S and thewage rate in the North is determined. Therefore, the equilibrium path is determinedby expectations of individuals. These results are stated as the following proposition.
Proposition 5. Suppose that the parameters satisfy the conditions of (39) and thatthe equation (40) have no real solution. Then there exist two equilibrium paths, thatthe economy evolves from Regime N to Regime S and that the economy remains inRegime N, and the equilibrium path is indeterminate.
Regime S : ṅ = 0 locus

wN

Boundary line

Regime N : ṅ = 0 locus

Regime S : w˙N = 0 locus

E ∗∗

∗∗
wN

Regime N : w˙N = 0 locus
E

1

∗

n
N
n∗∗ = fL S n̂1
Figure 6: The phase diagram when the equilibrium path is indeterminaten(0) ñ6

n̂2Conclusion

This paper proposes a dynamic North-South endogenous growth model in which
final good producers in the North determine to outsource the production of intermediate goods to either the North or the South. In this model, the final good21

producers are only in the North, where the wage rate is high, and they buy intermediate goods either from the North or from the South. When they outsource to
the North, the price of intermediate goods is expensive, whereas the management
cost is low. On the other hand, when they outsource to the South, the price of theintermediate goods is low, whereas the management cost is expensive. This modelshows that as the economy develops, the wage rate in the North becomes higher.
Therefore, the final good producers change the region to which they outsource production from the North to the South. We show that in the transition process, there
exists a time at which the final good producers change the location to which theyoutsource the production of intermediate inputs from the North to the South. This
paper proves that the steady state in the economy is a saddle point. In the steadystate, a decrease in the management cost paid by the final good producers to obtain
the intermediate goods from Southern firms causes an increase in the North of boththe number of firms and the wage rate in the North. Indeed, an increase in the
productivity of R&D raises the wage rate in the North. An increase in the laborendowment in the North also increases the number of firms.

Appendix
In this appendix, we derive that the steady states in Regime N and Regime S are
saddle points. The linearized system of (26) and (27) around the steady state isgiven by:

µ

ṅ
w˙N

¶

µ
=

b1 b2
b3 b4

¶µ

n − n∗∗
wN − wN

¶
,

where
¶
µ
¡
¢
(LN − 2f N n∗ ) 1 µ(1 − α)−
+ 1 LN − f N n∗ ,b1 =aa α(1 − µ)¢1n∗ ¡ Nb2 =L − f N n∗ ,∗a(1 − µ) wNµ¶∗µ(1 − α) wN
LN
b3 = −
ρ+,α(1 − µ) n∗
a
and

1
b4 =
1−µ

µ

22

LN
ρ+
a

¶
.

The determinant of the characteristic matrix is as follows:Nρ + La N ∗−f n < 0.
a(1 − µ)Thus, the determinant takes a negative value. This condition implies that the two
eigenvalues of the system have opposite signs and the steady state of Regime N isa point.
We next show that the steady state in Regime S displays saddle-path stability.
The linearized system of (31) and (32) around the steady state is:
µ

ṅ
w˙N

Ã

¶
=

N

L
−
0
a
³
´
∗
N
N
wN
ρ + La− n∗ ρ + La!µ

n − n∗∗
wN − wN¶,

The determinant of the characteristic matrix become
µ
¶
LN
LN
−
ρ+
< 0.
a
a
Thus, the determinant takes a negative value. This condition implies that the two
eigenvalues of the system have opposite signs and the steady state of Regime S is asaddle point.

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