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A PURBLY THEORETICAL STUDY ON BCONOMIC
GROWTH IN SMALL OPEN ECONOMIES
BY
THUY THI BICH DAO
Thesis presented for the degree of Doctor of Philosophy, Faculty of Economics,
University of Adelaide, Australia.
September 2000
ABSTRACT
This thesis
is a theoretical
study on economic growth
in small open
economies. The
motivation for economic growth theory is to explain the persistence of world economic
growth and the existence of large differences in cross-country income levels and growth
rates.
In order to explain
these facts, growth theory seeks to answer the question of what
factors determine the growth rate of an economy and how they can be influenced. Growth
theory has been developed to cover the issues in both closed and open economy contexts.
Our objective is to explore the open economy issues in the areas of international capital
movements, foreign investment and technology transfer in relation to economic growth. In
the study, we construct economic growth models in a small open economy context to study
the issues of convergence, the role of education, the role of foreign investment in technology
transfer and how government policies can influence the growth rate of an economy. This
thesis raises the interrelationships between foreign investment, technology transfer and
human capital accumulation
of the host countries, a topic not
adequately addressed in
previous literature.
The thesis is comprised of seven chapters where the original contributions are in four
chapters along with the introduction, literature review and conclusion chapters. Differential
equations and control theory are techniques used in the thesis. We use Mathcad Software
computer program to run simulations.
11
TABLE OF CONTBNTS
Chapter L: Introduction
1
2: Literature Review
7
l.Growth models in a closed economy context
8
1.1. Neoclassical growth models
8
1.2. Endogenous growth models
t4
2.Issues on economic growth in an open economy context
24
Chapter
Chapter 3: Capital Flows and Economic Growth in a Small
Open
30
Economy
l.Introduction
31
2. The models
32
2.L.The Solow-Swan open economy model
32
2.2.The extended Solow-Swan open economy model
37
2.2.1. The dynamics
42
2.2.2. The steady state
45
2.2.3. The transition: the speed of convergence
46
2.2.4. Comparative statics: the impact of changes in the saving rates
56
on the steady state variable
3. Conclusion
62
r11
chapter 4: Capital Flows, International Technology Transfer
65
and Economic Growth in a Small Open Economy
l.Introduction
66
2. The model
68
2.LThe dynamics
72
2.2.The steady state
16
2.3.The transition: the speed of convergence
18
2.2.4. Comparative statics: the impact of changes in the saving rates on the
83
steady state variable and the growth rate
3. Conclusion
89
Chapter 5: Optimal Foreign Borrowing, Physical and Human
97
Capital Accumulations and Technology Transfer
L.Introduction
92
2. The model
93
2.1. The optimal solution
98
2.2. The market solution
tol
2.3.The role of the government
115
3. Conclusion
I2I
Appendix A
r23
Appendix B
r24
1V
Chapter 6: Direct Foreign Investment, Technology Transfer
and
126
Economic Growth in a Small Open Economy
l.Introduction
121
2.Tl¡Le model
t29
2.1.
Autarþ economy
129
2.2. Open economy
135
3. Conclusion
t47
Chapter 7: Conclusion
150
REFERENCES
156
Chapter 1:
INTRODUCTION
Over long periods of time the world economy has experienced sustained growth in per
income and these growth rates show no tendency to decline. However, growth rates vary
greatly between countries; many countries achieve significant growth performances while the
growth rates of other countries are sluggish. In terms of levels of income, there also exist
(1995)
large differences between countries. To give some examples, Barro and Sala-i -Martin
reported that the real per capita gross domestic output (GDP) in the United States grew from
52244
in
1870 to $18258
in 1990, all
measured
in 1987 dollars. This increase coffesponds to
a growth rate of 1.75 percent per year. In the period from 1960 to 1990, many countries
achieved high growth rates, for example, South Korea with 6.7 percent per year while other
countries experienced very low growth rates such as -2.I percent per year for Iraq. In 1990,
the real per capita GDp of United States is $9174 compared to 5249 for Ethiopia or about 39
times difference (Barro and Sala-i -Martin, 1995).
Why does there exist large differences in income levels and growth performances among
different countries? What factors determine the growth rate of an economy and what can
influence those factors? These are the questions that growth theory tries to explain.
Among many different schools of thought in growth theory, neoclassical growth theory and
endogenous growth theory (also called new growth theory) are the dominant models. The
neoclassical growth theory has its boom during the period from the late 1950s to the 1960s.
This theory is often referred to as exogenous growth theory because it attributes technology
as an engine
of growth but leaves it unexplained. The theory is thus criticised as explaining
everything but growth. As distinct from the neoclassical growth theory, the new growth
theory endogenises the growth rate of technology into the economic system, so that this
theory has its term as endogenous growth theory.
2
It is often useful to start with the Solow-Swan
(1956) model as an well known model in
neoclassical growth theory. The main aspects of the Solow-Swan model is the neoclassical
form of production function with constant returns to scale but diminishing returns to
each
input, and an exogenous saving rate. The model predicts that different economies converge to
different steady state positions in the long run, dependent upon the saving rate and the rate of
population growth. A country with a higher saving rate and a lower population growth rate
ends up with a higher level of output per head
in the long run. Due to the assumption of
diminishing returns to capital, poor countries that have relatively lower capital per head enjoy
higher rates of returns to capital and higher growth rates. Thus poor countries tend to grow
faster to their steady states. Once in the steady state, each economy grows at the exogenous
rate of technology advance. Thus the model predicts convergence in the sense that poorer
countries initially grow faster, but have growth rates which slow down to that of the richer
countries over time.
However, we observe that per capita growth rate differences of the world economy persist
over time, and that poorer countries are not necessarily growing faster than richer ones. From
the mid 1980s, endogenous growth theory has attempted to explain this, beginning with the
works of Romer (1986) and Lucas (1988). This theory can explain the process in which an
economy generates its persistent growth. According to the theory, the key condition for
endogenous growth rests on the assumption of nondiminishing returns to all factors that can
be accumulated, taken together. As long as this condition is satisfied, endogenous growth is
possible.
In explaining the main factors of economic growth,
endogenous growth theory
attributes technology and human capital as engines of economic growth.
J
In one line of interest, starting with the work of Romer (1990), research on endogenous
growth attempts to explain the role of technology in the growth process. Abstract technology
has its own properties as
nonrival and nonexcludable. Technology is nonrival because the use
of technology in one activity by no way precludes its use in other activities in terms of
quality and quantity. Technology is nonexcludable if all firms can have access to the use of it.
However, the problem with the public good characteristic of technology is that since ex post
it can be available to all firms, ex ante there is no incentive for
a
firm to invent it. Technology
is costly to invent since time and resources must be allocated to this activity. In order to
provide an incentive in technology innovation, an inventing firm must exercise some degree
of monopoly power over its invented technology to capture returns on the technology. Thus
there must be some degree of partial excludable over technology. Copyright, government
intervention, law and order are some major factors that enforce it.
In another line of interest, Lucas (1988) focuses on the accumulation of human capital in
explaining the growth process. Human capital is defined as skills, knowledge and abilities
that are embodied in each individual. In difference to abstract technology, human capital is
rival and excludable. The use of it in one activity precludes its use in other activities. There
are several ways that an individual can acquire his or her own human capital. Human capital
can be accumulated through learning, education, training, on-the-job training, work
experience and so on. Since human capital
is an engine of growth, the importance
of
education and training as means of acquiring human capital are the main issues of concern.
The basic economic growth models of Solow-Swan (1956), Romer (1986,1990) and Lucas
(1988) consider growth in a closed economy context. This thesis explores growth theory in
4
an open economy context. Our interest is to study the effects of international capital
movements, foreign investment and technology transfer on economic growth of small open
economies. We seek to cover the issues of convergence, the role of education, the role of
foreign investment in technology transfer and how government policies can influence the
growth rate of an economy. There are several studies in this area dealing with different issues
in an open economy context. These are more fully discussed in Chapter 2. However,
those
studies do not consider the interrelationships between foreign investment, technology transfer
and human capital in relation to economic growth. This issue is the major focus of this thesis.
The next chapter comprises a more complete literature review of growth models in closed
and open economies. These models serve as the basic models for our study in the thesis. In
that chapter, the interested issues in an open economy context will be discussed. In Chapter 3,
we will study economic growth in a small open economy context using the extended SolowSwan model with human capital. The inclusion of human capital into the Solow-Swan model
improves the model's ability
in
explaining income differences among open economies.
However, we show that the model is still unable to explain cross-country differences in
growth rates.
In Chapter 4, we introduce technology transfer into the extended
Solow-Swan model of
Chapter 3. The enrichment of the model enables the model to be a type of endogenous growth
model. This model thus gives us a high potential
differences as well as growth rates.
In
in
explaining cross-country income
addition, this model gives a clearer picture in
explaining the convergence process and how it can be affected by economic policy.
5
Chapter 5 considers the case where foreign technology cannot be freely adopted by a poorer
country. There are many ways that a country can adopt foreign technology. Among them is
foreign investment. Foreign investment can act as a channel for technology transfer. In this
chapter we explore the problem of economic growth in a small open economy which hosts
foreign investment where we raise the interrelationships between foreign investment,
technology transfer and human capital accumulation of the host country.
In
Chapters
4 and 5 we assume that a small country must totally
depend on foreign
technology for its technological change. This assumption is relaxed in Chapter 6 where we
stress the idea that
country,
while direct foreign investment can enhance the growth rate of the host
it is not the only factor that determines the economic growth
rate. That is, the host
country does not rely totally on direct foreign investment for its technological progress.
Direct foreign investment contributes to the stock of technology in the host country which
helps to fasten the economic growth rate of the host country. But without direct foreign
investment, the country can grow at its endogenous growth rate. The model that we employ
in this chapter is the Lucas (1988) model where we make some extensions to study it in
a
small open economy. The objective of this chapter is to explain the process in which direct
foreign investment helps a less developed country catch up with the rest of the world in terms
of economic growth rate and income levels.
Finally, the conclusion of the thesis is given in Chapter 7 where we will summarise our main
findings and discuss the limitations of the study as well as give suggestions for further
studies.
6
Chapter
2z
LITERATURE RBVIEW
7
This chapter provides the literature review on economic growth theory in two main sections.
In section 1 we give a discussion on the development of economic growth theory starting
from the neoclassical growth models to the endogenous growth models in a closed economy
context. The purpose of this section is to give a descriptive picture of basic economic growth
models and also provide us some useful techniques that have been used in growth models. In
this section, the Solow-Swan (1956) model and the Lucas (1988) model will be discussed in
depth because we chose these models for our theoretical study in the thesis.
Section 2 then discusses the issues in an open economy context covering the related areas of
international capital movements, international trade, international technology creation and
diffusion, foreign investment and technology transfer in relation to economic growth. In this
section we will explain how our study fills in the literature of economic growth theory.
1,.
Growth models in a closed economy context
1.1. Neoclassical growth models.
During the periods from the 1950s to 1960s, neoclassical growth theory was developed and
dominated research on economic growth. The leading model in neoclassical growth theory is
the Solow-Swan (1956) model. The Solow-Swan (1956) model features an economy which is
populated by L(t) individuals where / denotes time. The labour force is equal to the size of the
population. There is a single good to be produced whose production function is assumed to
take a Cobb-Douglas form as
Y(t¡ =
n(rçt¡,A(t)LØ),
(1.1)
I
where
f(r) is output, K(r) is the stock of capital and A(r) is the level of technology or the
"effectiveness of labour" at time r. The term A(t)L(r) is then described as effective labour.
The production function is a well behaved neoclassical production function with constant
returns to scale in its two arguments: capital and effective labour. That is,
arguments by any nonnegative constant c then output
will
if
we multiply both
change by the same factor
F(cK(t),cA@t@)=cr(K7)'qØt@).
G.2)
In addition, the production function is assumed to satisfy the Inada conditions
Lim*-o F *(KG) A(t) L(r¡) =
-,
Limu-*Fu(KQ)A(t)L(r))=
o.
(1.3)
These conditions state that the marginal productivity of capital is very large when the stock
of capital is very small and it is very small when the capital stock is very large.
These
conditions are to ensure that the path of the economy does not diverge.
The dynamics of the model is described by the evolutions of the inputs into production
Labour and technology are assumed to grow at constant rates as
L(t)lL(t)=vt,
(r.4)
A(t)lA(t)=s,
(1.s)
where the dot above a variable denotes the change of the variable with respect to time.
The output of the economy can either be consumed or directly invested as capital. A main
assumption in the Solow-Swan model is an exogenous and constant saving rate. Suppose in
each period, the economy saves a fraction s
depreciates at the rate
of output in capital investment. Existing capital
õ so that the accumulation of capital
over time can be described as
9
(1.6)
K(t¡= sY(t)-6K(t)
Define nç¡= K(t)t(eçt¡rlt¡) and y(t)=Y(t)l(eçt¡rçt¡) as the stock of capital and
output per unit of effective labour respectively. From (1.1) the output per effective labour is
Y(t)
i(t)
A(t)
L(t)
1
A(t) L(t)
n(rçt¡, A(t)LØ)
(1.7)
.
By the constant returns to scale assumption (1'2) we have
r(xçt¡,
#6
eqt¡rçt¡) =
fet f(nçt¡,|= ¡(nft))
form
K(t)
r( A(t)L(t)
(1.8)
tn"n (1.7) and (1.8) give us the production function in an intensive
as
iG) =
¡(t
(1.e)
a>).
From equation (1.6) we can obtain the evolution of capital per effective labour
k(t¡ =
K(t)
A(t)L(1)
K(t) L(t) K(t) A(t)
A(t)L(t) L(t) A(t)L(t) A(t)
(1.10)
)
or equivalently
î,çt¡=si(r)- (n+ s+õ)tc(t)
(1.10')
'
The differential equation (1.10') is the key equation that describes the dynamics of the
economy. In this equation,
y(f ttl) is the actual investment
and
(n+
S
+ Ðk(t) is the break-
even inyestment or the amount of investment that needed to keep capital at its existing level.
When actual investment per unit of effective labour is more than break-even investment,
is rising and when actual investment per unit of effective labour is less than
investment, lr(r) is falling. The stock of capital f (r)
it
[(f)
break-even
unchanged when actual investment
10
just equals break-even investment. Figure 2.1 shows this dynamics (for the moment we
ignore the curve
r"¡(É1r;). It is there to serve
another purpose)
(n+
g+
6)k
sr"f (fr)
sf (k)
"i.
ki
k
k
Figure 2.1: The dynamics of the economy
As shown in Figure 2.I, the economy will converge to point A. Point A is the steady state of
the economy where k(t¡ = 0 and the stock of capital per unit of effective labour takes
constant value
of
of Ê.. Output per unit of effective
a
labour will be produced at a constant level
i. - Í (k.). The steady state stock of capital and output of the economy are
K(t) = A(I)LQ)Ê.
Y(t¡ = A(t)L(t)j.
The constancy
,
.
of Ê.
and
j-
implies that the steady state stock of capital and output will
grow at the rate equal to the sum of the growth rates in technology and labour force
as
I + n.
Thus the steady state capital and output per capita grow at the exogenous technological
growth rate g.
11
In the steady state /c1r¡ = 0 and thus É. is the solution of the equation
(1.r2)
.f (fr-) =(n+ g + õ)k
In equation (1.I2), the exogenous saving rate determines the steady state stock of capital per
unit of effective labour. As displayed in Figure 2.1, an increase in the saving rate will shift
rhe s/(fr(/)) upwarObutleavetheline
(n+g+Ðlc(t)
at its existing steady state at point A where
f
unchanged. Initially,theeconomyis
is equal to the existing É-. Ho*"uer, at this
level actual investment now exceeds break-even investment causing É to rise. This process
continues until the economy reaches the new steady state at point B with a higher value Êr-
'
Thus a higher saving rate raises the steady state stock of capital per unit of effective labour.
Since the per capita growth rate of the economy is exogenously determined by the exogenous
rate of technological progress, changes in the saving rate do not affect the long run growth
rate of the economy.
The Solow-Swan model explains that differences in saving rates among countries are the
factor that causes cross-country income differences.
'Wealthier countries are ones that save
more. Since the saving rates are different between different countries, each economy will
converge to its own steady state. The model then predicts conditional convergence. Due to
the assumption of diminishing returns to capital, the marginal productivity of capital and thus
the rate of return to capital in poorer countries must be relatively higher than that in richer
countries. Thus it then suggests that poorer countries should grow faster than richer ones and
finally reach their steady
states.
t2
Mankiw, Romer and Weil (1992) noticed the deficiency
in the Solow-Swan model
in
explaining cross-country income differences. In the Solow-Swan model with a conventional
value of capital's share, large income differences must require vast differences in saving rates
and rates
of population growth, implying vast
differences
in rates of returns to
capital.
in explaining
income
Mankiw et al (1992) raised the important role of human capital
differences among countries. They then introduced human capital into the Solow-Swan
model. In this extended Solow-Swan model with human capital, the production function is
assumed to take a specific Cobb-Douglas form as
y(t¡ = K(1)"
where
all
nØp(1tG)LQ))'-"-þ,
factors are the same as
(1.13)
in the Solow-Swan model except that capital
distinguished between physical capital
K(t)
and human capital Ë(r). Output
is
is used for
consumption and saving in physical capital and well as human capital. Mankiw et al (1992)
found that moderate changes
in the resources
accumulations may lead to large changes
devoted
to physical and human
capital
in output per worker. Thus this model has the
potential to greatly increase the ability to account for cross-country income differences.
However, the growth rate of the economy is still determined by the exogenous technological
progress which is left unexplained.
A main assumption in the Solow-Swan model is an exogenous saving rate. Cass-Koopmans
(1965) introduce the endogeneity of saving into the model but it does not help the problem of
exogenous growth. In their models, households save to spread their consumption optimally
and that their savings
will
respond to the available rates of return to capital. As long as the
marginal capital earns the return which is greater than the household's marginal willingness
to delay consumption, additional capital is accumulated. With a constant technology level,
a
higher capital per head implies a fall in the return on investment. Over time, a decreasing rate
13
of return to capital causes the incentive to accumulate capital to vanish. Thus the economy
must totally rely on exogenous technological progress to keep the rate of return to capital
away from falling and a continuous investment in capital.
Neoclassical growth theory comes up with the exogenous rate of technological progress
the determinant of growth
in
as
output per capita. Since technological progress occurs
arbitrarily, there is no policy that can affect
it and thus the growth
rate of the economy. In
explaining the long run growth rate of an economy, there is nothing to be said rather than
exogenous technological changes. For this reason the theory is criticised as unsatisfactory.
1.2 Endogenous growth models
Since the mid 1980s, endogenous growth theory has been developed. The theory tries to
endogenise the economic growth rate by factors inside the economic system. This theory
found that a crucial assumption for endogenous growth is nondiminishing returns to all
factors that can be accumulated. The AK model of Rebelo (1991) is a good example. In the
AKmodel the production function is
Y(t) =
AK(t),
where A is a constant factor,
(2.1)
f(/) is the output and K(Ð is capital. K
can be thought
of as all
types of inputs into the production which can be accumulated. The production function
displays constant returns to capital. The marginal product of capital is determined by
a
constant factor A as
MPu =
¡.
(2.2)
I4
Suppose the accumulation
of capital is made by the saving of an exogenous fraction s of
output. The stock of capital is changed by the investment in each period less the depreciation
in existing capital
as
kçr¡=sAK(t)-6K(t).
Q.3)
Equation (2.3) describes the dynamics of the economy. In this equation, the term 6K(t) is the
break-even investment or the amount needed to keep capital at its existing level and the term
sAK(t) is actual investment. Dividing both sides of equation (2.3)by K(/) gives us the growth
rate of capital
kft>tK(t¡=sÁ-ô.
(2'4)
As long as sA > ô, actual investment is greater than break-even investment causing the stock
of capital to grow. Figure 2.2 shows how the economy can generate growth in the long run.
sAK(r)
K(t)
(t)
K(t
K(t)
Figure 2.2zThe dynamics of theAKmodel
As shown in Figure 2.2, the difference between actual investment and break-even investment
is the
change
in the stock of
capital. The stock
of capital grows as more capital is
15
accumulated. In the AK model, the economy can generate endogenous growth in the long run'
The reason is that the AK model violates the neoclassical assumption of diminishing returns
to capital and assumes instead that there are constant returns to capital. Constant returns to
capital keep the incentives to invest
in capital from falling, resulting in a continuous
investment in capital and thus persistent growth.
The very early models of endogenous growth which attempt to endogenise the technological
process are referred to the work of Romer (1986, 1987, 1990). In his paper (Romer, 1986),
the source of technological progress is explained by the so call learning-by-doing. This
terminology
is
originated
by Arrow (1962) when he
argues that the improvement in
productivity occurs as a side effect of conventional economic activity, and not as a result of
deliberate efforts
in
research and development (R&D) activity. New technology or new
knowledge is created (tearning) as a side effect of the production of new capital (doing).Llke
Arrow (1962), Romer (1986) models the increase in knowledge as a function of the increase
in capital or the stock of knowledge as a function of the stock of capital. In the Romer (1986)
model, the production function displays diminishing returns to capital at the individual firm
since each firm sees the stock of knowledge as exogenously given. However, the industry
as
a whole production function displays nondiminishing returns to capital since the stock of
technology is determined by the industry stock of capital invested by all firms. Due to the
assumption of nondiminishing returns to capital, the economy is able to produce endogenous
growth. The constant returns to capital assumption enables the economy to generate a
constant steady state growth rate while the growth rate of the economy is explosive
if
there
exist increasing returns to capital.
16
The main and powerful source of technological progress is from research and development
activities. Grossman and Helpman (1991) argue that commercial research and development
present the main method
by which
business enterprises acquire technology
in modern,
industrialised economies. The technology innovation process is costly since vast resources
and efforts must be allocated to
R&D activities. In order for private firms to invest in these
activities, they must exercise some monopoly power over their inventing technology to
exclude the use of
it by other firms. Romer (1990)
has developed an endogenous growth
model which explains the creation of technology by monopoly firms engaging
in R&D
activities. In his model, technology innovation is assumed to be the introduction of new
goods. The contributions to modelling
R&D activities
as the source
of technological progress
are subsequently made by Grossman and Helpman (1991) and Aghion and
Howitt (1992)'In
their models, the innovation of technology is assumed to be the improvement of the product
quality. These models are referred to as R&D growth models.
In a quite different line of interest, Lucas (1983) focuses on human capital as an engine of
growth. Human capital is defined as the skills, knowledge and abilities that are embodied in
each individual. The concept of human capital and its role in explaining individual income
differences is far long studied by various authors (Becker 1964, Ben-Porath 1967, Thurow
lg71,Rosen I972,Mincer 1974, Blinders and Weiss 1976). Lucas brings human capital into
the context of economic growth and interprets that the level of development in each country
depends on the level of human capital possessed by each. The more developed countries have
higher levels of human capital than less developed ones. Thus the accumulation of human
capital becomes the crucial for economic growth process. Education and labour training
means
of acquiring human capital are then the major
issue
as
of concern. We now turn to
I7
provide a
full description of the Lucas (1988) model for it being the basic model that is
employed in our study in the thesis.
The Lucas (1988) model
The Lucas model describes a closed economy with a constant L identical and infinitely lived
individuals. Each individual is embodied with a level of human capital h(t) 1n period
r.
Human capital can be accumulated by investing time in learning activities. In each period,
each individual is endowed with 1 unit of time which can be spent in learning or working.
Suppose the individual allocates
chosen and
1
-
ty(t)
fraction of time to work where tt/(t) is endogenously
V/G) fraction of time to learn then the stock of human capital is accumulated
AS
ot\
h(t¡ =
õlt-
ry(t))h(t)
,
(3.1)
where ô is an exogenous parameter. The production function of human capital is postulated
to display constant returns to human capital. This is where Lucas argues that the Uzawa
(1965) model which is very similar to his model, cannot produce endogenous growth because
in that model there is a diminishing return to human capital. As pointed out by Lucas, since
human capital is an engine of growth, for the economy to generate persistent growth people
must have the incentive to invest in human capital in the long run. The returns to human
capital determine the incentive
to
accumulate
in human
capital. As long as there
are
nondecreasing returns to human capital, people keep investing in human capital and long run
growth is possible.
18
The economy produces a single good which can be consumed or invested as physical capital.
The goods production function is assumed to take a Cobb-Douglas form as
Y(t¡ =
K(t)"(trØhQ)t)'-"
The term
h(t)p
nçt¡a
.
creates externality
Q.2)
in the production. The externality is to capture the external
effect of human capital in the production. The argument for it is that smart workers raise the
efficiency of the working environment which benefits other co-workers. Due to the existence
of the externality in the production function, there will be two paths called the optimal
growth path and the competitive equilibrium path.
In the optimal path, the externality is
known to the social planner whereas in the equilibrium path the externality is unknown to
private sectors since firms and households are separate identities.
The accumulation of physical capital is described as
ke): K()"þye¡ttçt¡r)'-"
tt(t)o
- c(t)L,
(3.3)
where c(r) is per capita consumption and physical capital is not assumed to depreciate.
A feature of the model is the optimisation problem where
a representative
individual chooses
the optimal level of consumption and the allocation of time in each period to maximise his or
her lifetime utility. This is assembled with the Ramsey (1928) optimisation model where he
introduced the problem of dynamic consumer optimisation into economics. The individual's
lifetime utility is assumed to take the form of
u=j
(3.4)
0
where
p
is the discount factor and
o
is the risk aversion factor. The maximisation amounts
to the problem:
t9
Max
c(t) tl(t ,J
'0
st.
e -pl
L
c(t)t-" -l
dt
I-o
kç¡ = K(t)"þy@nçt¡r)'-" nG)' -c(t)L,
t 1,¡
= dU- w<t¡)nft¡.
We form the Hamiltonian expression as
,=,#+)",çt¡(rç)"(ty(t)h(t¡r),_"n1t¡p_c()r)+l,1t¡(a(t_wrt>)nrt¡)
l-o
where
Lr(t)
and
)"r(t)
are the shadow prices
of physical capital and human capital
respectively.
First order conditions yield
c(t)-" = [r(t),
)"r(t)(t- a)K(|"(y@h@r) " LhTt¡'*'
(3.s)
=
trr(t)&.(t)
(3.6)
The rate of change of )"r(t) is given by
i,çr¡ = pL,(t) - 7,(t)aK(t)"-tþyQ)h|)r)'-" nç¡a
(3.7)
In the optimal path, since the externality is known to the social planner the term h(t)þ is
taken into account in solving for the maximisation problem. The growth rate
tr"(t)= pLr(t)-Lr(t)(I-d+ tt)K(ù"(wçt¡r)'-"nçt)-o*p
of ),"(t) is
-Lr1t¡õ(t-v/Ø),
(3.8)
or by substituting (3.6) we have
20
)r(t) I )rçt¡
=
p- 6 - þ
I-u
(3.8')
6w(t)
In the equilibriumpath, the externality is unknown to private sectors so that the term h(t)p is
taken as given. The growth rate of
)"r(t) is written
as
l"(r) = pl27) - )"r(r)(r- a)K(t)" (tyçt¡r)'-" tt(t)-"t' - )"rçt¡6(t-
V/Q)),
(3.e)
or it is equivalently as by substituting (3.6)
).r(t)lAr(t)= p-ö
(3.9',)
The interest of the study is on the steady state which is defined as a path where human
capital, physical capital and consumption grow at constant rates and the time allocation t¿ is
constant.
In the steady state, physical capital and consumption are of the same type which
grow at the rate y and human capital grows at the rate rc. That is,
T = c(t) I c(t) = k(t) I k(t)
rc
= h(t) I h(t)
(3.10)
,
(3.11)
.
The growth rate of consumption is obtained by differentiating equation (3.5) with respect to
tlme
y = c(t) I c(t) = - ),(t) I Lr(t)
(3.12)
Substituting equation (3.7) into (3.I2) gives us the marginal productivity of physical capital
condition as
yo + p = aK(t)"-t (w@hØ
r)'-" r1r¡' .
(3.13)
2l
Differentiating equation (3.13) with respect to time and taking into account equations (3.10)
and (3.11) we have the per capita growth rate of consumption in terms of per capita growth
rate of human capital
v
I-u+p K
(3.r4)
L-q
To find the growth rate of human capital, one can see that by differentiating equation (3.6)
and substituting for equation (3.7) we have
)",(t)llr(t)=(q-o)y -(a- p)rc.
(3'15)
Equations (3.8') and (3.15) together give us the optimal growth rate of human capital as
(3.16)
The equitibrium growth rate of human capital is found by combining equations (3.9') and
(3.1s)
K
(6
p)(I- a)
=_.
o(I-a+p)-p
13.17\
Comparing equation (3.16) and (3.17) we see the difference between optimal and equilibrium
human capital growth rates
_K=_
K.þp
(3.19)
I-a+p ,
which is greater than zero or the optimal human capital growth rate is gfeater than the
equilibrium human capital growth rate.
In the Lucas model, due to the production externality assumption, the optimal growth rate of
the economy departs from the equilibrium growth rate with the former always greater than
the later. However, we note that this assumption is not crucial for the economy to generate
endogenous growth.
If
there
is no externality or lt = 0 the economy still produces
22
endogenous growth
with a balanced growth rate of y = (õ - p) lo . The Lucas
describes human capital as an engine
of growth. Thus the role of education
model
and labour
training is an important issue in the economic growth process.
Between the two categories; idea-based models which endogenise technological progress and
capital-based models which emphasise the investment
in
economic growth process, there are studies that attempt to
human capital
in
explaining
link the interactions
between
human capital and technological changes.
Chari and Hopenhayn (1991) construct a model of technological diffusion where
agents
invest in vintage-specific human capital to learn about exogenous technological changes.
Grossman and Helpman (1991, Ch. 5.2) endogenise both human capital and technological
change in the growth model. In the Grossman and Helpman model, labour is distinguished
between two types as skilled and unskilled labour. Human capital is embodied
in skilled
labour and to become a skilled labourer, an individual must forgone income from working
and spend time in education to acquire human capital. Technology is produced by private
firms engaging in R&D activities. The model produces endogenous growth with a constant
rate of economic growth, constant wage rates for skilled and unskilled labour, and a fixed
supply of skilled to unskilled workers. Eicher (1996) studies the interaction between human
capital and technology in an overlapping generation model. Skilled labour acquires human
capital by investing in education and technology is the product of education process. He finds
that higher rates of technological change and economic growth may be accompanied by a
higher relative wage but a lower relative supply of skilled to unskilled labour.
23
In difference to neoclassical growth models,
endogenous growth models explain that
persistent growth of an economy is the outcome of deliberate and intentional activities by
economic agents. The growth rate of the economy is dependent on factors that are inside the
economy and thus the growth rate can be controlled by various policies.
2. Issues on economic growth in an open economy context
Growth theory has expanded into the international economy context to cover the issues of
economic growth
in relation to
international capital movements, international trade,
international technology innovation and diffusion, foreign investment and technology
transfer.
There are several papers that study the effects
of
international capital movements on
economic growth. Milbourne (1997) and Benge and Wells (1998) analysed economic growth
of a small open economy using the Solow-Swan model. Their studies assume that a small
open economy faces an unlimited supply of the world's capital at the world interest rate 7 .
As pointed out by Milbourne (1991),
it is possible that
under perfect capital mobility, the
flows of capital will bring the economy immediately to its steady state. The initial jump
implies that there is no transition for the economy. The steady state capital per head
employed in the country is fixed by the country production technology and the world interest
rate.
A fixed stock of capital determines a fixed level of domestic output produced and the
saving rate has no effects on the domestic capital stock and output.
The Solow-Swan open economy model with perfect capital mobility predicts that the rate of
convergence for a small open economy would be infinite. However, Barro, Mankiw and Sala-
24
i
-Martin (1995) argue that this result conflicts sharply with the empirical evidence. The
Solow-Swan model treats
all types of capital as the same. In other words, it does not
distinguish between physical and human capital. Barro et al (1995) include human capital
into the Solow-Swan open economy model and set up the model in an optimisation context.
They found that perfect capital mobility will let the economy jump immediately to its steady
state and stay there for ever. However, the economy
steady state
if there is a credit constraint
will
undergo a transition toward the
imposed on the country. The credit constraint
prevails when the small open economy can borrow overseas to finance physical capital
investment but not human capital investment. They find that the rate of convergence for the
credit-constrained economy would not be infinite but
it is greater
than that for the closed
economy.
Since the study of Barro et al (1995) is set up
in an optimisation model of the extended
Solow-Swan open economy where the saving rate is endogenously determined, the model can
not predict how the saving rate can influence the rate of convergence. The impact of changes
in the saving rates on the convergence can be analysed in the extended Solow-Swan
open
economy model where saving rates are exogenously given. This issue, however, has not been
dealt with in the literature. It is then one of the interests that we pursue in our study.
The effects of international capital movements on an open economy in the various stages of
economic growth are investigated
pattems
in Onisuka (1914). In his paper, the long run growth
of the economy are discussed in terms of various
phases
of economic growth
characterised by levels of capital flows, indebtedness and domestic capital accumulation.
25
There are studies that analyse economic growth and perfect capital mobility in two-country
models. Ruffin (1979) constructs such a model in the context of Solow-Swan to assess the
effects of perfect capital mobility on economic growth of importing and exporting countries.
He finds that the steady state per capita income and capital under perfect capital mobility
exceed the autarky steady state solutions for both countries.
In addition, perfect
capital
mobility raises the steady state interest rate and lowers the steady state wage in the capital
exporting country compared to autarky, while the opposite holds in the capital importing
country. In another study, Wang (1990) introduces human capital into a two-country model to
analyse the properties
of steady state growth
rates existing
in two countries. The author
employs the Lucas (1988) type of growth model but instead assumes that human capital, as a
proxy of technology, grows at an exogenous rate with the rich country having a relatively
higher growth rate. He argues that perfect physical capital mobility allows physical capital to
flow from the rich to the poor country and such that the inflows of physical capital to the
poor country are embodied with technology transfer, resulting in the poor country growing at
the same rate with that of the rich.
Economic growth and international technology diffusion has been the focus of international
economic growth issues.
If international technological
gaps can explain differences in growth
rates across countries as suggested by the technology gap theory (Fagerberg, 1990) then the
diffusion of advanced technology to less developed countries would gradually close the gaps
in
economic growth rates. Technology diffusion can take various forms such as through
granting, licensing, franchising, trading, direct purchases
investment. Among different ways
of
or
through direct foreign
modelling international technology diffusion,
technology transfer via foreign investment probably occupies the dominant research agenda.
26
The role of foreign investment in technology transfer and its effects on economic growth of
host countries are found in various studies (Koizumi and Kopecky 1977,1980, Findlay 1978,
Ghosal 1982, Wang 1990, Wang and Blomstrom 1992, De Mello 1997, Walz
Borensztein, De-Gregorio and Lee 1998, Gupta 1938).
investment acts as a channel
1997,
It is well argued that foreign
for technology transfer which contributes to the economic
growth of the host country. Via foreign investment, the host country has access to foreign
advanced technology which enables
it to accelerate its own technology accumulation
and
thus growth.
Koizumi and Kopecky (1977) construct a model of international capital movements and
technology transfer
in a small open economy context to analyse the role of international
technology transfer.
In their model, technology transfer is
assumed
to take place when
foreign capital creates an externality in technology to the host country. In particular, the
technology level of the host country is assumed to be a function of the stock of foreign capital
per capita. Foreign capital and domestic capital are physically the same but foreign capital
imparts spillovers in the form of technological transfers. As a result, while foreign capital and
domestic capital are paid at the same world interest rate, the social marginal productivity of
foreign capital is higher than that of domestic capital. They found that changes in the saving
rate of the country can alter its level of capital intensity.
Findlay (1973) constructs a model of international technology transfer by international
corporations. In his model, the author stresses the importance of two effects which are called
the "relative backwardness" and the "contagious effect" in explaining the transfer of
technology. The idea of "relative backwardness", which was originally introduced by Veblen
27
(1915) and Gerschenkron (1962) and was later formalised in a technology transfer model by
Nelson and Phelps (1966), states that the larger the gap in technology between advanced and
backward countries, the faster the rate at which the backward country can catch up in
technology. The "contagion" idea stresses the importance
of personal
contacts. That is,
advanced technology is most effectively copied when there is personal contact between those
who already have the technology and those who eventually adopt it. In the Findlay's model,
foreign corporations are the carrier of new technology. By the "contagious effect", the rate of
technological change in the backward country is an increasing function of the relative extent
to which the activities of foreign firms pervade the local economy. This extent is measured
by the ratio of foreign-owned capital stock to domestic-owned capital stock. The economy
approaches the steady state where
it grows at the rate equal to the exogenous growth rate of
foreign technology.
In another study, De Mello (1997) models technology transfer via direct foreign investment
in such a way that the existence of direct foreign investment creates externalities in the stock
of technology of the host country. The stock of technology is assumed to be a function of
foreign-owned and domestic-owned physical capital stocks. He argues that the effect of direct
foreign investment on the growth performance of the host country is manyfold. In his model,
direct foreign investment is found to be a growth determining factor where a higher growth
rate of the economy is associated with a higher level of foreign investment.
In
addressing the question
of how direct foreign investment affects economic growth of
developing countries, Borensztein
et al (1998) proposes a model to describe that the
economic growth rates of developing countries are partly explained by a "catch-up" process
in the level of technology. That is, how well
a backward country can adopt and implement
28
new technology already in use in leading countries will determine the economic growth rate
of the country. In their model, technological progress takes the form of introducing new types
of intermediate goods available in the country. The existence of direct foreign investment
lowers the cost
of introducing new technology and thus raises the rate of
technological
change and economic growth. The model also captures the idea of "relative backwardness"
that is used in the Findlay (1973) model which says that the more backward in technology the
country is, the faster the growth rate the country can experience.
Wang and Blomstrom (1992) study technology transfer
in a game theoretics context'
Technology transfer is assumed to be a process when foreign subsidies of the multinational
corporations in the host country obtain foreign technology which is subject to diffusion to
domestic firms. Both foreign subsidised firms and domestic firms must incur costs of
technology adaptations. The strategic decisions between firms then determine the rate of
technology transfer.
These various studies
in technology transfer and economic growth do not raise and deal
adequately with the issue of the interrelationships between technology transfer via foreign
investment and human capital accumulation of the host countries. The objective of this thesis
is to
fill this gap. We argue that
the adaptation of foreign technology depends crucially on the
technology absorptive capacity of the host country. The technology absorptive capacity can
be captured by the levels of infrastructure, education and skills of labour possessed by the
host country. In narrow terms, we may think of human capital as a proxy for the technology
absorptive capacity.
In this
situation we
will
assess
how the interrelationships
between
foreign investment, technology transfer and human capital work in the growth process.
29
Chapter 3:
CAPITAL FLOWS AND BCONOMIC GRO\ryTH
IN A SMALL OPBN ECONOMY
30
l.INTRODUCTION
The growth model of Solow-Swan (1956), once applied in an open economy context, predicts
that under perfect capital mobility a small open economy can jump immediately to its steady
state and stays there forever, and thus there is no convergence.
In addition, the output of
the
economy is independent on its saving rate. Yet empirical studies with samples of open
economies (Mankiw, Romer and'Weil, 1992) find that the output of each open economy is a
function of its saving rate. Barro, Mankiw and Sala-i -Martin (1995) also argue that empirical
evidence shows convergence in open economies.
The present question is how can we explain these issues? The Solow-Swan model treats all
capital as of one type and ignores other factors such as the levels of technology employed,
infrastructure, and education in each country.
It thus does not distinguish
between physical
capital and human capital. Human capital is defined as skills embodied in labour. In reality
while physical capital can be perfectly mobile among countries, restrictions are often
imposed on international labour movements and thus there are some degrees of imperfect
human capital mobility.
In the Solow-Swan
open economy model,
if all countries
share the same production
technology then perfect capital mobility allows capital to flow from rich to poor countries
and enables poor countries to produce output at the same levels
with that of richer countries.
In reality, are physical capital flows consistent with the prediction of the Solow-Swan model?
The problem that poor countries may face is not only a shortage of physical capital but also a
shortage of human capital and poor levels of technology. The existence of a relatively lower
stock of human capital per head in poor countries may cause the marginal productivity of
31
physical capital in those countries to be lower than the world interest rate and thus discourage
physical capital to flow into the countries. This acts as a barrier to restrain poor countries to
produce output at high levels compared to richer ones.
The objective of this chapter is to provide an answer to the addressed question. In this chapter
we will study economic growth in a small open economy using the extended Solow-Swan
model with human capital.
It is shown that the inclusion of human capital
enriches the
Solow-Swan open economy model and gives us several interesting results. To make a cleat
comparison and address the improvement
in the results
obtained,
in section 2.1 we will
review the Solow-Swan open economy model and in section 2.2, the extended Solow-Swan
open economy model is presented. The conclusion
will
discuss these results with respect to
the addressed issues.
2. THE MODELS
2.1. The Solow-Swan open economy model
Milbourne (1997) and Benge and V/ells (1998) have studied economic growth in a small
open economy using the Solow-Swan model. Basically we consider a small open economy
which is populatedby L individuals. The population is assumed to grow at an exogenous rate
n. Time is continuous so that the growth of the population
is
Z(r) I L(t) = n. The labour
force is equal to the size of the population. There is a single good to be produced by means of
capital and labour. Suppose the production function takes a Cobb-Douglass form
Y(t¡ = K(t)"
L(t)'-",
as
(1.1)
32
where y(/) is the flow of output and K(r) is the stock of capital. For simplicity we assume that
there is absence of exogenous growth in technology.
Let y(t)=Y(t)/L(t) and k(r) = K(t)lL(t) be per capita output and capital respectively.
The production function in an intensive form is
y(t)=k(t)".
(1.1')
Perfect competition is assumed to exist so that capital is paid at its marginal productivity less
the depreciation rate
ô:
r(t)=uk(t)"-t-6.
G2)
The economy is small and it faces an unlimited stock of the world's capital at the exogenous
interest rate
7.
Suppose the economy starts at a given stock of capital per head which can
either be lower or greater relative to rest of the world. Due to the diminishing returns to
capital, a lower (higher) stock of capital implies that capital in the country is more (less)
productive and thus has a higher (lower) rate of return. As a result, the autarþ interest rate of
the economy is higher (lower) than the world interest rate. Upon trade in capital, perfect
capital mobility allows capital to flow into (out) the country to equalise the rate of return to
capital with the world interest rate.
Thc flows of capital will make the economy jump immediately to its steady state where the
stock of capital per head is fixed by the country production technology and the world interest
rate as
7
dk
*d-
t^
-Ò
(r.2')
or equivalently
J3
k
(1.3)
=(
In an open economy context, at any time the economy can either hold foreign debt or foreign
assets. Call Z(t) the stock of foreign debt (or foreign assets
if it has a negative
value) held by
the country at time r. Foreign debt is paid at the exogenous world interest rate so that income
accrued to foreigners
is VZ(t). 'We need to distinguish
between the national output and the
national income of the country. While the national output of the country is I(r), its national
income is
Y(t)
(1.4)
-72(t).
Out of their income, domestic residents are assumed to save an exogenous fraction s in
capital so that the aggregate domestic saving on capital is
s(r¡ =
'(r1r¡
-FzØ).
In each period,
(1.s)
(/) is the gross domestic investment
in capital to ensure that capital earns the
world interest rate. The stock of domestic capital is changed by the gross domestic
investment less the depreciation on existing capital as
K(t¡ = I(t) -
6K(t).
(1'6)
Whenever domestic saving falls short of domestic investment, the difference is financed by
overseas borrowing. Thus the accumulation of foreign debt is
2ç,¡ =
I(t) -'(r1r¡ -rz@)
.
(r'7)
Equations (1.6) and (1.7) together describe the dynamics of capital
aro
K(t)=so(r1r¡
-rzØ)+zî)-6K(t).
(1.8)
34
tet
z(t) = Z(t) I L(t) be the stock of foreign debt per head then from equation (1.8) we can
derive the evolution of capital in per capita terms as
(1.8',)
k(t¡ = z(t) + (n - 4s)z(t) + sy(r) - (ä + n)k(t)
Since the stock of capital per head is fixed at all time as given in equation (1.3), this implies
that
i(t)= 0 and output per head is produced at a constant
these into equation
level
of !*
= k*o. Substituting
(1.8') gives us the differential equation which describes the dynamics of
foreign debt
,(r)=-(n-Fs)z(t)-sk*o +(õ+n)k..
(1.9)
For the stock of foreign debt to converge to its steady state, the stability condition must
require that
n-
Zs
>
0.
(1.10)
The phase diagram for z is shown in Figure 3.1.
B
+
0
z
<-
Figure 3.1: The dynamics of foreign debt
35
The steady state stock of foreign debt is z* where ¿ = g
_-
z
sk*o
+(6 +n)k.
n-
(1.1 1)
rs
If zis lessthan z*, z ispositive andzisrising. If zexceeds z*, z isnegativeandzisfalling.
Thus regardless of where z starts, it finally converges to z*
Define a(t) = k.
.
- z(t) as wealth per capita. Substitutingfor z* from
(1.11) into the wealth
function gives us the steady state stock of wealth
(r)
_ sk*o - (ô + rs)kn-rs
(r.r2)
A positive value for wealth implies that sk*"
-
(ô + sr)fr. > 0
(1.13)
We now see how changes in the saving rate affect the steady state variables. Since the stock
of capital employed in the country is determined by the world interest rate as in equation
(1.3), changes in the saving rate have no effects on the capital stock. This also implies that
the output per capita produced by the economy is independent from its saving rate. The
saving rate, however, influences the wealth level and thus the foreign debt (assets) position of
the economy. Taking a partial derivative of equation
ãa¡*
(n." -rn.)@- rs) + r(sk." - (ô + rr)k.
^
)
(1.14)
_-t
ln- sr)-
We know fromequation (1.3) that k*o-r
k*o-t
(I.L2) with respect to s we have
>F or k*" -Fk. >o
=(7+õ)la.
Since
(r+ô) lq>7, it follows
that
(1.1s)
36
Conditions (1.10), (1.13) and (1.15) together imply that âo¡.
lâs>0 or an increase in the
saving rate must raise the stock of wealth. Since the stock of foreign debt (foreign assets) is
the difference between a fixed stock of capital employed and the stock of wealth, an increase
in the saving rate lowers (raises) the stock of foreign debt (foreign assets)'
The Slow-Swan open economy model suggests that under perfect capital mobility, the initial
jump in capital stock causes no transition for a small open economy and the economy
produces output at a fixed level regardless of its saving rate. Yet empirical studies show
convergence in open economies and the output of an open economy is a function of its saving
rate. The next section
will provide
an explanation to these issues.
2.2. Tllre extended Solow-Swan open economy model
The economy is of the same type as described in the previous section, except that capital is
distinguished between two types which are physical capital and human capital. Human
capital is the skills and knowledge of labour. The production function is assumed to take a
Cobb-Douglas form as
Y(t) = K(t)" HQ)p LQ¡r"-o
.
(2.1)
where K(r) is the stock of physical capital and H(t) is the stock of human capital.
cRemark:
If we assume perfect competition then the autarky interest rate of the economy
r,(t)
= aK(to)o-t
at
time /o is
H(t)p L(tr¡t-"-ø - u
3t
At time /o the country opens to the rest of the world. The degree of openness is applied to
physical capital only but not to labour and thus human capital. The economy is small and
rate
faces an unlimited stock of the world's physical capital at the exogenous world interest
F. We
assume a poor country has relatively lower stocks
of physical capital and human
by
capital per head compared to the rest of the world. The autarky interest rate is determined
physical
the marginal productivity of physical capital which in turn depends on the stocks of
capital and human capital of the country. A lower stock of physical capital in the country
implies that its marginal productivity of physical capital is relatively higher than that from the
lower
rest of the world due to diminishing returns to physical capital. However, an associated
The
stock of human capital depresses the country's marginal productivity of physical capital.
domination
of either effects will
determine the position
of the marginal productivity of
physical capital and thus the country's autarky interest rate. The autarky interest rate can
physical
either be less than, equal to or higher than the world interest rate. Thus under perfect
capital mobility, physical capital
will flow in or out the country
depending on its initial
interest rate. In other words, physical capital need not flow to a poor country'
The Solow-Swan model in an open economy context, however, suggests that perfect physical
capital mobility will let physical capital flow into a poor country which has an initially lower
capital per head compared to the rest of the world. In the present model, the existence of a
relatively lower stock of human capital per head in a poor country causes an uncertainty in
the direction of the flows of physical capital. This is the first different result obtained from
this model compared to the standard Solow-Swan open economy model.o
38
The accumulations of physical capital and foreign debt are of the same as described in the
Solow-Swan small open economy model. As given
physical capital can be expressed
o/
kç¡=
'u(r1r¡
in
equation (1.8), the dynamics of
as
-72Ø)+zQ)-6K(t),
(2'2)
where sK now presents the exogenous saving rate of physical capital.
The stock of human capital possessed by the country is the skills and knowledge which are
embodied in its residents. There are many ways that individuals can acquire human capital.
Among of them are education, training or job experience. As is standard in most growth
models, we are only considering education as means of acquiring knowledge. The richer the
resource allocated to education, the better the stock
acquire. For simplicity, as
of human capital that the society
in Mankiw et al (1992), we
can
assume that the accumulation of
human capital is governed by the resource devoted to education.
Let sr be an exogenous
fraction of income that spent on education so that
E(t¡ =""(r1r¡
- rzØ)
.
(2'3)
The stock of human capital is increased by the amount of the resource allocated to education
in each period. Thus the stock of human capital is assumed to evolve asl
itçr¡ = E(t¡ =
'"(r1r¡ -
rzØ)
(2'3')
where human capital is not assumed to depreciate.
t
This is not an optimization model where domestic residents can borrow overseas to finance their optimal level
of investment in human capital. While foreign investors can invest in physical capital, there is no incentive for
them to invest in human cãpital of the home country since foreigners cannot own domestic human capital and
thus cannot claim on domestic labour. For that reason all investments in human capital is made by domestic
residents out of their income.
39
Let k(t¡=K(t)lL(t), h(t¡=H(t)lL(t) and z(t)=z(t)lL(t) be per capita physical
capital, human capital and foreign debt respectively. We can write the production function in
an intensive
form
y(t) = k(t)" h(t)P
as
(2.I')
,
and the accumulations of physical and human capital are
a,
k(t¡=s.(y(r) -rzØ)+ z(t)+nz(t)- (ô+ n)k(t),
ttçr¡
=so (y(r)
-
¡z(t))
Define a(t) = k(t)
-
nh(t)
- z(t)
.
(2.4)
(2.s)
as the domestic non-human wealth per capita. From equation (2.4)
it follows that the individual non-human wealth changes according to
a\t) = k(t) - z(t¡ =r" (y(r) - 7k(t)) -
ã<(t)
-
(n
- s *v)a\t)
(2.6)
We proceed to derive the dynamics of the economy. At any time, the interest rate on physical
capital is equal to the world interest rate so that
7=qk(t)"-'hç¡o
-u.
(2.7)
Since equation (2.7) must be satisfied at all time, this constrains the relationship between
physical capital and human capital per capita to
k(t)=(+)
d-L
p
h(t)'"
(2.8)
For a given stock of human capital which is available in the country and the existing world
interest rate, physical capital
will flow into or out of the country in such a way that (2.8)
determines the stock of physical capital that is employed in the country.
40
Differentiating equation (2.8) with respect to time to derive the evolution of physical capital
AS
/.(/) =
r*Ò\
_t
[
d-l
u)
a+P-r .
R
*nØEne)
L-d
(2.e)
Substitutinefor h(t) from equation (2.5) into (2.9) we have
I
i 1,¡ =
(*)^ J- hØT;(,,
(
(2.r0)
rr,r - rz(ù) - nh())
which describes the evolution of physical capital in terms of output, stocks of physical
capital, human capital and foreign debt.
By rearranging terms we can write equation (2.8) equivalently
h(t¡=(+)
as
l-d
utçr¡o
(2.8',)
Also by substituting equation (2.8') into the production function (2.1') we can express the per
capita output as a function
*u
v(t)=- a
of physical capital
per capita alone at any time
(2.rr)
n(r)
Finally we can substitute fory(r) from (2.11),h(t) from (2.8') andz(t) into equation (2.10) to
derive the dynamics of physical capital per head as a function of two variables which are
physical capital and individual wealth
k(t) =
soB({r+Ðtu-F)
r+ál p (r- a)
_t
q)
'p
k(t)
.-l-o
+
7toþ
r*Ò\ þ
_t
(r- a)
d)
-(r-a-þ)
k(t)
P
ú)(t)
- Ér_*nU, . e.r2)
4l
The accumulation of individual wealth can be obtained by substituting for y(r) from (2.LI)
into equation (2.6)
( (¡+6
'
a(t)=["[
(2.r3)
- ¡ì-alrtrl-@-s*7)at(t)
)
)
"
2.2.I. The dynamics
Equations
(2.I2)
and
(2.I3) are basic equations of motion which describe the dynamics of the
system. We now proceed to construct a phase diagram to study the dynamics and the stability
of the system. In order to do so, we need to derive the curves fot
-õ
,sK
The a(t) =
0 locus is described
as a\t) =
a\t) = 0 and k(t) = g
n-
sKr
(2.14)
k(t)
fhe a(t)= 0 curve is a straight line passing through the origin. Depending on the sign of the
term
'.(+-l-'
nsKr
, the curve can either be in positive or negative quadrants. We
are
interested in the positive quadrant with positive values for wealth and capital since negative
values for wealth and capital will make no sense. Thus we need to impose the condition such
that
'.(+-l-'
nsKr
Suppose this condition is satisfied when
n-
(c1)
>0
,.(-+I) - ô > 0 and n^\d
so.>
0.
(c1')
s*F > 0 is the familiar stability condition in the Solow-Swan small open economy model
42
I-d
rrre i(r) = 0locus is described as ø(r)
=
(t -
Ç)orr,
.
snf
k(t) þ
.
(2.rs)
To find the shape of this curve we follow these steps. Take the first derivative of ar(r) with
respect to k(t)
1
âa(t)
7+ôì p
_t
ø)
n(l-
6
=I---r
dv
ã<(t)
7+
r-a-fl
k(t)
p
þt rF
þ
r-a-þ
ry=o
dk(t)
If
when
k=
k(t)> k", then
(r(1- a)+6)þsu
W>
Take the second derivatives
â2
a(t)
0 , and 1f k(t) <
ffi
to
of a¡Q) with respectto k(t)
n(1- a)(I- u - Ð(
ilr(t)'
k,then
þ"r7
r-u-2þ
k(t) f
29
It is obvious that k">0' To the left of k,'the iç'¡=0 curve is decreasing in k and to the
right of k, , the içr¡
=0
curve is increasin g in
k. fhe k(r) = 0 curve is minimised at k. ' The
phase diagram is given in Figure 3.2.
43
k=0
a
(Ð=0
A
k
Figure 3.2: The dynamic system
The steady state positions of the economy are at points 0 andA where
a(t)=k(t¡=0.
find the stability of the steady states we note that below the a\t) = 0 locus, Ø(t) > 0 or
is increasing and above
it a(t) < 0 or a(t)
To
at (t)
is decreasing. Below the k(t) = 0 locus, k(t) <0
or k(t) is falling and above it k(t) > 0 or ft(Ð is increasing. The arrows describing
dynamics are indicated in the diagram which shows that point
these
A is stable and point 0
is
unstable.
44
2.2.2. The steady state
In the absence of technological progress, the steady state is defined as a path where the stocks
of physical capital, human capital, output, wealth and foreign debt per capita are constant.
There are two steady states for this economy which are at point 0 and point A as displayed in
Figure 3.1. However, point 0 is unstable whereas point A is stable. Thus starting from
anywhere the economy will finally converge to point A. The steady state values at point A ate
calculated as
/-
lrn+ Òn- aÒr - uv-\)sH
an(n -s"r)(ir + õ) I a)''P
k
fl
1-a-þ
(2.16)
1-a
¡. =((7+6)lø)'Þ¡. n ,
y*
-d
-
(2.r7)
r*Ò
(2.18)
-2"
û)
'.(+-l-,
n-sKr
z* =k* -cÙ
.
d(õ + n) - s"(F + ô)
/-\
(2.re)
k*,
k*
(2.20)
\n-s*r)a
This is the long run outcome for the small open economy.
45
2.2.3. The transition: the speed of convergence
In this section we are interested in studying the transition of the open economy towards its
steady state, starting from the initial time when the country opens to the rest of the world.
The question we want to address is what is the speed of convergence and how long does this
take for the economy to reach the steady state. As suggested in Barro and Sala-i -Martin
(1995) and Romer (1996), we analyse the transitional dynamics by resorting to the method of
linear approximations around the steady state. As noted in section 2.2.I, the dynamics of the
system is described by the two equations (2.I2) and (2.13). Moreover, equations (2.12) and
(2.13) describe
a constant value
k(t)
and
ø(t)
as functions of
k(t) and a(t). In the steady state, ft and
of fr- and úD* respectively.
We take first-order Taylor approximations to (2.12) and (2.I3) around k =
.
a take
ditt)
k*
and
a = (I)*
iç'¡
=
ffi r=r..,-,.(ntù- m r=r.,,-,,('{') -'.)'
(2.2r)
àç¡
=
*
m r=r.,,=,.(rr,, - r.)
(2.22)
¿.) *
¿.) *
r=r.,,=,.(or,r-
Note that i1r¡ = dk(t) I ¿t =
alf U) - lr.l t dt
,çr¡
at.f I ú since ú¡* is constant. In addition, define
= da\t) I dt = dla@
-
alt<çt¡-lr.ltú=k7)lk.
and
since
.
k* is constant. Similarly,
dfa\t)-o.lldt=aqt¡!Ø*
so that we can write (2.2I)
and (2.22) as
kQ¡lk. =¿!!)
(or,>
k=k .a=a.
- o.) *#o
k=k,,a,=a.(ae)
- ø.) ,
(2.2r',)
46
(2.22',)
Using equations (2.12) and(2.I3) to calculate the derivatives
â
k(t)
ãlc(t) o=0,.,=,.
_(zB-t+
=
ø))(so @ + d)
t a-rtr)
((r + a) t o)''u çt*¡
(a+þ-l)Fsn
k(t)
âa(t) k=k,,a=a,
nþ
rL'_ r-a
âa(t)
&(t)
ã
(2.23)
þ
t-+(d+p-t)tlJ
(2.24)
='"((¡ +6)ta-7)-õ
(2.2s)
o=o',,=,.
a¡(t)
âa(t)
=@"
7t n
_
ç+1p+a_t)tn 'r
K
O*1tx-t)tgr¿*
@t'
â
as
= -(n
-
(2.26)
s*F).
k=k,,a=a,
Substitutin g for
k.
and ú)* from equations (2.16) and (2.I9) into equati on (2.23) and (2.24)
we have
aill
Ar(ù o-0.,,=,,
aift>
âot(t)
Let
â
k=k,
:=
(t-
þatr(n-'"u)
a)(rn + õn -
aù -
n(t-þ-a)
anr)
l-
(2.23',)
a
røtB(n- t"¡)
=
,ø=a. (1- @(rn + 6n - a& - mr)
k(t)
ãk(t)
-8,
o=0.,,=,,
ã
k(t)
ãa¡(t)
C,
k=k,,ora¡,
(2.24',)
âa(t)
ãlr(t)
D
o=o',,=,.
and
âa(t)
âro(t)
_E
k=k,.a=ar
whcrc B, C, D and E are equal to the right hand side of equations (2.23'), (2,24'), (2.25) and
(2.26) respectively. We can rewrite equations
(2.2I')
and
(2.22')
as
47
kçt¡l k. = n(k7) -
-
fr )
+ c(a4t¡
- ø.),
a4t¡lr. = o(nØ- ¿-)* E(a\t)Dividing both sides of (2.27)by k(t)
growth rates of k(t)
(2.21)
r,r.)
(2.28)
- k* and both sides of (2.28)by a(t) -
a)* to derive rhe
- fr* and o(t) - o)* as
(2.2e)
øft\!a.
(D(t)
k(t\ - k.
a(t) - a-
= Lt-------------
- (Ð"
We know that
k(t)-k*
steady state value.
measures the distance
of the physical capital stock at timet andits
Similarly, Ø(t) - to* measures the distance of the stock of wealth at time t
and its steady state value. Thus
converges to
(2.30)
Ì E
k(t)
- r) t(nç¡-f-)
its steady state value and (ø(r)
is the rate at which physical capiral
tò,@<r>-ø.) i,
rhe rare at which non-
human wealth converges to its steady state value. Suppose that physical capital and wealth
adjust with the same rate which we call
a
p
=(tç¡:
o) , U,r,>
-k.) = (rura
p
where
ù , @() - at.)
is thus the speed of convergence. Substituting
a(t)- a. pk(t)
-
k.
B
C
forp
(2.3r)
into (2.29) we have
(2.32)
Substitute (2.32) into (2.30) we have
p'-(B+E)l.t-CD+BE=0
(2.33)
48
The solutions for equation (2.33) are
pLet
(n + n)t ((a + E)' - 4(BE
-
,r))'''
(2.34)
2
p, and p,
be the two values
of p .For the economy to converge to its steady
state,
p
must take a negative value. Since output per head y is a linear function of k and the world
interest rate
r
as shown in equation
(2.1I), the convergence of k at the rate
also converges to its steady state value at the rate
p,
p implies that y
that is
(2.3s)
þ
The solution for this differential equation is
y(t)
- !* = c(h' + creh' ,
(2.36)
where C, and C, are constant. Rewriting equation (2.36) gives us the equation describing the
transition of the economy
as
y(t)=y* +Creh'+Creh'
(2.36',)
The two observed boundary conditions are:
when r = 0 then y(0) = yo,
(A)
y("")=y-
(B)
When
t=æ
then
From (A): C,
t
C, = lo
- !-
.
As indicated in the stability analysis in Figure 3.2 in section 2.2.I, the steady state at point A
is complete stable. A complete stability of the system implies that starting from anywhere the
economy will eventually converge to point A. In other words, any path will lead the economy
to its stable steady state. This implies that in the transition equation, the convergenc e rate
and
p,
ltl
must both take negative values. The transitional path is thus described as
49
y(t) = y* + Creh' + ()o
-
y*
-
Cr¡eh'
(2.36")
.
As an example, suppose a small open economy faces the world interest rate r = 0.03. The
population of the economy grows at the rate of n = 0.02. The production function intensities
of physical capital and human capital are assumed to take the values a= þ=I13. The
saving rates in physical and human capital are sK = sn = 0.2 and the depreciation rate of
physical capital
14
is
6 = 0.02. Applying formula (2.34) gives
= -0.016 and p" = -0.0089
two solutions for lt
as
.
The question arises as to how changes in the saving rates affect the speed of convergence.
Reading from equations (2.23') to (2.26) we note that the saving rate in human capital does
not enter the formula for B, C,
D
and
E
and thus
p
in equation (2.34). The saving rate in
physical capital, however, appears in the formula for B, C,
see the effect
D
and E
in
a complicated way. To
of the saving rate in physical capital on the speed of convergence we must
resort to a simulation2.
In the simulation method, numerical values are assigned to exogenous parameters. The values
are chosen as above except
for s,. To
show the impact of changes
in s^ on the speed of
convergence we assign all these values to exogenous parameters in equation (2.34). Letting
s¡( run from 0 to 0.6 we run the simulation on equation
(2.34). The result is reported by the
following figure
2
Mathcad program is used to run the simulation.
50
The speed of convergence
0
- 0.005
t¡1( s)
-0.01
P2( s)
-
-0.015
-0.02
0.1
o.2
0.3
0.4
0.5
s
The saving rate in physicalcapital
Figure 3.3: The impact of changes
This figure shows that as
in .s, on the speed of convergence
s, increases, one value of p
is falling and the other value of it is
rising in absolute values. This result may suggest that these two changes cancel each other
out and leave the general effect to be zero. However, since we do not know the values
of
C,
in the transition equation (2.36") we cannot give a definite conclusion about the effect of
changes
in the saving rate in physical capital on the
speed
of convergence. 'We come to
Proposition 3.1.
Proposition 3.1: The saving rate in human capital has no effect on the speed of convergence
while changes in the saving rate of physical capital have uncertain effects
on the speed of
convergence
In comparison to the standard Solow-Swan model in an open economy context we note that
the standard Solow-Swan model suggests that a small open economy can jump immediately
to its steady state with no transition and thus the speed of convergence would be infinity. Any
51
state or the speed of convergence'
change in the saving rate does not affect either the steady
can alway lend or borrow
This is because capital is perfectly mobile and thus the economy
of capital employed at a fixed level
capital at the world interest rate so that it keeps the stock
capital, the small open economy
irrespective of its savings. In the present model with human
does have
a transition towards the steady state and the rate of
convergence can be
by the saving rate in human capital'
determined. While the rate of convergence is unaffected
changes
in the saving rate of
physical capital have uncertain effects on the rate of
convefgence.
of the stocks of human
We close this section by illustrating diagrammatically the transition
from time
capital, physical capital and output per capita over time starting
economy starts at its autarky level
/o
'
Suppose the
h",kn and lo ãt time fo. At time /o when the country
is unchanged since people are not
opens to the rest of the world, the stock of human capital
capital is perfectly mobile so that
allowed to migrate in or out the country. However, physical
country's human
it will jump to a level where it is determined by the existing stock of the
capital and the world interest rate as given
situations: physical capital
in equation (2'8)'
will flow into or out of the country'
interest rate of the country and the world interest rate.
There
will be likely two
depending on the autarky
of course,
we do not rule out the
physical capital employed in the countfy
special case where there is no change in the stock of
interest rate.
since the autarky interest rate is just equal to the world
rate is higher than the world interest rate then capital
If
the autarky interest
will flow into the country'
otherwise
jump to a level which is determined by
physical capital will flow out. output per capita will
physical capital employed in the country'
the existing stock of human capital and the stock of
steady state' Figures 3'4 and 3'5
From then the economy undergoes the transition towards the
describe the transition.
52
h
h
hn
to
Time
k
k
to
v
Time
v
v
v
to
ro
Figure 3.4: The transition towards the steady state position. The case when
)
7 at to
53
h
Time
h
ho
t0
k
Time
k*
ka
ko
to
v
Time
v
!o
!o
to
F'igure 3.5: The transition towards the steady state position. The case when ro
17 at to'
54
In Figure 3.4, we consider the case when physical capital flows into the country at the time
when the country opens to the rest of the world. Figure 3.5 provides the case when physical
capital flows out the country at time /0. In both cases we assume that the steady state stocks
of human capital and physical capital are higher than their initial values.
Our interest is to know how long
it
takes for the economy to reach its steady state, starting
from the autarky level. We focus on the transition of fr towards its steady state value since the
dynamic behaviour of k is similar to that of y. As displayed in Figures 3.4 and 3.5, the
economy starts at its autarþ level at k". At time /o when the economy opens to the rest of
the world, physical capital will jump immediately
to ko which is a function of h,
After that, k will converge to its steady state value
and 7 .
k* at the rate ¡z . Thus the number of
years that must be taken for the economy to move half way to its steady state can be found as
to.r=-lnolp
Q'37)
n")
I@. -
where d =
k
-(n, - n.)
ka
Due to the initial jump in k from kn to ko at time /0, convergence would be either faster or
slower depending on the position
of /ro with respect to ko. The difference between the
autarky interest rate and the world interest rate
k" . As in the case in Figure 3.4 when rn
means
that
as shown
fro
-
kn
will determine the relative position of ko and
) 7 at time /0, the initial inflow of physical capital
) 0 so that convergence is faster. In another case when ro 17 at time fo
in Figure 3.5, the initial outflow of physical capital implies that
fro
-
kn <
0
or
convergence will be slower.
55
the steady state
2.2.4. Comparative statics: The impact of changes in the saving rates on
variables
'We
rates of the
note from section2.2.2 that all steady state variables depend on the saving
steady state
country. Thus policies that change the saving rates can have influence on the
state variables'
variables. We now see how changes in the saving rates can alter the steady
per capita
To see the impact of changes in the saving rates on the steady state physical capital
we take the partial derivative
of ft- with
to sr and s"
respect
p
dk*
âs K
( 7n+
7P
r-a-þ
&t- aff - an|)s,
d-l
r-a-þ
(2.38)
(n- s*7)r-"-P ,
*(e+õ¡ta)''P
p
dk*_
ât,
p
( 7n+ &t- aff - an7
t-q- þ øt(n-s*r)((r+Ðla )"
By condition
(cl'): n-
S
l-a
or-a-þ
(2.3e)
sul >0.
From equation (2.16), a positive
7n+ õn-
r-a-þ
k-
implies that
(2.40)
q& - uF >0.
Thus äk-
lâs*>0
and ãk.
lãs, >0 or an increase in the saving
rate of either type of
an increase
capital causes a higher stock of physical capital employed per worker. similarly,
in su or s"
also raises the stock of human capital per head as
dh* r(L- a) (r+ó) la)r-"+
1-a- þ
âs K
(rn+ õn- aff - æt|)so
Øt
-'-o ^
l-d-ß
z(,.-l)+þ
' (n-sKl)* t-a-þ >0'
(2.41)
56
I-d
dh* _
âto
(r-
ø)
(r+6)la
r-a-þ
r-a-þ
These results imply
that
k
m¡)s r) r-a-þ
(¡n + õn - a&
s*D
þ
sor-"-P >O
-
(2.42)
)
-*4n-
and
h are complements. An increase in k (ot h) due to an increase
in the saving rate is accompanied with an increase in h (or k)'
human
Since the domestic output depends positively on the stocks of physical capital and
capital
capital, an increase in the saving rate of either type of capital causes higher stocks of
which lead to a higher output produced. Taking the partial derivative
,s¡(
or
of Ø* with respect
to
.sn rwe show that individual wealth increases as a result of an increase in the saving
rates:
p
âu¡.
(rn+ õn- q& - m7)su
ãs K
*(fr+õ¡ta)''P
f#l'r(+ I
n- sKr )t-"-P
a]{.-,",)
P
âo¡*
âs H
(rn+ õn- q& - ut|) ,-"+
*(e+6¡ta)''P
a-l
r-a-þ
+
2(d-r)+þ
zT+a-r
(2.43)
r-a-þ
.
a-l
Borffi(n- s*r)*o
t-a- þ
["(# I
,]'0,
(2.44)
by conditions (c1') (page 42) and (2'40).
proposition 3.2: An increase in the saving rate in physical capital or human capital raises the
steady state stocks
of physical capital, human capital and domestic wealth per capita.
57
We explain these results intuitively
An increase in s *
An increase in the saving rate in physical capital will immediately raise domestic wealth.
Initially domestic output per capita is unchanged due to fixed stocks of capital per head' A
higher domestic wealth accompanied by an unchanged stock of employed physical capital
implies that the stock of foreign debt (foreign assets) must be reduced (increased). For
a
given level of domestic output and a lower (higher) stock of foreign debt (foreign assets), the
domestic income must be higher. since an exogenous fraction
of
domestic income is
on the
allocated to education, an increase in domestic income means more resources are spent
of
accumulation of human capital resulting in a higher stock of human capital' A higher stock
human capital causes physical capital
to be more productive leaving the marginal
productivity of physical capital well above the existing world interest rate.
higher rate of return to physical capital
will
A
relatively
attract physical capital to flow into the country
and thus raise the stock of physical capital per head which
is employed in the
country'
Domestic output can be produced at a higher level. This process keeps going until the
capital'
economy reaches a ne\ry steady state with higher stocks of human capital and physical
at
Figure 3.6 shows this effect diagrammatically. Initially the economy is on the steady state
point A.
A
rise
in
.rK makes the curve tçr¡ = g steeper but leaves the curve
ilt¡
=g
point A.
unchanged. Initially k is constantbtst a¡ is rising which moves the economy above
Theeconomyissomewhereabovenel<(t)=0curvecausingkstarttorise.Thereafter,both
58
@ andfr increase and the economy reaches the new steady state at point B with higher levels
of physical capital and wealth per capita.
k=0
û)
B.
Ø=0
a¡=0
1
k
0
Figure 3.6: An increase
An increase in
in
,s*
s,
An increase in so immediately raises the resource allocated to education resulting in
a higher
stock of human capital. Since human capital cannot flow into or out of the country, a higher
stock
of
human capital makes physical capital more productive and thus increases the
marginal productivity of physical capital above the world interest rate. This causes more
physical capital to be employed in the home country and thus a higher level of domestic
59
output is produced. This process continues until the economy reaches a new steady state with
higher stocks of physical capital and human capital.
Figure 3.7 shows this result. An increase
does not affect the
in so makes
curve ,çr¡ = g . Initially,
ø
the curve k(t¡ =
0 become flatter but it
is unchanged but fr is increasing due to the
inflows of physical capital to the country to take the advantage of a higher rate of return to
physical capital. The economy is placed away from point A and at somewhere below the
ú)(t) =
a
0 curve causing
a
starl to rise.
After that both
ø
and ft rise and the economy reaches
new steady state at point B with higher levels of physical capital and wealth per capita.
(r)
k-
k =O
Ø=0
A
0
-|'
k
Figure 3.7: An increase
in
,srt
60
Finally we see how changes in the saving rates affect the foreign debt (foreign
assets)
position of the country. Taking the partial derivative of z- with respect to s,. we have
i*
l-o
(
#=^f-rt
/1 ^,\=
I
.m@@+n)-srt'+al)J,
+õ)(n-,*7)ú
(2'4s)
þ
1-a-þ
at>0.
where A =
The steady state stock of foreign debt (foreign assets) is determined in equation (2.20).
Equation (2.20) suggests that
z*
if z* > 0, it is foreign debt and if z* < 0, it is foreign assets' If
>o then (aç6+n)-src1r+ô¡)>0,
Equation (2.45) follows that
if
and
z* < 0 then
if
ù.
z*
<0 then (açõ+n)-s*(7+á¡)<0.
I ãsu <0
or z* is decreasing as sK
increasing. However, since z* takes a negative value, this then implies that a fall
its value increases in absolute terms. Thus a higher
Intuitively, a higher
s*
s,
is
in z* means
raises the stock of foreign assets.
raises the stock of physical capital employed and also the stock of
wealth as displayed in Figure 3.6. The stock of foreign assets is the difference between the
stock of physical capital and the stock of wealth. An increase
in su results in the stock of
wealth raises by an amount which is more than an increase in the stock of physical capital
employed in the country which leads to a rise in the stock of foreign assets.
Taking the partial derivative
of
z-
with respect to so we have
p
( 7n+&t-aff-mF)
æt(n
-s*r¡((r
l-a-þ
zþ-r+d
S, l-a-P
(2.46)
+ õ¡ I a)''þ
6t
z* is foreign
assets when
it takes a negative value as suggested in equation (2.20) or
(a@+n)-soþ +Ð)<0. It thus follows that ù. lâs, <0 or z*
increases.
decreases
In absolute values, this implies that a higher so raises the stock of foreign
as
sH
assets.
3. CONCLUSION
The impact of changes in the saving rates on the steady state variables can be summarised in
the following table
Open economy
Extended open economy Solow-Swan model
Solow-Swan model
ds>0
ds*>0
dsr>o
dy.
0
+
+
dk.
0
+
+
dh*
na
+
+
dz. (foreign assets)
+
+
+
dú)
+
+
+
na stands for not applicable.
In
comparison with the standard Solow-Swan open economy model, there are several
clifferences that can be obtained from the extendcd Solow-Swan open economy model.
Firstly, in the Solow-Swan model without human capital,
it is always
possible that under
perfect capital mobility a small economy which faces an infinite supply of the world capital
62
can
jump immediately to its steady state position without
it
transition. In the extended Solow-
it
must take time for the economy to
adjusts its formation
of human capital and the stock of
Swan model, this outcome cannot be obtained since
undergo the transition when
a
physical capital towards the steady state. The speed of convergence is found to be definite but
it is unaffected by the saving rate in human
capital while changes in the saving rate in
physical capital have uncertain effects on the speed of convergence.
Secondly, the existence of a poor stock of human capital accompanied with a lower stock of
physical capital per capita
in a poor country may not make its marginal productivity of
physical capital be higher than the world interest rate. Thus under perfect physical capital
mobility, there is an uncertainty in the direction of physical capital movements; that
is,
physical capital need not flow to a poor country. Thirdly, during the adjustment, the available
stock
of human capital governs the stock of physical capital which is employed in
the
country. A country that has a richer stock of human capital employs more physical capital
and thus experiences a higher growth performance. Finally, while in the Solow-Swan open
economy model the steady state output per capita is fixed, in the extended Solow-Swan open
economy model the output per capita can be controlled by the government which
uses
policies that change the saving behaviour of private agents'
The final different result deserves some explanations. In the Solow-Swan open economy
model, physical capital per head is the only factor that matters in the production function and
it is mobile. Capital flows freely so that its rate of return is always equal to the world interest
rate. This determines the per capita capital that is employed in the country and thus the level
of domestic output. Policies that change the stock of capital owned by domestic residents
63
cannot alter the marginal productivity
of capital and so the capital labour ratio that is
employed in the country
In the extended Solow-Swan open economy model, human capital is another factor of
production function and more importantly,
it is immobile. The immobility of human
the
capital
factor means that the stock of human capital that is possessed by the country is also equal to
the stock of human capital that is employed by it. Since human capital cannot flow, policies
that result in a change in the stock of human capital can affect the marginal productivity of
human capital and physicalcapital. This effect creates a gap between the rate of return on
physical capital and the world interest rate causing physical capital to flow in or out the
country to close the gap. The economy reaches a new equilibrium'with different ratios of
capital to labour employed and thus a different level
produced. From this argument, desired ratio
of
domestic output per capita is
of domestic capital to labour and level of
domestic output can always be obtained.
64
Chapter
4z
CAPITAL FLOWS, INTERNATIONAL TECHNOLOGY
TRANSFER AND ECONOMIC GROWTH IN A SMALL
OPEN ECONOMY
65
l.INTRODUCTION
Among its many objectives, growth theory tries to explain why there exist large differences
in income levels and growth performances in different countries, and what can be done to
close these gaps. The Solow-Swan (1956) model explains that countries with different saving
rates have different income levels
in the long run. Mankiw, Romer and Weil
(1992),
however, explained that the Solow-Swan model has a deficiency in explaining the crosscountry income differences. In the Solow-Swan model with a conventional value of capital's
share, large income differences among countries are only explained by vast differences in
saving rates. Mankiw, Romer and Weil (1992) noticed the role of human capital which
consists of the abilities, skills and knowledge of workers. By introducing human capital into
the Solow-Swan model, they are able to improve the model's ability to account for cross-
country differences.
In the extended
Solow-Swan model with human capital, moderate
changes in the saving rates can lead to large changes in output per worker'
The Solow-Swan model, applied to an open economy with perfect capital mobility, shows
that for a given technology level, the stock of capital and thus output of a small open
economy are fixed by the world interest rate. Changes in the saving rate of the country cannot
alter the stock of capital employed and output produced
change the level
in the country though they
of wealth. Beside the Solow-Swan model in an open economy
can
context
predicts that perfect capital mobility allows a poor country to jump immediately to a position
where
it
can produce the same output level as a richer country. These do not happen in
reality. The question is can the extended Solow-Swan model with human capital explain the
issues better?
studied
In Chapter 3 we used the extended Solow-Swan model with human capital and
it in an open economy context. In the absence of technological
progress, we found
66
that each economy reaches its steady state where the steady state income and wealth are
functions of its saving rates. A higher saving rate in human capital directly raises the stock of
human capital while a higher saving rate in physical capital indirectly raises the stock of
human capital via the wealth effect. Since human capital is assumed not to flow out or into
the country, in both cases, a higher stock of human capital leads to a higher output produced
and the stock
of wealth. The model thus can explain the differences in income levels and the
level of outputs produced among small open economies; richer countries are the one that
have higher saving rates. However, due to the absence of technological progress, the model
falls into the type of exogenous growth and thus it is unable to explain why countries have
different growth rates.
'Why
are growth rates different among open economies? For many countries, the openness to
the rest of the world allows them to have access to world technology. So why do different
countries which have access to world technology grow differently? There is an argument on
the grounds that the difficulty that such countries face is not the lack of access to advanced
technology but lack of abilities to use that technology (Romer, 1996).In other words, the
degree of world technology absorption depends critically on each country's level of human
capital. A country with a richer stock of human capital is able to adopt a more sophisticated
level of technology and experiences a faster growth. To explore this issue, in this chapter we
develop the extended Solow-Swan model with human capital incorporating technology
adoption in an open economy context. The model is the extension of the model in Chapter
3
where we introduce international technology diffusion into the previous model. The purpose
of this chapter is to explain why different growth rates may exist in different countries' In the
next section the model with its detailed study is presented. The conclusion to summarise the
results is given in the last section.
67
2. THE MODEL
We consider a model of a small open economy which is similar to that in Chapter 3 except
there is technological progress in this model. The labour force is equal to the size of the
population which is constantt. There is a single good to be produced whose production uses
physical capital K(r), human capital Ë(r), labour
L
and the technology level
A(r) according to
the Cobb-Douglas production function
y(t¡ = K(t)" u(ùP(t@L)r-"-o
(1)
,
where the term A(t)L is defined as effective labour.
The country has access to unlimited physical capital at the world interest rate 7 ' While
physical capital is assumed to be perfectly mobile, labour and thus human capital cannot
migrate from the country. perfect mobility of physical capital
will
assure that physical capital
earns the world interest rate at any time. At time / the country can either hold the stock of
foreign
debt
Z(t) or foreign
assets
if
Z(t) has a negative value. As in Chapter 3, we have
basically the same equations describing the economy's accumulations of physical capital and
human capital as
o/
kç¡ = ro (r1r¡ - rzØ)
itlr¡ = r"(r1r¡ -
+ zQ)
-
DK(t)
,
(2)
rzØ),
where .rK,,r¡l and
â are the exogenous
(3)
domestic saving rates
in physical capital,
human
capital and the depreciation rate ofphysical capital respectively.
twe
Since the growth rate of labour plays no role in determining the per capita growth rate of the economy,
does not alter the
assume a zeio growth rate of labóur for simplicity. Non-zero growth rate of labour, however,
basic results of the model.
I
68
'We
assume that the
Now we introduce international technology diffusion into the model.
openness to the world allows the country to have free access
grows at the exogenous rate
g*.
to world technology which
However, how well world technology can be adopted
depends critically on the country's technology absorption.
In other words, while world
technology is free to obtain, it can only be used in the country
if
the country has an adequate
skill level to handle it. The level of human capital possessed by the country acts as a proxy
for the country's technology absorption. Thus we
assume that changes
technology level is determined by its general level of human capital
as
(Ir(t) t L,
Àçr¡ =
where
in the country's
(4)
is the exogenous technology absorptive parameter. Equivalently, the rate of change
f
in technology can be written
A(t)
A(t)
as
H(t)
5
(4',)
A(t)L'
The economy has access to the full stock of world technology with the growth rate of 9".
Depending on the economy's activities, the country's rate of technology acquisition can be at
any value up to the maximum
Derine
9".
î(t)=Y(t)/(eçt¡r), t (r)= K(t)t(eçt¡r), nQ)= H(t)t(t'ç¡r)
2(t) = Z(t)
l(eçt¡f)
ana
as output, physical capital, human capital and foreign debt per unit
of
effective labour respectively. V/e can write the production function in an intensive form as
î(r)= n(t)"n1)P
,
(1')
and the evolutions of physical capital, human capital and technology are
69
o(,)
=,*(î(r) -r2e)) - a,r,>
,(r) =r"(î(r)
u]t,,,
[*.
-r2e))-frnu,
*i(,).*or,
(2')
(3',)
,
(4")
A(t)tA(t)=€h(t)
Let îoçt¡ = nØ - ¿(t) be the domestic
non-human wealth per effictive labour. The
accumulation of non-human wealth can be derived from equation (2') as
tt^'
îrç,¡ = nr,> -âQ)=,"(î(rl
- îîa)) - a,o't
[*
- r,-Jr,,,
(s)
Under perfect physical capital mobility, physical capital must earn the world interest rate at
any time so that
¡=aî(ù"-'û.ç¡n-U,
(6)
or equivalently we have a relationship between physical capital and human capital
per
effective labour at all time as
L(r) =
f
r+â\ "-1^
h(t)'"
_t
[ u)
(7)
This equation says that the stock of physical capital to be employed in the country in each
period is determined by the available stock of human capital in the country and the existing
world interest rate.
Differentiating equation (7) with respect to time we derive the evolution of physical capital
'^(
rrr)
r+ô-ì
_t
=
d)
[
d-l
n
a+ß-1
T\nç¡ '-"
as
o
nG)
'
(8)
70
Substitutin
gfor ûçt¡ from equation (3') into (8) we have
'oa+B-t('l
o(,)=(+)^ -Ê--na>ï#[,,{r,,, -72(ù)-Prrur),
(e)
which describes the evolution of physical capital in terms of output, stocks of physical
capital, human capital, foreign debt and the rate of technological progress. To derive the
dynamics
of physical capital as a function of physical capital and domestic
non-human
wealth alone we first rearrange terms in equation (7) to obtain
I
(r+ô\7^
t-d
"
iìqt¡=l;)-kçt¡o
(7')
Substituting equation (7') into the intensive production function (1') gives us
i(r) =
-a
rIÒ
^
(10)
-k(t)
Finallywecan substitutefor
i(r) from(10), á1r¡ fro* (7'),2(t)
utO å1r¡ lA(t) from(4")
into (9) to obtain
îqt¡ = tu
+-)oaf.î *,,,fi(-"
p
l-a
tL(Ll!\F
"1-øt a ) oç,¡'.î
-1
)
a+B-l
trfr>
P ô(t)
(1 1)
.
The accumulation of non-human wealth per effictive labour can be obtained by substituting
for î(r) from (10) and A(r) I A(t) from (4") into (5)
àç,¡=[,.(#
I
a]rr,r
[{-"
);0,,,î-,"-]r,,,
(r2)
We proceed to display the dynamics of the system.
7t
2.I.
The dynamics
The dynamics of the system are described by two differential equations (11) and (12)
as
functions of physical capital and domestic wealth per effective labour.2 The dynamics of the
system can be displayed by a phase diagram. The phase diagram is constructed by two curves
aa
t çt¡ =0
and ãçt¡ = g .
.^2
rne ¿(r) = 0 locus is described as ár(r) =(+)'
When îçt¡ = g,
îoçt¡ =
5
snf
o<aT1+ )+ ¡û(r). (r¡)
0. To derive the shape of this
curve, firstly, we take the first
derivative of îo(t) with respect to ft(r)
âô(t)
ãk(t)
2€(r- q) tr*Ò\
sr7þ
l")
uo1,¡T"-(#-)
2
þ
2(r-d)
s,þ(rçt- q) +
P
2aÉ(r- a)
..
dô:(t)>o.and¡nrt< k,
^ then
If--.
k(t) > ft. then =*rU
Take the second derivatives
â'ô1t¡
ãk(t)'
Thus the
)
r\þt
nç¡ = 0 locus
ffi
<O
of ã(t) with respect to t1r¡
zÉ(t- Ø( 2-2a- B)
s
âîo(t\
Z
(-")
is minimise d
2,-d),1
'nG) f '2g.
at î,, where [.
is clearly greater than zerc.
îù
'The ,eason for us to choose to display the dynamics of the system in two variables É and
dynamics'
the
to
study
clear
way
and
very
easiest
it
is
the
variables is because
rather than other
72
The ô)(t) = 0 locus is describe d as îo(t¡ =
'.(+-l-,
/-
ô\-
1
(/)
(r4)
l-d
'\4ry9)unç¡u
a) " -sxl
When
Êçt¡ =
O
, îo(t)= 0. Take
the first derivative
,)
âô(t)
I
r+â\ =p
_t
d)
dk(t)
(r+õ
Ir '"[ o -î I uto
)ãçt¡
of îo(t) with respectto t (t)
then
u9"\t'
dk(t)
.o
l-a
nØu
+s*7
or îo(t) is strictly decreasing in É(r). sln..
=0 when Ê1t¡ =0 then tne îo(t) = 0 locus is on the negative quadrants
negative wealth or physical capital.
strictly increasing ¡n lc(t) tt
"n
,, ,-(-+-l-á<0
u
)
then
ry>O
ã<(t)
with either
or ár(r)
is
å(r) = 0 locus is on the positive quadrant, starting from the
origin, with positive wealth and physical capital. We restrict our interest in the positive
quadrant so that we need to impose the condition
,..('*u-rl-ô<0.
^\u )
(c1)
The steady state position of the system is defined as where O(r) = ,îrçr¡
equations (13) and (14) we can obtain the steady state f
Equating
- as a solution of the equation
a
3
;p^ *p
3(l-a)
k
=0.
(s K
.sH
r+âl ß^k
2(t-d)
p
î.1r' +,fli- = o
")
(1
s)
73
Equation (15) always has a solution f
-
= 0. Except for Ê- = 0, equation (15) can either has
no solution, a unique solution or multiple solutions. Thus we likely have three cases which
are illustrated as
a)
in the following figure
a)
k=O
0
k
0
Case
1
k
ct)
0
Ct)
0
E
e{,
k
0
Case2
74
a\
t=o t
(t)
c w
0
6
-1,
k
0
Case 3
Figure 4.L: The dynamic sYstem
Case
l
corresponds to a situation where there is no solution other than zero. In Case2, except
zero there is a unique solution at pointA. There are multiple steady states in Case 3. In Figure
4.1, the stability of the system is observed as follows. Below
tir¡ i,
falling and above
tft" ¿(r) = 0 locus, [1r¡ < 0 o.
,, lr(r¡> 0 or t(r) i. rising. For the ,î,çr¡ =0
positive quadrant with positive values for tît(t) anA
locus to be in
t1r¡, the condition (c1)
a
has to be
imposed which then implies that the denominator in the right hand side of equation (14) must
be negative. Thus below
Afr>
r 0 or
,n" ,îr(r) = 0 locus, àØ .0 or îo(t) is decreasing
and above it
îo(t) is increasing. The arrows describe the stability of the system. In Case
2,
75
point A is unstable so that starting from any where the economy will either converge to zero
or infinity. In Case 3, points
A
and
C
are unstable whereas
point B is conditional stable.
Around the neighbourhood of point B, there are paths called the saddle path where the
economy converges to point
B
and there are paths where the economy
is moving away from
point B. The poverty trap may exist in this case. Depending on the initial position, the
economy can either converge
per effective
labour. É1r¡
to
a steady state
"onu"rges
to
zero
with a high or a low value of physical capital
ifits initial value is sufficiently low and to a high
level at point B when its initial value is sufficiently high.
As an example, suppose a small open economy faces the world interest rate of F = 0.03. We
assume the economy employing the production function which has physical capital and
human capital intensities a = þ = L I 3. Physical capital is assumed to depreciate at the rate
of ô = 0.02 while human capital
the economy takes the value
does not depreciate. The technology absorptive parameter in
of ( =0.01.
The economy's saving rate in physical capital and
human capital are sK = s¡r = 0.2. Given these values, equation (15) gives us three solutions
which ur" îr*
=0,
Êr* =12.16I and
fr- =59.678.
have multiple steady states. Point B corresponding
This example falls into Case 3 where we
to Ér- is conditional
stable.
2.2. The steady state
If
the economy reaches a steady state,
it is where
output, physical capital, human capital,
wealth and foreign debt per effective labour arc
i.
,Ê. ,û.* ,ôr* and
2. respectively.
Ê.
variables can be expressed as functions
all
constant. Call these values
is the solution of equation (15). Other steady
as
state
of É- as
76
1
h
^*
Y
-d
l-a
p^ *þ,
(16)
k
(-")
f *Ò ¡*
(17)
=-K,
-6
a
(t)
(18)
(r) =
1
r+á\ p
4 ")
l-d
k.1t¡ o - sr/
1
z
k -A
=
rtÒ
l-a+P
r+á\ þ *þ
_t
k
4 a)
na
J.-
(1e)
I
4*)
7^ *þ
l-d
k
sxf
Let y(t)=Y(t)lL, k(t¡=K(t)lL, h(t¡=H(t)lL, and z(t¡=Z(t)lL be per capita
output, physical capital, human capital, and foreign debt respectively then we have
î(t) = y(t) / A(t), nO) = k(t) t A(t),
of !.
The constancy
,Ê*
ûQ) = h(t) I
,û* and 2- in the
A(t)
and
2(t) = z(t) I A(t)
.
steady state implies that per capita output, physical
capital, human capital, and foreign debt grow at the same rate as the rate of technological
progress, or
y = y(t) | y(t) = k(t) I k(t) = h(t) I h(t) = z(t) I z(t) = A(t) I A(t).
From (4') the steady state growth rate of technology is
A(t) I A(t¡ =
(20)
€h
Substituting (16) into (20) we have the growth rate of the economy in the steady state as
I
v
4+)
p^
k
l-d
p
(2r)
77
where É- is the solution of equation (15). Thus this small open economy can generate
endogenous growth with the steady state growth rate as given in equation
(zI).It
should be
clear that the growth rate of technology acquisition determines the growth rate of the
economy. To retain our assumption of small open economy absorbing world technology we
must impose the restrictions on exogenous parameters to ensure that the growth rate of the
country's technology is always less than or at least equal to the exogenous growth rate of the
world technology
9".
2.3. The transition: the speed of convergence
In this section our objective is to study the convergence of the economy to its steady state and
how the convergence rate can be affected by policy changes. The dynamic system in Figure
4.1 shows that there are steady states which are unstable or conditional stable.
If the economy
converges to a steady state, the steady state must be conditional stable such as at point B in
Case 3. As
is given in section 2.I, the dynamics of the system is described by the two
differential equations (11) and (12) which are functions of É1r¡ and tît(t). The economy will
converge to its steady state where
î
and
îo
takeconstant values
of Ê-
and
îa. respectively.
During the transition towards its steady state, the economy converges at the rate
lt.
To find
the rate of convergence, we can employ the formula which has been developed in Chapter 3
u*3
3
is
Notice the similarity in this model and the model in Chapter 3. In Chapter 3, the dynamics of the system
described by two differential equations
constant values
of k* and ú)*
k
and @ as functions of k and l0 . In the steady state k and A) Ãke
respectively. In the model ofthis chapter
The dynamics of the sysrem is described by two differential equations
f
k and Ø
an¿
ãi
are replaced
by
Ê
and îo
as function,
of É
and îO
*
as f
and ô* respectively. The procedure to find the speed of convergence is
with their steady state values
thus similar to that in Chapter 3 which gives us the following equation (22).
78
þ=
(B + E)t (1r + E)'
- 4(BE -
CD)
v2
(22)
2
where
p
D_
â
-
sr(2þ + ø -\( (r+ô) -aF)
a(I- a)
k(t\
--------)--:
ãk(t) u.,a=a.
(r+alì
[-"
âk(t\
v-
sr7þ
)
a+p-l
+----;-
I
r*Ò\ p"
n.Tîu. - 6@-a+I) _t
k
(1-
I
D
--------:--:-
L_
p
a
ãk(t) ¿=r,a=a,
dtt¡(t)
þ^
É(r- a)
aArt¡
-
þ
a)
a)
d )
âô¡G\
---------)--:
l-d
(r + ô)7 ^ þ+a-t
l x* þ
=-l
^^
1-ø\
d(D(t) t,=r,.a=a.
rì -
+
[-")
k
r-d-fl
p
a
= fsx
k=k,.ô)=ô),
Note that the steady state É- is the solution of equation (15) and the steady state
ô-
is
determined by equation (18). The quadratic form in equation (22) usually gives us two values
for
p,
call
it p,
and
p".With the speed of convergeîce 14 and p,
the transition of the
economy can be expressed as
î(t)=it* +Creh'+Creh',
where C, and C,
are constant.
(23)
As shown in Figure 4.I in section 2.I, we are interested in the
steady state positions to which the economy can converge. Such a steady state is at point B in
Case 3 of Figure 4.1. Since point B is conditional stable, on the saddle path the economy
converges to point
B
and
off the saddle path the economy is moving away from point
Correspondingly, the convergence rate
p
B.
should be such that one value of it is negative and
79
If
the
economy is to converge to point B, it must be on the saddle path.In the saddle path, p =
ltt,
another is positive. Suppose
Cz =
4
takes a negative value and Lt'2 takes a positive value.
0. To find the value for C, we need 1 boundary condition. We know that
î(0) = !o
if
r = 0 then
and thus
Cr=îo-î.
If t = oo then i(-)
=
!-
i(r) = i- +(io - î,.)",u
. The
transition of the economy can be expressed
as
(23',)
.
As an example, we consider the economy as given in the example in section 2.1 where
F=0,03,a= þ=I13,õ=0.02,É=0.0t and s, -sH =0.2. Applying the formula (22) we
find the convergence rate of the economy to its steady state at point B as 14 =-0'18
and
in the
Itz = 0.0056. For the economy to converge to its steady state, the economy must be
saddle
path andconverges
at the rate 14 = -0'18
.
As in Chapter 3 we want to know how changes in the savings rate affect the speed of
convergence. In order to do so we must resort to simulatiottsa. The reason to use simulations
is that the speed of convergence in equation (22) is a function
solution
of É- which in terms is the
of non-linear equation (15). The non-linear equation (15) cannot be solved
analytically.
a
Mathcad program is used to run the simulation.
80
Simulations
Changes
in
su.
To
see
how changes
in s" affect p
we assign the given example values to all
exogenous parameters in equation (22)but let so run from 0 to 0.8 and run the simulation on
equation (22).The result is given in Figure 4.2
The speed of convergence
0
F.1(
sk)
p2( sk)
-0.1
-
-o.2
-0.3
0
0.2
0.4
0.6
0.8
sk
The saving rate in physical capital
Figure 4.2: The impact of changes
in su on the speed of convergence
Figure 4.2 shows that as s" changes from 0 to 0.8, the negative value of the convergence rate
p, is fluctuating
around -0.2.In other words, changes in the saving rate in physical capital
have insignificant effects on the speed of convergence.
81
Changes
in sr. 'We now assign the given example
values
for exogenous parameters
to
equation (22)but this time let so run from 0.01 to 0.8 and run the simulation on equation
(22).The impact of changes in
s' on the speed of convergence is reported in Figure 4.3
The speed of convergence
02
0
p1(sh)
p2( sh)
-o.2
-
-0,4
-0.6
0.6
o.4
0.2
0.8
sh
The saving rate in human capital
Figure 4.3: The impact of changes
in s"
on the speed of convergence
In Figure 4.3,it can be seen that as s¡r increases, the value of
p
increases
in absolute value.
This result suggests that a higher saving rate in human capital significantly raises the speed of
convergence causing the economy to converge faster to its steady state. The results in Figures
4.2 and4.3leadto Proposition 4.1.
Proposition 4.1: An increase in the saving rate in human capital quickens the convergence to
the steady state while the speed of convergence is almost unaffected by an increase in the
saving rate in physical capital.
82
2.4. Comparative statics: the impact of changes in the saving rates on the steady state
variables and the growth rate.
To
see
how changes in the saving rates affect the steady state per ffictive labour output,
physical capital, human capital and domestic wealth as well as the steady state growth rate
we must resort to simulations5.
Simulations
Changes
in s*. 'We consider
the economy which has been studied so far. The values for
exogenous parameters are given as F= 0.03,
A= þ=113, 6=O.02,€=0'01 and s" =0'2'
To see the impact of changes in the saving rate in physical capital on the steady state physical
capital per effective labour, we assign all values for exogenous parameters to equation (15)
and
let
sK run from 0 to 0.8 and run the simulation on that equation
3
7+ä\ p^ *3(1-d) -,
k
")
þ
Ét*
'Ín
lr.
ôl
[ø)
L
2
ß ^ 2(l-d)
þ
|
4# l(-") o.î *ô=o
7
(1s)
The result is reported in Figure 4.4
5
Again, the reason to use simulations is that we have a complicated system of non-linear equations (15) and
(21) which cannot be analysed analytically.
83
The steady state stock of physicalcapital
per effective labour
64
62
k( sk)
60
58
o.2
0
0.6
0.4
0.8
sk
The saving rate in physicalcapital
Figure 4.4: The impact of changes
in s"
on the steady state physical capital pet effective labour
It is clear from Figure 4.4 that a higher saving rate in physical capital raises the steady
physical capital per effective labour. The appearance
of this relationship looks
state
linear,
however this is because changes in the steady state physical capital per effective labour arc
very small as the saving rate in physical capital changes. We explain this result intuitively.
An increase in the saving rate in physical capital witl immediately raise the stock of nonhuman wealth per effective labour.Initially, the stock of physical capital and thus domestic
output per effective labour are fixed.
accompanied
A
higher stock
of wealth per ffictive
by an initial fixed domestic output per effective labour imply that
labour
domestic
income per effective labour must be higher. Since an exogenous fraction of domestic income
per effective labour
is allocated to education, a higher domestic income per ffictive labour
causes more resources to be spent on education which raises the stock of human capital per
effictive labour.
A
higher stock
of
human capital causes physical capital
to be more
productive leaving the marginal productivity of physical capital well above the world interest
rate. As a result, physical capital
will flow into the country and thus raise the stock of
84
physical capital per effective labour.
A higher domestic output per effective labour
can be
produced. This process keeps going until the economy reaches a new steady state with higher
stocks of physical capital and human capital per effective labour.
To find the impact of changes
in s.
on the steady state growth rate
y, we assign the given
values for exogenous parameters to equations (21) and (15). Let sK run from 0 to 0.8 , we run
the simulations on equations (21) and (15)
I
v
4
where
n^l-o
r*Òì 'k-p
_t
(2r)
d)
ft-
is the solution of equation (15). The impact of changes
growth rate
y
in s,
on the steady state
is reported as follows
The steady state growth rate
0.135
0.13
y( sk) 0 125
o.t2
0.1
l5
0
0.2
0.6
0.4
0.8
sk
The saving rate in physicalcapital
Figure 4.5: The impact of changes
in s"
on the steady state growth rate
Figure 4.5 shows that the steady state growth rate of the economy increases as the saving rate
in physical capital increases6. The reason for this is obvious. Since the growth rate of the
6
Once again, the relatively small absolute effect on the growth rate makes the curve appear linear
85
economy is determined by the stock of human capital per effective labour, a higher saving
rate in physical capital leads to a higher steady state human capital per effective labour via a
wealth effect which results in a higher steady state growth rate.
Changes
in
s
u . Similarly, we can see how changes
in the saving rate in human capital affect
the steady state variables and the growth rate. Now we let s rc = 0.2 and assign all values for
other exogenous parameters as the same as above except
.sr1
for
s'
to equation (15). We allow
run from 0.01 to 0.8 and run the simulation on equation (15)
3
p
n.v?'+(+)
2
2( l-d\
P^ -t
=
k
p
þ--l
r-d
^
P +ä=0
k.
We report the following result.
The steady state stock of physicalcapital
per effective labour
100
80
k(sh)
60
40
20
0
0.2
0.4
0.6
0.8
sh
The saving rate in human capital
Figure 4.6: The impact of changes
in .to on the steady state physical
capital per effective labour
As Figure 4.6 shows, a higher saving rate in human capital gives rise to a higher stock of
physical capital per effectiv¿ labour. The explanation is that an increase in the saving rate in
human capital immediately raises the stock of human capital per effective labour. A higher
86
stock
of
human capital makes physical capital more productive causing the marginal
productivity of physical capital to rise above the world interest rate. As a result physical
capital will flow in the country to capture a higher rate of return. A higher stock of physical
capital accompanied by a higher stock of human capital per effective labour give rise to
a
higher output per effective labour. This process ends when the economy reaches a new steady
state with higher stocks of physical capital and human capital per effective labour. This effect
is much more pronounced than changes
in s"
because
it has two effects:
a direct effect on
human capital accumulation and an indirect effect via wealth.
Finally, to find the impact of changes
src
in so on the steady
state growth rate T , we assign
= 0.2 and the given values for exogenous parameters to equations (21) and (15). We run a
simulation on equation (2I) and (15) by letting sH run from 0.01 to 0.8
I
B^
where f
-
1-d
(2r)
p
is the solution of equation (15). Figure 4.7 displays the result.
The steady state growth rate
o4
03
y(sh)
02
0.1
0
0
0.2
0.4
0.6
0.8
sh
The saving rate in human capital
Figure 4.7: The impact of changes
in .t"
on the steady state growth rate
87
The result shows that an increase in the saving rate in human capital has a favourable effect
on the steady state growth rate of the economy. We come to Proposition 4.2.
Proposition 4.2: Countries with higher saving rates enjoy higher long run growth rates as
well
as higher income levels.
Comparing all results in Figures 4.4 to 4.7 we note that changes in the saving rate in human
capital have significant effects on the steady state physical capital per effictive labour and the
growth rate relative to changes in the saving rate in physical capital. While changes in s"
have such insignificant effects on the growth rate, a change in the growth rate of 0.11 to 0.13
as sK raises from 0 to 0.8, changes
in s"
have relatively large effects on the growth rate, a
change of around 0.01 to 0.3 in the growth rate as
s,
raises from 0.01 to 0.8. The reason for
this is that the growth rate of the economy is determined by the stock of human capital per
effective labour. While an increase in the saving rate in human capital has a direct effect in
raising the stock of human capital, an increase in the saving rate in physical capital has an
indirect effect in raising the stock of human capital through the wealth effect. As a result, an
increase in the saving rate in physical capital has less effect on the stock of human capital per
effictive labour and thus the growth rate. Proposition 4.3 follows.
Proposition 4.3: In a small open economy context with perfect physical capital mobility,
a
one percent increase in the saving rate in human capital raises the economic growth rate more
than which can be obtained by increasing one percent in the saving rate in physical capital
88
3. CONCLUSION
In this chapter we developed the extended Solow-Swan model incorporating international
technology and study it in an open economy context. This model is thus the extension of the
model employed in Chapter 3. The enrichment of the model gives us several interesting
results. Firstly, we can explain that under perfect physical capital mobility, different small
open economies facing the world interest rate and free access to world technology can have
different growth rates due to different saving rates in those countries. Countries with higher
saving rates can grow quicker and enjoy higher income levels and wealth.
Secondly, while an increase in either saving rates in physical capital or human capital can
raise the steady state growth rate, their effects on the growth rate are significantly different.
The growth rate increases pronouncedly as the saving rate in human capital increases but the
increase in the growth rate is relatively less significant as the saving rate in physical capital
increases. This result suggests that a small open economy should more quickly apply its
saving to human capital than it does with physical capital.
Finally, in relation to the convergence issue we note that in the model of Chapter 3 the speed
of convergence is independent on the saving rate in human capital. The saving rate in
physical capital while
it
affects the speed
of
convergence, appears
to do so in
such
insignificant magnitudes. In this chapter, the saving rate in physical capital affects the speed
of convergence but again its effect is very small. The saving rate in human capital, however,
has a large effect on the speed of convergence and the transition process.
89
perfectly mobile
To explain the last result, in both models we assume that physical capital is
while human capital is not. The stock of human capital
is
determined
by the
saving
out of the country'
behaviours of domestic residents since human capital cannot flow into or
capital to be at a
Perfect mobility of physical capital, however, allows the stock of physical
available at a time.
level determined by the world interest rate and the stock of human capital
of human capital.
The process of convergence is thus tied down to the adjustment of the stock
the stock of
The saving rates of human capital and physical capital both have effects on
in human capital is much
human capital but we already know that the effect of the saving rate
expect that the
bigger than that of the saving rate in physical capital. Given these, we would
speed of convergence is much affected by changes
in the saving rate in human capital than it
of the saving rate
would be with physical capital. As expected, both models find little effects
in Chapter
in physical capital on the speed of convergence. Due to its enrichment, the model
4
have findings which can explain the effects
of changes in the saving rate of human capital
on the speed of convergence'
However,
In this chapter we assume that the country has free access to the world technology.
technology at no
in reality it is not often the case. The country cannot usually acquire foreign
can act as
cost. Among different ways to adopt foreign technology, foreign investment
a
in the economic
channel for technology transfer. Thus foreign investment can play a role
5'
growth process of the host country. This issue is the subject of the following Chapter
90
Chapter 5:
OPTIMAL FOREIGN BORROWING, PHYSICAL AND
HUMAN CAPITAL ACCUMULATIONS AND
TECHNOLOGY TRANSFER
9T
l.INTRODUCTION
Foreign investment has played a significant role in economic growth of developing countries.
Foreign investment not only serves as a private source of finance for economic development
of the host country but more importantly it acts as a main channel for the transfer of
technology. Via foreign investment, developing countries can have access
to
advanced
technology which has been developed and used in developed countries.
The degree to which developing countries can adopt advanced technology, however, depends
on their absorptive technological capabilities. These capabilities may be captured by the
levels
of infrastructure, education and human capital in
each country. Human capital is
defined as skills, knowledge and abilities which are embodied in workers. The use of more
sophisticated technologies often requires higher levels of human skills. In order for foreign
firms to upgrade new technology frequently, the host country must develop and thus be able
to supply an adequate level of human capital or the appropriate and comparable skills needed.
Thus investment in education and labour training are crucial.
There are several studies that have analysed the growth performance
of a small open
economy that hosts foreign investment (Koizumi and Kopecky L917, Findlay 1978,
Borensztein, De-Gregorio and Lee 1998 and Gupta 1998). Koizumi and Kopecky (1977)
developed an exogenous growth model which assumes that technology transfer depends on
the extent of foreign ownership in physical capital. This assumption is also made in Gupta
(1998). In the Findlay (1978) model, technology transfer is assumed to depend on the relative
foreign-domestic ownership in physical capital and the technological gap. Borensztein et al
(1998) assume that technology transfer takes the form of the introduction of new goods'
92
These models
all
emphasise the role
of
in
foreign investment
technology transfer on
economic growth of the host country. However, they are not concerned with the role of
human capital in the technology transfer function'
There are clearly interactions between technology transfer and human capital accumulation in
the growth process. Human capital complements technology transfer in the sense that
a
higher level of human capital makes it possible for firms to introduce superior technology. In
another way, technology complements human capital accumulation since continuous
improvements in technology keep the marginal productivity of human capital from falling.
Nondecreasing returns to human capital cause people to have an incentive to keep investing
in human capital which results in a persistent growth.
The objective of this chapter is to extend the line of interest
economic growth
in studying the problem of
in a small economy that hosts foreign investment. We develop
endogenous growth model
in an optimisation context to explore the interactions
an
between
foreign investment, technology transfer and human capital accumulation of the host country'
In the next section, the model is presented. Two versions of the centralised and decentralised
economy are analysed. The conclusion is given in the last section.
2.
THE MODEL
A small open economy is populated by constant L identical and infinitely lived individuals'
The economy produces a single good which can be consumed or directly invested as physical
capital. The good is produced by means of technology, physical capital and human capital
according to a Cobb-Douglas type as
93
Y(t¡ = A(t)K(t)" H,(t)P
(1)
,
whereA(r)isthelevelof technology,K(t) isthestockof physicalcapitaland I/,(r) isthe
stock of human capital which is supplied to the production of goods. The labour force is
equal to the size of the population
Human c apit al
ac c umulation
Human capital is the skills and knowledge which are embodied in each individual. Human
capital can be accumulated by investing time in learning activities or education. We assume
that each individual has h(r) units of human capital at time l. In each period, each individual
is endowed with one unit of time which he or she can supply to work or to learn. Suppose the
individual allocates ry(t) fraction of his or her time to work and the other
1
- tt/(t)
fraction
of time to learn where ty(t) is an endogenous choice factor. The evolution of the individual
stock of human capital is assumed to be
ct\
h(t¡ =
6lt- ry(t))hçt¡,
Q)
where
á is the exogenous
effective learning parameter. Since the individual allocates ty(t)
fraction of his or her time to work then he or she supplies yt(t)h(t) units of human capital to
the goods industry in each period. Thus the stock of human capital which is employed by the
goods industry is
(3)
H,(t) = V/G)h(t)L.
Substituting (3) into (1) we can write the production function as
Y(t¡= A(t)K(t)"(w@h(ùL)þ
.
(1')
94
Let y(t) = y(t) I
L
and
k(t) = K(t) lL
be per capita output and physical capital respectively
then the production function in an intensive form is
y(t) = r:*Pu
Phy
sic
aI
c ap
(1")
A1)k(t)"(ttØw(r))8.
it al ac
c
umul ati on
The economy is small relative to the rest of the world and it faces unlimited access to the
world's physical capital. We assume that there is no perfect physical capital mobility and the
economy has access to foreign borrowing via foreign investment only. Let Z(t) be the stock
of foreign-owned physical capital or foreign debt held by the country at time l. Foreign
physical capital must be paid at the exogenous world interest rate 7.
'We
need to distinguish
is
between the national output and the national income of the country. The national output
y(/) which is determined by equation (1). The national income is the difference between the
national output and the income accrued to foreigners. In each period, the total payment to
foreigners is
Y(t)
.Z(t)
and thus the national income of the country is
(4)
-72(t).
Income of domestic residents is spent on consumption C(r) and investment in physical capital
I
physical capital the amount of
n0) . Thus in each period domestic residents invest in
InQ) =Y(t) -FZ(t)
Call
I
r(r)
-
C(t)
.
(5)
the flow of foreign investment in period r. The stock of foreign-owned physical
capital is changed by the flow of foreign investment in each period so that its stock is evolved
AS
Z(t¡ = I r(t).
(6)
95
The sum of investments made by domestic residents and foreigners is the national investment
in physical capital
I(t¡ = IdQ)+ I rQ).
Q)
The stock of physical capital which is employed in the country is accumulated by the national
investment in each period so that its evolution can be described as
(8)
K(t¡=I(t¡=IrQ)+Y(t)-rZ(t)-C(t).
In equations (6) and (8) we assume for simplicity that physical capital does not depreciate.
t-et
z(t)=Z(t)lL,irT)=I¡(t)tL
andc(t)=C(t)lLbepercapitastockof foreigndebt,
the flow of foreign investment and the domestic consumption respectively then
the
accumulations of foreign debt and physical capital in per capita terms are
(6',)
,t, ¡=ir(t),
k(t) = ir (t) + y(t)
T e chnol
o
gy t ran
-
sfe r
rz(t)
-
c(t)
(8',)
.
funct i on
We now introduce the technology transfer function into the model. We assume that foreign
investment acts as a channel for technology transfer as
it brings foreign
technology to the
host country. Suppose that via foreign investment the country has access to the stock of world
technology which grows at an exogenous rate
g". Given this, the rate of technology
acquisition by the host country is constrained to be less than or at least equal to the world
technology growth rate. The extent of technology transfer is assumed to be
foreign ownership
in
physical capital which
is
measured
by per capita
a-
function of
foreign-owned
96
physical capital or foreign debt z(t).This assumption is in line with Koizumi and Kopecky
(1917) when they argue that technology transfer occurs when foreign investors provide
advanced technology, training and discussion.
In addition, how much of technology can be transferred to the country also
depends on the
country's absorptive capacity. Here, we stress the important role of the internationally
immobile factor which is the stock of human capital. The stock of human capital can act as a
proxy for the absorptive technology capacity of the host country. A country with
a
high level
of human capital provides an adequate level of infrastructure which enables it to absorb high
level of foreign technology. Combining the assumptions, the technology function can
be
expressed as
A(t¡ = f (z(t),hØ)
which has the properties
investment
of
âA(t) t ù"(t) >
0
and âA(t) I ãh(t) > 0 or the more foreign
in the country the more foreign technology can be transferred. Similarly, the
better the stock of human capital that the country possesses the more foreign technology can
be absorbed. 'We assume further that the technology transfer function takes a Cobb-Douglas
form
A(t¡ = (z{t)'t
{ùr)' ,
(9)
Substituting (9) into (1") we can write the production function
y(t) = lÍ*þ-t z(t)q k(t)"
h(t)Pn+Þ
¡[ç¡Þ
as
(10)
The term h(t)P is the direct contribution of human capital to output and the terrn h(t)Pa is its
indirect contribution to output via its assistance in technology accumulation. The term z(t)q
is the technology brought in by foreign investment or the indirect contribution of foreign
97
capital to output via technology transfer. For the economy to generate endogenous growth,
the nondiminishing returns to all factors that can be accumulated must be imposed on the
production function. Such a condition requires that
(0+p)ry+d+þ>L
(c1)
2.1. Tlae optimal solution
'We
assume that the economy is controlled by a social planner whose objective is to maximise
the lifetime utility of its residents. Utility of each individual is assumed to derive from
consumption only. The lifetime utility of each individual is described as
u
=i"-,
(1.1)
dt
0
where
p
is the discount factor and
o
is the risk aversion factor. In order to achieve
the
objective, the planner can adjust the level of consumption in each period, the time allocated
to learning activities and the flow of foreign investment. The optimisation problem then
amounts to the following
Max
c(t),i ¡
st.
(t),VG\
ï
0
a(t) = ir(t)
k(t) = i r (t) + y(t) - rz(t) - c(t)
t 1,¡ =
y
(t) =
d(t-
yçt¡)nçt¡
L"* P-t AG)k 1t¡" (ttçt¡ty 1t¡)P
A(r) = (zçù'n{ùr)'
.
98
We form the Hamiltonian expression as
+
Lrç¡(i r 0) + L"*pu z1)q k(t)"
where
lr,lz
and
)",
h(t)þn*Þ ¡p'ç¡¡Þ
are the shadow prices
- rz(t)- r(r)) + )"r()õ(t- v/Ø)h7)
of foreign debt, physical capital and human
capital in terms of consumption respectively
First order conditions yield
c(t)-" = trr(t),
(r.2)
trrçt¡ =
(1.3)
-trr(t),
Lr(t)þIi*P-t zçt¡ø kQ)"
).r7t¡ =
lr(r)
plr(t)
h(t)Pn+Ê-r ¡yç¡)P-t =
lr(t)õ
(1.4)
,
-.7r(t)0r¡L"+Þ-trç¡¡ena kQ)" h(t)Pn+Þ 1yçt¡Þ + ).r(t)T
= plz7)
- )"r(t)aL"*þ-tz(t)q kQ)"+h0)"*þ V(t)þ ,
irçr¡ = ptr3e)
- LrØ(pry + p)t:*þ-'a(ùon ke)" h(t¡rn+ø-r
,
(1.s)
(1.6)
wQ)P
- Lrç)õ(t- v¡Ø) .Q.t¡
The transversality conditions are
Lim,-*lr(t)e-ø z(t) = 0,
(1.8)
Lim,-*).r(t)e- ø kçt¡ = g,
(1.e)
Lim,-*)"r(t)e-
ø
hlt¡ = g .
(1.10)
Equation (1.2) says that the optimal choice of consumption and investment in physical capital
is guided by the condition where the shadow price of physical capital is equal to the marginal
utility of consumption. The optimal foreign borrowing requires that the shadow prices of
99
foreign debt and physical capital are equal in absolute value, as displayed in equation (1.3).
They have opposite signs to reflect the fact that the planner has disutility while holding
foreign debt. Equation (1.4) describes the optimal allocation of time between working and
learning. Equations (1.5) to (1.7) express the rates of change in the shadow prices of foreign
debt, physical capital and human capital respectively. The transversality condition (1.8)
ensures national intertemporal solvency. That is, all foreign debt must be paid as time goes to
infinity. Conditions (1.9) and (1.10) imply that at infinity, physical capital and human capital
have zero values.
The steady state analysis
Our interest is to examine the properties solutions of the economy in the steady state. The
steady state is defined as a path where t¿ is constant and all per capita output, consumption,
physical capital, foreign debt and human capital grow at constant rates. Call these growth
rates y y,T ,,T *,T
,
and
r
respectively. We proceed to find these growth rates.
Taking logs and differentiating first-order condition (1.2) with respect to time we have the
growth rate of consumption
1
Y"=c(t)lc(t)
The growth rate
ir(r)
o
as
(1.1 1)
)"r(t) I Lr(t)
of 7, is obtained from (1.6) as
t )"r1t¡ = p
-
qL"*þ-t z(t)q k(t)"n h(t)un+Þ
,e
.
(1.6',)
Substituting equation (1.6') into equation (1.11) gives us the condition for the marginal
productivity of physical capital
100
t-þ
1t
y,o
+
p = dL"'þ-t z(t)q k(t)"-'h(t¡un*ø ,u
(1.1
,
¡;ùt
or equivalently we have
y,+e-r r7¡¡on
k¡)"-t
h(t)p,t*þ tlrþ =
T#
.
(
LI2')
Since the shadow prices of physical capital and foreign debt grow at the same rate
as
is
suggested in equation (1.3), from (1.11) the growth rate of the shadow price of foreign debt
determined by the growth rate of consumption as
'
).r(t) I Lr(t) = -cY,
(1.13)
.
Substituting (1.3) into (1.5) to derive the growth rute of 7,
ir(r)ttr (t)= p-r +0r[Í*P-tz(t)q-'k(t)"h(t)un+f ,Þ.
(1'5')
using equation (1.5',) to eliminat" irçr¡ I ).r(t) in equation (1.13) we have
-
cy
"
-- p
- r + hr¡Lo+f-'z(t)q k(t)"-t h(t)þn*P ty(t)
p k(t)
(1.14)
z(t)
By substituting equation (1.12') into equation (1.14) we derive the ratio of foreign debt to the
physical capital stock employed in the country
as
(1.1s)
Since the right hand side of equation (1.15) is constant in the steady state, this implies that
the
ratio z(t) I k(t) is constant or foreign debt and physical capital grow
at the same rate as
(1'16)
T,=Yt '
The steady state growth rate of physical capital is obtained by dividing both sides of equation
(8') by k(t)
,r =t#+
rÍ*þ-,2(t)q k(t),uh(t¡Þ*unwu
--# #.
(8")
101
From equation (6') the steady state growth rate of foreign debt is
(6")
T, = z(t) I z(t¡ = i, (t) I z(t)
Substituting equations
(6"), (I.L2')
and (1.15) into equation
(8") to derive the steady state
ratio of consumption to physical capital
c(t)
y,o+ p
k(t)
d
1+
-r)
r-y,o-p -Yr
ert(v,
.
(1.17)
Notice that the constancy of the terms in the right hand side of equation (1.17) implies that
consumption and physical capital per capita grow at the same rate in the steady state or
T"=Tt' '
(1'18)
Combining equations (1.16) and (1.18) together we see that consumption, physical capital
and foreign debt growth at the same rate in the steady state. Call this common growth fate y
where
T=Tr=Tt=T,
(1.le)
The steady state growth rate of human capital per capita can be obtained by differentiating
equation (1.I2) with respect to time and taking account of equation (1.19)
K=I-a-0r7 Y
w+þ
(t.20)
A condition imposed on exogenous parameters to ensure the growth rates of human capital
and physical capital are of the same sign is
L-a-0q>0.
(c2)
Finally, the production function in equation (10) together with equations (1.19) and (1.20)
imply that the growth rate of output per capita is equal to the common growth rate of per
capita consumption, physical capital and foreign debt
Tr=T
(1.21)
to2
It is clear from (2) that the steady state growth rate of human capital is
rc
=
õ(I-
V)
.
(2')
Differentiating first-order condition (1.4) we have the relationship between the growth rates
of the shadow prices of physical capital and human capital
.a
7,(t) I ).,(t)+(Ort* o)y +(prt* P-t)* = )'Q) t Lr(t).
(r.22)
Using (1.11) to eliminate Lr(t) I )"r(t) in equation (I.22)
ir(t) t )"r(t) = (ert * o - o)y * (pry + B The growth rate
).r(t)l ).rçt¡=
r)rc
(r.23)
of 2, is also derived by substituting (1.4) into (1.7)
p-ô-
lu,
(t.7')
Finally we substitute for ty fuomequation (2') into equation (1.7') and equate (I.7') to (L23)
to obtain the steady state growth rate of human capital per capita
(r.24)
From (L.20), the steady state per capita growth rate of physical capital and consumption is
(t.2s)
A restriction must impose on the exogenous parameters in such a way that assures a positive
growth rate. To retain the assumption of a small economy absorbing world technology, the
growth rate of the economy must be less than or at least equal to the world technology growth
rate
g*.
103
Finally, the transversality conditions (1.8)-(1.10) must be satisfied. Given the steady
growth rates
of ),,
and
)", in equations (1.11) and (1.13), and the growth
rate
state
of )', in
equation (1.23) we have
),rçt¡=lr,o"-*,
and
h(t) = ho€o
trr(t)=lz,€-4,Lr(t)=),r,oe((Ø*o-o)v+(Pq+þ-1)*)',
z(t)-zo€l, k(t)=kref
(I.26)
,
where År,o,7r,o,.7r',zs,ko and ho arc the constant initial values of variables in the steady
state. Substituting (L.26) into the transversality conditions we have
Lim,-*),rlzo"({t-o)r-e)'
- o,
Lim,-*),r.okorßt-o)r-ol'=0,
Lim,-.*),r.ohorkt-">r-n\t
=g.
(1.8')
(1.9')
(1.10')
The transversality conditions are satisfied when
(I-o)y-p<O.
Substituting
for y from equation (1.25) into condition (I.27) we
Q.27)
have the constraint on
exogenous parameters as
\fu, * P)õ - pp r- o)
-p<0.
Bo +l- a- þ- ert
(c3)
As shown in equations (1.24) and (I.25), the growth rates of human capital and physical
capital do not depend on the exogenous rate of return to foreign capital. These growth rates
are functions of the economy's exogenous parameters. Thus depending on these exogenous
parameter values, the optimal growth rate of the economy can be different. This result then
suggests that different economies of the same type can experience different growth rates.
ro4
The steady state ratio of foreign debt to the total stock of physical capital employed in the
country is, however, affected by the world interest rate as shown in equation (1.15)
z(t)
Kù=
erfyo
+ P)
"(¡-ay-p)'
where y is determined in equation (I.25) which is independent on
¡. It is clear that a higher
interest rate on foreign physical capital lowers the long run proportion of foreign debt held in
the country. We come to Proposition 5.1.
Proposition 5.1: The world interest rate does not affect the steady state growth rate of the
economy but it determines the relative ownership in physical capital of the country. A higher
world interest rate lowers the proportion of foreign debt held in the country and visa versa.
Up to this point we have constructed a model of economic growth in a small open economy
to explain the role of foreign investment as a growth determining factor. We raised the
interrelationships between technology transfer via foreign investment and the human capital
accumulation
of the host country. Technology is a driving force of
economic growth.
Improvements in technology brought in by foreign investment keep the marginal productivity
of human capital and physical capital away from falling and thus the incentives to invest in
capital. As a result, a sustained growth in the long run is possible. The command rate of
return to foreign investment, while it affects the level of income per head, does not influence
the economic growth rate of the host country. This may suggest that the host country should
not see high returns to foreign investment as a barrier to host foreign investment as long
as
foreign investment provides the source of technological changes.
105
For the rest of this chapter, we study the special case when
It=I-7 and1=I-a-þ.
(1'28)
Substituting condition (1.28) into equation (1.20) gives us T =
K or the growth
rate of
consumption is equal to the growth rate of human capital. Thus in this case the economy
generates a steady state balanced growth path where output, consumption, physical capital,
foreign debt and human capital per capita grow at the same rate. Call this growth fate
y
o
where
a
a
a
"
y, = y(t) | y(t) = c(t) I c(t) = k(t) I k(t) = z(t) | z(t¡ = h(t) I h(t)
.
The common growth rate of the economy is obtained by substituting condition (1'28) into
equation (I.25)
vo =
ô(1-exl
(1-
-a-þ)+þ(6-p)
dX1
(r.2e)
- q- þ)+ þo
The optimal allocation of time between working and learning is
Vo=
p- 6)
a(tr- ØG-q-P)*"Þ)
þ(6o
+
(1.30)
The condition imposedon ty, is 0 < Vl"
<L which constraints the exogenous parameters
as
(1.31)
õo+p-ô>0
The growth equation (1.29) suggests that the higher the share of technology intensity in the
production function (higher r:.) and the higher the share of human capital intensity in the
technology transfer function (smaller
0),
the higher the optimal growth ratel. "Smartef"
countries (higher learning effective parameter
IA
detail calculation is provided in Appendix
á) grow faster. Countries which
have higher
A
106
discount rates
p
and risk aversion factors
ø
experience lower growth. These results are
summarised in Table 5.1
da>0
dþ>o
dn>o
+
dy,
d0>0
d6>0
do>0
dp>
o
+
Table 5.1: The effects of exogenous parameters on the growth rate
2. 2. Tl¡e market solution
In this section we analyse the decentralised economy so as to see if there is any difference
between the centralised and decentralised solutions. The decentralised economy is described
by two agents: private competitive profit maximising firms and domestic households. Firms
can employ physical capital from overseas as well as from domestic residents. Domestic
households supply physical capital and labour to domestic firms in return for goods.
Firms
Private firms employ physical capital, human capital and labour to produce goods according
to a Cobb-Douglas production function as given in equation (1)
Y(t¡ = A(t)K(t)" H,Q)þ
where
K(r) and H,(t)
,
are the stocks of physical capital and human capital supplying to
firms. The stock of physical capital employed by firms is the sum of foreign-owned and
domestic-owned physical capital
rcl
(2'l)
K(t¡=2,(t)+KoQ),
where
Z,(t) is the stock of foreign
physical capital and Ko(/) is the stock of domestic
physical capital. We assume that firms care what type of physical capital they employ since
they know how foreign investment influences the technology transfer function. Foreign
investment provides the source of technology and firms must rely on foreign investment for
its technological change. Suppose the stock of technology is a function of the stocks of
foreign physical capital and domestic human capital per unit of labour
(9')
A(t¡ = (z{t)' lr{t)'-o)1-d-þ ,
where
z(t)
and
h(t)
are the stock
of foreign debt and human capital per unit of
labour
respectively. In knowing the technology transfer function, firms can derive their production
function
as
Y(t¡ = z,(t¡e<r-"-Þ' KO)" H,{t)o-q$-a-f)+þ L,(t)d+P-t
where Z,(t) =
z(t)L,(t), H,(t)
=
(1')
,
h(t)t (r) and L,(r)
is the total units of labour employed
by firms. Firms must pay foreign physical capital at the world interest rate. Since domestic
physical capital and labour are supplied to domestic firms only, firms pay these factors at the
ongoing rate ro(r) on physical capital and the wage rate w(t) on human capital. Perfect
competition
is
assumed
to exit so that each factor is paid according to its marginal
productivity
ú,(t)"*Pt
qL, (t)
+
d+
P
Z(t¡eo-"-O' K(t)"-t H,(t¡Þ*o-e)Q-d-P) =
-r Z (t¡e o-
"
-
U' K
(t) " -t H, (t¡ Ê * t
e)o-
0(I- d - P)L,(t)o*þ-'Z(t)eo-,_P)-t K(t)"
(P
* ç- 0)Q- d - Ð)L,Q)"*þ-'
d-
rd(t) ,
(2.2)
þ)
H,(t¡f*<t-e)o-o-fl)
Q3)
- 7'
z(t)o(l-a-P\ KQ)" H,(t¡d-t*<t-e)Q-a-þ) =
wG)'
Q'4)
108
In equation (2.2), the rate of return to domestic physical capital is determined by its marginal
productivity. Equation (2.3) is the condition for the employment of foreign physical capital.
Since an increase in the stock of foreign physical capital raises output through its direct
effect, it also raises the level of technology employed by firms which gives an indirect effect
on the increase in total output. At the optimal choice, firms employ foreign physical capital
up to the point where the marginal productivity of physical capital plus an increment in the
stock of technology is equal to the world interest rate or
(2.3',)
MP*+A(t¡=7
which
is
equivalent
to equation (2.3). Equation (2.4)
describes the
profit maximising
condition for the employment of labour. As far as the firm is concerned, there is no
externality
in the production function since the effect of the technology
transfer is
internalised in the firm's decision making.
Households
Households own domestic physical capital and human capital which are supplied to firms at
competitive market rates. In each period, each individual supplies r¿(r) units of time to work
so that his or her labour income
is w(t)ttt(t)h(t) . The total income of an individual is the sum
of the incomes from physical capital and labour as
y,(t)=roQ)koî)+w(t)tt¡Q)h(t).
Q5)
This income is spent on consumption and saving in physical capital, resulting in the stock of
physical capital being evolved as
koQ) = roQ)koî) + w(t)tt¡Øh(t)
- c(t)
Q'6)
109
where physical capital is not assumed to depreciate. In each period, the individual invests
I-
Vr(t) units of time to learn. The accumulation of the stock of human capital is
or\
h(t¡ =
a$-
ylt¡)nç¡.
(2)
The objective of each individual is to maximise the lifetime utility by choosing the level of
consumption c(t) and the time allocation tt/Q), subject to the budget constraints. This
amounts to the following problem:
Max J
c(t) rlt()
st.
0
kn
(t) = ro (t)k o Q) + w(t)ry (t)h(t) - c(t)
iç,¡ = d$- ylt¡)nçt¡
.
The Hamiltonian expression is
cft\t-"
I
J=Y+p,Q)(roQ)koQ)+w(t)w0)h(t)_c1r¡)+¡l,()6(t_v/@)hQ)
l-o
where
p,
and,
ltz
aÍe the shadow prices
of physical capital and human capital in terms of
consumption respectively.
First order conditions yield
(2.7)
c(t)-" = ltr(t),
ltr(t)w(t) = ltrõ
ltr(t) = pltL(t) - ro(t)ltr(t)
i,Q)
= ptt27)
(2.8)
,
,
- w(t)ttt(t)tt,(t) - õ(t- yçt¡)p,çt¡
(2.e)
(2.r0)
110
Equation (2.7) describes the optimal choice between consumption and saving' The optimal
(2.9)
time allocation between working and learning is given in equation (2'8). From equation
we have the growth rate
of Pr as
ltr(t) | ltr(t) = p- r¿(t).
(2.g',)
Substituting equation (2.8) into equation (2'10) to derive the growth tate of p"
(2.r0')
þr(t)llt"(t)= p-õ
Taking logs and differentiating both sides of equation (2.7) with respect to time we have the
growth rate of consumPtion
as
i(t) t c(t) = -)i,Q) t p,(t),
(2.tr)
which is equal to
(2.12)
i(,)tr(r)=(roQ)-o)to
by (2.9').
The market is cleared when the stocks of raw labour, human capital and physical capital
supplied to the goods industrY are
L,(t) = V/(t)L
(2.r3)
,
H,(t) = h(t)L,(t)
(2.t4)
,
K(t¡ = z(t)L,(t) + ko(t)L
.
(2.rs)
111
The steady state analYsis
There exists a steady state balanced growth path where per capita output, consumption,
stocks of physical capital, human capital and foreign debt grow at the same rate and t¿ is
constant. The growth rate of the economy is
aaat'
y" =
y(t)ly(t)=k(t) lk(t)=h(t)th(t)= a(t)lz(t)=c(t) lc(t)
'
(2.16)
The market steady state growth rate and its steady state ratio of foreign debt to physical
capiral are to be found. Equations (2.2)-(2.4) and (2.16) imply that in the steady state, the
interest rate on domestic physical capital
ro
and the wage tate
w are constant. From
the
individual optimal time allocation condition (2.8), the constancy of the wage rate implies that
the shadow prices of human capital and physical capital
will grow
at the same rate. That is,
ltr(t)l ttr(t)= ltr(t)l ltr(t)
(2.r7)
Thus by equating (2.9') to (2.10') we find that the domestic learning productivity parameter
determines the steady state interest rate on domestic physical capital
(2'18)
r¿=õ.
Equation (2.t}),once substituted into equation (2.I2), gives us the steady state growth rate of
the economy as
y"=(õ-p)to
Q'te)
The optimal time allocation is obtained by equating equations (2.19) and (2)
V" = (6o + p- 8) I õo
.
Q.2O)
tt2
Define
I
= z(t) I
k(t)
as the steady state per capita
ratio of foreign debt to physical capital.
Substitutingfor rn from (2.18) into (2.2) gives us the steady state marginal productivity of
physical capital condition
al,o+Þ-rt,(t¡eo-ø-U' K(t)"-'H,(t)þ*0-ext-"-Ð
-6
(2.2')
Substiture (2.2') into (2.3)
0(l-a-
p)L,"*p-rz,{t)tQ-"-Ð-rKG)" H,çt¡Þ*<'-t)(r-d-þ)
(2.3")
=7-6
which is the payment accrued to foreign physical capital for technological change. The ratio
of foreign debt to the stock of physical capital employed by firms is obtained by dividing
(2.3")by (2.2')
Z,(t) 6eG-a-p)
K(t)
a(¡ -
(2.2r)
d)
By definition Z,(t) = z(t)L,(r). And by (2.I3),
Z,(t)
is also equal to z(t)W(t)L.
K(t) is the
total physical capital stock which is employed in the economy so that the per capita physical
capital employed is k(l) = K(t) / Z. Substituting for
Z,(r) and K(/) into (2.2I) to derive the
ratio of foreign debt to the stock of physical capital in per capita terms
z(t)
" k(t)
t
=-
6eQ- a- þ
v¡alr - 6)
Q'22)
Finally we can substitute for ry from (2.20) into (2.22) to have the steady state ratio
62o0(Ir=@
a
-þ
which depends on the world interest rate.
7
(2.23)
It is obvious from equation
(2.23) that a higher
world interest rate lowers the steady state ratio 7 and visa versa. Reading from equations
(2.19) and (2.23) we come to Proposition 5.2'
113
proposition 5.2: The world interest rate does not affect the market growth rate of the
economy though
it
influences the steady state per capita ratio of foreign debt to physical
capital. A lower world interest rate improves the position of foreign debt held by the country
and visa versa.
A comparison of the optimal growth rate and the market growth rate
From equation (I.29) the optimal growth rate is
To
ô( I-Ø(t-a-þ)+þ(õ-P),
(1-dX1 -a-þ)+þo
and the market growth rate is given in equation (2.19) as
r"
=:þ-
p)
The difference between these growth rates is
-ØQ-u- þ)(õo+ P-õ)
Yo-Y"= o(Þo+ (1ØG- a - B)
(1
condition (1.31):
õo+p-ó'>0
implies that
(2.24)
y"-T,)0
or the optimal growth rate
is
greater than the market growth rate'
The intuition behind this result is that in the centralised version, the social planner has perfect
information about the externality via technology transfer so that the optimisation problem
gives the maximum growth rate. In the market solution, firms and households are separate
agents. While the technology transfer is known to firms, the external effect of human capital
via technology transfer on the production function is unknown to households so that it is not
taken into account by households in their decision making on the accumulation of human
capital. As a result, a lower growth rate exists in the decentralised version.
IT4
In this section we have shown that we do not need a production externality to have the
market growth rate to be lower than the optimal growth rate as
in Lucas
(1988)' The
externality created by technology transfer is known to firms. It is the human capital choice
externality in households' decisions that gives this result.
2.3. The role of the government
The question is why the market growth rate is less than the optimal growth rate and what can
be done to close the gap? In comparing the time allocation to the learning activities in the
centralised and decentralised versions we note that in the centralised version it is
.,- !'o - 6(1-axl -q-þ)+þ(õ-p)
L-U
(1.30')
alrr- ØG-a- þ)+ þo]
and in the decentralised version the time allocation to the learning activities is
6-p
.
r-w"=
6o
(2.20')
The difference between these two terms ts
(r-w")-(t-w") =
(1-axl
-q-
þ)(õo+ p-õ)
ao(tr-ØG-q-þ¡+þo)
(3.1)
which is clearly greatï than zero by condition (1.31). This suggests that in the decentralised
version, less than optimal time has been allocated to the learning activities and as a result less
human capital
is accumulated in each period which leads to a lower growth rate. In
the
centralised version, the planner takes into account the effect of the accumulation of human
capital on the technology transfer. This is not taken account of in the decentralised case.
115
A lower than the optimal growth rate in the decentralised version calls for the intervention of
the government. In the decentralised version, the role of the government is to use its
appropriate tax and subsidy policies to help the decentralised economy to obtain its optimal
growth rate. Since less time has been allocated to the learning activities, the government
should encourage people to invest more time in these activities. The government can do so by
subsidising the learning activities. The decentralised version is now described by three agents
which are the government, private firms and households.
The government
Suppose that for each hour of learning, the government gives a subsidy at the rate
each individual spends
so
(r) =
VJt)
of to ' If
hours of time to learn then he or she will receive a subsidy of
rotyr(t)h(t)
Q2)
from the government. Thus the total cost incurred to the government is
so
(r) = tnwtî)h(t)L
(3'2')
.
The government can finance its budget from the tax revenue. There are different ways that
the government can tax such as output tax, consumption tax or income tax.
example when the government imposes a tax rate
of Ík
orr
'We
consider one
the earning of domestic physical
capital. The purpose of introducing tax and subsidy is to get the decentralised growth rate
back to the optimal one. The total tax revenue accrued to the government is thus
RoQ) = rt
r¿(t)KoQ).
(3'3)
We assume that the government runs a balanced budget in each period Ro(t) = Sc
(/)
so that
the government budget constraint is
roro7)KaT)=tnVrî)h(t)L'
(3'4)
116
Firms
decisions
Firms are described as the same as in the market solution in section 2'2.The firms'
are reproduced here for convenience
aL,(t)"*Ê-r z(t¡eo-o-u, K(t)"-t H,çt¡Þ*t'-t)(1-q-þ) =
ú,
+
(t) "* þ u Z Q¡e o- "-
0(l-
(P
d
"
-t
K (t)"
- þ)L,(t)"*þ-t
H,
çt¡
zQ¡e<r-"-fDu
Ê*
rd(t)
(2.2)
,
o- e)0- d - þ)
(2.3)
K(t)" H,çt¡Ê*o-e)o-o-þ)
- 7'
* O- ØG- d - ø)L,(t)"*p-t Z(t¡eo "-u' K(t)" H,(t¡Ê-r*<t-0)(t-d-P) = wG)
(2.4)
Households
problem
The usual optimisation of a representative household amounts to the following
M*T e-e,
(t),tt/(r)Jo
st.
c(t)'-"
-r dt
I- O
inçr¡ =
(t- ,o)roQ)koQ) + wry(t)h(t)
+ coty'(t)h(t)
-
c(t)
iç'¡ = 6tYr(t)h(t)
tt/(t)+Vr(t)=1
where
ry(t)
and
tyr(t)
.
arc the time allocated to working and learning respectively. The tax
rate Í o and the subsidy rate
tn
are exogenously given to the household'
The Hamiltonian exPression is
LI7
First order conditions yield
(3.s)
c(t)-" = ltr(t),
p,Q)(wQ)
i,Ø,
-
ro) =
(3.6)
þ,(t)6 ,
/tr(t) = p-(r- to)ro|)
ir(,), p"(t) = p-
6
(3.7)
,
- w(t),l!'- -
(3.8)
ro
Equation (3.5) describes the optimal choice of consumption where equation (3.6) is the
optimal allocation of time between learning and working. The growth rates of the shadow
prices of physical capital and human capital in terms of consumption are given in equations
(3.7) and (3.8).
Equations (3.5) and (3.7) give us the per capita consumption growth rate as
i(,) t r(,)=
ro)roQ)
[(t-
-
ol o
(3.e)
.
The market solution
The steady state market solution is obtained as when the economy grows at a constant rate of
aaaa'
y", = y(t) I y(t) = k(t) I k(t) =h(t) I h(t¡ = z(t) I z(t) = c(t) I
c(t).
(3.10)
Equations (3.10) and(2.2)-(2.4)thenimply that the rate of return to domestic physical capital
ro andthe wage rate w are constant. From (3.9)
and (3.10), the steady state growth rate of the
economy is
r, =l$-
ro)ro
-
ol
o
.
(3'e')
118
In the steady state the shadow prices of human capital and physical capital,
pr(t)
and ltz(t)
,
grow at the same rate. Equating equations (3.7) and (3.8) gives us the after tax return on
domestic physical capital
/\6r
lL-ro)ro
=õ.ft,
(3.11)
Substituting (3.11) into (3.9') we can write the growth rate of the economy equivalently
læ,--n -pllo.
I
y",=lõ+
w-to
I
I
as
Q.l2)
Reading from the growth rate equations (3.9') and (3.12) we note that
%=-fn.o,
dî* o
ù",
õ,v
âro ( w-în o
2
>0,
or a higher tax rate lowers the growth rate while a higher subsidy rate raises the growth rate.
Tax has a distortion effect on the accumulation of physical capital. A higher tax causes
a
lower rate of physical capital accumulation which leads to a lower growth rate. However,
a
higher rate of subsidy raises the rate of human capital accumulation and thus the growth rate.
Via their effects on the accumulations of physical capital and human capital, the tax
and
subsidy rates can influence the growth rate of the economy.
The market solutions for this economy are described by five equations in five unknowns
ro
,w,l[,T ¡,
and.
t o as follows2
åw
2
(3.13)
A detailed calculation is provided in Appendix B.
119
* =lþ+ (1 - Ø(l- - /ii',**P'a"ll¡- a - P)feo-d-P\ ,;" (, - ,o)-"'-"-Þt ,"*'_t}ø*<t-ext-"+t
"
(3.r4)
(3.1s)
ro(t- v) =
Ít
r¿v[lo''u-"-0,(atr - o -
ø)'
I:x+þ-tr¿eo-"-ø>'(r
- ,o)-' ,ø+Þ-l þ+Q-ØQ-&-þ) _
(3.16)
6G-ØQ-d-þ)+þ(õ-P)
_
(t-
ro)o
-,
(3.r7)
o
(1- áXl - a,- þ) + þo
The steady state interest rate on domestic physical capital is determined in equation (3.13).
Equation (3.I4) formalises the function of the wage rate. The optimal allocation of time
between learning and working
is obtained in equation (3.15). The government
balanced
budget is in equation (3.16). Equation (3.17) is the government's objective equation' The
objective of the government is to bring the market growth rate to equal the optimal growth
rate. Thus by equating y", in(3.9') to
y,
in(I.29) we have equation (3.I7).
The system of five equations (3.13)-(3.17)
will
solve for the five unknowns which then give
us the optimal tax rate and subsidy rate that the government can impose in order to achieve
the optimal growth rate for the decentralised economy. We come to the final proposition.
r20
Proposition 5.3: In order to achieve the optimal growth rate the government should subsidise
the learning activities which they may finance from tax revenue.
The intuition for it is obvious. The acquisition of human capital and foreign technology is the
driving force of economic growth. The more stock of human capital that the country
possesses, the better foreign technology
it
can absorb. In the market solution without
government, due to the households' unawareness of the external effect of human capital on
technology transfer, less time has been invested in the learning activities which lowers the
human capital accumulation progress and thus the economic growth rate' The role of the
government is to encourage people to spend more time on the learning activities in order to
boost the rate of human capital accumulation and thus the economic growth rate. An
appropriate policy is that the government subsidises the learning activities and it can finance
its budget by using the tax revenue.
3. CONCLUSION
In this chapter we study the economic growth performance of a small open economy which
hosts foreign investment. In the model, we raise the interactions between technology transfer
via foreign investment and the human capital accumulation of the host country. The issue is
that while the host country depends on foreign investment for its technological change, the
degree of technology absorption is constrained by the stock of human capital in the country.
Human capital can be accumulated and the more stock
possesses, the higher level
of
human capital the country
of technology it can obtain from foreign investment' Technology
transfer is modelled as a function of foreign physical capital and domestic human capital. In
order to assign a clear role of foreign physical capital on technology transfer, in this model
LzI
we assumed that while the country can rent foreign physical capital it can never lend physical
capital overseas.
abandoned
In
other words, the perfect physical capital mobility assumption is
in this model. Two versions of the economy, the centralised version with
the
optimal growth rate and the decentralised version with the market growth rate ate studied.
'We
found that the steady state growth rates of the centralised and decentralised economy are
independent on the world interest rate though the world interest rate determines the steady
state per capita ratio
of foreign debt to physical capital. Since the optimal growth rate of the
economy is a function of exogenous parameters, depending on the values of these parameters
the economy can grow at different rates. This result then suggests that different small open
economies of the same type can experience differences in growth rates. Due to the existence
of the externality, the market growth rate is lower than the optimal growth rate which calls
for the intervention of the government. The government can subsidise the learning activities
in order to raise the market growth rate to equal the optimal growth rate. Finally, what we
learn from this model is that we do not need a production externality to have the market
growth rate to be lower than the optimal growth rate as in Lucas (1988). The human capital
choice externality in the model can give this result.
t22
APPENDIX A
The effects of exogenous parameter values on the optimal steady state growth rate
The optimal growth ratei
Y
o=
Condition (1.31): õo + p- ô > 0.
To find how changes in exogenous parameter values affect the optimal steady state growth
rate, we take partial derivative
of T, with each parameter.
The results are reported as
d1/ -1
#=#Ur-e¡(õo+p-ô) .0,
d1, -1
=#,t
ïË
ù,
-
ØG- a¡(6o + P-ä)
1
ârt
p-ó')
(.)' Bç-o¡(õo+
ùo
-1
de
(.)'
'o,
to,
þ(r-a-Ð(6o+p-ô)<0,
%=åt,t -Ø(r-d-Ð*þ),0,
k#
Áô(1- Ø(1
- a - þ) + þ(õ- r)) < o,
+=-|o..o
where (.) =
(1
- ØG- a-
B) + po > 0
t23
APPENDIX B
Equation (3.13) is obtained from equation (3.12). Equation (3.I4) is the expression for the
wage rate which is derived from equation(2.4).In equation (2.4),the steady state ratios of
foreign debt to total physical capital Z,(t) t K(t) , and the human capital to the physical
capital H,(t) I K(t) need to be found. Equations (2.2) and (2.3) together give us
expression for Z,(t) I
K(t)
as
Z,(t) ro0(l-a-P)
K(t)
the
(B1)
o(r - r,)
which can be substituted back to equation (2.2) to obtain the steady state ratio H,(t) I K(t)
I
ry
K(t)
(g(1
=lorrr-o-,l)-r
\ \
L
-a-
t-d-þ rdt-,(t-"-Ð (7
L'
F))-t<'"-u'
u
\
"'
-
ro)e(t-"-zt
,r"-øfø-<r-etrv"+t
.
(82)
Substiture (81) and (82) into equation (2,4)we obtain equation (3.14). In the steady state,
human capital and consumption grow at the same rate. From equation (2), the growth rate of
human capital is
nity nØ =
õí- v)
.
(83)
Thus by equating equations (B3) and equation (3.9') we derive the optimal allocation of time
r¿ as described in equation (3.15). Finally, the government budget constraint (3.4) gives
us
equation (3. 1 6). The government budget constraint (3 .4) can be written in per capita terms as
k,(t)
To(l- V) = ttr¿;Ø
(84)
124
The stock of physical capital employed by firms is the sum of foreign physical capital and
domestic physical capital
as K(t)
= Z,(t) +
Ko(t). The ratio of the stock of
domestic
physical capital to the stock of human capital that employed by firms is
:+=
Í(?, -39
H,(t) H,(t)
H,(t)'
(Bs)
From equation (2.2) we can derive the steady state ratio of foreign debt to the stock of human
capital supplied to firms as a function
z,(t)
of K(r) I H,(t)
0(t-d-p\
_
H,(t)
(86)
o(Lvr)ruu
Substituting (86) into (B5) we have
rd
(87)
o(L,/r)"*u'
Since the stock of domestic physical capital
capital supplied to firms
is K, (t) = ko(t)L
is .F/,(t)=yrh(t)L,we
and the stock of human
can derive the per capitaratio of domestic
physical capital to human capital
ko(t)
h(t)
-
(88)
o(Lr{)"*u'
Finally we can substitute (82) and (88) into the government budget constraint (84) to obtain
equation (3.16).
t25
Chapter 6:
DIRECT FOREIGN INVESTMENT, TECHNOLOGY
TRANSFER AND BCONOMIC GROWTH IN A SMALL
OPEN ECONOMY
t26
l.INTRODUCTION
Direct foreign investment (DFI) has become an important source of private external finance
for developing countries and developing countries seek such investment to accelerate their
development efforts. The study of Fernandez-Arias and Montiel (1996) on capital inflows to
developing countries since 1970s shows that capital keeps flowing to developing countries
and there is a shift away from debt instruments to equity instruments.
In the period 1973-
1981, capital inflows are mainly in the form of private bank loans directed to the public
sector and from the early 1990s capital flows take largely the form
of direct foreign
investment and portfolio investment.
Borensztein, De-Gregorio and Lee (1998) use data on DFI flows from industrial countries to
69 developing countries over the two decades from 1970-1989. They reported that there is a
favourable effect of direct foreign investment on economic growth of developing countries.
They argue that DFI is the main channel for the transfer of technology which contributes to
economic growth of the host countries.
The present issue is why does direct foreign investment flow to developing countries and
how does it contribute to economic growth of developing countries?
Direct foreign investment, in narrow terms, is defined as investment made by multinational
business enterprises in foreign countries to control assets and manage production activities in
those countries (Mallampally and Sauvant, 1999). With debt instruments, foreign investors
lend capital to domestic firms at a certain interest rate and have no control over firms. With
t27
direct foreign investment, equity gives foreign investors ownership which allows them to run
the firms
Foreign entrepreneurs from developed countries have access to advanced technology. Direct
foreign investment allows them to employ modern technology in the host country which
gives them the possibility to increase output at a given set of inputs. In other words, advanced
technology makes their investment more productive than domestic investment and thus
promises a better rate of return to investment. Foreign firms want to exploit the profitability
that is given to them due to the employment of advanced technology. As long as there is a
gap
in technology, foreign firms are induced to enter the domestic market to take this
advantage and thus the inflow of foreign capital takes the form of direct foreign investment.
Direct foreign investment brings with
it
a package of capital, advanced technology and
foreign management expertise. By introducing new technology to the host country, DFI acts
as a channel
for the transfer of world technology. Thus it helps to build up the host country's
stock of technology which enhances the long run growth rate of the country.
Our objective in this chapter is to develop an endogenous growth model which can describes
the addressed issue. That is, the model should explain two things. Firstly, the existence of
DFI is motivated by the reaping off profit
employment
overseas which
of superior technology and management
investment acts as a growth-enhancing factor since
transfer"
An
endogenous growth model
is
made possible by the
expertise. Secondly, direct foreign
it is a major channel for technology
of the Lucas
(1988) type
will be
employed.
Technology transfer via direct foreign investment is then introduced to the model and we
study its impact on economic growth of the host country. In the next section, two versions of
an autarky economy and an open economy with direct foreign investment will be discussed.
r28
the role
The difference in the growth rates of the economy under two regimes will highlight
of direct foreign investment.
2. THE MODEL
2.1. Autarky economY
lived
Consider a small economy which is populated by a constant L identical and infinitely
individuals. There is a single good to be produced and there are two factors of production
which are physical capital and human capital. The single good can be consumed or directly
in
invested as physical capital. Human capital is the skills or knowledge which are embodied
each
individual and can be accumulated by investing time in learning activities. Let h(t) be
capital is
the individual stock of human capital at time r so that the aggregate stock of human
H(t)=¡ç¡¡¡.
The economy is described by two agents which are private firms and households. Households
physical
own human capital and physical capital which they rent out to firms. Firms employ
unit
capital and labour to produce goods. In each period, each individual is endowed with one
of time which can either be spent on working or learning. Suppose the individual supplies
fraction ttt(t) of his or her time to work and allocates the other fraction
1
-
a
V/(t) of time to
as
create new knowledge. New human capital or new knowledge is assumed to evolve
iI,¡ = d$-
ylt¡)nçt¡,
(1'1)
where ô is the indigenous technology creativity capability parameter or simply the endowed
capability to create new knowledge.
129
Firms
Private firms employ physical capital and labour to produce goods according to a CobbDouglas production function
(r'2)
Y(t)=AK(t)"þy(t¡nçt¡t)'-",
where A is an exogenous parameter which describes the managerial skills of firms or the
management methods that firms employ to convert inputs into output. K(t) is the aggtegate
stock of physical capital and Vt(t)h(r)L is the aggregate stock of human capital supplied to
firms at time r. The production function can be rewritten
as
Y(t¡ = h(t)u" AK(ù"(w(t)L)' " ,
so that the term
h(t)
can be interpreted as an
indicator of the technology level at time r. We
may think that technology is created by individuals who possess some levels of human
capital skills. But new technology also creates new human capital as people must acquire
new skills to handle the new technology. Technology can be assumed to be intangible
knowledge and
in
order to use
Knowledge is embodied
in
it,
knowledge
of it must be embodied in
individuals.
each individual and becomes the individual's human capital.
Viewing it this way, human capital, knowledge and technology are used interchangeably' To
make it clear we note that while the country's stock of human capital is H(t)=hç¡¡L, its stock
of knowledge or technology is
Define
y(t)=Y(t)lL
and
/¿(r).
k(t¡=K(t)lL astheoutputpercapitaandphysicalcapitalper
capita respectively then the production function in an intensive form is
130
(r'2')
Ak(t)"(w(t¡tt1¡)'-"
y(t) =
profit maximising' The
Perfect competition is assumed to exist among firms and firms are
productivity:
profit maximising condition requires that firms pay each factor their marginal
(1'3)
r(t)=aAk(t)"-'(tyçt¡hç))'-",
w(t) =
(1'4)
(t- a)Ak(t)"(tr(t)h(r))- ,
capital'
where r(r) is the interest rate and w(f) is the wage paid to one unit of human
Households
market
Households earn income from supplying labour and physical capital at competitive
the wage rate as
rates. From the individual point of view, while taking the interest rate and
physical capital
given, a higher income is realised when the individual owns higher stocks of
(1.1). Physical capital
and human capital. Human capital can be accumulated according to
capital can be
can be accumulated by saving from income. The accumulation of physical
described as
içr¡=r(t)k(t)+w(t)ttt(t)h(t)-c(t),
(1'5)
to depreciate. The
where c(r) is the consumption at time / and physical capital is not assumed
utility of an individual is assumed to derive from consumption only. We
assume that the
lifetime utility of the individual is
u
-r
=i,-' c(t)t-"
I-o
(1.6)
0
where
p is the discount factor and o
is the risk aversion coefficient. The objective of each
individual is to maximise his or her lifetime utility. The individual does so by choosing
131
consumption and how much of time to work and learn in each period. This amounts to the
following problem:
Max
c(t),tttu)
st.
k(t¡ = r(t)k(t) + w(t)tt¡(t)h(t) - c(t)
nç¡ =
dl-
ylt¡)nç¡.
The Hamiltonian expression is
r = 9T=
),
where
).,()(r
+
),,
and
(t) k (t) +
w
(t)
ty
(t)h(t)
- c Ø)
+
t,
çt¡(a$
-y
çt¡)nçt¡)
arc the shadow prices of physical capital and human capital in terms of
consumption respectively
First order conditions yield
c(t)-" = ¿r(t)
(1.7)
,
)"rw(t) = 121)6
Lr(t)=
l,(,)
(1.8)
,
p4Ø-r(tfl"(t),
= pLr(t)
- w(t)W|)),(t) - 6(t- yçt¡lrçt¡)
(1.e)
(1.10)
Equation (1.7) describes the optimal choice of consumption, and the optimal allocation of
time between learning and working is given in equation (1.8). From (1.9) we have the growth
rate
of ),,
as
7,(t) I hQ) = p-
r(t).
(1.9')
r32
Substitute (1.8) to (1.10) to have the growth rute
of 7,
(1.10')
)"r(t) / )"rçt¡ = P- 6
The usual growth rate of consumption can be obtained as
i(,) t ,(,) =(rQ) - p) |
(1.1 1)
".
The steady state analysis
The steady state is described as a balanced growth path when income, consumption, physical
capital and human capital per capita grow at the same and constant rate and the time
allocation r¿ is constant. Call g" the autarky growth rate of the economy where
g, = y(t) I y(t) = k(t) I k(t) = h(t) I h(t¡ = c(t) I c(t)
.3),
(r.12)
.4) and (I.I2) imply that in the steady state, the interest rate r and the wage
Equations
(1
rate w are
constant.Let p(t) = )"r(t) I ),r(t) be the shadow price of physical capital in terms
(1
of human capital. From equation (1.8), the constancy of the wage rate implies that the steady
state
p is constant or
),r(t)
and
)"r(r) grow at the same rate
)"r(t) I trr(t) = ).r(t) I ).r(t)
(1.13)
Equating equation (1.9') to (1.10') we have the autarky interest rate of the economy
rn =
6,
(1'14)
and thus from (1.1 1) the autarky long run growth rate of the economy is
g" = (õ - p) I o.
(1.15)
r33
So far we just described the Lucas model without externality for a closed economy. However,
we have shown that the autarky growth rate of the economy is determined by the autarky
interest rate which is equal to the indigenous technology creativity capability parameter á.
If
people in all countries share the same taste and preferences which are captured by factors
p
and
o
except for the endowed factor
ô then this result implies two things. Firstly,
the better
the indigenous technology creativity capability the country possesses, the higher growth rate
the country experiences. In other words, a "smart" country can grow faster. Secondly, a high
growth country also has a high interest rate.
Suppose two countries start at the same wealth level but experience different growth rates
then in the long run a richer country is the one that possesses a higher growth rate. The
second result then implies that a lower interest rate exists
in
a
poorer country. We explain the
reason for this. Human capital or knowledge is an engine of growth. The accumulation of
new knowledge determines the growth rate of the economy. However, the capability to create
new knowledge is different in each country due to the fact that people are different so that
isolated countries experience different growth rates. In the long run, a poorer country which
experiences a relatively lower growth rate has relatively lower stocks of physical capital and
human capital. The rate
of return to
physical capital
is determined by the marginal
productivity of physical capital which in turn depends on the available stocks of physical
capital and human capital. The marginal productivity of physical capital displays diminishing
returns to physical capital alone implying that the marginal productivity of physical capital in
the poor country is relatively higher due to a lower stock of physical capital per
However, an associated lower stock
of human capital per head depresses the
head,
marginal
productivity of physical capital. Thus the domination of either effect will determine the
position of the interest rate in the poor country. In this model, it turns out to be the case that
t34
the long run shortage of human capital always dominates the shortage of physical capital
causing a lower marginal productivity of physical capital and thus a lower interest rate in the
poor country.
2.2. Open economy
We assume that at the time the economy opens to the rest of the world, the economy is in its
long run autarky steady state with the economic growth rate
of g, . The degree of openness is
applied to physical capital only and people cannot migrate from the country. The economy is
small and takes the world interest rate
r
as exogenously given. We also assume that people
in the world share the same taste and preferences so that the world growth rate
g*=(7-Ðlo.
g"
is
Q'I)
Suppose that the autarky interest rate of the economy is less than the world interest rate or
tn = õ <
F which implies that the autarþ
steady state growth rate of the country is relatively
smaller than the world growth rate. Due to the autarky technology growth rate of the country
being less than the world technology growth rate gn 1 Bw, initially there exists alarge gap in
technology levels between the country and the rest of the world.
We assume that there is no perfect physical capital mobility. The country, while it can borrow
foreign physical capital, can never lend physical capital overseas. Since the home interest rate
is less than the world interest rate, there is no inflow of portfolio capital to the country'
However, the initial technological gap creates an incentive for foreign firms to enter the
domestic market as they seek opportunities to reap the returns to better technology. Direct
foreign investment occurs when foreign firms bring with them a package of foreign capital
135
has
and foreign advanced technology to the host country. We also assume that the country
access to foreign borrowing via direct foreign investment only.
produce
The economy has two sectors: the foreign sector and the domestic sector which both
foreigners
the same goods. Foreign firms are assumed to be totally owned and controlled by
While the
whereas domestic firms are totally owned and controlled by domestic residents.
of
foreign sector has access to an unlimited supply of the world physical capital, the supply
physical capital to the domestic sector is constrained to the stock of physical capital owned
by domestic residents. Domestic labour is employed by both sectors. At time r, the labour
force is comprised of
L identical individuals each with h(t) units of human
capital. Each
in
individual allocates a fraction WQ) of his or her time to work where rl¡(r) is the same as
section 2.1 of autarky economy, so that
ttt(t)h(t)L is the
aggtegate stock of domestic human
capital supplied to all firms in each period. Out of the labour force, foreign firms employ
Lr O) individuals and Ln(r) individuals are employed by domestic firms
L= Lr(t)+ Ld(t).
Let Lr(t) = ç(t)L where
ç(t) is an endogenous parameter then LnQ) =$- ççt¡)t ' m"
aggregate stock of human capital employed in the foreign sector is
H
(t) =
r Q) = ry(t)h(t)Lr
and in the domestic sector
Hne) = ty(t)h(t)LnQ)
ç(t)ty(t)h(t)L
,
(2'2)
it is
=(t-
ç<,>)w(t)h(t)L
(2.3)
136
foreign sector
The
Foreign firms implement production technology T to convert inputs into output.
'We
assume
that T>A to capture the scope for which foreign firms possess better managerial skills and
have access to foreign advanced technology. The production function of foreign firms is
Yr(t)=TKrU)" Hr(t)'"
,
(2.4)
where KrQ) is the stock of foreign-owned physical capital and Hr(r) is the stock of
domestic human capital supplied to foreign firms. Substitute
!¡(t)=YrG)lL
and
kr1)=KrQ)lL
foreign sector in an intensive form
!¡
(t) =
Tk
r G)" (aQ)ty(t)hQ))'-"
for H, (r) from (2.2) and let
then we can write the production function of the
as
.
(2.4')
The condition T>A is a factor that distinguishes foreign firms from domestic firms. This
condition indicates that foreign firms are in a better technology position than domestic firms.
As long as this condition is satisfied, foreign firms are induced to enter the domestic market.
Foreign firms have access to an unlimited supply of the world physical capital. For them to
have an incentive to carry on the production in the host country, they would expect the rate
of
return to their investment to be higher than they would obtain from elsewhere. This is to reap
off the return to superior technology that they introduce to the host country. Suppose that the
command rate of return to foreign investment
is r, where it is greater than the world interest
rate
rÍ)r
This condition provides ex-ante incentive for direct foreign investment.
(2.s)
rî
can be thought
of
the rate of return to a package of foreign capital and foreign technology
r3l
The foreign sector will employ physical capital and human capital up to the point where the
rate of return to each factor equals to its marginal productivity
rr=drkr(t)"-'(çU)ty(t)hQ))'-",
w,(t) = (1- a)Tk,(ù"(çU)w@h@)-"
(2.6)
.
(2.1)
Now we highlight the role of foreign firms as the main channel for the transfer of foreign
technology. As mentioned in section
2.I of autarky economy, in this model, technology or
intangible knowledge and human capital are the same. While the country's stock of human
capital
is
,F1(/)
=h(t)L, its level of technology is h(t). Foreign technology is unknown
domestic residents unless there
is personal
to
contact between foreigners and domestic
residents. Direct foreign investment brings domestic residents into contact with foreigner
technology. Technology transfer occurs when foreign technology is embodied in domestic
workers through training and skills acquisition provided by foreign firms. We wish to express
formally the rate of foreign technology transfer. In it simplest form, as in Findlay (1978), we
take the relative extent to which direct foreign investment pervades the local market as
a
proxy for the rate of technology transfer. This extent is expressed as an increasing function of
the ratio of foreign physical capital to domestic physical capital as
/\
It(t) = ttlkrØ / kn@),
where
âtt(t)
(2.8)
>0
a(t ,ft> t tc,@)
This is to say the more foreign firms in the industry (as measured by the proportion of
physical capital investments), the more chance domestic workers have contact with foreign
firms and the more chance domestic workers acquire foreign knowledge. Alternatively, the
138
more foreign firms in the industry, the cheaper and easier for foreign firms to introduce new
technology and thus the higher the rate
of technology transfer. Strictly
speaking, only
workers who are employed by foreign firms can acquire foreign knowledge. In the long run
workers are mobile between sectors and labour mobility creates spillovers
technology
to domestic firms and benefits the whole economy. In the
of
advanced
aggregate
it
is
reasonable to assume that all workers have equally benefit from gaining foreign knowledge'
Via direct foreign investment, the accumulation of foreign technology by the host country is
h(t¡=tt(t)h(t).
Q.9)
To satisfy the assumption of the technology transfer function (2.8), there is a clear distinction
between foreign physical capital and domestic physical capital. Foreign capital, while
it
is
physically the same as domestic capital, is accompanied with foreign advanced technology.
Equation (2.8) suggests that a higher stock of domestic physical capital, with a given stock of
foreign physical capital, will lower the rate of foreign technology transfer. Since technology
and human capital are used interchangeably,
it then follows from equation (2.9) that a lower
acquisition of foreign technology means a lower acquisition of human capital.
The domestic sector
Domestic firms employ domestic-owned physical capital KoQ) and domestic labour Hn(t)
to produce goods according to the production function
YoQ) =
AK|Q)" HnG)'-".
Subsritute for Hn(r) from (2.3) andlet yo7) =Y¿(t) I
(2.10)
L and knG) = Ko(t) / L we deriïe the
intensive form of the domestic sector's production function
as
r39
yoe) = ennçt¡"($-
(2.r0')
çØ)wG)hQ))'-".
The supply of physical capital to the domestic sector is constrained to the stock of domesticowned physical capital. The domestic sector pays the factors of production according to their
marginal productivity
rn|) = aAknQ)"-'((r- çra)wçt¡nçt¡)'-"
woT) = (1- a)Akn(f )"((1
,
(2'tr)
- çØ)wØhØ)
Q.r2)
The foreign and domestic sectors compete for labour so that the labour market is cleared
when both sectors pay each unit of human capital at the same price. We call w(t) the market
wage at time
t.
Equalising equations (2.7) and (2.12) to derive the labour market equilibrium
condition at all time
(t- a)rk, (ù" (çØw@h@)-" = (1 - u) Ako(r)" ((1 - çØ)wØhØ)
,
(2'r3)
which gives us the allocation of labour between sectors
rt"
Ð
k,t(t)
(2.t4)
krG)
Dividing (2.11) by (2.6)
.JI (t-
l-a
ççt¡)rr,{t)
_ rn(t)
(2.rs)
rr
ç(t)kn(t)
and substituting (2.I4) into (2.15) we have the interest rate paid to domestic physical capital
at any time
(
A.\''"
rt
,rt)=lT)
Note that A<T or r¿
(2.16)
1rÍ
140
Households
Households are indifferent between working in each sector when the wage paid per unit of
human capital is the same in both sectors. Income of the household is the sum of earnings
from physical capital and working, which amounts to
y,(t)=ro7)kaî)+w(t)ty(t)h(t)'
Q'17)
This income is spent on consumption and saving in physical capital so that the individual
stock of physical capital is evolved as
(2.18)
i n çr¡ = ro G)k a Q) + w(t)tt¡ Q)h(t) - c(t)
The accumulation of the stock of new knowledge incorporating technology transfer is now
described as
t çr¡ = dû- vlt¡)hT) + p(t)h(t) ,
where the
term
t¿-
(z're)
yçt¡)hG) is the new knowledge which is created by domestic residents
through domestic education or learning activities; and the tetm
p(t)h(t) is the foreign
knowledge acquired by domestic residents through technology transfer by foreign firms.
Thus the technology accumulation of the host country is the sum of indigenous technology
and foreign technology.
lt(t)
acts as an externality on the knowledge accumulation.
It
says
that the more foreign firms (or DFI) exist in the country, the more foreign knowledge that
domestic residents can acquire.Il
determined by the parameter
technological progress.If
p(t)
ty(t)=0, the growth rate of
and the country
lt(t)=Q,
knowledge is exogenously
is totally dependent on foreign firms for
the country is back to autarky with no direct foreign
investment in the countrY.
r4t
The individual maximises the lifetime utility by choosing consumption and the allocation of
time to work and learn in each period subject to his or her budget constraints
Max e-ø
c(t ) ttt() J
c(t)t-"
l-o
0
st.
as
-I
içr¡ = roT)k(t) + w(t)tt/(t)h(t) - c(t)
t
çr¡
=
d$-
Yç¡)hQ) + t"t(t)h(t)
'
The Hamiltonian expression is
t
=
ÚY
where
A,()(roe)ko(t) + w(t)ttt(t)h(t) -c1r¡)
l-o
+
2,
),,
and
are
1",çt¡(a$- v¡Q>)nrtl + ¡t(t)h@)
+
the shadow prices of physical capital and human capital in terms of
consumption respectivelY
First order conditions yield
c(t)-"
: trr(t)
(2.20)
,
)"r(t)w(t) = lz(t)õ
Àr(t) = plr(t)
-
(2.2r)
,
rn(t)),r(t)
l"(r) = pl,(t) - w(t)w|)1,(t) -(A(t The growth rate
,l.r(r) t
(2.22)
,
v/@)
+
¡tçt¡))""çt¡
of 2, is obtained from equation (2.22)
lr(t) = p- rn|)
as
(2.22',)
.
Substitute (2.21) into (2.23) to derive the growth tate
i,(r)tA,(t)=p-(6+pØ)
(2.23)
of
.1',
(2.23',)
r42
Combining (2.20) and (2.22') we have the growth rate of consumption as
c(t) t c(t) = (rnQ)
(2.24)
- O) t o .
The steady state analYsis
The steady state is defined as a balanced growth path where
ty and (P are constant
and the
Sectof
domestic sector output, domestic physical capital, human capital, consumption, foreign
go the
output and foreign physical capital all grow at the same and constant rate. We call
open economy growth rate where
aat'
k¿(t) I knî) = k¡(t) I kr(t) = h(t) I h(t¡ = c(t) I c(t) '
Bo = toe) I y¿(t) = i ¡(t) | y¡(t) =
(2.2s)
In the steady state the shadow prices of physical capital and human capital, )"r(t) and Lr(t)
grow at the same rate
ir(t) t t,7t¡ = lrrr>, trr(t)
.
(2.26)
Substitute (2.22') and (2.23') into (2'26) we have
r¿=õ*p¿.
(2'27)
For any p > 0, or there is direct foreign investment in the country, ro ) 6 = r" , that is the
the
interest rate on domestic physical capital under an open economy regime is greater than
autarky interest rate. In other words, the inflow of foreign physical capital does not depress
the
the rate of return to domestic physical capital but rather enhances it. The reason is that
inflow of foreign physical capital is accompanied with the inflow of foreign technology. The
inflow of foreign technology adds to the accumulation of the home country stock of
period
technology and thus the stock of human capital. A richer stock of human capital in any
makes domestic physical capital more productive. Since physical capital
is paid at its
143
marginal productivity, a higher marginal productivity of physical capital results in a higher
interest rate
Substitutine Q.l6) into (2.27) gives us
"
(p*
(2.28)
6)
or the rate of return to foreign investment is determined by the rate of technology transfer p
.
The higher the rate of technology transfer, the better the return to investment that foreign
firms can obtain. Substituting ro ftom equation (2.27) into equation (2.24) we have
growth rate of the economy
It-
Bo
the
as
=;(ô + þ- p).
\
(2.2e)
Reading from the growth equation (2.29) we first note that DFI (with its proxy as
in the growth equation. DFI acts
as a growth-enhancing factor
raises the growth rate of the economy.
If
in the
p)
appears
sense that higher DFI
there is no DF[, the country's growth rate is back to
its autarky level. We come to Proposition 6.1.
Proposition 6.1; Direct foreign investment raises the steady state growth rate of the economy
above its autarky level
The reason is obvious. Direct foreign investment acts as an agent for the transfer of foreign
advanced technology to the host country. Since technology is an engine of growth, foreign
knowledge adds to the stock of domestic technology which results in a higher growth rate of
the economy.
t44
We now define the "catch-up" period and the "long run" steady state. The "catch-up" period
is described by multiple steady state paths where the country can grow at different growth
rates. The "long run" steady state is described as the steady state path where the growth rate
of the country is the same as the rest of the world. The growth equation (2.29) and equation
(2.28) give us the following propositions.
Proposition 6.2: During the catch-up period
o The higher the rate of technology transfer
(a
higher
¡t)
, the larger the proportion of DFI in
the country and the higher the growth rate the country can experience. That is, a country
which hosts more DFI tends to grow faster'
o The larger the gap in technology levels, the more incentives for foreign firms to enter the
domestic market since a larger technological gap gives a possibility for a higher rate of
technology transfer which results in a higher rate of return to foreign investment.
Proposition 6.3: In the "long run" steady state, the growth rate of the country is the same
as
r,
is
the rest of the
world gor=
g*
and thus the rate
of return to direct foreign investment
endogenously determined.
Let us explain the intuition behind these propositions. We assumed that at the time when the
country opens to the rest of the world, the country is in its long run steady state autarky
growth path. Since the autarky growth rate of technology in the country is less than the long
run growth rate of the world technology, initially there exists a large technological gap
between the country and the rest of the world. This gap in technology
will induce foreign
r45
firms to enter the domestic market to take the advantage of superior technology. The catch-up
period is described as when the country closes the initial gap in technology levels.
Foreign firms bring new technology to the host country. In the earlier stages of development,
the large gap in technology between the country and the rest of the world provides an ample
rate of technology transfer. The rate of technology accumulation in the country can be
expected to be higher than the growth rate
of world technology due to the catch up in
technology levels. During this period the economy enjoys excessive growth which can be
higher than the world growth rate or go" 2
g, where go.
is the growth rate of the economy
during the catch-up periods. Equations (2.I) and(2.29) then give us
p+6>7.
Q.30)
Since the rate of return to foreign investment is determined as
follows that
r, > 7 or there is scope for foreign firms
investment than from elsewhere.
as a result, the proportion
tltgh r,
in equation (2.28), it
then
to earn a better rate of return to their
attracts more foreign firms to the local market and
of direct foreign investment to domestic investment will be high.
The country experiences large inflows of direct foreign investment. This period ends when
the economy reaches a position where the initial gap in technology level is closed.
It is the
"long run" steady state.
In the "long run" steady
state,
DFI will exist if there is a difference in the long run growth
rate of technology between the country and the rest of the world.
If
the country's indigenous
technology growth rate (or the rate of technology created by the country) falls behind the
world technology growth rate, DFI acts as a channel for technology transfer to fill this gap. If
the country's indigenous technology growth rate is equal to or greater than the world
t46
technology growth rate then there is no DFI in the country since foreign firms are no better in
technology than local firms
Since õ <
r
by assumption, the indigenous technology growth rate of the country is less than
the world technology growth rate. The country must rely on DFI to
fill
the gap in technology
growth rates. In the long run, the maximum rate of technology accumulation that the country
can obtain via direct foreign investment is equal to the world technology growth rate. In other
words, the rate of technology transfer is constrained by the world technology growth rate.
Thus by equating go,
l.¿
to gw we have the "long run" technology transfer rate as
r 6
(2.31)
This determines the "long run" steady state return to direct foreign investment
,, =(T / A)''"7,
(2.32)
which is higher than the world interest rate.
3. CONCLUSION
In this chapter we construct a growth
model to explain the inflows
of direct
foreign
investment to a developing country and its impact on economic growth of the host country. In
our model, we make two major assumptions. Firstly, we explain that the incentive for foreign
investors to undertake direct investment in the host country does not necessarily come from
the favourable interest rate differentials but rather from the technological gap that exists
between the host country and the rest of the world. Direct foreign investment is motivated by
the reaping off profit overseas which is made possible by the employment of superior
141
technology and management expertise. Viewing
it this way, direct foreign
investment still
flows to a developing country which has an initially lower interest rate
The second assumption provides the role for direct foreign investment as a channel for
technology transfer. Direct foreign investment complements human capital via technology
transfer. As foreign firms enter the domestic market, they provide local workers with the
skills to handle new technology. Direct foreign investment complements the expansion of
domestic investment in physical capital via spillovers of advanced technology to domestic
firms. As domestic firms employ workers who were previously employed and trained by
foreign firms, better trained workers make physical capital more productive resulting in
a
higher marginal productivity of physical capital and thus the rate of return to physical capital.
A higher interest rate on domestic physical capital induces domestic residents to invest more
in physical capital. As a result, the country can experience a higher growth rate
Given these modelling assumptions, our results show that there is scope for mutual benefits
to both foreign firms and the host country. Foreign firms can enjoy a higher rate of return to
their investment than the world interest rate while the growth rate of the host country can be
higher than it would be without direct foreign investment
This endogenous extended Lucas' growth model can also explain the position of the country
toward DFI in the long run. Depending on its indigenous technology capability relative to the
rest of the world, the country can either be hosting inward DFI, no DFI or having outward
DFI. It hosts inward DFI
if its indigenous
technology growth rate is less than the world
technology growth rate. DFI does not exist in the country
if
there is no gap in technology
148
growth rates. The country takes outward DFI when its growth rate of technology is higher
than that of the rest of the world.
This model is simple but it does describe the issues that are of concern. There are periods in
which developing countries grow at faster rates than the average world growth rate. In the
earlier stages of development, the poor technology condition of a developing country attract
large inflows of DFI since foreign firms are more effective than domestic firms in terms of
technology. High rates of return to both foreign and domestic investments are realised as the
result of the employment of modern technology. The more backward the country is, the
larger the share of DFI and the higher the growth rate of the country. We can expect the
country to grow rapidly as it closes the gaps in technology levels and international incomes
and then converges to the common world growth rate
in the long run. Similarly, we can
expect there is a large share of DFI with high rates of return at first, then the share of DFI
falls and finally reaches
a
long run determined level. This conclusion rests on the assumption
about the role of DFI as an agent for the transfer of world technology to the host country.
During the catch-up process, DFI has its role in closing the initial gap in technology levels
while in the long run steady state DFI exits to close the gap in technology growth rates.
t49
Chapter
7z
CONLUSION
150
This thesis studied the interrelationships between technology transfer, foreign investment and
human capital
in the growth context of small open economies, a topic not adequately
addressed in previous literature.
A starting point was an open economy version of the Solow-
Swan (1956) model. This model applied in an open economy context predicts that
if
small
open economies share the same production technology then under perfect capital mobility,
the output levels produced in those countries are the same and fixed by the world interest
rate. In other words, regardless of original rich or poor, perfect capital mobility
will allow
those small open economies to produce the same output levels. In such a model, a small open
economy jumps immediately to its steady state position and there is no transition. Differences
in the saving rates cannot change the stock of capital employed and thus output produced in
the country though they can change the wealth level.
The empirical study of Mankiw, Romer and Weil (1992) finds that the output of each open
economy is a function of its saving rate. Beside, Barro, Mankiw and Sala-i -Matin (1995)
argue that empirical evidence shows convergence in open economies. In Chapter 3 we used
the extended Solow-Swan model with human capital and studied it in a small open economy
context. We found that in the absence of technological progress and under perfect physical
capital mobility, the steady state output of a small opàn economy is not only determined by
the world interest rate but also the saving rates of the country. A higher level of output exists
in
a country
with higher saving rates in physical capital and human capital. Thus the output
level of the economy can be influenced by policies that change its saving behaviour. In
addition, our model shows that there is a transition for a small open economy towards its
steady state. These rationalise the empirical results.
151
For many countries, the openness to the rest of the world allows them to have access to world
technology which has great effects on their economic growth performances. In Chapter 4 we
introduced international technology adoption into the extended Solow-Swan open economy
model of Chapter 3. We assumed that while a small economy has free access to world
technology, how much
of the world technology can be absorbed to the country
depends
critically on the country's level of human capital. Physical capital is perfectly mobile but
human capital is not. The enrichment of the model gains significant results. This model is
able to produce endogenous growth and thus can explain growth differences among small
open economies. Countries with higher saving rates in physical capital and human capital
enjoy higher growth rates and income levels. However, increases in the saving rate in human
capital have relatively larger effects on raising the growth rate while changes in the saving
rate in physical capital have insignificant effects on the growth rate.
In relation to the speed of convergence, this model predicts that an increase in the saving rate
in human capital clearly raises the speed of convergence. The speed of convergence is almost
unaffected by changes in the saving rate of physical capital. In this model, human capital is
an internationally immobile factor which plays an important role in explaining differences in
economic growth rates and the speed of convergence.
In reality, countries cannot usually acquire foreign technology at no cost. Among different
ways to adopt foreign technology, foreign investment can act as a channel for technology
transfer. In Chapter 5, we studied the economic growth of a small open economy which hosts
foreign investment. Technology transfer is assumed to depend on the stock of foreign owned
physical capital and also the technology absorptive capacity of the host country, which is
measured
by the country's stock of human capital. The model was set up in an optimisation
r52
problem where the country can choose the optimal level of foreign borrowing, physical and
human capital accumulations and thus technology transfer'
We found that the exogenous interest rate paid to foreign capital has no influence on the
steady state growth rates of the centralised and decentralised economy, though this interest
rate determines the extent of ownership in physical capital in the country. This model can
explain that different small open economies of the same type can obtain quite different
growth rates. Due to the existence of the externality in human capital choice, the market
growth rate is less than the optimal growth rate which calls for the intervention of the
government. In order to achieve the optimal growth rate, the government needs to subsidise
learning activities to raise the speed of the human capital accumulation. Finally, we learnt
that we do not need a production externality to have the market growth rate to be lower than
the optimal growth rate as in Romer (1986) or Lucas (1988). The human capital externality
choice in the model can give this result.
The role of direct foreign investment in technology transfer and its impacts on econonuc
growth of the host country is the subject of Chapter 6. In difference to Chapter 4 and 5 where
we assumed that a small economy must totally depend on foreign technology for its
technological change, in Chapter 6 we stressed the idea that direct foreign investment acts
a growth enhancing factor but is not the solely growth determining factor.
In
as
the model we
explained that the incentive for foreign investors to undertake direct investment comes from
the technological gap that exists between the host country and the rest of the world. Direct
foreign investment is motivated by reaping profit overseas which is made possible by the
employment of superior technology and management expertise. The results showed that there
is scope for mutual benefits to both foreign firms and the host country. Foreign firms
can
153
enjoy a higher rate of return to their investment than the world interest rate while the growth
rate of the host country can be higher than it would be without direct foreign investment. This
model suggests that there are periods in which developing countries grow at faster rates than
the average world growth rate. We can expect the country grows rapidly as it closes the gaps
in technology and international incomes and then converges to the conìmon world growth
rate in the long run.
Our main contribution to the study of economic growth theory is that we consider the
interrelationships between foreign investment, technology transfer and human capital
accumulation in modelling economic growth to explain the growth performances of small
open economies. In the study, we cover the issues in convergence, the role of education,
government policy and the impact of foreign investment on the economic growth rate.
The major limitations of the study are the ignorance of risk. In the models, the assumptions of
certain returns to assets (physical capital) and human capital are made. Since certain returns
to physical capital can be reasonably obtained by hedging against risk, this is hard to be done
with human capital because of moral hazard.
In our study, technology and human capital are assumed to be complement in the sense that
an increase in technology raises the marginal productivity of human capital and thus the
returns to human capital. Higher returns to human capital induce people to invest more in
human capital which raises the rate of human capital accumulation. A higher stock of human
capital makes it possible for the economy to increase its rate of technological progress since
people have adequate skills and abilities to handle new technology. Thus, technological
changes have favourable effects on human capital accumulation and thus the growth rate of
154
the economy. However, we ignored the negative effect of technological changes on human
capital. New technology may render existing skills irrelevant. This negative effect may
discourage people to invest in human capital today because they foresee that their skills will
be irrelevant sometime in the future when technology changes. In another way, a country
may be reluctant to acquire foreign new technology since
it is costly to undergo the labour
training process. As a result a lower economic growth is realised.
Other non-economic factors such as political risk are also ignored in our study. In reality
political risk can influence the flows of capital. Foreign investment may not flow into
countries with unstable political policies, wars for example, even though high retums to
capital existing in those countries because foreign investors fear of investment loss in the
countries.
A straight extension for future studies is the inclusion of uncertain returns to physical capital
and/or human capital. Uncertainty
will
add complexity to the models however.
In a different
approach to study the phases of economic growth in a small open economy, we may abandon
the assumption of perfect capital mobility by assuming that the interest rate faced by the
small open economy is a function of its ratio of foreign debt to capital employed. Finally,
further research should aim to expand the study on the effects of foreign investment on
economic growth of the host country into the case where technology transfer via foreign
investment has both positive and negative effects on the host country's human capital
accumulation.
155
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