Description
The purpose of this paper is to examine global monetary economic growth with free trade.
It develops a multi-country monetary growth model with capital accumulation to provide some
insights into complexity of economic globalization with free trade and financial markets.
Journal of Financial Economic Policy
Global growth, international trade patterns and national inflation policies with capital
accumulation in a multi-country economy
Wei-Bin Zhang
Article information:
To cite this document:
Wei-Bin Zhang, (2010),"Global growth, international trade patterns and national inflation policies with capital
accumulation in a multi-country economy", J ournal of Financial Economic Policy, Vol. 2 Iss 1 pp. 60 - 79
Permanent link to this document:http://dx.doi.org/10.1108/17576381011055343
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Muhammad Tahir, Toseef Azid, (2015),"The relationship between international trade openness and
economic growth in the developing economies: Some new dimensions", J ournal of Chinese Economic and
Foreign Trade Studies, Vol. 8 Iss 2 pp. 123-139http://dx.doi.org/10.1108/J CEFTS-02-2015-0004
Daniel Lederman, (2013),"International trade and inclusive growth: a primer", Indian Growth and
Development Review, Vol. 6 Iss 1 pp. 88-112http://dx.doi.org/10.1108/17538251311329568
H. Aydin Okuyan, Alper Ozun, Erman Erbaykal, (2012),"Trade openness and economic growth: further
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Global growth, international trade
patterns and national in?ation
policies with capital accumulation
in a multi-country economy
Wei-Bin Zhang
Ritsumeikan Asia Paci?c University, Beppu-shi, Japan
Abstract
Purpose – The purpose of this paper is to examine global monetary economic growth with free trade.
It develops a multi-country monetary growth model with capital accumulation to provide some
insights into complexity of economic globalization with free trade and ?nancial markets.
Design/methodology/approach – The real aspects of the model is based on the neoclassical
growth theory and monetary aspects of the model are based on the money-in-utility approach.
The behavior of households is based on an alternative approach. The paper shows that the dynamics
of the J-country world economy can be described by 2J-dimensional differential equations.
Findings – This paper simulates equilibrium and motion of the global economy with three,
developed, newly industrializing, and developing countries and Cobb-Douglas production functions.
As the global monetary economic system is unstable, the perfectly competitive world economy may
either experience unlimited growth or economic crisis. Because of the choice of the initial conditions
and the parameters, the simulation demonstrates a situation of global economic declination. This
paper also demonstrates, for instance, that the global economy worsens off as the developed economy
reduces its propensity to save or increases its in?ation policy.
Social implications – This paper also tries to provide some possible implications of our model for
the recent economic crisis. A policy implication of the results is that as global economies with free
trade and ?nancial markets are possibly structurally unstable and the global economy may suffer
from economic declination, government interventions, and co-operation among countries are necessary
for global sustainable development.
Originality/value – The paper offers insights into the linkage between national monetary policies
and global economic growth.
Keywords Economic growth, Globalization, International trade, Capital growth, Money, In?ation
Paper type Research paper
1. Introduction
Since international capital markets have been liberalized about two decades ago,
capital ?ows among countries have increased greatly. There is now a historically
unprecedented percentage of the world’s ?nancial capital ?ows among nations
(Obstfeld and Taylor, 2004; Gozzi et al., 2009). Free capital transactions in capital and
?nance among national economies have been sometimes considered as growth
opportunities and sometimes perceived as sources of ?nancial stability and crises
(Stiglitz, 2000; Levin, 2001). As described by Tamborini (2010):
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1757-6385.htm
JEL classi?cation – F11, F30, O42
The author is grateful for the constructive comments of two anonymous referees and a
grant-in-aid from the Zengin Foundation for Studies on Economics and Finance.
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Vol. 2 No. 1, 2010
pp. 60-79
qEmerald Group Publishing Limited
1757-6385
DOI 10.1108/17576381011055343
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[. . .] since the turn of the century the most developed countries in the world have been prone
to repeated ?nancial crises of major order of magnitude and widespread macro-consequences.
While the depth, extent and duration of the latest event – the gigantic “subprime” mortgage
market failure in the USA – are still largely hidden from view, it is already clear that the
theoretical system of the Great Moderation has been seriously shaken by the abrupt demise of
those historical conditions.
Here, the age of great moderation is referred to the sustained growth and employment
with low and stable in?ation that blessed most of the world economy throughout the
1990s. The search for the reasons of these economic crises is still afoot. Evidently, it is
not easy to theoretically explain the issues related to economic growth and global
economic crisis as we need an analytical framework for multiple countries with
economic mechanisms of growth and ?nancial markets and determination of trade
patterns. As far as I know, there are only a few economic models, which can examine
global economic growth with money for any number of national economies on the basis
of microeconomic foundation for behavior of ?rms and individuals[1]. The purpose
of this study is to propose a multi-country monetary growth model with capital
accumulation and free trade in capital and goods. Our model is a synthesis of the
neoclassical international growth theory and the monetary growth model with the
money-in-utility (MIU) approach with an alternative approach to household behavior.
As early as 1984, Findlay (1984) pointed out that one topic that was almost entirely
absent from the pure theory of international trade was any consideration of the
connection between economic growth and international trade in classical literature of
economic theory. Almost all the trade models developed before the 1960s are static in the
sense that the supplies of factors of production are given and do not vary over time.
Trade models with capital movements are originated by MacDougall (1960) and Kemp
(1961), but these models were limited to static and one-commodity frameworks.
A dynamic model, which takes account of accumulating capital stocks is initially
developed by Oniki and Uzawa (1965) and others, in terms of the two-country, two-good,
two-factor model of trade. The Oniki-Uzawa model is developed within the framework of
neoclassical growth theory. The model is primarily concerned with the process of world
capital accumulation and distribution. Since the publication of Oniki and Uzawa’s paper
on theory of trade and economic growth, various trade models with endogenous capital
have been proposed Deardorff (1973), Ruf?n (1979), Findlay (1984), Frenkel and Razin
(1987), Eaton (1987), Brecher et al. (2002), Nishimura and Shimomra (2002) and Sorger
(2003). It is well-known that dynamic-optimization models with capital accumulation are
associated with analytical dif?culties. This study applies an alternative approach to
consumer behavior. As far as the real aspects of the model are concerned, this paper
studies capital accumulation and distribution with the neoclassical growth approach.
This study is concerned with effects of money on global economic growth. Many of
the most intriguing and important questions in dynamic economic analysis involve
money. It is generally agreed that modern analysis of dynamic interaction of in?ation
and capital formation begins with Tobin’s seminal contribution in 1965. Tobin (1965)
deals with an isolated economy in which “outside money” competes with real capital in
the portfolios of agents within the framework of the Solow model[2]. In Tobin’s
approach, a monetary economy has a real sector exactly like that in the Solow growth
model, so that the monetary nature of the model depends on how money is introduced
into the model. A monetary economy is characterized by that prices are expressed
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in money, transactions require money, and ?nancial wealth can be held in the form of
money or ?nancial instruments competing with money. Mundell (1968, Chapter 18)
proposes a model of international transmission effects of monetary and ?scal policy
shocks in a two-country version of what is now known as the Mundell-Fleming model.
The model shows that under ?oating exchange rates, positive monetary policy
innovations tend to have a “beggar-thy-neighbor” effect, raising domestic output and
reducing foreign output through the effects of real depreciation. On the other hand, ?scal
policy shocks tend to increase output in both countries. Extended versions of the model
have been frequently used to study problems of international macroeconomic policy
coordination. But it has been pointed that the Mundell-Fleming model (and many of its
extensions) failed to specify the underlying preferences and technology. To analyze
short-run macroeconomics in the open economy, Turnovsky (1979) move beyond
the Mundell-Fleming model toward a dynamic utility-maximizing framework[3]. We
introduce money into the international trade model with the MIU function approach.
The approach was used initially by Patinkin (1965), Sidrauski (1967a) and Friedman
(1969). In this approach, money is held because it yields some services and the way to
model it is to enter real balances directly into the utility function[4]. Sidrauski (1967b)
made a benchmark contribution to monetary economics, challenging Tobin’s
non-neutrality result. He found that money is superneutral in steady-state comparison
and changes in the in?ation rate have no effect on all the real variables in the economy[5].
Nevertheless, it has become evident that his results are dependent on the speci?c set-up
of the model. For instance, the choice of Ramsey’s version of an in?nite horizon economy
is essential for money to be superneutral. Moreover, the superneutrality in Sidrauski’s
model is no more held if leisure is introduced into the utility function. As observed by
Wang and Yip (1992), the direction of the non-superneutrality result is related to the
signs of the cross-partial derivatives of the utility function with respect to consumption,
leisure, and real balances.
This paper is to synthesize the neoclassical trade growth model and the monetary
growth theory with the MIUapproach within a compact framework. This paper can also
be considered as a generalization of a multi-regional growth model (without money) by
Zhang (2007) and monetary growth models with the MIU approach by Zhang (2009).
This paper is organized as follows. Section 2 de?nes the multi-country model with
money and capital accumulation. Section 3 shows that the dynamics of the world
economy with J countries can be described by 2J-dimensional differential equations. The
result of this section is proved in Appendix. Section 4 simulates the motion of the
three-country world economy. Section 5 examines the effects of changes in a country’s
technology, propensity to save, and in?ation policy. Section 6 concludes the study.
2. The multi-country trade model with capital accumulation
In describing economic production, we follow the neoclassical trade framework.
It is assumed that there is only one (durable) good in the global economy under
consideration[6]. Most aspects of production sectors in our model are similar to the
neo-classical one-sector growth model[7]. Production sectors or ?rms use capital and
labor. Exchanges take place in perfectly competitive markets. Production sectors sell
their product to households or to other sectors and households sell their labor and assets
to production sectors. Factor markets work well; factors are inelastically supplied and
the available factors are fully utilized at every moment. A national economy has two
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assets, domestic money and traded capital goods. The economy consists of consumers,
?rms and the government. The foreign price of traded goods is given in the world
market. The domestic residents may hold two assets, domestic money and traded goods.
We neglect transport cost, customs, or any other possible impediments to trade. We have
perfect mobility of goods. For each good the law of one price holds. We have absolute
purchasing power parity, which means that, measured in the same currency, the same
basket of goods costs the same at home and abroad.
The system consists of multiple countries, indexed by j ¼ 1, . . . ,J. Perfect competition
is assumedtoprevail ingoodmarkets bothwithineachcountryandbetweenthe countries,
and commodities are traded without any barriers such as transport costs or tariffs.
We assume that there is no migration between the countries and the labor markets are
perfectly competitive within each country. Each country has a ?xed labor force, N
j
,
( j ¼ 1, . . . ,J). Let prices be measured in terms of the commodity. We denote wage
and interest rates by w
j
(t) and r
j
(t), respectively, in the jth country. In the free trade
system, the interest rate is identical throughout the world economy, i.e. r(t) ¼ r
j
(t).
We assume that capital markets operate frictionless. The government levies no
taxes. Money is introduced by assuming that a central bank of country j distributes at
no cost to the population a per capita amount of ?at money, M
j
(t) . 0[8]. We assume
that each country’s money is a nontradable asset. The scheme according to which the
money stock evolves over time is deterministic and known to all agents. With m
j
being
the constant net growth rate of the money stock, M
j
(t) evolves over time according:
_
M
j
ðtÞm
j
M
j
ðtÞ; m
j
. 0:
At t the government brings m
j
M
j
(t) additional units of money per capita into circulation
in order to ?nance all government expenditures via seigniorage. Let m
j
(t) stand for
the real value of money per capita measured in units of the output good, that is,
m
j
(t) ¼ M
j
(t)/P
j
(t). The government expenditure in real terms per capita, t
j
(t), is given by:
t
j
ðtÞ ¼
_
M
j
ðtÞ
P
j
ðtÞ
¼
m
j
M
j
ðtÞ
P
j
ðtÞ
¼ m
j
m
j
ðtÞ:
The representative household receives m
j
m
j
(t) units of paper money fromthe government
through a “helicopter drop,” also considered to be independent of his money holdings.
First, we describe behavior of the production sections. We use production functions
to describe the physical facts of a given technology. We assume that there are only two
productive factors, capital K
j
(t) and labor N
j
at each point of time t. The production
functions are given by:
F
j
ðK
j
ðtÞ; N
j
Þ; j ¼ 1; . . . ; J ;
where F
j
are the output of country j. Assume F
j
to be neoclassical. We have:
f
j
ðtÞ ¼ f
j
ðk
j
ðtÞÞ; f
j
ðtÞ ;
F
j
ðtÞ
N
j
; k
j
ðtÞ ;
K
j
ðtÞ
N
j
:
The functions f
j
have the following properties:
.
f
j
(0) ¼ 0;
.
f
j
are increasing, strictly concave on R
þ
, and C
2
on R
þ þ
; f
0
j
ðk
j
Þ ¼ 0 and
f
00
j
ðkÞ , 0; and
.
lim
k
j
!0
f
0
j
ðk
j
Þ ¼ 1 and lim
k
j
!þ1
f
0
j
ðk
j
Þ ¼ 0:
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Markets are competitive; thus labor and capital earn their marginal products, and ?rms
earn zero pro?ts. The rate of interest r(t) and wage rates w
j
(t) are determined by
markets. Hence, for any individual ?rm r(t) and w
j
(t) are given at each point of time.
The production sector chooses the two variables K
j
(t) and N
j
to maximize its pro?t.
The marginal conditions are given by:
r þd
kj
¼ f
0
j
ðk
j
Þ; w
j
ðtÞ ¼ f
j
ðk
j
Þ 2k
j
f
0
j
ðk
j
Þ; ð1Þ
where d
kj
is the depreciation rate of physical capital in country j.
Different from the optimal growth theory in which utility de?ned over future
consumption streams is used, we do not explicitly specify how consumers depreciate
future utility resulted from consuming goods and services. We assume that we can ?nd
preference structure of consumers over consumption and saving at the current state.
We assume that we can observe each consumer’s preference structure over consumption
levels of goods, services, and saving, rather than an “aggregated utility” derived
from consuming services and goods over the future. This study uses the approach to
consumers’ behavior proposedbyZhang[9]. First, we note that inthe monetary economy,
the personal wealth, a
j
(t) is the sum of the real money, m
j
(t), and non-monetary wealth,
k
j
ðtÞ; held by the representative consumer in country j. Let p
j
(t) stand for the in?ation
rate. The current income per capita, y
j
(t), in country j is given by:
y
j
ðtÞ ¼ rðtÞ
k
j
ðtÞ þ w
j
ðtÞ 2p
j
ðtÞm
j
ðtÞ þm
j
m
j
ðtÞ; j ¼ 1; . . .J ; ð2Þ
where r
k
j
is the interest payment, w
j
is the wage payment, p
j
m
j
is the real cost of holding
money, and m
j
m
j
is the real value of paper money from the government. The disposable
income of the representative consumer is equal to the consumer’s current income and
wealth. That is:
^ y
j
ðtÞ ¼ y
j
ðtÞ þ a
j
ðtÞ: ð3Þ
We assume that selling and buying wealth can be conducted instantaneously without
any transaction cost. The total value of wealth that a consumer of country j can sell to
purchase goods and to save is obviously equal to a
j
(t). At each point of time, a consumer
distributes the total available budget among real money balances, m
j
(t), saving, s
j
(t),
and consumption of goods, c
j
(t). The budget constraint is given by:
ð1 þ rðtÞÞm
j
ðtÞ þ c
j
ðtÞ þ s
j
ðtÞ ¼ ^ y
j
ðtÞ:
From this equation and equations (2) and (3), we have:
ðp
j
þ rÞm
j
þ c
j
þ s
j
¼ y
j
; ð1 þ rÞ
k
j
þ w
j
þm
j
m
j
ð4Þ
where we omit time in the expression. We assume that utility is dependent on the saving,
the real money and consumption level. The inclusion of real money in the utility function
is base on the MIU approach[10]. The utility level of the typical consumer in country j is
represented by:
U
j
ðtÞ ¼ u
j
m
1
0j
j
ðtÞc
j
0j
j
ðtÞs
l
0j
j
ðtÞ; 1
0j
; j
0j
; l
0j
. 0; ð5Þ
in which 1
0j
, j
0j
, and l
0j
are a typical person’s elasticity of utility with regard to real
money balances, commodity and savings. We call 1
0j
, j
0j
and l
0j
propensities to hold
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money, to consume lot size, goods, and to hold wealth (save), respectively. Maximizing
U
j
(t) subject to budget constraints (4) yields:
ðp
j
þ rÞm
j
¼ 1
j
y
j
; c
j
¼ j
j
y
j
; s
j
¼ l
j
y
j
; ð6Þ
where:
1
j
; r
j
1
0j
; j
j
; r
j
j
0j
; l
j
; r
j
l
0j
; r
j
;
1
1
0j
þj
0j
þl
0j
:
According to the de?nitions of s
j
(t), the wealth accumulation of the representative person
in country j is given by:
_ a
j
ðtÞ ¼ s
j
ðtÞ 2a
j
ðtÞ: ð7Þ
This equation simply says that the change in wealth is equal to the saving minus the
dissaving. According to the de?nitions of the real money and in?ation rate, we have:
_ m
j
ðtÞ ¼ ðm
j
2p
j
ðtÞÞm
j
ðtÞ: ð8Þ
The total capital stocks employed by all the ?rms in the world is equal to the total
non-monetary wealth owned by all the countries. That is:
KðtÞ ¼
X
J
j¼1
K
j
ðtÞ ¼
X
J
j¼1
k
j
ðtÞN
j
: ð9Þ
We now describe trade balances of the countries. As money is held only by the domestic
households, we see that if K
j
2
k
j
N
j
. ð,Þ0, country j is in trade de?cit (trade surplus);
if K
j
2
k
j
N
j
¼ 0, country j is in trade balance. We introduce the following variables to
measure trade balances:
E
j
ðtÞ ; ð
k
j
ðtÞ 2k
j
ðtÞÞN
j
;
~
E
j
ðtÞ ; ð
k
j
ðtÞ 2k
j
ðtÞÞrðtÞN
j
:
We have thus built the model, which explains the endogenous accumulation of capital
and the international distribution of capital in the world economy in which the domestic
markets of each country are perfectly competitive, international product and capital
markets are freely mobile and labor is internationally immobile. We now examine the
properties of the system.
3. The world economic dynamics
In order to describe the world economic dynamics, it is necessary to show how to solve
the differential equations with given initial conditions. This section ?rst shows that the
dynamics of the world economy can be described by 2J-dimensional differential
equations. The following lemma is proved in the Appendix.
Lemma 1. The dynamics of the world economy is given by the following
2J-dimensional differential equations:
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_ m
j
¼ C
j
ðk
1
; {
k}; m
j
Þ; j ¼ 1; 2; . . . ; J
_
k
1
¼ C
1
ðk
1
; {
k}; m
1
Þ;
_
k
j
¼ C
j
ðk
1
; {
k}; m
j
Þ; j ¼ 2; . . . ; J ;
with m
j
(t), k
1
(t) and
k
j
ðtÞ as variables. For any given positive values of m
j
(t), k
1
(t), and
k
j
ðtÞ, at any point of time, the other variables are uniquely determined by the following
procedure: p
j
(t) by ð6Þ !
k
1
ðtÞ by Að4Þ !a
j
ðtÞ ¼
k
j
ðtÞ þ m
j
ðtÞ !k
j
ðtÞ; j ¼ 2; . . . ; J by
Að1Þ !f
j
¼ f
j
ðk
j
Þ !rðtÞ and w
j
(t) by ð1Þ ! y
j
ðtÞ by Að5Þ !c
j
ðtÞ and s
j
(t) by
Að6Þ !F
j
ðtÞ ¼ N
j
f
j
ðtÞ.
This result is important for us to analyze the dynamic properties of the world
economy. It guarantees that once the production functions are speci?ed and the initial
conditions are given, we can use computer to simulate the motion of the system.
Although we may analyze behavior of the 2J-dimensional differential equations, it is
dif?cult to explicitly interpret results. Following the computing procedure given in
Lemma 1, we will simulate the model to illustrate motion of the system. Before
simulating the motion of the system, we ?nd equilibrium of the dynamic system.
By equations (8) and (7), at equilibriumwe have s
j
¼ a
j
and p
j
¼ m
j
. By this and equation
(6), we have:
ðm
j
þ rÞm
j
¼ 1
j
y
j
: ð10Þ
From equations (6), (10) and s
j
¼ a
j
, we have:
ð1 2l
j
m
j
Þm
j
¼ ðl
j
þl
j
r 21Þ
k
j
þl
j
w
j
;
k
j
¼
l
j
1
j
ðm
j
þ rÞ 21
m
j
; ð11Þ
where we use a
j
¼
k
j
þ m
j
and the de?nition of y
j
. From equation (11), we solve:
m
j
¼ w
j
ðk
1
Þ ; l
j
þl
j
r 2l
j
m
j
2ðl
j
þl
j
r 21Þðm
j
þ rÞ
l
j
1
j
21
l
j
w
j
: ð12Þ
where we use the fact that r and w
j
are functions of k
1
. From equations (11) and (12),
we have:
k
j
¼ w
j
ðk
1
Þ ;
l
j
1
j
ðm
j
þ rÞ 21
w
j
ðk
1
Þ: ð13Þ
From equations (A3), (13) and (9), we have:
wðk
1
Þ ;
X
J
j¼1
f
j
ðk
1
ÞN
j
2
X
J
j¼1
w
j
ðk
1
ÞN
j
¼ 0: ð14Þ
Lemma 2. An equilibrium point of the dynamic world economy is be determined by
the following procedure: p
j
¼ m
j
!k
1
by ð14Þ !k
2
and k
3
by ðA
1
Þ !
k
j
by ð13Þ !m
j
by ð12Þ !a
j
¼
k
j
þ m
j
!f
j
¼ f
j
ðk
j
Þ !r and w
j
by ð1Þ ! y
j
by ðA5Þ !c
j
and s
j
by ð6Þ !F
j
¼ N
j
f
j
:
The number of equilibrium points is equal to the number of meaningful solutions of
nonlinear equation (14).
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4. Simulating motion of the world economy with three countries
As it is dif?cult to prove properties of the nonlinear dynamic system, we will follow the
procedure given in Lemma 1 to simulate motion of the trade system. We specify the
production functions as follows:
F
j
ðtÞ ¼ A
j
K
a
j
j
ðtÞN
b
j
j
; a
j
þb
j
¼ 1; a
j
; b
j
. 0; j ¼ 1; . . . ; J
where A
j
is country j’s productivity and a
j
is a positive parameter. From equation (A2)
and f
j
¼ A
j
k
a
j
j
; we have:
f
j
ðk
1
Þ ¼
a
1
A
1
k
2b
1
1
2d
j
a
j
A
j
!
21=b
j
; j ¼ 2; . . . ; J ;
f
j
ðk
1
Þ ¼ A
j
b
j
f
a
j
j
ðk
1
Þ; j ¼ 1; . . . ; J :
By equations (A4)-(A6), we obtain:
L ¼
X
J
j¼1
n
j
f
j
ðk
1
Þ 2
X
J
j¼2
n
j
k
j
;
L
1
¼ f
r
ðk
1
ÞL þ
f
1
ðk
1
Þ; L
j
¼ f
r
ðk
1
Þ
k
j
þ
f
j
ðk
1
Þ; j ¼ 2; . . . ; J ;
L
j
¼
1
j
L
j
m
j
2a
1
A
1
k
2b
1
1
þ m
j
; j ¼ 1; . . . ; J ;
where f
1
(k
1
) ¼ k
1
and f
r
ðk
1
Þ ¼ 1 þa
1
A
1
k
2b
1
1
2d
k1
: From equations (A7), (A9)
and (A11), we have:
_ m
j
¼ C
j
¼ ðm
j
2
L
j
Þm
j
;
_
k
j
¼
C
j
¼ l
j
L
j
þl
j
m
j
m
j
2
k
j
2m
j
2C
j
; j ¼ 2; . . . ; J ;
_
k
1
¼
C
1
¼
X
J
j¼2
n
j
C
j1
þl
1
L
1
þl
1
m
1
m
1
2L 2m
1
2C
1
!
X
J
j¼1
n
j
f
0
j
" #
21
:
ð15Þ
Using differential equations (15) and the initial conditions, we determine the values of
m
j
(t), k
1
(t), and
k
j
ðtÞ at any point of time. Then following the procedure in Lemma 1, we
can determine the values of all the other variables in the world economy. First
following Lemma 2, we try to ?nd equilibrium. We specify the parameters as follows:
N
1
N
2
N
3
0
B
B
B
@
1
C
C
C
A
¼
2
3
4
0
B
B
B
@
1
C
C
C
A
;
A
1
A
2
A
3
0
B
B
B
@
1
C
C
C
A
¼
8
4
2
0
B
B
B
@
1
C
C
C
A
;
l
01
l
02
l
03
0
B
B
B
@
1
C
C
C
A
¼
0:8
0:77
0:75
0
B
B
B
@
1
C
C
C
A
;
j
01
j
02
j
03
0
B
B
B
@
1
C
C
C
A
¼
0:10
0:13
0:15
0
B
B
B
@
1
C
C
C
A
;
1
01
1
02
1
03
0
B
B
B
@
1
C
C
C
A
¼
0:02
0:02
0:02
0
B
B
B
@
1
C
C
C
A
;
m
1
m
2
m
3
0
B
B
B
@
1
C
C
C
A
¼
0:03
0:04
0:05
0
B
B
B
@
1
C
C
C
A
;
a
1
a
2
a
3
0
B
B
B
@
1
C
C
C
A
¼
1=3
0:3
1=3
0
B
B
B
@
1
C
C
C
A
;
d
k1
d
k2
d
k3
0
B
B
B
@
1
C
C
C
A
¼
0:05
0:04
0:05
0
B
B
B
@
1
C
C
C
A
;
Country 1 has the highest level of productivity and highest propensity to save. Its
population is less than that of Country 2. Country 2’s level of productivity is the second,
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next to Country 1’s. Country 3 has the largest population and the lowest levels of
productivity and propensity to save. Countries 1, 2, and 3 in?ation policy parameters
are, respectively, 3, 4 and 5 percent. We term Country 1 as developed country, Country 2
as newly industrialized country, and Country 3 developing country. The three
economies have the equal propensity to hold money. As shown in Figure 1, equation
(14) has a unique positive solution.
An equilibrium point is given by setting _ m
j
¼ 0 and
_
k
j
¼ 0: In equilibrium point, we
have p
j
¼ m
j
: Following Lemma 2, we calculate the equilibrium values of the global
economy. The simulation results are summarized in (16):
F ¼ 117:15; C ¼ 39:99; K ¼ 270:65; r ¼ 0:089;
k
1
k
2
k
3
0
B
B
B
@
1
C
C
C
A
¼
84:51
19:79
10:56
0
B
B
B
@
1
C
C
C
A
;
f
1
f
2
f
3
0
B
B
B
@
1
C
C
C
A
¼
35:11
9:80
4:39
0
B
B
B
@
1
C
C
C
A
;
w
1
w
2
w
3
0
B
B
B
@
1
C
C
C
A
¼
23:41
6:86
2:93
0
B
B
B
@
1
C
C
C
A
;
k
1
k
2
k
3
0
B
B
B
@
1
C
C
C
A
¼
78:79
23:76
10:45
0
B
B
B
@
1
C
C
C
A
;
m
1
m
2
m
3
0
B
B
B
@
1
C
C
C
A
¼
21:07
6:02
2:49
0
B
B
B
@
1
C
C
C
A
;
c
1
c
2
c
3
0
B
B
B
@
1
C
C
C
A
¼
10:10
3:94
1:99
0
B
B
B
@
1
C
C
C
A
;
F
1
F
2
F
3
0
B
B
B
@
1
C
C
C
A
¼
70:21
29:39
17:55
0
B
B
B
@
1
C
C
C
A
;
K
1
K
2
K
3
0
B
B
B
@
1
C
C
C
A
¼
169:02
59:37
42:25
0
B
B
B
@
1
C
C
C
A
;
K
1
K
2
K
3
0
B
B
B
@
1
C
C
C
A
¼
157:58
71:28
41:28
0
B
B
B
@
1
C
C
C
A
;
E
1
E
2
E
3
0
B
B
B
@
1
C
C
C
A
¼
21:01
1:05
20:04
0
B
B
B
@
1
C
C
C
A
;
ð16Þ
Figure 1.
A unique solution
of equation (14)
18.5 19.0 19.5 20.0 20.5 21.0
6
4
2
2
4
6
j (k
1
)
k
1
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in which:
F ;
X
3
j¼1
F
j
; C ;
X
3
j¼1
C
j
; E
j
; rð
K
j
2K
j
Þ:
Countries 1 and 3 are in trade de?cit and Country 2 in trade surplus. The per-capita
levels of wealth and consumption and wage rate in the developed economy are much
higher than the corresponding variables in the developing economy. The differences
result from the developed country’s higher levels of productivity and the propensity to
save. To see differences in the income and wealth among the world economies, we
calculate the following variables:
^
N
1
^
N
2
^
N
3
0
B
B
B
@
1
C
C
C
A
¼
2=9
3=9
4=9
0
B
B
B
@
1
C
C
C
A
;
^
F
1
^
F
2
^
F
3
0
B
B
B
@
1
C
C
C
A
¼
59:9%
25:1%
15:0%
0
B
B
B
@
1
C
C
C
A
;
^
K
1
^
K
2
^
K
3
0
B
B
B
@
1
C
C
C
A
¼
62:5%
21:9%
15:6%
0
B
B
B
@
1
C
C
C
A
;
^
K
1
^
K
2
^
K
3
0
B
B
B
B
@
1
C
C
C
C
A
¼
58:2%
26:3%
15:4%
0
B
B
B
@
1
C
C
C
A
;
^
C
1
^
C
2
^
C
3
0
B
B
B
@
1
C
C
C
A
¼
50:5%
29:6%
19:9%
0
B
B
B
@
1
C
C
C
A
;
where a variable x
j
with circum?ex, ^ x
j
, denotes country j’s share of the corresponding
variable in the world economy. We see that irrespective of its small population size, the
global shares of the output, capital used, wealth, and consumption of the developed
economy are, respectively, 59.9, 58.2, 66.1, 58.2, and 50.5 percent. The developing
economy has 44.4 percent of the world population its global shares of output, capital
used, wealth, and consumption are, respectively, only 15, 15.6, 15.4, and 19.9 percent.
We examined the equilibrium structure of the global economy. It is important to
know the motion of global economy when it starts from a state away from the
equilibrium. As we have shown by Lemma 1 how to follow the dynamic processes,
it is straightforward to simulation the motion. We simulate the dynamics with the
parameter values speci?ed as the same for the results in equation (16) and the
following initial conditions:
m
1
ð0Þ ¼ 21; m
2
ð0Þ ¼ 7; m
3
ð0Þ ¼ 2:5; k
1
ð0Þ ¼ 80;
k
2
ð0Þ ¼ 15;
k
3
ð0Þ ¼ 6:
The simulation results are shown in Figure 2. We observe that the variables do not
approach to their equilibrium values over time. This occurs as the monetary economic
system is unstable[11]. The system does not converge to the equilibrium point and goes
economically downward. During the study period, all the national economies suffer as
time passes. The global output, capital and consumption all fall down over time. In a
well-connected global economy with free capital ?ows, all the countries are experiencing
the economic decline at the same time. Although our model is built on some strict
assumptions, our analysis shows the possibility of global economic crisis among the
countries with free trade. As trades among countries have become increasingly
liberated in the contemporary world and economic information have also become
increasingly available among the countries partly due to wide spread of computer,
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the main assumptions about highly aggregated variables in our model are perhaps
not very strict.
5. Comparative dynamic analysis
The previous section identi?es the unique equilibrium of the global economy and
demonstrates that the global economy is unstable. We demonstrated that the monetary
economic growth with globally liberalized capital markets may experience global crisis
without government intervention[12]. It is important to ask questions such as how a
developing economy like India or China may affect the global economy as its
technology is improved or population is enlarged; or how the global trade patterns may
be affected as technologies are further improved or propensities to save are increased
in developed economies like the USA or Japan. Many people are now concerned with
what will happen to the global economies if the US Government raises the in?ation
policy in the current economic crisis. This section examines impact of changes in some
parameters on dynamic processes of the global economic system.
First, we examine the case that all the parameters, except Country 1’s productivity,
A
1
, are the same as in equation (15). We increase the productivity level A
1
from 8 to 8.2.
Figure 2.
The motion of
the global economy
0.5 1.0 1.5 2.0
50
60
70
80
90
0.5 1.0 1.5 2.0
15
20
25
30
35
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
100
120
140
160
180
(a) (b) (c)
10
15
20
25
4
6
8
20
30
40
50
5
10
15
0.5
1.5
2.0
2.5
3.0
3.5
4.0
15
20
25
30
35
40
k
1
F
1
F
2
F
3
k
2
k
–
2
k
3
k
–
3
f
1
f
2
f
3
c
2
c
3
m
1
m
2
m
3
w
2
w
3
E
1
E
2
E
3
w
1
k
–
1
K
1
K
2
K
3
K
–
1
K
–
2
K
–
3
c
1
t t t
t t t
(d) (e) (f)
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
(g) (h) (i)
t t t
(j) (k) (l) (m)
t
t
t
t
0.5 1.0 1.5 2.0
0.12
0.14
0.16
0.18
0.5 1.0 1.5 2.0
4
2
2
4
6
0.5 1.0 1.5 2.0
20
30
35
0.5 1.0 1.5 2.0
150
200
250
r
C
K
F
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The simulation results are shown in Figure 3. In the plots, a variable
Dx
j
ðtÞ stands for
the change rate of the variable x
j
ðtÞ in percentage due to changes in the parameter
value from A
10
( ¼ 8 in this case) to A
1
( ¼ 8.2). That is:
Dx
j
ðtÞ ;
x
j
ðt; A
1
Þ 2x
j
ðt; A
10
Þ
jx
j
ðt; A
10
Þj
£ 100;
where x
j
(t; A
1
) stands for the value of the variable x
j
with the parameter value A
1
at
time t and x
j
(t; A
10
) stands for the value of the variable x
j
with the parameter value A
10
at time t. We will use the symbol
D with the same meaning when we analyze other
parameters. As the developed economy improves its technology, its total output,
product per worker and wage tend are increased over time; the corresponding variables
of the other two economies are reduced. The levels of real money of the three economies
are increased. Although the global total output is increased, the global capital stock
and consumption are reduced. The rate of interest is increased over time as the
developed economy improves its technology. If we interpret possible implications for
the contemporary global economies, this implies that a technological advance in,
for instance, the USA, will help the USA to be out of the economic crisis, but not be
helpful for the Japanese and Chinese economies.
Figure 3.
An improvement in
the developed country’s
productivity
0.5 1.0 1.5
0.5 1.0 1.5
2.0
2.0
1
1
2
0.5
0.5
1.0
1.5
2.0
2.5
0.5 1.0 1.5 2.0
1.5
1.0
0.5
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l) (m)
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
7
6
5
4
3
2
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
2.5
2.0
1.5
1.0
0.5
1.0
0.5 1.0 1.5 2.0
7
6
5
4
3
2
4
3
2
1
2.0
1.5
1.0
0.5
1.0
4
3
2
1
0.5 1.0 1.5 2.0
3.3
3.4
3.5
3.6
3.7
3.8
0.5 1.0 1.5 2.0
0.20
0.15
0.10
0.05
0.00
0.05
0.10
0.5 1.0 1.5 2.0
0.55
0.50
0.45
0.40
0.35
0.30
0.5 1.0 1.5 2.0
1.5
1.0
0.5
0.5
1.0
?
–
F
1
?
–
F
2
?
–
F
3
?
–
k
2
?
–
k
3
?
–
r
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?
–
m
2
?
–
m
3
?E
2
?E
1
?E
3
?
–
c
2
?
–
c
3
?
–
F ?
–
C
?
–
K
?
–
K
2
?
–
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?
–
K
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?
–
m
1
?
–
c
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?
–
k
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?
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k
–
1
?
–
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–
2
?
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k
–
3
?
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2
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It is interesting to note that as the developingeconomy improves its technology, the global
effects are different from the effects brought about by the technological change of the
developed economy. Figure 4 shows the effects of a rise of the developing economy’s total
productivity, A
3
, from2 to 2.1. We see that all the economies bene?t fromthe technological
change during the study period[13]. If we interpret possible implications for the
contemporary global economies, this implies that a technological advance in, for instance,
China or India, will help the global economy to be out of the economic crisis.
Another important question is how the global economy may be affected if the
developed economy saves less. Figure 5 show the effects of a fall in the developed
economy’s propensity to save, l
1
, from 0.8 to 0.75. We see that the global economy
worsen offs as the developed economy reduces its propensity to save. If we interpret
possible implications for the contemporary global economies, this implies that if the US
consumers save even less than before, it would be more dif?cult for the world to be out
of the economic crisis.
There is a large amount of theoretical work with disparate modeling approaches,
which demonstrates that ef?ciency in a monetary is inconsistent with in?ationary
policy (see for instance, Boel and Camera, 2009). Nevertheless, there are only a few
theoretical models of international monetary growth with capital accumulation to
Figure 4.
An improvement in the
developing country’s
productivity
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
0.5 1.0 1.5 2.0
0.5
0.5
1.0
1.5
0.5
1.0
1.5
2.0
(a) (b) (c)
(d) (e) (f)
(g)
(j) (k) (l) (m)
(h) (i)
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
0.5 1.0 1.5 2.0
0.2
0.2
0.4
0.6
0.5
1.0
1.5
5
10
15
2
4
6
8
10
5
10
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0.5 1.0 1.5 2.0
1.5
1.0
0.5
0.5 1.0 1.5 2.0
0.3
0.2
0.1
0.1
0.2
0.3
0.5 1.0 1.5 2.0
1.5
2.0
2.5
0.5 1.0 1.5 2.0
2.0
2.5
3.0
t
t
t
t
t
t
t t t
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study the role of in?ationary policies on national as well as international economies.
Stiglitz (2000) points out that ?nancial liberalization may trigger ?nancial instability,
which tends to be detrimental for both investment in physical capital and productivity.
Although we do not take account of holding of foreign money in our model, this does
not mean that national monetary policy will not affect global economies. In fact, as
money is not neutral for a national economy in our approach (Zhang, 2009), it is
reasonable to expect that a change in the national monetary policy will affect global
economies. We now examine effects of changes in the in?ation policy on the global
economy. Figure 6 show the effects of a rise in the developed economy’s in?ation policy
parameter, m
1
, from 0.03 to 0.04. We see that the global economy worsen offs (except
that each economy increases its real money holdings) as the developed economy
increases its in?ation policy. If we interpret possible implications for the contemporary
global economies, this implies that if the US Government prints more money, not to say
the other countries will not be able to be out of the economic crisis, the US economy will
suffer more in the near future.
6. Conclusions
This paper proposed a multi-country monetary growth model with capital
accumulation. We simulated the motion of model with the Cobb-Douglas production
Figure 5.
A reduction in the
developed country’s
propensity to save
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
30
25
20
15
10
5
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
25
20
15
10
5
30
25
20
15
10
5
(a) (b) (c)
(d) (e) (f)
(g)
(j) (k) (l) (m)
(h) (i)
t
t
t
25
20
15
10
5
10
5
25
20
15
10
5
25
20
15
10
5
10
8
6
4
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25
20
15
10
5
0.0 0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0
5
10
15
20
25
30
0.5 1.0 1.5 2.0
0.6
0.4
0.2
0.2
0.4
15
10
5
25
20
15
10
5
t
t
t
t
t
t
t
t
t
t
?
–
F
1
?
–
F
2
?
–
F
3
?
–
r
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?
–
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1
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–
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2
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3
?
–
m
1
?
–
m
2
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m
3
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c
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C
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k
1
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k
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?E
3
?E
2
?E
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k
–
1
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functions and demonstrated effects of changes in the parameters. As the global
monetary economic system is unstable, the world economy may either experience
unlimited growth or economic crisis. Our simulation demonstrated a situation of global
economic declination. It should be noted that the global economy without money will
converge to its long-term equilibrium. The introduction of endogenous money makes
the system unstable. We also carry out comparative dynamic analysis with regard to
some parameters. We demonstrated, for instance, that the global economy worsens off
as the developed economy reduces its propensity to save or increases its in?ation
policy. If we interpret possible implications for the contemporary global economies,
these conclusions imply that if the US Government prints more money or the US
consumers save less, not to say the other countries would not be able to be out of the
economic crisis, the US economy would suffer more in the near future.
Our model is built on the basis of the two main approaches in macroeconomics.
The global economic growth is due to the two economic forces, capital accumulation and
money, under national as well as international perfect competition in goods markets.
The model shows that the perfectly competitive growth economy is unstable and its
“great moderation” is not sustainable without government intervention in the long-term.
Indeed, monetary global economic growth with trade is complicated. There are many
other factors which we do not take account of in this initial stage for understanding
Figure 6.
An increase in the
developed country’s
in?ation policy
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
1.2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5
0.5
1.0
1.5
2.0
1.2
1.0
0.8
0.6
0.4
0.2
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l) (m)
1.0
0.8
0.6
0.4
0.2
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
t
t
t
t
t
t
1.0
0.8
0.6
0.4
0.2
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
0.5 1.0 1.5 2.0
0.02
0.01
0.00
0.01
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5
0.4
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
t
t
t
t
t
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c
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–
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?
–
F
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?
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= ?
–
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= ?
–
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= ?
–
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the mechanism of growth and trade pattern of monetary global economies. Our
approach dealt with only a simple situation as far as monetary economics and ?nancial
policies are concerned. It is signi?cant to take account of differences in speeds of
?nancial variables when examining global economic trade and growth (Obstfeld and
Rogoff, 1998; Buiter, 2007; Aghion et al., 2009). Adirection extension or generalization of
our approach is to examine global economic growth with different exchange rate
regimes. For instance, some economies, like China, are characterized by relatively
in?exible exchange rate systems, while some other economies, like Japan and the USA,
adopt relatively ?exible exchange rate systems. Evidently, it is important to take
account of the situation that households can hold foreign currencies. As addressed by
Coeurdacier et al. (2010), although capital ?ows are liberalized among OECD countries,
equity home bias remains sizeable (Goldberg and Tille, 2009; Patton, 2006). We did not
take account of the possibility of home bias (or bias for a special country) in our
approach. Further research should examine global as well as domestic implications of
the bias. We assumed that each country has one production sector and a homogeneous
population. As shown in Zhang (2009), it is conceptually not dif?cult to extend our model
to multiple sectors and heterogeneous households in each country, though some
analytical dif?culties may arise. Given the rich literature of contemporary ?nancial
economics, it is reasonable to expect that the effects of monetary policies may vary,
depending on the structures of international and national ?nancial institutions.
Notes
1. Most of the general equilibrium models with free international capital and bond markets are
concerned with endowment economies where there is production or capital accumulation.
It should be noted that there are empirical studies on issues related to trade, growth, and
?nancial markets. See for instance, Edison et al. (2002), Glick et al. (2006), Galindo et al. (2007)
and Bon?glioli (2008). Although these empirical studies identify relations among various
economic as well as institutional variables, most of these studies lack theoretical economic
foundation.
2. Outside money is the part of money stock, which is issued by the government.
3. Obstfeld and Rogoff (1998) also develop some other dynamic models for open economies.
4. Eden (2005, Chapter 2) for the reasons why money is introduced into the utility function.
5. Superneutrality of money means that the growth rate of money has no effect on the real
equilibrium.
6. The assumption of single homogeneous goods is well-accepted in the neoclassical growth
trade theory. See, for instance, Ikeda and Ono (1992).
7. The neoclassical growth theory is referred to Burmeister and Dobell (1970).
8. The government may distribute the money in various ways and there are different
mechanisms for the government to issue money. As systematically demonstrated by Zhang
(2009), it is possible to build the model according to speci?ed characters of the monetary
system. For instance, an important way for the government to intervene the markets is
through controlling nominal rates of interest.
9. A detailed explanation of the approach and its relations to the traditional approaches to
consumer behavior is referred to Zhang (2009).
10. For the literature on the MIU approach, see for instance, Wang and Yip (1992), Turnovsky
(2000) and Zhang (2009).
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11. As shown in Zhang (2009), the model for a single national economy (which can be interpreted
as the case that multiple economies are identical in all aspects) has a saddle point. The author
tried to simulate the motion with various combinations of initial conditions and found out
that system does not converge to the equilibrium point.
12. As demonstrated in Zhang (2008), if the role of money is neglected and the global economy
includes only “real variables,” then the global economic growth model is characterized of a
unique stable equilibrium. The introduction of money causes instabilities in the trade model.
It should be remarked that in future research, we may introduce governments’ intervention
in international trade or/and ?nancial markets.
13. The insights obtained here should be interpreted with caution as the system is unstable.
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Tamborini, R. (2010), “Monetary policy with investment-saving imbalances”, Metroeconomica,
March, 4.
Tobin, J. (1965), “Money and economic growth”, Econometrica, Vol. 33, pp. 671-84.
Turnovsky, S.J. (1979), “Optimal monetary and ?scal policies in an open dynamic economy”,
Scandinavian Journal of Economics, Vol. 71, pp. 400-14.
Turnovsky, S.J. (2000), Methods of Macroeconomic Dynamics, MIT Press, Cambridge, MA.
Wang, P. and Yip, C.K. (1992), “Examining the long-run effect on economic growth”,
Journal of Macroeconomics, Vol. 14, pp. 359-69.
Capital
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Zhang, W.B. (2007), “Growth and agglomeration of a small-open multi-regional economy”,
The Journal of Economic Integration, Vol. 22, pp. 515-49.
Zhang, W.B. (2008), International Trade Theory: Capital, Knowledge, Economic Structure,
Money and Prices over Time and Space, Springer, Berlin.
Zhang, W.B. (2009), Monetary Growth Theory: Money, Interest, Prices, Capital, Knowledge,
and Economic Structure over Time and Space, Routledge, London.
Further reading
Bekaert, G., Harvey, C. and Lundblad, C. (2005), “Does ?nancial liberalization spur growth?”,
Journal of Financial Economics, Vol. 77, pp. 3-55.
Dornbusch, R. (1976), “Expectations and exchange rate dynamics”, Journal of Political Economy,
Vol. 84, pp. 1161-76.
Appendix
The Appendix proves Lemma 1. First, from equation (1) we obtain:
f
0
j
ðk
j
Þ ¼ f
0
1
ðk
1
Þ 2d
j
; j ; 2; . . . ; J ; ðA1Þ
where d
j
; d
k1
2d
kj
: If f
0
1
ðk
1
Þ 2d
j
. 0 for all j ¼ 2, . . . , J and given k
1
(t) . 0, then the equations
determine unique relations between k
j
and k
1
, denoted by:
k
j
¼ f
j
ðk
1
Þ; j ¼ 1; . . . ; J ; ðA2Þ
where f
1
(k
1
) ¼ k
1
. From equations (A1), we have:
f
00
j
ðk
j
Þ
dk
j
dk
1
¼ f
00
1
ðk
1
Þ; j ¼ 2; . . . ; J :
As f
"
j
ðk
j
Þ # 0; j ¼ 1; . . . ; J ; we see that dk
j
=dk
1
$ 0; j ¼ 2; . . . ; J : That is, f
0
j
ðk
1
Þ $ 0:
Hence, for any given k
1
ðtÞ . 0; we determine k
j
ðtÞ; j ¼ 2; . . . ; J as unique functions of k
1
(t).
From equation (1), we determine the wage rates as functions of k
1
(t) as follows:
w
j
ðtÞ ¼
f
j
ðk
1
Þ ; f
j
ðf
j
ðk
1
ÞÞ 2f
j
ðk
1
Þf
0
j
ðf
j
ðk
1
ÞÞ; j ¼ 1; . . . ; J : ðA3Þ
We can rewrite equation (9) as:
X
J
j¼1
k
j
ðtÞN
j
¼
X
J
j¼1
k
j
ðtÞN
j
:
Insert equation (A2) into the above equation:
k
1
ðtÞ ¼ Lðk
1
; {
k}Þ ;
X
J
j¼1
n
j
f
j
ðk
1
Þ 2
X
J
j¼2
n
j
k
j
; ðA4Þ
in which n
j
; N
j
/N
1
and {
kðtÞ} ; ð
k
2
ðtÞ; . . . ;
k
J
ðtÞÞ: We see that Country 1’s per capita physical
wealth,
k
1
ðtÞ; can be expressed as a unique function of Country 1’s capital intensity and the other
countries’ per capita physical wealth {
kðtÞ} at any point of time.
From equations (1) and (A3) and the de?nitions of y
j
; we have:
y
j
¼ L
j
ðk
1
; {
k}Þ þm
j
m
j
; j ¼ 1; . . .J ; ðA5Þ
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where:
L
1
ðk
1
; {
k}Þ ; f
r
ðk
1
ÞLðk
1
; {
k}Þ þ
f
1
ðk
1
Þ;
L
j
ðk
1
; {
k}Þ ; f
r
ðk
1
Þ
k
j
þ
f
j
ðk
1
Þ; j ¼ 2; . . .J ;
f
r
ðk
1
Þ ; 1 þ f
0
1
ðk
1
Þ 2d
k1
:
From equations (A5) and ðp
j
þ rÞm
j
¼ 1
j
y
j
in equations (6), we have:
p
j
¼
L
j
ðk
1
; {
k}; m
j
Þ ;
1
j
L
j
ðk
1
; {
k}Þ
m
j
2f
0
1
ðk
1
Þ þ m
j
; j ¼ 1; . . . ; J ; ðA6Þ
where m
j
; 1
j
m
j
þd
k1
: These equations show that a country’s in?ation rate can be expressed as
a function of the global distribution of capital stocks and its real money per capita. Substituting
equation (A6) into equation (8) yields:
_ m
j
¼ C
j
ðk
1
; {
k}; m
j
Þ ; ðm
j
2
L
j
ðk
1
; {
k}; m
j
ÞÞm
j
: ðA7Þ
Insert s
j
¼ l
j
y
j
and equation (A5) in equation (7):
_
k
1
¼ l
1
L
1
ðk
1
; {
k}Þ þl
1
m
1
m
1
2Lðk
1
; {
k}Þ 2m
1
2C
1
ðk
1
; {
k}; m
1
Þ; ðA8Þ
_
k
j
¼ C
j
ðk
1
; {
k}; m
j
Þ ; l
j
L
j
þl
j
m
j
m
j
2
k
j
2m
j
2C
j
; j ¼ 2; . . . ; J ; ðA9Þ
where we also use equation (A4). Taking derivatives of equation (A4) with respect to time yields:
_
k
1
¼
X
J
j¼1
n
j
f
0
j
ðk
1
Þ
" #
_
k
1
2
X
J
j¼2
n
j
C
j
ðk
1
; {
k}; m
j
Þ; ðA10Þ
where, we use equations (A9). Equalizing the right-hand sides of equations (A8) and (A10) yields:
_
k
1
¼ C
1
ðk
1
; {
k}; m
1
Þ ;
X
J
j¼2
n
j
C
j1
þl
1
L
1
þl
1
m
1
m
1
2L2m
1
2C
1
!
X
J
j¼1
n
j
f
0
j
" #
21
: ðA11Þ
In summary, we have thus proved Lemma 1.
About the author
Wei-Bin Zhang PhD (Sweden) is a Professor in Ritsumeikan Asia Paci?c University, Japan.
His main research ?elds are nonlinear economic dynamics, growth theory, trade theory,
East Asian economic development, Chinese philosophy, and ethics. He has published more than
100 academic articles and authorized 20 academic books in English by international publishing
houses. Wei-Bin Zhang can be contacted at: [email protected]
Capital
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79
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doc_402556452.pdf
The purpose of this paper is to examine global monetary economic growth with free trade.
It develops a multi-country monetary growth model with capital accumulation to provide some
insights into complexity of economic globalization with free trade and financial markets.
Journal of Financial Economic Policy
Global growth, international trade patterns and national inflation policies with capital
accumulation in a multi-country economy
Wei-Bin Zhang
Article information:
To cite this document:
Wei-Bin Zhang, (2010),"Global growth, international trade patterns and national inflation policies with capital
accumulation in a multi-country economy", J ournal of Financial Economic Policy, Vol. 2 Iss 1 pp. 60 - 79
Permanent link to this document:http://dx.doi.org/10.1108/17576381011055343
Downloaded on: 24 January 2016, At: 21:37 (PT)
References: this document contains references to 44 other documents.
To copy this document: [email protected]
The fulltext of this document has been downloaded 210 times since 2010*
Users who downloaded this article also downloaded:
Muhammad Tahir, Toseef Azid, (2015),"The relationship between international trade openness and
economic growth in the developing economies: Some new dimensions", J ournal of Chinese Economic and
Foreign Trade Studies, Vol. 8 Iss 2 pp. 123-139http://dx.doi.org/10.1108/J CEFTS-02-2015-0004
Daniel Lederman, (2013),"International trade and inclusive growth: a primer", Indian Growth and
Development Review, Vol. 6 Iss 1 pp. 88-112http://dx.doi.org/10.1108/17538251311329568
H. Aydin Okuyan, Alper Ozun, Erman Erbaykal, (2012),"Trade openness and economic growth: further
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Global growth, international trade
patterns and national in?ation
policies with capital accumulation
in a multi-country economy
Wei-Bin Zhang
Ritsumeikan Asia Paci?c University, Beppu-shi, Japan
Abstract
Purpose – The purpose of this paper is to examine global monetary economic growth with free trade.
It develops a multi-country monetary growth model with capital accumulation to provide some
insights into complexity of economic globalization with free trade and ?nancial markets.
Design/methodology/approach – The real aspects of the model is based on the neoclassical
growth theory and monetary aspects of the model are based on the money-in-utility approach.
The behavior of households is based on an alternative approach. The paper shows that the dynamics
of the J-country world economy can be described by 2J-dimensional differential equations.
Findings – This paper simulates equilibrium and motion of the global economy with three,
developed, newly industrializing, and developing countries and Cobb-Douglas production functions.
As the global monetary economic system is unstable, the perfectly competitive world economy may
either experience unlimited growth or economic crisis. Because of the choice of the initial conditions
and the parameters, the simulation demonstrates a situation of global economic declination. This
paper also demonstrates, for instance, that the global economy worsens off as the developed economy
reduces its propensity to save or increases its in?ation policy.
Social implications – This paper also tries to provide some possible implications of our model for
the recent economic crisis. A policy implication of the results is that as global economies with free
trade and ?nancial markets are possibly structurally unstable and the global economy may suffer
from economic declination, government interventions, and co-operation among countries are necessary
for global sustainable development.
Originality/value – The paper offers insights into the linkage between national monetary policies
and global economic growth.
Keywords Economic growth, Globalization, International trade, Capital growth, Money, In?ation
Paper type Research paper
1. Introduction
Since international capital markets have been liberalized about two decades ago,
capital ?ows among countries have increased greatly. There is now a historically
unprecedented percentage of the world’s ?nancial capital ?ows among nations
(Obstfeld and Taylor, 2004; Gozzi et al., 2009). Free capital transactions in capital and
?nance among national economies have been sometimes considered as growth
opportunities and sometimes perceived as sources of ?nancial stability and crises
(Stiglitz, 2000; Levin, 2001). As described by Tamborini (2010):
The current issue and full text archive of this journal is available at
www.emeraldinsight.com/1757-6385.htm
JEL classi?cation – F11, F30, O42
The author is grateful for the constructive comments of two anonymous referees and a
grant-in-aid from the Zengin Foundation for Studies on Economics and Finance.
JFEP
2,1
60
Journal of Financial Economic Policy
Vol. 2 No. 1, 2010
pp. 60-79
qEmerald Group Publishing Limited
1757-6385
DOI 10.1108/17576381011055343
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[. . .] since the turn of the century the most developed countries in the world have been prone
to repeated ?nancial crises of major order of magnitude and widespread macro-consequences.
While the depth, extent and duration of the latest event – the gigantic “subprime” mortgage
market failure in the USA – are still largely hidden from view, it is already clear that the
theoretical system of the Great Moderation has been seriously shaken by the abrupt demise of
those historical conditions.
Here, the age of great moderation is referred to the sustained growth and employment
with low and stable in?ation that blessed most of the world economy throughout the
1990s. The search for the reasons of these economic crises is still afoot. Evidently, it is
not easy to theoretically explain the issues related to economic growth and global
economic crisis as we need an analytical framework for multiple countries with
economic mechanisms of growth and ?nancial markets and determination of trade
patterns. As far as I know, there are only a few economic models, which can examine
global economic growth with money for any number of national economies on the basis
of microeconomic foundation for behavior of ?rms and individuals[1]. The purpose
of this study is to propose a multi-country monetary growth model with capital
accumulation and free trade in capital and goods. Our model is a synthesis of the
neoclassical international growth theory and the monetary growth model with the
money-in-utility (MIU) approach with an alternative approach to household behavior.
As early as 1984, Findlay (1984) pointed out that one topic that was almost entirely
absent from the pure theory of international trade was any consideration of the
connection between economic growth and international trade in classical literature of
economic theory. Almost all the trade models developed before the 1960s are static in the
sense that the supplies of factors of production are given and do not vary over time.
Trade models with capital movements are originated by MacDougall (1960) and Kemp
(1961), but these models were limited to static and one-commodity frameworks.
A dynamic model, which takes account of accumulating capital stocks is initially
developed by Oniki and Uzawa (1965) and others, in terms of the two-country, two-good,
two-factor model of trade. The Oniki-Uzawa model is developed within the framework of
neoclassical growth theory. The model is primarily concerned with the process of world
capital accumulation and distribution. Since the publication of Oniki and Uzawa’s paper
on theory of trade and economic growth, various trade models with endogenous capital
have been proposed Deardorff (1973), Ruf?n (1979), Findlay (1984), Frenkel and Razin
(1987), Eaton (1987), Brecher et al. (2002), Nishimura and Shimomra (2002) and Sorger
(2003). It is well-known that dynamic-optimization models with capital accumulation are
associated with analytical dif?culties. This study applies an alternative approach to
consumer behavior. As far as the real aspects of the model are concerned, this paper
studies capital accumulation and distribution with the neoclassical growth approach.
This study is concerned with effects of money on global economic growth. Many of
the most intriguing and important questions in dynamic economic analysis involve
money. It is generally agreed that modern analysis of dynamic interaction of in?ation
and capital formation begins with Tobin’s seminal contribution in 1965. Tobin (1965)
deals with an isolated economy in which “outside money” competes with real capital in
the portfolios of agents within the framework of the Solow model[2]. In Tobin’s
approach, a monetary economy has a real sector exactly like that in the Solow growth
model, so that the monetary nature of the model depends on how money is introduced
into the model. A monetary economy is characterized by that prices are expressed
Capital
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in money, transactions require money, and ?nancial wealth can be held in the form of
money or ?nancial instruments competing with money. Mundell (1968, Chapter 18)
proposes a model of international transmission effects of monetary and ?scal policy
shocks in a two-country version of what is now known as the Mundell-Fleming model.
The model shows that under ?oating exchange rates, positive monetary policy
innovations tend to have a “beggar-thy-neighbor” effect, raising domestic output and
reducing foreign output through the effects of real depreciation. On the other hand, ?scal
policy shocks tend to increase output in both countries. Extended versions of the model
have been frequently used to study problems of international macroeconomic policy
coordination. But it has been pointed that the Mundell-Fleming model (and many of its
extensions) failed to specify the underlying preferences and technology. To analyze
short-run macroeconomics in the open economy, Turnovsky (1979) move beyond
the Mundell-Fleming model toward a dynamic utility-maximizing framework[3]. We
introduce money into the international trade model with the MIU function approach.
The approach was used initially by Patinkin (1965), Sidrauski (1967a) and Friedman
(1969). In this approach, money is held because it yields some services and the way to
model it is to enter real balances directly into the utility function[4]. Sidrauski (1967b)
made a benchmark contribution to monetary economics, challenging Tobin’s
non-neutrality result. He found that money is superneutral in steady-state comparison
and changes in the in?ation rate have no effect on all the real variables in the economy[5].
Nevertheless, it has become evident that his results are dependent on the speci?c set-up
of the model. For instance, the choice of Ramsey’s version of an in?nite horizon economy
is essential for money to be superneutral. Moreover, the superneutrality in Sidrauski’s
model is no more held if leisure is introduced into the utility function. As observed by
Wang and Yip (1992), the direction of the non-superneutrality result is related to the
signs of the cross-partial derivatives of the utility function with respect to consumption,
leisure, and real balances.
This paper is to synthesize the neoclassical trade growth model and the monetary
growth theory with the MIUapproach within a compact framework. This paper can also
be considered as a generalization of a multi-regional growth model (without money) by
Zhang (2007) and monetary growth models with the MIU approach by Zhang (2009).
This paper is organized as follows. Section 2 de?nes the multi-country model with
money and capital accumulation. Section 3 shows that the dynamics of the world
economy with J countries can be described by 2J-dimensional differential equations. The
result of this section is proved in Appendix. Section 4 simulates the motion of the
three-country world economy. Section 5 examines the effects of changes in a country’s
technology, propensity to save, and in?ation policy. Section 6 concludes the study.
2. The multi-country trade model with capital accumulation
In describing economic production, we follow the neoclassical trade framework.
It is assumed that there is only one (durable) good in the global economy under
consideration[6]. Most aspects of production sectors in our model are similar to the
neo-classical one-sector growth model[7]. Production sectors or ?rms use capital and
labor. Exchanges take place in perfectly competitive markets. Production sectors sell
their product to households or to other sectors and households sell their labor and assets
to production sectors. Factor markets work well; factors are inelastically supplied and
the available factors are fully utilized at every moment. A national economy has two
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assets, domestic money and traded capital goods. The economy consists of consumers,
?rms and the government. The foreign price of traded goods is given in the world
market. The domestic residents may hold two assets, domestic money and traded goods.
We neglect transport cost, customs, or any other possible impediments to trade. We have
perfect mobility of goods. For each good the law of one price holds. We have absolute
purchasing power parity, which means that, measured in the same currency, the same
basket of goods costs the same at home and abroad.
The system consists of multiple countries, indexed by j ¼ 1, . . . ,J. Perfect competition
is assumedtoprevail ingoodmarkets bothwithineachcountryandbetweenthe countries,
and commodities are traded without any barriers such as transport costs or tariffs.
We assume that there is no migration between the countries and the labor markets are
perfectly competitive within each country. Each country has a ?xed labor force, N
j
,
( j ¼ 1, . . . ,J). Let prices be measured in terms of the commodity. We denote wage
and interest rates by w
j
(t) and r
j
(t), respectively, in the jth country. In the free trade
system, the interest rate is identical throughout the world economy, i.e. r(t) ¼ r
j
(t).
We assume that capital markets operate frictionless. The government levies no
taxes. Money is introduced by assuming that a central bank of country j distributes at
no cost to the population a per capita amount of ?at money, M
j
(t) . 0[8]. We assume
that each country’s money is a nontradable asset. The scheme according to which the
money stock evolves over time is deterministic and known to all agents. With m
j
being
the constant net growth rate of the money stock, M
j
(t) evolves over time according:
_
M
j
ðtÞm
j
M
j
ðtÞ; m
j
. 0:
At t the government brings m
j
M
j
(t) additional units of money per capita into circulation
in order to ?nance all government expenditures via seigniorage. Let m
j
(t) stand for
the real value of money per capita measured in units of the output good, that is,
m
j
(t) ¼ M
j
(t)/P
j
(t). The government expenditure in real terms per capita, t
j
(t), is given by:
t
j
ðtÞ ¼
_
M
j
ðtÞ
P
j
ðtÞ
¼
m
j
M
j
ðtÞ
P
j
ðtÞ
¼ m
j
m
j
ðtÞ:
The representative household receives m
j
m
j
(t) units of paper money fromthe government
through a “helicopter drop,” also considered to be independent of his money holdings.
First, we describe behavior of the production sections. We use production functions
to describe the physical facts of a given technology. We assume that there are only two
productive factors, capital K
j
(t) and labor N
j
at each point of time t. The production
functions are given by:
F
j
ðK
j
ðtÞ; N
j
Þ; j ¼ 1; . . . ; J ;
where F
j
are the output of country j. Assume F
j
to be neoclassical. We have:
f
j
ðtÞ ¼ f
j
ðk
j
ðtÞÞ; f
j
ðtÞ ;
F
j
ðtÞ
N
j
; k
j
ðtÞ ;
K
j
ðtÞ
N
j
:
The functions f
j
have the following properties:
.
f
j
(0) ¼ 0;
.
f
j
are increasing, strictly concave on R
þ
, and C
2
on R
þ þ
; f
0
j
ðk
j
Þ ¼ 0 and
f
00
j
ðkÞ , 0; and
.
lim
k
j
!0
f
0
j
ðk
j
Þ ¼ 1 and lim
k
j
!þ1
f
0
j
ðk
j
Þ ¼ 0:
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Markets are competitive; thus labor and capital earn their marginal products, and ?rms
earn zero pro?ts. The rate of interest r(t) and wage rates w
j
(t) are determined by
markets. Hence, for any individual ?rm r(t) and w
j
(t) are given at each point of time.
The production sector chooses the two variables K
j
(t) and N
j
to maximize its pro?t.
The marginal conditions are given by:
r þd
kj
¼ f
0
j
ðk
j
Þ; w
j
ðtÞ ¼ f
j
ðk
j
Þ 2k
j
f
0
j
ðk
j
Þ; ð1Þ
where d
kj
is the depreciation rate of physical capital in country j.
Different from the optimal growth theory in which utility de?ned over future
consumption streams is used, we do not explicitly specify how consumers depreciate
future utility resulted from consuming goods and services. We assume that we can ?nd
preference structure of consumers over consumption and saving at the current state.
We assume that we can observe each consumer’s preference structure over consumption
levels of goods, services, and saving, rather than an “aggregated utility” derived
from consuming services and goods over the future. This study uses the approach to
consumers’ behavior proposedbyZhang[9]. First, we note that inthe monetary economy,
the personal wealth, a
j
(t) is the sum of the real money, m
j
(t), and non-monetary wealth,
k
j
ðtÞ; held by the representative consumer in country j. Let p
j
(t) stand for the in?ation
rate. The current income per capita, y
j
(t), in country j is given by:
y
j
ðtÞ ¼ rðtÞ
k
j
ðtÞ þ w
j
ðtÞ 2p
j
ðtÞm
j
ðtÞ þm
j
m
j
ðtÞ; j ¼ 1; . . .J ; ð2Þ
where r
k
j
is the interest payment, w
j
is the wage payment, p
j
m
j
is the real cost of holding
money, and m
j
m
j
is the real value of paper money from the government. The disposable
income of the representative consumer is equal to the consumer’s current income and
wealth. That is:
^ y
j
ðtÞ ¼ y
j
ðtÞ þ a
j
ðtÞ: ð3Þ
We assume that selling and buying wealth can be conducted instantaneously without
any transaction cost. The total value of wealth that a consumer of country j can sell to
purchase goods and to save is obviously equal to a
j
(t). At each point of time, a consumer
distributes the total available budget among real money balances, m
j
(t), saving, s
j
(t),
and consumption of goods, c
j
(t). The budget constraint is given by:
ð1 þ rðtÞÞm
j
ðtÞ þ c
j
ðtÞ þ s
j
ðtÞ ¼ ^ y
j
ðtÞ:
From this equation and equations (2) and (3), we have:
ðp
j
þ rÞm
j
þ c
j
þ s
j
¼ y
j
; ð1 þ rÞ
k
j
þ w
j
þm
j
m
j
ð4Þ
where we omit time in the expression. We assume that utility is dependent on the saving,
the real money and consumption level. The inclusion of real money in the utility function
is base on the MIU approach[10]. The utility level of the typical consumer in country j is
represented by:
U
j
ðtÞ ¼ u
j
m
1
0j
j
ðtÞc
j
0j
j
ðtÞs
l
0j
j
ðtÞ; 1
0j
; j
0j
; l
0j
. 0; ð5Þ
in which 1
0j
, j
0j
, and l
0j
are a typical person’s elasticity of utility with regard to real
money balances, commodity and savings. We call 1
0j
, j
0j
and l
0j
propensities to hold
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money, to consume lot size, goods, and to hold wealth (save), respectively. Maximizing
U
j
(t) subject to budget constraints (4) yields:
ðp
j
þ rÞm
j
¼ 1
j
y
j
; c
j
¼ j
j
y
j
; s
j
¼ l
j
y
j
; ð6Þ
where:
1
j
; r
j
1
0j
; j
j
; r
j
j
0j
; l
j
; r
j
l
0j
; r
j
;
1
1
0j
þj
0j
þl
0j
:
According to the de?nitions of s
j
(t), the wealth accumulation of the representative person
in country j is given by:
_ a
j
ðtÞ ¼ s
j
ðtÞ 2a
j
ðtÞ: ð7Þ
This equation simply says that the change in wealth is equal to the saving minus the
dissaving. According to the de?nitions of the real money and in?ation rate, we have:
_ m
j
ðtÞ ¼ ðm
j
2p
j
ðtÞÞm
j
ðtÞ: ð8Þ
The total capital stocks employed by all the ?rms in the world is equal to the total
non-monetary wealth owned by all the countries. That is:
KðtÞ ¼
X
J
j¼1
K
j
ðtÞ ¼
X
J
j¼1
k
j
ðtÞN
j
: ð9Þ
We now describe trade balances of the countries. As money is held only by the domestic
households, we see that if K
j
2
k
j
N
j
. ð,Þ0, country j is in trade de?cit (trade surplus);
if K
j
2
k
j
N
j
¼ 0, country j is in trade balance. We introduce the following variables to
measure trade balances:
E
j
ðtÞ ; ð
k
j
ðtÞ 2k
j
ðtÞÞN
j
;
~
E
j
ðtÞ ; ð
k
j
ðtÞ 2k
j
ðtÞÞrðtÞN
j
:
We have thus built the model, which explains the endogenous accumulation of capital
and the international distribution of capital in the world economy in which the domestic
markets of each country are perfectly competitive, international product and capital
markets are freely mobile and labor is internationally immobile. We now examine the
properties of the system.
3. The world economic dynamics
In order to describe the world economic dynamics, it is necessary to show how to solve
the differential equations with given initial conditions. This section ?rst shows that the
dynamics of the world economy can be described by 2J-dimensional differential
equations. The following lemma is proved in the Appendix.
Lemma 1. The dynamics of the world economy is given by the following
2J-dimensional differential equations:
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_ m
j
¼ C
j
ðk
1
; {
k}; m
j
Þ; j ¼ 1; 2; . . . ; J
_
k
1
¼ C
1
ðk
1
; {
k}; m
1
Þ;
_
k
j
¼ C
j
ðk
1
; {
k}; m
j
Þ; j ¼ 2; . . . ; J ;
with m
j
(t), k
1
(t) and
k
j
ðtÞ as variables. For any given positive values of m
j
(t), k
1
(t), and
k
j
ðtÞ, at any point of time, the other variables are uniquely determined by the following
procedure: p
j
(t) by ð6Þ !
k
1
ðtÞ by Að4Þ !a
j
ðtÞ ¼
k
j
ðtÞ þ m
j
ðtÞ !k
j
ðtÞ; j ¼ 2; . . . ; J by
Að1Þ !f
j
¼ f
j
ðk
j
Þ !rðtÞ and w
j
(t) by ð1Þ ! y
j
ðtÞ by Að5Þ !c
j
ðtÞ and s
j
(t) by
Að6Þ !F
j
ðtÞ ¼ N
j
f
j
ðtÞ.
This result is important for us to analyze the dynamic properties of the world
economy. It guarantees that once the production functions are speci?ed and the initial
conditions are given, we can use computer to simulate the motion of the system.
Although we may analyze behavior of the 2J-dimensional differential equations, it is
dif?cult to explicitly interpret results. Following the computing procedure given in
Lemma 1, we will simulate the model to illustrate motion of the system. Before
simulating the motion of the system, we ?nd equilibrium of the dynamic system.
By equations (8) and (7), at equilibriumwe have s
j
¼ a
j
and p
j
¼ m
j
. By this and equation
(6), we have:
ðm
j
þ rÞm
j
¼ 1
j
y
j
: ð10Þ
From equations (6), (10) and s
j
¼ a
j
, we have:
ð1 2l
j
m
j
Þm
j
¼ ðl
j
þl
j
r 21Þ
k
j
þl
j
w
j
;
k
j
¼
l
j
1
j
ðm
j
þ rÞ 21
m
j
; ð11Þ
where we use a
j
¼
k
j
þ m
j
and the de?nition of y
j
. From equation (11), we solve:
m
j
¼ w
j
ðk
1
Þ ; l
j
þl
j
r 2l
j
m
j
2ðl
j
þl
j
r 21Þðm
j
þ rÞ
l
j
1
j
21
l
j
w
j
: ð12Þ
where we use the fact that r and w
j
are functions of k
1
. From equations (11) and (12),
we have:
k
j
¼ w
j
ðk
1
Þ ;
l
j
1
j
ðm
j
þ rÞ 21
w
j
ðk
1
Þ: ð13Þ
From equations (A3), (13) and (9), we have:
wðk
1
Þ ;
X
J
j¼1
f
j
ðk
1
ÞN
j
2
X
J
j¼1
w
j
ðk
1
ÞN
j
¼ 0: ð14Þ
Lemma 2. An equilibrium point of the dynamic world economy is be determined by
the following procedure: p
j
¼ m
j
!k
1
by ð14Þ !k
2
and k
3
by ðA
1
Þ !
k
j
by ð13Þ !m
j
by ð12Þ !a
j
¼
k
j
þ m
j
!f
j
¼ f
j
ðk
j
Þ !r and w
j
by ð1Þ ! y
j
by ðA5Þ !c
j
and s
j
by ð6Þ !F
j
¼ N
j
f
j
:
The number of equilibrium points is equal to the number of meaningful solutions of
nonlinear equation (14).
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4. Simulating motion of the world economy with three countries
As it is dif?cult to prove properties of the nonlinear dynamic system, we will follow the
procedure given in Lemma 1 to simulate motion of the trade system. We specify the
production functions as follows:
F
j
ðtÞ ¼ A
j
K
a
j
j
ðtÞN
b
j
j
; a
j
þb
j
¼ 1; a
j
; b
j
. 0; j ¼ 1; . . . ; J
where A
j
is country j’s productivity and a
j
is a positive parameter. From equation (A2)
and f
j
¼ A
j
k
a
j
j
; we have:
f
j
ðk
1
Þ ¼
a
1
A
1
k
2b
1
1
2d
j
a
j
A
j
!
21=b
j
; j ¼ 2; . . . ; J ;
f
j
ðk
1
Þ ¼ A
j
b
j
f
a
j
j
ðk
1
Þ; j ¼ 1; . . . ; J :
By equations (A4)-(A6), we obtain:
L ¼
X
J
j¼1
n
j
f
j
ðk
1
Þ 2
X
J
j¼2
n
j
k
j
;
L
1
¼ f
r
ðk
1
ÞL þ
f
1
ðk
1
Þ; L
j
¼ f
r
ðk
1
Þ
k
j
þ
f
j
ðk
1
Þ; j ¼ 2; . . . ; J ;
L
j
¼
1
j
L
j
m
j
2a
1
A
1
k
2b
1
1
þ m
j
; j ¼ 1; . . . ; J ;
where f
1
(k
1
) ¼ k
1
and f
r
ðk
1
Þ ¼ 1 þa
1
A
1
k
2b
1
1
2d
k1
: From equations (A7), (A9)
and (A11), we have:
_ m
j
¼ C
j
¼ ðm
j
2
L
j
Þm
j
;
_
k
j
¼
C
j
¼ l
j
L
j
þl
j
m
j
m
j
2
k
j
2m
j
2C
j
; j ¼ 2; . . . ; J ;
_
k
1
¼
C
1
¼
X
J
j¼2
n
j
C
j1
þl
1
L
1
þl
1
m
1
m
1
2L 2m
1
2C
1
!
X
J
j¼1
n
j
f
0
j
" #
21
:
ð15Þ
Using differential equations (15) and the initial conditions, we determine the values of
m
j
(t), k
1
(t), and
k
j
ðtÞ at any point of time. Then following the procedure in Lemma 1, we
can determine the values of all the other variables in the world economy. First
following Lemma 2, we try to ?nd equilibrium. We specify the parameters as follows:
N
1
N
2
N
3
0
B
B
B
@
1
C
C
C
A
¼
2
3
4
0
B
B
B
@
1
C
C
C
A
;
A
1
A
2
A
3
0
B
B
B
@
1
C
C
C
A
¼
8
4
2
0
B
B
B
@
1
C
C
C
A
;
l
01
l
02
l
03
0
B
B
B
@
1
C
C
C
A
¼
0:8
0:77
0:75
0
B
B
B
@
1
C
C
C
A
;
j
01
j
02
j
03
0
B
B
B
@
1
C
C
C
A
¼
0:10
0:13
0:15
0
B
B
B
@
1
C
C
C
A
;
1
01
1
02
1
03
0
B
B
B
@
1
C
C
C
A
¼
0:02
0:02
0:02
0
B
B
B
@
1
C
C
C
A
;
m
1
m
2
m
3
0
B
B
B
@
1
C
C
C
A
¼
0:03
0:04
0:05
0
B
B
B
@
1
C
C
C
A
;
a
1
a
2
a
3
0
B
B
B
@
1
C
C
C
A
¼
1=3
0:3
1=3
0
B
B
B
@
1
C
C
C
A
;
d
k1
d
k2
d
k3
0
B
B
B
@
1
C
C
C
A
¼
0:05
0:04
0:05
0
B
B
B
@
1
C
C
C
A
;
Country 1 has the highest level of productivity and highest propensity to save. Its
population is less than that of Country 2. Country 2’s level of productivity is the second,
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next to Country 1’s. Country 3 has the largest population and the lowest levels of
productivity and propensity to save. Countries 1, 2, and 3 in?ation policy parameters
are, respectively, 3, 4 and 5 percent. We term Country 1 as developed country, Country 2
as newly industrialized country, and Country 3 developing country. The three
economies have the equal propensity to hold money. As shown in Figure 1, equation
(14) has a unique positive solution.
An equilibrium point is given by setting _ m
j
¼ 0 and
_
k
j
¼ 0: In equilibrium point, we
have p
j
¼ m
j
: Following Lemma 2, we calculate the equilibrium values of the global
economy. The simulation results are summarized in (16):
F ¼ 117:15; C ¼ 39:99; K ¼ 270:65; r ¼ 0:089;
k
1
k
2
k
3
0
B
B
B
@
1
C
C
C
A
¼
84:51
19:79
10:56
0
B
B
B
@
1
C
C
C
A
;
f
1
f
2
f
3
0
B
B
B
@
1
C
C
C
A
¼
35:11
9:80
4:39
0
B
B
B
@
1
C
C
C
A
;
w
1
w
2
w
3
0
B
B
B
@
1
C
C
C
A
¼
23:41
6:86
2:93
0
B
B
B
@
1
C
C
C
A
;
k
1
k
2
k
3
0
B
B
B
@
1
C
C
C
A
¼
78:79
23:76
10:45
0
B
B
B
@
1
C
C
C
A
;
m
1
m
2
m
3
0
B
B
B
@
1
C
C
C
A
¼
21:07
6:02
2:49
0
B
B
B
@
1
C
C
C
A
;
c
1
c
2
c
3
0
B
B
B
@
1
C
C
C
A
¼
10:10
3:94
1:99
0
B
B
B
@
1
C
C
C
A
;
F
1
F
2
F
3
0
B
B
B
@
1
C
C
C
A
¼
70:21
29:39
17:55
0
B
B
B
@
1
C
C
C
A
;
K
1
K
2
K
3
0
B
B
B
@
1
C
C
C
A
¼
169:02
59:37
42:25
0
B
B
B
@
1
C
C
C
A
;
K
1
K
2
K
3
0
B
B
B
@
1
C
C
C
A
¼
157:58
71:28
41:28
0
B
B
B
@
1
C
C
C
A
;
E
1
E
2
E
3
0
B
B
B
@
1
C
C
C
A
¼
21:01
1:05
20:04
0
B
B
B
@
1
C
C
C
A
;
ð16Þ
Figure 1.
A unique solution
of equation (14)
18.5 19.0 19.5 20.0 20.5 21.0
6
4
2
2
4
6
j (k
1
)
k
1
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2
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6
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in which:
F ;
X
3
j¼1
F
j
; C ;
X
3
j¼1
C
j
; E
j
; rð
K
j
2K
j
Þ:
Countries 1 and 3 are in trade de?cit and Country 2 in trade surplus. The per-capita
levels of wealth and consumption and wage rate in the developed economy are much
higher than the corresponding variables in the developing economy. The differences
result from the developed country’s higher levels of productivity and the propensity to
save. To see differences in the income and wealth among the world economies, we
calculate the following variables:
^
N
1
^
N
2
^
N
3
0
B
B
B
@
1
C
C
C
A
¼
2=9
3=9
4=9
0
B
B
B
@
1
C
C
C
A
;
^
F
1
^
F
2
^
F
3
0
B
B
B
@
1
C
C
C
A
¼
59:9%
25:1%
15:0%
0
B
B
B
@
1
C
C
C
A
;
^
K
1
^
K
2
^
K
3
0
B
B
B
@
1
C
C
C
A
¼
62:5%
21:9%
15:6%
0
B
B
B
@
1
C
C
C
A
;
^
K
1
^
K
2
^
K
3
0
B
B
B
B
@
1
C
C
C
C
A
¼
58:2%
26:3%
15:4%
0
B
B
B
@
1
C
C
C
A
;
^
C
1
^
C
2
^
C
3
0
B
B
B
@
1
C
C
C
A
¼
50:5%
29:6%
19:9%
0
B
B
B
@
1
C
C
C
A
;
where a variable x
j
with circum?ex, ^ x
j
, denotes country j’s share of the corresponding
variable in the world economy. We see that irrespective of its small population size, the
global shares of the output, capital used, wealth, and consumption of the developed
economy are, respectively, 59.9, 58.2, 66.1, 58.2, and 50.5 percent. The developing
economy has 44.4 percent of the world population its global shares of output, capital
used, wealth, and consumption are, respectively, only 15, 15.6, 15.4, and 19.9 percent.
We examined the equilibrium structure of the global economy. It is important to
know the motion of global economy when it starts from a state away from the
equilibrium. As we have shown by Lemma 1 how to follow the dynamic processes,
it is straightforward to simulation the motion. We simulate the dynamics with the
parameter values speci?ed as the same for the results in equation (16) and the
following initial conditions:
m
1
ð0Þ ¼ 21; m
2
ð0Þ ¼ 7; m
3
ð0Þ ¼ 2:5; k
1
ð0Þ ¼ 80;
k
2
ð0Þ ¼ 15;
k
3
ð0Þ ¼ 6:
The simulation results are shown in Figure 2. We observe that the variables do not
approach to their equilibrium values over time. This occurs as the monetary economic
system is unstable[11]. The system does not converge to the equilibrium point and goes
economically downward. During the study period, all the national economies suffer as
time passes. The global output, capital and consumption all fall down over time. In a
well-connected global economy with free capital ?ows, all the countries are experiencing
the economic decline at the same time. Although our model is built on some strict
assumptions, our analysis shows the possibility of global economic crisis among the
countries with free trade. As trades among countries have become increasingly
liberated in the contemporary world and economic information have also become
increasingly available among the countries partly due to wide spread of computer,
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the main assumptions about highly aggregated variables in our model are perhaps
not very strict.
5. Comparative dynamic analysis
The previous section identi?es the unique equilibrium of the global economy and
demonstrates that the global economy is unstable. We demonstrated that the monetary
economic growth with globally liberalized capital markets may experience global crisis
without government intervention[12]. It is important to ask questions such as how a
developing economy like India or China may affect the global economy as its
technology is improved or population is enlarged; or how the global trade patterns may
be affected as technologies are further improved or propensities to save are increased
in developed economies like the USA or Japan. Many people are now concerned with
what will happen to the global economies if the US Government raises the in?ation
policy in the current economic crisis. This section examines impact of changes in some
parameters on dynamic processes of the global economic system.
First, we examine the case that all the parameters, except Country 1’s productivity,
A
1
, are the same as in equation (15). We increase the productivity level A
1
from 8 to 8.2.
Figure 2.
The motion of
the global economy
0.5 1.0 1.5 2.0
50
60
70
80
90
0.5 1.0 1.5 2.0
15
20
25
30
35
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
100
120
140
160
180
(a) (b) (c)
10
15
20
25
4
6
8
20
30
40
50
5
10
15
0.5
1.5
2.0
2.5
3.0
3.5
4.0
15
20
25
30
35
40
k
1
F
1
F
2
F
3
k
2
k
–
2
k
3
k
–
3
f
1
f
2
f
3
c
2
c
3
m
1
m
2
m
3
w
2
w
3
E
1
E
2
E
3
w
1
k
–
1
K
1
K
2
K
3
K
–
1
K
–
2
K
–
3
c
1
t t t
t t t
(d) (e) (f)
0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
(g) (h) (i)
t t t
(j) (k) (l) (m)
t
t
t
t
0.5 1.0 1.5 2.0
0.12
0.14
0.16
0.18
0.5 1.0 1.5 2.0
4
2
2
4
6
0.5 1.0 1.5 2.0
20
30
35
0.5 1.0 1.5 2.0
150
200
250
r
C
K
F
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The simulation results are shown in Figure 3. In the plots, a variable
Dx
j
ðtÞ stands for
the change rate of the variable x
j
ðtÞ in percentage due to changes in the parameter
value from A
10
( ¼ 8 in this case) to A
1
( ¼ 8.2). That is:
Dx
j
ðtÞ ;
x
j
ðt; A
1
Þ 2x
j
ðt; A
10
Þ
jx
j
ðt; A
10
Þj
£ 100;
where x
j
(t; A
1
) stands for the value of the variable x
j
with the parameter value A
1
at
time t and x
j
(t; A
10
) stands for the value of the variable x
j
with the parameter value A
10
at time t. We will use the symbol
D with the same meaning when we analyze other
parameters. As the developed economy improves its technology, its total output,
product per worker and wage tend are increased over time; the corresponding variables
of the other two economies are reduced. The levels of real money of the three economies
are increased. Although the global total output is increased, the global capital stock
and consumption are reduced. The rate of interest is increased over time as the
developed economy improves its technology. If we interpret possible implications for
the contemporary global economies, this implies that a technological advance in,
for instance, the USA, will help the USA to be out of the economic crisis, but not be
helpful for the Japanese and Chinese economies.
Figure 3.
An improvement in
the developed country’s
productivity
0.5 1.0 1.5
0.5 1.0 1.5
2.0
2.0
1
1
2
0.5
0.5
1.0
1.5
2.0
2.5
0.5 1.0 1.5 2.0
1.5
1.0
0.5
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l) (m)
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
7
6
5
4
3
2
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
2.5
2.0
1.5
1.0
0.5
1.0
0.5 1.0 1.5 2.0
7
6
5
4
3
2
4
3
2
1
2.0
1.5
1.0
0.5
1.0
4
3
2
1
0.5 1.0 1.5 2.0
3.3
3.4
3.5
3.6
3.7
3.8
0.5 1.0 1.5 2.0
0.20
0.15
0.10
0.05
0.00
0.05
0.10
0.5 1.0 1.5 2.0
0.55
0.50
0.45
0.40
0.35
0.30
0.5 1.0 1.5 2.0
1.5
1.0
0.5
0.5
1.0
?
–
F
1
?
–
F
2
?
–
F
3
?
–
k
2
?
–
k
3
?
–
r
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?
–
m
2
?
–
m
3
?E
2
?E
1
?E
3
?
–
c
2
?
–
c
3
?
–
F ?
–
C
?
–
K
?
–
K
2
?
–
K
3
?
–
K
1
?
–
m
1
?
–
c
1
?
–
k
1
?
–
k
–
1
?
–
k
–
2
?
–
k
–
3
?
–
K
–
1
?
–
K
–
2
?
–
K
–
3
t
t
t
t
t
t
t
t
t
t
t
t
t
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It is interesting to note that as the developingeconomy improves its technology, the global
effects are different from the effects brought about by the technological change of the
developed economy. Figure 4 shows the effects of a rise of the developing economy’s total
productivity, A
3
, from2 to 2.1. We see that all the economies bene?t fromthe technological
change during the study period[13]. If we interpret possible implications for the
contemporary global economies, this implies that a technological advance in, for instance,
China or India, will help the global economy to be out of the economic crisis.
Another important question is how the global economy may be affected if the
developed economy saves less. Figure 5 show the effects of a fall in the developed
economy’s propensity to save, l
1
, from 0.8 to 0.75. We see that the global economy
worsen offs as the developed economy reduces its propensity to save. If we interpret
possible implications for the contemporary global economies, this implies that if the US
consumers save even less than before, it would be more dif?cult for the world to be out
of the economic crisis.
There is a large amount of theoretical work with disparate modeling approaches,
which demonstrates that ef?ciency in a monetary is inconsistent with in?ationary
policy (see for instance, Boel and Camera, 2009). Nevertheless, there are only a few
theoretical models of international monetary growth with capital accumulation to
Figure 4.
An improvement in the
developing country’s
productivity
0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
2.0
0.5 1.0 1.5 2.0
0.5
0.5
1.0
1.5
0.5
1.0
1.5
2.0
(a) (b) (c)
(d) (e) (f)
(g)
(j) (k) (l) (m)
(h) (i)
0.0 0.5 1.0 1.5 2.0
0.0 0.5 1.0 1.5 2.0
0.5
1.0
1.5
0.5 1.0 1.5 2.0
0.2
0.2
0.4
0.6
0.5
1.0
1.5
5
10
15
2
4
6
8
10
5
10
15
0.5 1.0 1.5 2.0
1.5
1.0
0.5
0.5 1.0 1.5 2.0
0.3
0.2
0.1
0.1
0.2
0.3
0.5 1.0 1.5 2.0
1.5
2.0
2.5
0.5 1.0 1.5 2.0
2.0
2.5
3.0
t
t
t
t
t
t
t t t
t
t
t
t
?
–
F
1
?
–
F
2
?
–
F
3
?
–
r
?
–
C ?
–
K
?
–
F
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?E
1
?E
2
?E
3
?
–
K
1
?
–
K
2
?
–
K
3
?
–
m
1
?
–
m
2
?
–
m
3
?
–
c
1
?
–
c
2
?
–
c
3
?
–
k
1
?
–
k
2
?
–
k
3
?
–
k
–
1
?
–
k
–
2
?
–
k
–
3
?
–
K
–
1
?
–
K
–
2
?
–
K
–
3
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study the role of in?ationary policies on national as well as international economies.
Stiglitz (2000) points out that ?nancial liberalization may trigger ?nancial instability,
which tends to be detrimental for both investment in physical capital and productivity.
Although we do not take account of holding of foreign money in our model, this does
not mean that national monetary policy will not affect global economies. In fact, as
money is not neutral for a national economy in our approach (Zhang, 2009), it is
reasonable to expect that a change in the national monetary policy will affect global
economies. We now examine effects of changes in the in?ation policy on the global
economy. Figure 6 show the effects of a rise in the developed economy’s in?ation policy
parameter, m
1
, from 0.03 to 0.04. We see that the global economy worsen offs (except
that each economy increases its real money holdings) as the developed economy
increases its in?ation policy. If we interpret possible implications for the contemporary
global economies, this implies that if the US Government prints more money, not to say
the other countries will not be able to be out of the economic crisis, the US economy will
suffer more in the near future.
6. Conclusions
This paper proposed a multi-country monetary growth model with capital
accumulation. We simulated the motion of model with the Cobb-Douglas production
Figure 5.
A reduction in the
developed country’s
propensity to save
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
30
25
20
15
10
5
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
25
20
15
10
5
30
25
20
15
10
5
(a) (b) (c)
(d) (e) (f)
(g)
(j) (k) (l) (m)
(h) (i)
t
t
t
25
20
15
10
5
10
5
25
20
15
10
5
25
20
15
10
5
10
8
6
4
4
25
20
15
10
5
0.0 0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0
5
10
15
20
25
30
0.5 1.0 1.5 2.0
0.6
0.4
0.2
0.2
0.4
15
10
5
25
20
15
10
5
t
t
t
t
t
t
t
t
t
t
?
–
F
1
?
–
F
2
?
–
F
3
?
–
r
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?
–
K
1
?
–
K
2
?
–
K
3
?
–
m
1
?
–
m
2
?
–
m
3
?
–
c
1
?
–
c
2
?
–
c
3
?
–
C
?
–
K
?
–
F
?
–
k
1
?
–
k
2
?
–
k
3
?E
3
?E
2
?E
1
?
–
k
–
1
?
–
k
–
2
?
–
k
–
3
?
–
K
–
1
?
–
K
–
2
?
–
K
–
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functions and demonstrated effects of changes in the parameters. As the global
monetary economic system is unstable, the world economy may either experience
unlimited growth or economic crisis. Our simulation demonstrated a situation of global
economic declination. It should be noted that the global economy without money will
converge to its long-term equilibrium. The introduction of endogenous money makes
the system unstable. We also carry out comparative dynamic analysis with regard to
some parameters. We demonstrated, for instance, that the global economy worsens off
as the developed economy reduces its propensity to save or increases its in?ation
policy. If we interpret possible implications for the contemporary global economies,
these conclusions imply that if the US Government prints more money or the US
consumers save less, not to say the other countries would not be able to be out of the
economic crisis, the US economy would suffer more in the near future.
Our model is built on the basis of the two main approaches in macroeconomics.
The global economic growth is due to the two economic forces, capital accumulation and
money, under national as well as international perfect competition in goods markets.
The model shows that the perfectly competitive growth economy is unstable and its
“great moderation” is not sustainable without government intervention in the long-term.
Indeed, monetary global economic growth with trade is complicated. There are many
other factors which we do not take account of in this initial stage for understanding
Figure 6.
An increase in the
developed country’s
in?ation policy
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
1.2
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5
0.5
1.0
1.5
2.0
1.2
1.0
0.8
0.6
0.4
0.2
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l) (m)
1.0
0.8
0.6
0.4
0.2
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
t
t
t
t
t
t
1.0
0.8
0.6
0.4
0.2
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
0.5 1.0 1.5 2.0
0.0
0.2
0.4
0.6
0.8
0.5 1.0 1.5 2.0
0.02
0.01
0.00
0.01
0.5 1.0 1.5 2.0
0.5 1.0 1.5 2.0
0.5
0.4
0.3
0.2
0.1
0.1
1.0
0.8
0.6
0.4
0.2
t
t
t
t
t
t
t
3
c
?
–
F
1
?
–
F
2
?
–
F
3
?
–
f
1
= ?
–
w
1
?
–
f
2
= ?
–
w
2
?
–
f
3
= ?
–
w
3
?
–
K
1
?
–
K
2
?
–
K
3
?
–
K
?
–
F
?
–
m
1
?
–
m
2
?
–
m
3
?
–
C
?
–
c
1
?
–
c
2
?
–
k
1
?
–
k
2
?
–
k
3
?E
2
?E
3
?E
1
?
–
r
?
–
k
–
1
?
–
k
–
2
?
–
k
–
3
?
–
K
–
1
?
–
K
–
2
?
–
K
–
3
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the mechanism of growth and trade pattern of monetary global economies. Our
approach dealt with only a simple situation as far as monetary economics and ?nancial
policies are concerned. It is signi?cant to take account of differences in speeds of
?nancial variables when examining global economic trade and growth (Obstfeld and
Rogoff, 1998; Buiter, 2007; Aghion et al., 2009). Adirection extension or generalization of
our approach is to examine global economic growth with different exchange rate
regimes. For instance, some economies, like China, are characterized by relatively
in?exible exchange rate systems, while some other economies, like Japan and the USA,
adopt relatively ?exible exchange rate systems. Evidently, it is important to take
account of the situation that households can hold foreign currencies. As addressed by
Coeurdacier et al. (2010), although capital ?ows are liberalized among OECD countries,
equity home bias remains sizeable (Goldberg and Tille, 2009; Patton, 2006). We did not
take account of the possibility of home bias (or bias for a special country) in our
approach. Further research should examine global as well as domestic implications of
the bias. We assumed that each country has one production sector and a homogeneous
population. As shown in Zhang (2009), it is conceptually not dif?cult to extend our model
to multiple sectors and heterogeneous households in each country, though some
analytical dif?culties may arise. Given the rich literature of contemporary ?nancial
economics, it is reasonable to expect that the effects of monetary policies may vary,
depending on the structures of international and national ?nancial institutions.
Notes
1. Most of the general equilibrium models with free international capital and bond markets are
concerned with endowment economies where there is production or capital accumulation.
It should be noted that there are empirical studies on issues related to trade, growth, and
?nancial markets. See for instance, Edison et al. (2002), Glick et al. (2006), Galindo et al. (2007)
and Bon?glioli (2008). Although these empirical studies identify relations among various
economic as well as institutional variables, most of these studies lack theoretical economic
foundation.
2. Outside money is the part of money stock, which is issued by the government.
3. Obstfeld and Rogoff (1998) also develop some other dynamic models for open economies.
4. Eden (2005, Chapter 2) for the reasons why money is introduced into the utility function.
5. Superneutrality of money means that the growth rate of money has no effect on the real
equilibrium.
6. The assumption of single homogeneous goods is well-accepted in the neoclassical growth
trade theory. See, for instance, Ikeda and Ono (1992).
7. The neoclassical growth theory is referred to Burmeister and Dobell (1970).
8. The government may distribute the money in various ways and there are different
mechanisms for the government to issue money. As systematically demonstrated by Zhang
(2009), it is possible to build the model according to speci?ed characters of the monetary
system. For instance, an important way for the government to intervene the markets is
through controlling nominal rates of interest.
9. A detailed explanation of the approach and its relations to the traditional approaches to
consumer behavior is referred to Zhang (2009).
10. For the literature on the MIU approach, see for instance, Wang and Yip (1992), Turnovsky
(2000) and Zhang (2009).
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11. As shown in Zhang (2009), the model for a single national economy (which can be interpreted
as the case that multiple economies are identical in all aspects) has a saddle point. The author
tried to simulate the motion with various combinations of initial conditions and found out
that system does not converge to the equilibrium point.
12. As demonstrated in Zhang (2008), if the role of money is neglected and the global economy
includes only “real variables,” then the global economic growth model is characterized of a
unique stable equilibrium. The introduction of money causes instabilities in the trade model.
It should be remarked that in future research, we may introduce governments’ intervention
in international trade or/and ?nancial markets.
13. The insights obtained here should be interpreted with caution as the system is unstable.
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Vol. 84, pp. 1161-76.
Appendix
The Appendix proves Lemma 1. First, from equation (1) we obtain:
f
0
j
ðk
j
Þ ¼ f
0
1
ðk
1
Þ 2d
j
; j ; 2; . . . ; J ; ðA1Þ
where d
j
; d
k1
2d
kj
: If f
0
1
ðk
1
Þ 2d
j
. 0 for all j ¼ 2, . . . , J and given k
1
(t) . 0, then the equations
determine unique relations between k
j
and k
1
, denoted by:
k
j
¼ f
j
ðk
1
Þ; j ¼ 1; . . . ; J ; ðA2Þ
where f
1
(k
1
) ¼ k
1
. From equations (A1), we have:
f
00
j
ðk
j
Þ
dk
j
dk
1
¼ f
00
1
ðk
1
Þ; j ¼ 2; . . . ; J :
As f
"
j
ðk
j
Þ # 0; j ¼ 1; . . . ; J ; we see that dk
j
=dk
1
$ 0; j ¼ 2; . . . ; J : That is, f
0
j
ðk
1
Þ $ 0:
Hence, for any given k
1
ðtÞ . 0; we determine k
j
ðtÞ; j ¼ 2; . . . ; J as unique functions of k
1
(t).
From equation (1), we determine the wage rates as functions of k
1
(t) as follows:
w
j
ðtÞ ¼
f
j
ðk
1
Þ ; f
j
ðf
j
ðk
1
ÞÞ 2f
j
ðk
1
Þf
0
j
ðf
j
ðk
1
ÞÞ; j ¼ 1; . . . ; J : ðA3Þ
We can rewrite equation (9) as:
X
J
j¼1
k
j
ðtÞN
j
¼
X
J
j¼1
k
j
ðtÞN
j
:
Insert equation (A2) into the above equation:
k
1
ðtÞ ¼ Lðk
1
; {
k}Þ ;
X
J
j¼1
n
j
f
j
ðk
1
Þ 2
X
J
j¼2
n
j
k
j
; ðA4Þ
in which n
j
; N
j
/N
1
and {
kðtÞ} ; ð
k
2
ðtÞ; . . . ;
k
J
ðtÞÞ: We see that Country 1’s per capita physical
wealth,
k
1
ðtÞ; can be expressed as a unique function of Country 1’s capital intensity and the other
countries’ per capita physical wealth {
kðtÞ} at any point of time.
From equations (1) and (A3) and the de?nitions of y
j
; we have:
y
j
¼ L
j
ðk
1
; {
k}Þ þm
j
m
j
; j ¼ 1; . . .J ; ðA5Þ
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where:
L
1
ðk
1
; {
k}Þ ; f
r
ðk
1
ÞLðk
1
; {
k}Þ þ
f
1
ðk
1
Þ;
L
j
ðk
1
; {
k}Þ ; f
r
ðk
1
Þ
k
j
þ
f
j
ðk
1
Þ; j ¼ 2; . . .J ;
f
r
ðk
1
Þ ; 1 þ f
0
1
ðk
1
Þ 2d
k1
:
From equations (A5) and ðp
j
þ rÞm
j
¼ 1
j
y
j
in equations (6), we have:
p
j
¼
L
j
ðk
1
; {
k}; m
j
Þ ;
1
j
L
j
ðk
1
; {
k}Þ
m
j
2f
0
1
ðk
1
Þ þ m
j
; j ¼ 1; . . . ; J ; ðA6Þ
where m
j
; 1
j
m
j
þd
k1
: These equations show that a country’s in?ation rate can be expressed as
a function of the global distribution of capital stocks and its real money per capita. Substituting
equation (A6) into equation (8) yields:
_ m
j
¼ C
j
ðk
1
; {
k}; m
j
Þ ; ðm
j
2
L
j
ðk
1
; {
k}; m
j
ÞÞm
j
: ðA7Þ
Insert s
j
¼ l
j
y
j
and equation (A5) in equation (7):
_
k
1
¼ l
1
L
1
ðk
1
; {
k}Þ þl
1
m
1
m
1
2Lðk
1
; {
k}Þ 2m
1
2C
1
ðk
1
; {
k}; m
1
Þ; ðA8Þ
_
k
j
¼ C
j
ðk
1
; {
k}; m
j
Þ ; l
j
L
j
þl
j
m
j
m
j
2
k
j
2m
j
2C
j
; j ¼ 2; . . . ; J ; ðA9Þ
where we also use equation (A4). Taking derivatives of equation (A4) with respect to time yields:
_
k
1
¼
X
J
j¼1
n
j
f
0
j
ðk
1
Þ
" #
_
k
1
2
X
J
j¼2
n
j
C
j
ðk
1
; {
k}; m
j
Þ; ðA10Þ
where, we use equations (A9). Equalizing the right-hand sides of equations (A8) and (A10) yields:
_
k
1
¼ C
1
ðk
1
; {
k}; m
1
Þ ;
X
J
j¼2
n
j
C
j1
þl
1
L
1
þl
1
m
1
m
1
2L2m
1
2C
1
!
X
J
j¼1
n
j
f
0
j
" #
21
: ðA11Þ
In summary, we have thus proved Lemma 1.
About the author
Wei-Bin Zhang PhD (Sweden) is a Professor in Ritsumeikan Asia Paci?c University, Japan.
His main research ?elds are nonlinear economic dynamics, growth theory, trade theory,
East Asian economic development, Chinese philosophy, and ethics. He has published more than
100 academic articles and authorized 20 academic books in English by international publishing
houses. Wei-Bin Zhang can be contacted at: [email protected]
Capital
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