Discussion Paper No. 238
North-South Asymmetry in Returns to Scale,
Uneven Development, and the Population Puzzle

Hiroaki Sasaki
September 2008

TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANNorth-South Asymmetry in Returns to Scale,
Uneven Development, and the Population Puzzle
Hiroaki SASAKI∗
September 2008

Abstract
This paper develops a model of North-South trade and economic development to analyze how an increase in the growth rate of population affects the growth rate of real
income per capita. We assume that the North is characterized by an increasing-returnsto-scale technology while the South is characterized by a decreasing-returns-to-scaletechnology. The main results are as follows: (i) an increase in the growth rate of population in the South decreases the growth rate of its own income per capita; (ii) a rise in
the growth rate of population in the North either increases or decreases the growth rateof its own income per capita depending on conditions; (iii) population growth in one
country raises the growth rate of income per capita in the other country; and (iv) eventhe decreasing-returns South can experience a positive growth rate of income per capita
if a continuous improvement in the terms of trade is larger than a threshold value.
Keywords: North-South trade; economic development; population growth; non-scale
growth; terms of trade
JEL Classification: F43; O11; O41∗

Graduate School of Economics and Management, Tohoku University, 27-1 Kawauchi, Aoba-ku, Sendai
980-8576, Japan. E-mail: hsasaki@hotmail.co.jp11

Introduction

This paper attempts to give a probable explanation to what is called the population puzzle
by combining a traditional model of North-South trade and economic development with amodern model of endogenous growth.1) The word “the population puzzle” is termed by
Goodfriend and McDermott (1995), who point out that conclusions obtained from non-scale
growth models, a class of endogenous growth models, are not consistent with empiricalfindings. Let us explain this issue below.
First, with respect to empirical findings, Goodfriend and McDermott (1995) refer to results of Kuznets (1973). Using postwar cross-country data for 63 countries, Kuznets (1973)shows that there is a statistically significant negative relationship between the growth rate of
population and that of output per capita. Kuznets also shows that if the sample is divided
into developed and developing countries, the correlation between the two growth rates becomes statistically insignificant. That is, the relationship between the growth of populationand that of output per capita is not so unambiguous.
Second, we explain non-scale growth models. Since Jones (1995) challenged the scaleeffects of the endogenous growth model, non-scale growth models have gained attention inthis field. In the scale-growth model, the growth rate of output per capita along the balanced
growth path depends positively on the size of population. That is, the larger the size of population, the faster a country grows, which, however, seems counterfactual. Then, Jones (1995)
attempts to remove the scale-effects and presents a non-scale growth model, from which thefollowing conclusion is obtained: the growth rate of output per capita depends positively
on the rate of population growth, and not on the size of population. That is, the higher thegrowth rate of population, the faster a country grows.2) However, this conclusion also seems
inconsistent with the findings of Kuznets (1973) aside from the period considered. In thissense, Goodfriend and McDermott (1995) point out that population-driven models of growthmust confront the population puzzle.
Existing non-scale growth models are built under a closed economy setting. However,
the real world is an open economy, and accordingly conclusions obtained from closed economy models would not be applicable to the real world as they are. Therefore, we need an
open economy model to apply the implication to the real world. In this respect, Christiaans
1) Traditional North-South models here mean a model that considers some asymmetries between developed
and developing countries. See, for instance, Findlay (1980), Taylor (1981), Molana and Vines (1989), Darity(1990), and Sarkar (1997, 2001). Chui, Levine, Murshed, and Pearlman (2002) comprehensively survey theliterature that combines growth and trade into models of North-South interaction. They deal with a combinationof new trade and new (endogenous) growth models.
2) For a systematic exposition concerning scale effects and non-scale growth, see Jones (1999) and Christiaans (2004).

2

(2003) develops a small open economy model of non-scale growth that considers imported
intermediate goods and exogenous export demand.3) The conclusion is that the relationship between the growth rate of population and that of output per capita differs dependingon whether export goods of a country is price elastic or not. This model can be termed as
the modified non-scale growth model in the sense that the unconditional positive correlation
between the growth rate of population and that of output per capita is modified.4)
A model presented in this paper belongs to the class of modified non-scale growth models and is constructed under an open economy setting. Unlike Christiaans (2003), however,
the model is a two-country model: two large countries engage in trade with each other.
It is not appropriate that we apply the same specification to the developed and developing
countries that have different structures. Thus, we develop a model of North-South trade andeconomic development that takes into account various asymmetries between the developed
North and the developing South. Using it, we examine how the growth rate of population ineach country affects the growth rates of income per capita in both countries.
Our model is based on Conway and Darity (1991), who develop a model of North-South
trade and economic development along the lines of Kaldor’s idea. Specifically, they modela situation where the North is characterized by increasing returns to scale while the Southis characterized by decreasing returns to scale. Then, they analyze what consequences theasymmetry in returns to scale has for the growth rate of capital in each country, the terms oftrade, and so forth. In this paper, we extend Conway and Darity’s model in some respects.
First, we assume that the Northern imports from the South are used for both consumption
and intermediate inputs while the Southern imports from the North are used for both consumption and investment. In Conway and Darity’s model, the Northern imports from theSouth are devoted only to consumption, which leads to the conclusion that the growth rateof capital stock in the North depends only on the Northern factors while the growth rate ofcapital stock in the South depends on both Northern and Southern factors.5) That is, there
is no interdependence between the two growth rates. In our model, on the other hand, theNorth-South interdependence arises because the Southern good is used as input in the production of the Northern good. This idea is based on Dutt (1996). Second, we employ a3) Christiaans (2003) introduces a learning-by-doing effect into Bardhan and Lewis’s (1970) model to endogenize the rate of economic growth.
4) For modified non-scale growth models, see also Christiaans (2008) and Sasaki (2008). These papers deal
with the relationship between trade specialization processes and industrialization under a small open economy
setting.
5) Conway and Darity (1991) conclude that the growth rate of output in the North (South) is determined only
by factors in the North (South). Note, however, that their definition of the long-run equilibrium is not correctand if the definition is corrected, our statement above is obtained. They define the long-run equilibrium asa situation in which capital-output ratio will be constant. However, when defining capital-output ratio in theSouth, they do not consider the terms-of-trade, which leads to the incorrect definition of capital-output ratio.

3

dynamic optimization technique to solve the model. In Conway and Darity’s model, wage
income is entirely consumed and profit income is entirely invested. In this paper, we do notdifferentiate workers from capitalists and treat the choice between consumption and saving
as a dynamic optimization problem of a representative agent. Finally, we refer to other differences between Conway and Darity’s and our models. In Conway and Darity’s model, real
wages in the short run are fixed both in the North and in the South. In the North, firms set theprice with a mark-up on unit labor costs while in the South employment is determined by theequalization of the value of marginal products of labor and the nominal wage. In the longrun of Conway and Darity’s model, full employment is assumed and the real wages in bothcountries are adjusted so as to maintain full employment. In our model, in contrast, goods
and factors markets in both countries are perfectly competitive and the marginal principle
prevails both in the short run and in the long run.6)
The paper not only gives a probable explanation to the population puzzle, but also contributes to some issues in this field.
First, we contribute to the issue of the relationship between the terms of trade movement
and economic development. Using a North-South model of endogenous growth, Felbermayr(2007) shows that even if the South specializes in a technologically stagnant sector, it canachieve a sustainable growth owing to endogenous, continuous improvements in the termof-trade. Álvarez-Albelo and Perera-Tallo (2008) also reach a similar result using a different
two-country endogenous growth model.7) As will be explained later, our model shows that
if the rate of endogenous improvement in the Southern terms of trade is large, the growthrate of real income per capita in the South can be positive even if the Southern production is
described by a decreasing-returns-to-scale technology. As a practical matter, there is an empirical observation that developing countries face a continuous improvement of their termsof trade. Felbermayr (2007), stated above, provides the following two observations: exports
of developing countries to OECD countries are heavily biased toward consumption goods
while their imports from OECD countries are mainly made up by investment goods; and theprice of investment goods relative to consumption goods has been falling over time. Combining these two observations, we reach the conclusion that the terms of trade of developing
countries continuously improves relative to developed countries.8)
Second, we contribute to the issue of two-country endogenous growth models. A large
6) The assumption that both agricultural and industrial markets are competitive and producers act as a pricetaker is also adopted by Dutt (1992).
7) In their model, not a final good but a intermediate good is traded. One country is characterized by a LucasRebelo type model while the other country is characterized by the neoclassical growth model. The terms oftrade of the latter country permanently improves along the balanced growth path.
8) This phenomenon seems to contradict the Prebisch-Singer hypothesis. However, results of existing studies
on this hypothesis are so diverse that the hypothesis is not necessarily an established proposition.

4

literature on two-country endogenous growth models focuses its attention on the balanced
growth path and does not examine the dynamic stability. Ben-David and Loewy (2000), forexample, develop a two-country endogenous growth model that considers knowledge as afactor of production and international spillovers of knowledge. In their model, each country specializes in production of a single good. They focus their analysis on the balanced
growth path and do not examine the transitional dynamics. Felbermayr (2007) also confines
the analysis to the balanced growth path equilibrium. Osang and Pereira (1997) present a
two-country model of endogenous growth based on the Uzawa-Lucas human capital model.
In this model as well as our model, the terms of trade between two countries continuously
changes along the balanced growth path. However, they do not consider the dynamic stability. In contrast to these studies, we examine the dynamic stability and show the saddle-pathstability of the model using a numerical method.
The remainder of the paper consists of the following five sections. Section 2 presents thebasic framework of our model. Section 3 obtains the growth rates of endogenous variableson the balanced growth path. Section 4 derives the growth rate of income per capita in eachcountry and investigates the relationship between population growth and income growth.
Section 5 numerically shows the saddle-path stability of the long-run equilibrium. Section
6 concludes the paper.

2 The model
Consider a world that consists of the North, a developed country, and the South, a developing
country. Each country completely specializes in production of a single good. The Northerngood is used for consumption and investment in both countries. The Southern good, in
contrast, is used for consumption in both countries and for intermediate input in the North.
That is, we assume that the North exports a final consumption-cum-investment good to theSouth while the South exports a final consumption-cum-intermediate good to the North.
Note that in the North value of total production differs from total value added due to theexistence of imported intermediate input.
In this paper we focus on a competitive equilibrium path. As will be explained below,
in the North, there exist externalities arising from capital accumulation. Therefore, a competitive equilibrium path diverges from an optimal path in which a social planner internalizes externalities. Steady states values of scale-adjusted variables are larger in the centrally
planned case than in the decentralized case. However, growth rates on the balanced growth
path are equal in both cases.

52.1 Firms
The North produces the N good according to the following Cobb-Douglas production function:
XN = AN KN1−α−β LαN M β ,0 < α < 1, 0 < β < 1, α + β < 1,(1)

where XN is total production, AN external effects, KN capital stock, LN employment, andM imported intermediate goods. If AN is regarded as an exogenous variable, the Northern
production is constant returns to scale. Externalities due to capital accumulation are given
by as follows:
AN = KNα .

(2)

We assume Marshallian externality in the following analysis. Accordingly, profit maximizing firms regard AN as given exogenously. Substituting equation (2) into equation (1), we
can rewrite the production function as follows:
XN = KN1−β LαN M β ,(3)which shows increasing returns to scale in KN , LN , and M.
The South produces the S good according to the following constant-returns-to-scale
Cobb-Douglas production function:
YS = KS1−a−b LSa T b ,0 < a < 1, 0 < b < 1, a + b < 1,(4)

where YS is output, KS capital stock, LS employment, and T land input. Suppose that supplyof land is fixed. Then, we can normalized it to T = 1. From this, equation (4) can be
rewritten as follows:
YS = KS1−a−b LSa ,(5)which shows decreasing returns to scale in KS and LS .
Let p ≡ pS /pN be the Southern terms of trade relative to the North with the N goodbeing the numéraire. Then, profits of firms in the North and South are respectively given byΠN = XN −rN KN −wN LN − pM and ΠS = pYS −rS KS −wS LS −qT , where wi denotes wage incountry i, ri rental rate of capital, and q rental rate of land: all prices are measured in terms

6

of the N good. From profit maximizing conditions, we obtain the following relations:αXN
βXN(1 − α − β)XN
, rN =
, p=
,
LN
M
KN
p(1 − a − b)YS
pbYS
paYS
South : wS =, rS =
, q=
.
LS
KS
T
North : wN =Note that in the derivation above, AN is treated as an exogenous variable by firms in the
North.

2.2 Consumers
Consumers in both countries choose the flow of consumption to maximize the present discounted value of life time utility. We assume that instantaneous utility is given by the loglinear function.
Z ∞
(6)
UN =
[γ ln C NS + (1 − γ) ln C NN ] exp(−ρN t) dt,
0
Z ∞
US =
[γ ln CSS + (1 − γ) ln CSN ] exp(−ρS t) dt,(7)0

where C NS , for instance, denotes the consumption of the S good in the North, γ a parameter
governing an expenditure share for the S good, and ρi the rate of time preference, which cantake a different value in each country.
The budget constraints are as follows.9)
K̇N = rN KN + wN LN − C NN − pC NS ,(8)K̇S = rS KS + wS LS + qT − CSN − pCSS .

(9)

A dot over a variable denotes the time derivative of the variable: for example, K̇N =dKN (t)/dt. For simplicity t is omitted unless it is needed. Since the total income in the Northis given by wN LN + rN KN = (1 − β)XN and that in the South is given by wS LS + rS KS + qT =pYS , equations of motion for capital stock respectively lead toK̇N = (1 − β)XN − C NN − pC NS ,(10)K̇S = pYS − CSN − pCSS .

(11)9) We assume that a domestic asset market in each country is competitive but an international asset market
does not exist. Therefore, in each country, the rental rate of capital and the rate of interest are equalized throughasset arbitrage.

7

From equations (6), (7), (8), and (9), current-value Hamiltonian functions for the Northand the South are respectively given by
HN = γ ln C NS + (1 − γ) ln C NN + λN (rN KN + wN LN − C NN − pC NS ),(12)

HS = γ ln CSS + (1 − γ) ln CSN + λS (rS KS + wS LS + qT − CSN − pCSS ),(13)

where λi is the co-state variable. The first-order necessary conditions are given by as follows:γ− pλN = 0,C NS
1−γ− λN = 0,
C NN

(14)
(15)

λN rN = ρN λN − λ̇N ,γ− pλS = 0,CSS
1−γ
− λS = 0,
CSN

(16)

λS rS = ρS λS − λ̇S .

(19)

(17)
(18)

In addition, we need the transversality condition.
lim λi (t)Ki (t) exp(−ρi t) = 0,t→+∞i = N, S .

(20)

From equations (16) and (19), we have λ̇i (t)/λi (t) = ρi − ri , and consequently λi (t) =hR tiλi (0) exp 0 {ρi − ri (τ)} dτ , which is substituted into equation (20).
" Z t#
lim λi (0)Ki (t) exp −ri (τ) dτ = 0.

t→+∞

(21)

0

2.3 Market clearing conditions
Let us describe market clearing conditions for both goods. Taking into account the factthat investment in the South depends entirely on imports from the North while intermediate
input in the North depends entirely on imports from the South, we can write market clearing
conditions for both goods as follows:
XN = C NN + CSN + IN + IS ,(22)

YS = C NS + CSS + M.

(23)

8

The trade balance condition is given by
pC NS + pM = CSN + IS .

3

(24)

Balanced growth path

This section derives the balanced growth path (BGP). The BGP in the present paper is asituation where all variables grow at constant rates, which are not necessarily the same.
Using equations (15), (16), (18) and (19), we obtain two Euler equations for consumption. From equations (10) and (11), we obtain two equations of motion for capital stock.
These four equations are presented below.10)
Ċ NN

XN
= (1 − α − β)− ρN ,
N
KN
CN
ĊSN
pYS
= (1 − a − b)− ρS ,
N
KS
CS
K̇N
XN
1 C NN= (1 − β)−,KN
KN 1 − γ KN
pYS
1 CSN
K̇S
=
−
.
KS
KS
1 − γ KS

(25)
(26)
(27)
(28)

With these equations, let us derive the BGP growth rates of variables. In what follows wedenote the growth rate of variable z as gz ≡ ż(t)/z(t).
To begin with, for consumption to grow at a constant rate, we need gXN = gKN andg p + gYS = gKS from equations (25) and (26). When gXN = gKN , the output-capital ratio in
the North, XN /KN , becomes constant. Substituting M = βXN /p into the production function(1), we can rewrite the output-capital ratio as follows:αββXN= β 1−β LN1−β p− 1−β .
KN

(29)

Along the BGP, the right-hand side of equation (29) will be constant, that is, the rate ofchange in the right-hand side will be zero. Therefore, along the BGP, the following relation
is obtained.
g∗p =αnN > 0.
β(30)10) Euler equations for C NS and CSS are not needed directly because these two equations can be derived fromthe Euler equations for C NS and CSN with given p.

9An asterisk (∗) denotes the BGP value of a variable. Equation (30) implies that the terms oftrade of the South (the North) improve (deteriorate) continuously along the BGP. In short,
our model describes a situation where technical progress occurs in the North whereas does
not occur in the South. Technical progress in the North decreases the relative price of theNorthern good, so that the Southern terms of trade improve over time. Note that g∗p isdetermined only by the parameters of the North and that the larger the population growth of
the North, the higher the improvement rate of the South becomes. Let us focus on β. The
parameter β corresponds to the ratio of imports of intermediate input to total production.
Therefore, a large β implies a high degree of the Northern dependence on the Southernintermediate good. As the Northern dependence gets larger, the rate of improvement in theSouthern terms of trade becomes smaller.
Next, for capital stock to grow at a constant rate, we require gCNN = gKN and gCSN = gKS
from equations (27) and (28) provided that the output-capital ratios in both countries areconstant. In this case, C NN and CSN must grow at the same rate; otherwise, world consumption
for the N good does not grow at a constant rate. From this observation we have gCNN = gCSN ,so that gKN = gKS , that is, along the BGP capital stocks in both countries grow at the samerate. Applying gKS = gKN = gXN to g p + gYS = gKS , we obtain g∗p + g∗YS = g∗XN along the BGP.
Using these results for the production functions in both countries, we finally get the growthrate of capital stock in each country as follows:g∗KN = g∗KS = φnN + ψnS ,where φ ≡

a
α
, ψ≡
.
β(a + b)a+b

(31)

These growth rates depend on the growth rates of population and the parameters of the production functions. Using (31), we can obtain the growth rates of other endogenous variables.
Finally, let us demonstrate that the transversality condition (21) holds. As has beenshown above, along the BGP, g∗KN = g∗KS = φnN + ψnS holds. After some calculations, the
BGP rental rates of capital in both countries are given by rN∗ = ρN + φnN + ψnS and rS∗ =ρS +φnN +ψnS . From these observations we obtain rN∗ −g∗KN = ρN > 0 and rS∗ −g∗KS = ρS > 0.
Applying these results to equation (21), we can see that the transversality condition holds.

4

Population growth and the growth of income per capita

We here focus on the BGP growth rate of real income per capita, g∗yi (i = N, S ). For this purpose we have to obtain real national income, which in turn requires an appropriate definitionof the consumption price index.
Let pC denote the price index that is consistent with the expenditure minimizing problem.
10

γ
−γ
−(1−γ) γ
Then, the price index is given by pC = γ−γ (1 − γ)−(1−γ) p1−γ
p . Since
N pS = γ (1 − γ)both countries face the common relative price, p, and preferences are same, the price index
is also same in both countries. Note that the relation g pC = γg p holds between the priceindex and the terms of trade.
From the analysis above, nominal national incomes in the North and the South are XN −pM and pYS , respectively. Both are measured in terms of the N good. Accordingly, real
national incomes in the North and the South are respectively given by (XN − pM)/pC andpYS /pC . Using these real national incomes, we can compute the growth rates of real incomes
per capita along the BGP as follows:aα[1 − γ(a + b)] − β(a + b)
nN +
nS ,
β(a + b)a+b
α[1 − γ(a + b)]
b
nN −
nS .
g∗yS =β(a + b)a+bg∗yN =(32)(33)

Note that when nN = nS , we have g∗yN = g∗yS , which suggests that the difference between g∗yN
and g∗yS arises from the difference between nN and nS . Therefore, the growth rates of real
national incomes are identical.11) From equations (32) and (33), we obtain the following
propositions:12)
Proposition 1(i) A rise in the rate of population growth in the North increases or decreases the BGP
a+b
growth rate of its own real income per capita according as αβ ≷ 1−γ(a+b).
(ii) A rise in the rate of population growth in the South decreases the BGP growth rate ofits own real income per capita.
(iii) A rise in the rate of population growth in one country leads to a rise in the BGP growthrate of real income per capita in the other country.
(iv) Provided that the BGP rate of improvement in the Southern terms of trade is large,
the BGP growth rate of real income per capita in the South can be positive even if theSouthern production is described by a decreasing-returns-to-scale technology.
11) Note that the growth rate of real national income is equal to that of aggregate consumption. Aggregate
consumption is defined as C N ≡ (C NN )1−γ (C NS )γ in the North and CS ≡ (CSN )1−γ (CSS )γ in the South. Let E N and
ES be the nominal expenditure in the North and that in the South, respectively. Then, we have E N = pC C N
and ES = pC CS .
12) It is possible that the growth rate of real income per capita in the North becomes negative dependingon parametric conditions. As will be shown in section 5.2, however, our numerical examples show that we
probably have g∗yN > 0.

11

(v) Even if a country’s own rate of population growth is zero, the BGP growth rate of real
income per capita of the country can be positive.
Propositions 1-(ii) and 1-(iii) are easily checked by observing the coefficients of nN and nS
in equations (32) and (33).
On Proposition 1-(i): The coefficient of nN in g∗yN is positive or negative according as
α[1 − γ(a + b)] − β(a + b) ≷ 0, which is equivalent to the condition in Proposition 1-(i). Given
the parameters of the South, the larger α and/or the smaller β, the larger α/β becomes, sothat the coefficient of nN in g∗yN is likely to become positive. As has been stated above, αmeasures the degree of externalities due to capital accumulation and β measures the degreeof the Northern dependence on the Southern intermediate good.
On Proposition 1-(iv): Rearranging the condition g∗yS ≷ 0 and using g∗p = αβ nN along the
BGP, we obtain the following relation:
g∗p ≷

bnS
.
1 − γ(a + b)

(34)

This means that if the rate of improvement in the Southern terms of trade is larger than theright-hand side, then the growth rate of real income per capita is positive.
On Proposition 1-(v): Substituting nN = 0 and nS = 0 into g∗yN and g∗yS , respectively, wehaveanS > 0,a+bα[1 − γ(a + b)]g∗yS =
nN > 0.
β(a + b)g∗yN =(35)(36)

In this case, the Northern growth rate is determined only by the Southern factors while theSouthern growth rate is determined only by the Northern factors. In existing nons-scale
growth models for closed economy, the growth rate of real income per capita will be zero
when population growth is zero. In our model, however, the growth rate of income per capita
is positive in each country due to the presence of international trade.
Here, referring to the existing literature, we add an explanation to the real national income used in the present paper. Temple (2005) points out that in calculating real national
income in an open economy setting, there is an important distinction between a GDP priceindex and a cost-of-living index because the structure of consumption and that of production
can be different. When measuring economic welfare, we have to use the cost-of-living index, which is nothing less than the price index pC in our model. This distinction corresponds
to the difference between real GDP (gross domestic product) and real GDI (gross domestic
income) in the National Accounts. Real GDI is equal to real GDP plus the trading gains
12(or less trading losses) resulting from changes in the terms of trade. The importance of this
distinction is emphasized by Álvarez-Albelo and Perera-Tallo (2007) and Kohli (2004). Let
us consider this issue in our model.
First, national income measured in terms of the N good is given by (1 − β)XN , pYS , andthe growth rate of the national income becomes gXN , g p + gYS . Second, national income
measured in terms of the good produced in each country is given by (1 − β)XN , YS , andthe growth rate of the national income becomes gXN , gYS . Third, national income measured
in terms of the cost-of-living index, that is, our real national income (real GDI) is givenby (1 − β)XN /pC , pYS /pC , and the growth rate of the national income becomes g∗XN − γg∗p ,g∗YS + (1 − γ)g∗p .
In the first measure, the growth rates of both countries are equal because g∗XN = g∗p + g∗YS
along the BGP. In the second measure, with g∗p > 0, we have g∗XN > g∗YS . In the third
measure, the growth rates of both countries are equal because g∗XN = g∗p + g∗YS . From this we
can see that the first measure, compared with the third measure, overestimates the growth
rates of both countries by γg∗p and that the second measure, compared with the third measure,
overestimates the growth rate of the North by γg∗p and underestimates the growth rate of theSouth by (1 − γ)g∗p .

5

Existence and stability of the long-run equilibrium

This section investigates the existence and the local stability of a steady state. Unless there
exists a path converging to the steady state, analysis along the BGP will lose its importance. However, little literature on two-country endogenous growth models has dealt withthis issue. Farmer and Lahiri (2005), for instance, investigate multiple equilibria and indeterminacy using a two-country endogenous growth model. In their model, however, ahomogeneous good is produced in both countries, which implies that the terms of trade between two countries are always unity. A lot of existing studies confine their analysis to the
BGP. In the first place, there exists little literature that investigates the dynamic stability of
two-country neoclassical growth models. Brecher, Chen, and Choudhri (2005) is one of the
few studies that exist. The main reason for this is that in the two-country setting, we havea lot of endogenous variables, so that we have to analyze a system of differential equationsof high order. From our model we obtain a system of four differential equations. Analyticaltreatment of the system is troublesome, and accordingly numerical simulation is used.13)

13) Arnold (2006) analytically proves that there exists a unique path converging to a steady state (saddle-pathstability) in Jones’s (1995) model, which produces a system of four differential equations.

13

5.1 Existence of the equilibrium
Because KN , KS , C NN , and CSN keep on growing along the BGP, we cannot investigate thedynamic stability of the system without modifications. For this reason, we introduce thefollowing scale-adjusted variables with consideration for the BGP growth rates of the abovementioned variables and the terms of trade:14)
C NN
CSN
N
N
π ≡ α , kN ≡ φ ψ , kS ≡ φ ψ , cN ≡ φ ψ , cS ≡ φ ψ .
L N LS
L N LS
L N LS
L N LS
LNβ
p

KN

KS

(37)

[Figure 1 to be inserted here]
Here, we show that the newly defined terms of trade, π, can be expressed as a functionof the other scale-adjusted variables. Expressing the market clearing condition for the S
good (23) by the scale-adjusted variables, we obtain the following modified market clearing
condition:
πkS1−a−b −β1γ(cNN + cSN ) = β 1−β π− 1−β kN ,
1−γ

(38)

where cSN and cSS are transformed into cNN and cSN , respectively, with the use of equations (14),(15), (17), and (18). In equation (38), the left-hand side (LHS) and right-hand side (RHS)
can be regarded as functions of π, which are drawn in Figure 1. As Figure 1 shows, theintersection of LHS and RHS uniquely determines π. Therefore, the scale-adjusted terms oftrade π lead to a function of the other scale-adjusted variables.
π = π(kN , kS , cNN , cSN ),
+

−

+

(39)

+

where a sign below a variable show the sign of the corresponding partial derivative of thevariable.15)
14) See Eicher and Turnovsky (1999, 2001) for a technique based on scale-adjusted variables.
15) Specifically, each partial derivative is given by as follows:1γβ(1 − a − b)πkS−a−b∂π∂π
β 1−β π− 1−β
∂π
∂π
1−γ
= N =
=
> 0,
=−
< 0,
> 0,
N
∂kN
∆
∂kS
∆
∆
∂cN
∂cS
where
2−β

∆ ≡ kS1−a−b +1β 1−β − 1−βπkN > 0.
1−β14Substituting equation (37) into equations (25)–(28), we obtain the following differential
equations for the scale-adjusted variables:
"

#
γ cNN
k̇N = kN · (1 − β)β π−− φnN − ψnS ,1 − γ kN
"
#
γ cSN
−a−b
− φnN − ψnS ,k̇S = kS · πkS −1 − γ kSββċNN = cNN · (1 − α − β)β 1−β π− 1−β − ρN − φnN − ψnS ,hiċSN = cSN · (1 − a − b)πkS−a−b − ρS − φnN − ψnS .
β
1−β

β
− 1−β

(40)
(41)
(42)
(43)

The steady state to the above system is a situation where k̇N = k̇S = ċNN = ċSN = 0, fromwhich the following steady state values are uniquely determined:!− 1−βρN + φnN + ψnS β
π =β
,
1−α−β
"
#− 1
ρS + φnN + ψnS a+b∗,
kS =(1 − a − b)π∗
1 − γ ρS + (a + b)(φnN + ψnS ) ∗cSN∗ =·· kS ,
γ1−a−b(1 − α − β)(φnN + ψnS )kN∗ =· kS∗ ,
ρN + (α + β)(φnN + ψnS )1 − γ (1 − β)ρN + α(φnN + ψnS ) ∗·· kN .
cN∗
N =
γ
1−α−β
∗

(44)
(45)
(46)
(47)
(48)

Since 1 − α − β > 0 and 1 − a − b > 0, all steady state values are positive.

5.2 Stability of the equilibrium
In our system of differential equations, cNN and cSN are jump variables while kN and kS are
predetermined variables. Therefore, if the Jacobian matrix corresponding to the systemof differential equations has two positive eigenvalues and two negative eigenvalues, thenthe steady state obtained above is saddle-path stable.16) Here, we refer to both positive(negative) real eigenvalues and complex eigenvalues with positive (negative) real parts aspositive (negative) eigenvalues. To know signs of eigenvalues analytically is very difficult,16) If the number of jump variables are equal to that of positive eigenvalues, a system of differential equations
is saddle-path stable. If the number of positive eigenvalues exceeds that of jump variables, the system isunstable. If, in contrast, the number of jump variables exceeds that of positive eigenvalues, the system is inindeterminacy.

15and consequently we resort to a numerical simulation.17) Let J be the Jacobian matrix of oursystem. Then, J is a 4 × 4 matrix, and each element is presented in Appendix A.
We confine our numerical analysis to the following four cases: Cases 1, 2, 3, and 4.
These four cases are chosen to produce Proposition 1 obtained above and shown in Table
1: g∗yN > 0 (mentioned in footnote 12), g∗yS ≷ 0 (Proposition 1-(iv)), and ∂g∗yN /∂nN ≷ 0(Proposition 1-(i)). In what follows, we present combinations of parameters corresponding
to these four cases, and confirm whether there exists a unique path converging to the steadystate. Table 2 shows a combination of parameters.
[Tables 1 and 2 to be inserted here]
Table 3 presents results of numerical analysis based on the values of the parameters in
Table 2. Here, χ1 , . . . , χ4 denote eigenvalues. In every case we have two positive eigenvalues
and two negative eigenvalues. Therefore, the steady state is characterized by saddle-pathstability.
[Table 3 to be inserted here]
Let us demonstrate how the scale-adjusted capital stock in each country converges tothe steady state value with an initial value being given. To this end, we need to solve thesystem of differential equations using the eigenvalues and eigenvectors thus obtained. This
procedure is shown in Appendix B. For simplicity, let the initial values be KN (0) = KS (0) =1 and LN (0) = LS (0) = 1. Then, the initial values of scale-adjusted capital stock are alsokN (0) = kS (0) = 1. Case 2 is adopted for example. Figures 2 and 3 show the dynamics ofkN and kS , respectively: the horizontal axis measures t while the vertical axis denotes kN ,kS . Both figures show that starting from the initial value the scale-adjusted capital stock
converges to the steady state value. In Case 2, we have kN∗ = 16.2936 and kS∗ = 64.8417,
which might be counterintuitive because the capital stock in the developed North is smaller
than that in the developing South. However, kN and kS are scale-adjusted capital stock, andnot capital stock per capita. When, for example, t = 1000, the capital stock per capita in the
North and that in the South are given by KN /LN = 9.86 × 1027 and KS /LS = 1.78 × 1024 ,respectively, which clearly shows that KN /LN > KS /LS .
[Figures 2 and 3 to be inserted here]17) For numerical calculations, we use Mathematica 4 of Wolfram Research Inc.

16

6

Concluding remarks

In this paper we have united a traditional model of North-South trade and economic development and a modern model of endogenous growth. Using the model, we have investigated
how a rise in the growth rate of population affects the growth rate of real income per capita
in each country. The following are the main results: (i) an increase in the growth rateof population in the South decreases the growth rate of its own income per capita; (ii) a
rise in the growth rate of population in the North either increases or decreases the growthrate of its own income per capita depending on conditions; (iii) population growth in one
country raises the growth rate of income per capita in the other country; and (iv) even thedecreasing-returns South can experience a positive growth rate of income per capita if acontinuous improvement in the terms of trade is larger than a threshold value. As these results show, in developed countries the correlation between the growth rate of population andthat of income per capita can be either positive or negative while in developing countries
the correlation is negative. Therefore, empirical analysis lumping developed and developing
countries together will yield an ambiguous correlation between the two growth rates.
Let us add an explanation to the fourth result. It is true that the South can experience asustainable growth in per capita income depending on conditions. Even in this case, however, the growth rate of income per capita in the South is necessarily lower than that in the
North if the growth rate of population in the South is higher than that in the North. For thisreason, if the initial level of income per capita in the South is lower than that in the North, theSouth cannot catch up with the North in its income level. Therefore, uneven development
still remains though the Southern terms of trade continue to improve.
Our model does not consider international capital mobility. Along the BGP, if the ratesof time preference are equal in both countries, the rental rates of capital are equalized, thatis, rN∗ = rS∗ , so that capital does not move between the two countries. Off the BGP, however,
we have rN∗ , rS∗ even if ρN = ρS , and consequently capital can move between the countries
in search of higher returns. The rental rates of capital off the BGP depend on the scaleadjusted terms of trade, π(t). Therefore, in order to investigate capital movement, we haveto examine the dynamics of π(t) in detail. Analytical treatment of the capital movement will
be very difficult because the dynamics of π(t) are implicitly determined by the dynamics ofthe other variables, kN , kS , cNN , and cSN . In this case, numerical simulation will be useful.

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20

Appendix A
The following are elements of the Jacobian matrix evaluated the BGP:
"
#
cNN
1
1
γ
∂k̇N
∂π
−
= kN −β 1−β π 1−β
+
,
J11 =
∂kN
∂kN 1 − γ kN2
"
#
1 ∂π
1
∂k̇N
− 1−β
1−β
J12 =
= kN −β π
> 0,
∂kS
∂kS
#
"
1
1
∂k̇N
∂π
1
γ
−
J13 = N = kN −β 1−β π 1−β N −< 0,∂cN∂cN 1 − γ kN
"
#
1
1
∂k̇N
∂π
− 1−β
1−β
J14 = N = kN −β π
< 0,
∂cS
∂cSN
"
#
∂k̇S
−a−b ∂π= kS kS
> 0,
J21 =
∂kN
∂kN
"
#
∂k̇S
γ cSN
−a−b−1−a−b ∂π
J22 == kS (−a − b)kS+,
π + kS∂kS∂kS 1 − γ kS2"#∂k̇S−a−b ∂π
J23 = N = kS kS
> 0,
∂cN
∂cNN
"
#
∂k̇S
γ 1
−a−b ∂π
J24 = N = kS kS−,∂cS∂cSN 1 − γ kS"!
#
ββ−2 ∂π
∂ċNN
β
N
J31 =
= cN (1 − α − β)β 1−β −
π 1−β
< 0,
∂kN
1−β
∂kN
"
!
#
β
β−2 ∂π∂ċNN
βN= cN (1 − α − β)β 1−β −
π 1−β
J32 =
> 0,
∂kS
1−β
∂kS
"
!
#
β
β−2 ∂π
∂ċNN
β
N
1−β
1−β
π
< 0,
−
J33 = N = cN (1 − α − β)β
1−β
∂cN
∂cNN
"
!
#
β
β−2 ∂π∂ċNNβN
J34 = N = cN (1 − α − β)β 1−β −π 1−β N < 0,
1−β
∂cS
∂cS
"
#
N
∂ċ
∂π
> 0,
J41 = S = cSN (1 − a − b)kS−a−b
∂kN
∂kN
#
"
∂ċSN
N
−a−b−1−a−b ∂π
J42 == cS (−a − b)(1 − a − b)πkS< 0,+ (1 − a − b)kS
∂kS
∂kS
"
#
∂ċSN
N
−a−b ∂π
J43 = N = cS (1 − a − b)kS
> 0,
∂cN
∂cNN
#
"
∂ċSN
N
−a−b ∂π
J44 = N = cS (1 − a − b)kS
> 0.
∂cS
∂cSN

21

(49)
(50)
(51)
(52)
(53)
(54)
(55)
(56)
(57)
(58)
(59)
(60)
(61)
(62)
(63)
(64)

Appendix B
We solve the system of differential equations:
 
 


b12 b11  kN (t) − kN∗   


 


 kS (t) − kS∗ b21 = B1 eχ1 t   + B2 eχ2 t b22  ,b32 b31 cN (t) − cN∗ N  


 N



 
N∗ 
N
b42
b41
cS (t) − cS

(65)

where χ1 and χ2 denote negative eigenvalues, b11 , . . . , b41 eigenvectors corresponding to χ1 ,and b12 , . . . , b42 eigenvectors corresponding to χ2 . B1 and B2 are constants which should bedetermined by initial conditions.
In our model, the initial values of kN and kS are historically given but those of cNN and cSN
are determined so that they will be located on the saddle path. Let us show the procedure in
the following. At t = 0, the following relations hold:


 
 
 kN (0) − kN∗ b11 b12   ∗ 
b21   kS (0) − kS   + B2 b22  .
=
B
 N

1


b31 b32 cN (0) − cN∗N    N
cS (0) − cSN∗
b41
b42

(66)

Here, kN (0) and kS (0) are given exogenously and kN∗ , kS∗ , b11 , b12 , b21 , and b22 are also known
to us. From this, the two upper equations in equation (66) constitute a system of equations
with respect to B1 and B2 :kN (0) − kN∗ = B1 b11 + B2 b12 ,(67)

kS (0) − kS∗ = B1 b21 + B2 b22 .

(68)

Solving the equations above, we can determine B1 and B2 .
Substituting B1 and B2 obtained now into the two lower equations in equation (66) andN∗
considering that cN∗
N , cS , b31 , b32 , b41 , and b42 are already known, we obtain the system of
equations with respect to cNN (0) and cSN (0):cNN (0) − cN∗
N = B1 b31 + B2 b32 ,(69)

cSN (0) − cSN∗ = B1 b41 + B2 b42(70)

From this, cNN (0) and cSN (0) are determined.

22

Figures and Tables
RHS
LHS

O

π

π∗

Figure 1: Determination of the scale-adjusted terms of trade

16

60
50

14

40
12

30
20

10

10
200 400

600

800 1000 1200 1400
200 400 600 800 1000 1200 1400

Figure 2: Dynamics of kN

Figure 3: Dynamics of kS

23

Table 1: Four possible combinations of g∗yN , g∗yS , and ∂g∗yN /∂nN

Case 1Case 2
Case 3
Case 4

g∗yN
+
+
+
+

g∗yS
−
+
+
−

∂g∗yN /∂nN−+−+

Table 2: Numerical values of the parameters

Case 1Case 2
Case 3
Case 4

γ = 2/5, ρN = 1/50, ρS = 1/25, nN = 1/50, nS = 3/100
a = 1/3, b = 1/3, α = 1/3, β = 1/2
a = 1/3, b = 1/3, α = 1/2, β = 1/4
γ = 1/5, ρN = 1/50, ρS = 1/25, nN = 1/100, nS = 1/50
a = 1/2, b = 1/4, α = 1/4, β = 1/3
a = 1/4, b = 1/2, α = 1/3, β = 1/3

Table 3: Eigenvalues of the Jacobian matrix in Cases 1–4

Case 1Case 2
Case 3
Case 4

χ1−0.024 − 0.014i−0.074−0.023−0.020χ2−0.024 + 0.014i−0.0047−0.00058−0.00069

24

χ3
χ4
0.064 0.13
0.089 0.21
0.036 0.066
0.045 0.069