

Department of Social Systems and Management
Discussion Paper Series

No.1298

Does International Knowledge Spillover Always Lead toa Positive Trickle Down?

byShiro KUWAHARA

July 2012

UNIVERSITY OF TSUKUBA
Tsukuba, Ibaraki 305-8573
JAPAN

Does international knowledge spillover always
lead to a positive trickle down?∗
Shiro Kuwahara †August 16, 2011

Abstract
This paper demonstrates the negative effects of positive international
knowledge spillovers on economic growth. In other words, we obtain thepossibility that educational investment for human capital is crowded out under global economic growth. To this end, we assume the phenomenon ofinternational knowledge spillover, effects of population growth on humancapital accumulation, and non-unity intertemporal elasticity of substitution
in an endogenous growth model along the lines developed by Arnold. Thismodel comprises R&D activities along the lines proposed by Jones and human capital accumulation along the lines proposed by Uzawa and Lucas.
The results show that even if international spillover increases, low-growthtraps without human capital investment emerge in some cases, for example,
an economy with a large intertemporal elasticity of substitution and a highpopulation growth rate.

Keywords : R&D-based growth; human capital accumulation; international
knowledge spillover
JEL Classification

O11, O34, O41∗ The author thanks Akira Momota and Katsunori Yamada for their useful comments, as well as

an anonymous referee for contributing valuable insights. This study is also financially supportedby the Japan Society for the Promotion of Science (Wakate B).
† University of Tsukuba, 1-1-1 Tennodai Tsukuba, Ibaraki, 305-8573 Japan, Tel.:+29-8536206, Fax: +29-853-6206, E-mail: kuwahara@sk.tsukuba.ac.jp1

1

Introduction

Economists have widely acknowledged the trickle-down effect, that is, positiveeffects proceeding from developed countries to neighboring developing countries.
The development of endogenous growth theory (Lucas 1988, Romer 1990, Grossman and Helpman 1991, Aghion and Howitt 1992) stimulated many empiricaland theoretical studies on international knowledge spillovers. For example, Coeand Helpman (1995) empirically show a positive R&D spillover among developed countries (OECD countries and Israel), and Coe, Grossman, and Hoffmaister (1997) obtain a similar result between north and south countries. Theoretical works on these positive effects include Aghion and Bolton (1997), an R&Dbased growth model with the trickle-down effect, and Holod and Reed (2009), an
Uzawa-Lucas type growth model with regional externality.
However, such spillovers do not always have positive effects. In reality, someunderdeveloped countries are caught in poverty traps in the modern economic
growth process, which is characterized by integration with the global economy.
This has been substantiated by some empirical works (see Easterly 1994; Quah
1996, 1997) that report the polarization of the world economy into the rich andthe poor. Furthermore, in this growth process, the school enrollment rate of someunderdeveloped countries has decreased despite (positive) economic growth. For
instance, a decline of school enrollment rates of 19 countries1 can be observedin the data set created by Momota (2009). This is despite the fact that the GDP
growth rates of 6 countries in this group (Grenada, Myanmar, the Republic ofEquatorial Guinea, the Federation of St. Kitts and Nevis, Bahrain, and the Islamic
Republic of Iran) are above the world average.
Against this background, this paper aims to theoretically demonstrate the possibility of negative effects of international spillovers by using an endogenous
growth model with R&D activities through international knowledge spillover andhuman capital accumulation through education. This paper basically follows
the endogenous growth model proposed by Arnold (1998), which comprises two
growth engines: R&D activities and human capital accumulation. The former
is executed by using the Jones-type innovation function (Jones 1995a, 1995b),that is, a function with decreasing returns on existing knowledge and research
1 These countries are the Republic of Congo, Solomon Islands, the United Arab Emirates,

Grenada, the Republic of Kenya, Myanmar, the Republic of Equatorial Guinea, Georgia, Central African Republic, Gabon, Mongolia, the Kyrgyz Republic, the Federation of St. Kitts andNevis, Jamaica, Latvia, the Republic of Azerbaijan, Estonia, Bahrain, and the Islamic Republic of
Iran. These countries have a decline of over 3% in their net primary school enrollment rates, andthe figure of 3% was stipulated in order to eliminate statistical errors. Hence, under the simple
criterion of a decline of more than 0% in the net primary school enrollment rate, more countries
and regions would fall under the above category.

2

input. The latter is executed through the linear human capital accumulation function (Uzawa 1965, Lucas 1988), which we term the Uzawa-type human capital
accumulation function.
The Arnold model was developed on the basis of the following discussions.
When Romer (1990) developed an early-stage endogenous growth model, he considered a linear relationship between R&D success and R&D inputs, which heassumed as human resources and knowledge respectively. This linear relationshipcaptures the non-decreasing returns of knowledge but has theoretical2 and empirical defects. The original model implies that a country with a large population
exhibits a higher growth rate, but this implication is not supported by empiricalevidence. This property of the early endogenous growth models is called the scaleeffect of population. With regard to this defect, Jones (1995a, 1995b) empiricallyshowed that knowledge creation functions have the property of decreasing returns,
and proposed the Cobb-Douglas type R&D function as a desirable form of an
R&D function (i.e., Jones technology). In Jones’s model, inputs comprises labor(which grows at an exogenously given population growth rate) and knowledge.
Jones demonstrated that this arrangement can link population growth and endogenous technological progress, thereby satisfying empirical results. However, Jones
technology immediately leads to another defect; that is, long-run growth is related
to the exogenous population growth rate, which is termed as semi-endogenous
property and is a rebound to the Solow model (1957) in a sense. Arnold (1998)
avoids this defect by assuming that the human resource input to R&D is humancapital, which is endogenously accumulated through the Uzawa-type human capital accumulation function. In this structure, Arnold (1998) introduces international spillover by considering the knowledge input in the R&D function as thesum of domestic and foreign knowledge accumulation, and considers a small open
economy with regard to knowledge. In order words, the domestic knowledge accumulation is so relatively scattered that the dynamics of knowledge input in R&D
can be considered to obey the dynamics of foreign knowledge.
Thus, there are two primary sources of long-run growth in the present model:
foreign knowledge growth and domestic human capital accumulation. This structure immediately yields the following two points: First, the long-run growth of thiseconomy is partially exogenous; it depends on exogenously given (foreign) technological progress like in the Solow (1957) model. However, because our model
explicitly contains (domestic) R&D and human capital investment, incorporating
microfundation into accumulation activities enables us to analyze the mechanismof the (developing) country’s endogenous selection of the regime with or without education as a reaction of (exogenous) knowledge spillover. Thus, the model
2 Theoretically, this model cannot contain incessant population growth at its original formationin the steady state.

3

can be said to include the endogenous determination of long-run growth at certain level. Second, because incorporating Jones technology into the Romer model
makes the human resource input of R&D the critical growth engine, the main
growth engine of present study is the human capital accumulated by the UzawaLucas technology.
However, the present model differs from the basic model of Arnold (1998)(i.e., the closed economy model) with respect to three added factors: international
knowledge spillover, the effects of population growth on human capital accumulation, and non-unity intertemporal elasticity of substitution (considered as Assumptions A, H, and U, respectively, in the next section). We derive the results asfollows: First, we introduce international knowledge spillover in the same manner as introduced in the extension of Arnold (1998), where the world knowledge
stock is the sum of the knowledge of the world’s countries, and the home country
is small; therefore, the contribution of this country’s accumulated knowledge tothe stock of world knowledge is negligible in the long run. Thus, this arrangement is appropriate for analyzing the dynamics of a developing country. Second,
our model explicitly captures the effects of population growth on human capital
accumulation. With regard to Jones-type R&D technology, the positive growthof input factors is necessary for a long-run steady growth path (SGP). In Arnold(1998), non-educational investment cannot be a steady-state equilibrium, whereas
the population growth effects in our model facilitate the analysis of the steadystate with non-educational investment. Third, we assume non-unity intertemporalelasticity of substitution in our model. Consequently, the human capital accumulation rate and GDP growth rate are determined simultaneously and not separately3 .
Further, the increase in knowledge spillover changes the GDP and human capital accumulation rates. Under some parameter conditions with low intertemporalelasticity of substitution, the growth regime shifts from one with positive educational investment to one with no educational investment. In some cases, likecountries with a sufficiently high population growth, such an increment decreases
the GDP growth rate.
Further, as the present study shares its concerns and critical growth engine
with Holod and Reed (2009), this work can be considered an important preceding
research. The main differences between the present study and Holod and Reed(2009) are (i) the latter analyze the spillover as an externality of the production
function but model domestic knowledge spillover and foreign spillover separately,
whereas the present study explicitly introduces an R&D structure, and as statedabove, the scale of domestic knowledge spillover is assumed to be negligible. (ii)
The main result of Holod and Reed (2009) is the positive effects of spillovers;
however, this result does not conflict with the present one as the aims of the two
3 See the Appendix A for an explanation of these terms.

4studies differ. Holod and Reed (2009) focus on positive regional externality, so theinner solution case is implicitly presumed, whereas the present study focuses on
the possibility of negative spillover and thereby elaborates on the corner solution.
The rest of this paper is organized as follows. The model is constructed in Section 2. Steady states are derived in Section 3. Some growth patterns are analyzed
in Section 4. Finally, the conclusion is presented in Section 5.

2

The Model

Adopting the production structure followed in Romer (1990), this paper studies
three sectors, final goods, intermediate goods, and R&D, and demonstrates therole of two types of knowledge capital: patents and human capital. Final goods
are produced by employing human capital in the final goods sector (denoted as
HY ) and by clusters of intermediate goods. New intermediate goods are developed in the R&D sector and protected by permanent patents, by which the firmthat developed the intermediate goods obtains monopoly profits. We can regard
the cluster of patents as (domestic) knowledge accumulation, specifically, written
or formal (domestic) knowledge, or simply knowledge. We denote this knowledge
as A. R&D is carried out by employing human capital in the R&D sector (denoted
as HA ) and by using existing knowledge A. The utilization of knowledge–that is,the developed blueprints of intermediate goods–is prohibited only with regard to
the production of the intermediate good. Thus, the R&D firm can freely use thisknowledge in the process of R&D activities, whereas this firm must procure human capital in the labor market. Further, human capital can be used for education,that is, investment to create new human capital, which is denoted as HH . Thus,
under this setup, human capital is an excludable asset held by the household. By
representing the aggregate human capital as H, we impose the resource constraint
for human capital H = HY + HH + HA .
In the R&D process, we introduce international knowledge spillover. Following the extension of Arnold (1998), the input of freely used knowledge in the R&D
function is specified not by domestic knowledge accumulation, but by worldwide
knowledge accumulation; that is, we assume an open economy with international
knowledge accumulation. For simplicity, we omit any international transaction of
goods or immigration. Thus, all economies are linked with the world economy
only by knowledge. An economy can use the entire world knowledge stock, andthe knowledge created by that economy becomes a part of this stock. Furthermore, we assume that the country in our analyses is sufficiently small; therefore,
this country’s contribution to the world knowledge stock is also sufficiently small.
This arrangement implies the following two points. First, since we are concernedwith the dynamics of developing countries, we assume the case of a small country
5with regard to knowledge accumulation. Second, as we are also interested in therole of knowledge spillover, we have omitted other international factors for simplification. We assume that time is continuous, and the final good is used as thenuméraire.

2.1

Production

The structure of final goods is arranged as follows: These goods are used for
consumption or as inputs in the production of intermediate goods. Final goods areproduced by employing human capital, and from a cluster of intermediate goods.
In this economy, the scale of the cluster, that is, the variety of the cluster indexedby A, can be regarded as knowledge. Thus, the production function of the final
goods sector is
Y = HY1−! X,

0 < ! < 1,(1)

where Y and X denote the final goods product and intermediate goods input respectively4 . The last is defined byX :=! A(t)
0x( j)! d j,

where A and x( j) denotes number of varieties and j’s intermediate goods input
respectively.
The final goods sector is assumed to be perfectly competitive; therefore, firms
operate under the equating marginal cost and factor price, as shown below:

"Y
= w,
" HY

and

"Y
= p( j)," x( j)(2)where w and p( j) denote the wage rate of human capital and the price of theintermediate goods of sector j, respectively.
Intermediate goods are used only in the production of final goods, and one unitof the intermediate good is assumed to be produced by # (> 1) units of the final
good. The profit of an intermediate firm with index j is given by$ ( j) = p( j)xt (p( j)) − # xt (p( j)),(3)

where p( j) is the price of j’s sector intermediate goods. The intermediate goods
are assumed to be monopolized; that is, the firms set the price of these goods
4 The time index is omitted to ease the burden of notation until we consider dynamics in thelatter part of this paper.

6

for profit maximization. By solving the optimization problem of the intermediate
goods firm given in (3), with the optimal conditions in the final goods sector (2),the following conditions are obtained:"!2
x( j) =#

# 1−1!

HY ,#p( j) = ,!

$ ( j) =$%1− 1 # xi .
!

(4)

From (1), x( j) in (4) and the definition of X, we obtain
"

!2
Y=
#

# 1−!!

AHY .

(5)Since the final goods are consumed as either intermediate goods or consumption
goods, (4) and (5) yield
"
#
! A
!2
C =Y −
x( j)d j = 1 −Y (t).
(6)#0
Since ! 2 < 1 < # , a positive consumption is ensured. The profit of the intermediate goods firm can be obtained asY$ = ! (1 − ! ) .
A

2.2

(7)

R&D Activities and International Spillovers

R&D is established as the process of creating new varieties, and the term of thepatents of a new variety is assumed to be permanent. To eliminate the scale effects, we assume Jones technology, so that the creation of a new variety exhibits
diminishing returns to scale. Furthermore, we assume that the stock of knowledge
is the world knowledge stock, which is given for this (small) country. Thus, theknowledge function is explained as follows.
Assumption A

Assume the following R&D function:&Ȧ = BA% HA ,& ∈ (0, 1),% ∈ [0, 1),(8)where Ż ≡ dZ
dt is the time derivative of the variable Z, and A is the world knowledge stock. We assume A ≡ '( ∈S A( , where S denotes the set of all countries
in the world. Thus, the ith country’s knowledge stock A affects the dynamics of
A; however, since we assume that the analyzed country is small, the contributionof the knowledge accumulation of the country to the world knowledge stock A is
negligible.
7

Free entry into R&D equates the aggregate cost and profit; this provides thefollowing equations:vȦ = wHA ,(9)
therefore,

1−&

H
v = A % w,
(10)BA
where the valueof R&D equals the present value of perpetual monopoly prof& ) − & t r(s)ds
its, v ≡ 0 e 0$ (t)dt. The perfect mobility of human capital between final
goods production and R&D sectors equates the wage rates of human capital between these two sectors. We denote this common wage rate as w.
The term of a patent for a newly created variety is assumed to be permanent.
Hence, the value of R&D (denoted as&v) can& be designated as the present value) − t* r(s)ds M
of perpetual monopoly profits: v(t) ≡ t e$ (* )d * . Differentiating thisequation with respect to time yields the well-known no-arbitrage condition, which
is given as(11)r(t)v(t) = $ (t) + v̇(t).

2.3 Dynamic Optimization of the Household and Human Capital Accumulation
In the model, the population is denoted by N, and grows at a positive constantexogenous rate n > 0 (specifically, Ṅ(t) = nN(t)). The scale of the representative
household is normalized to a unit of the population. Population growth dilutes therepresentative household’s asset holdings, which includes financial assets (equity
issued by firms that lead to innovation in new intermediate goods) and humanassets (human capital accumulated by conscious educational investment for theselection between education and work).
My study follows the model of Arnold (1999), wherein one knife-edge assumption is eliminated; that is, the Romer-type linear R&D function is replacedby the Cobb-Douglas-type non-linear one. However, Arnold (1999) introduced
another knife-edge condition, which is the log-linear utility function. We loosen
this linearity, and in the latter part of this paper, this non-log-linear utility function
is shown to be necessary to yield the main results of the model.
Assumption U! )0

The utility of the representative household is given byc(t)1−+ − 1 −, t
e dt,
1−+

+>

(
& '
≡ + ; + ∈ (0, 1) ,
1+&

(12)

where c, , , and + represent the per capita consumption, subjective discount rate,and inverse of the elasticity of intertemporal substitution, respectively. Despite
8

loosening the log-linearity assumption, we still have to impose the domain of theinverse of the elasticity of intertemporal substitution, + ≥ + . This assumption implies that the intertemporal substitution is not very high and simplifies our analysis
in the latter part of this paper.
The budget constraint in the per capita form is given by,'(ȧ(t) = r(t)a(t) + w(t) uY (t) + uA (t) h(t) − c(t) − n a(t),(13)

where a, uY , uA , and h represent the per capita asset holding, allocation rate ofhuman capital to final goods production, allocation rate of human capital to R&D
activities, and per capita human capital, respectively. Note that uY = HY /H anduA = HA /H hold under the assumption of the representative agent. Further, notethat because (uY + uA )h is the rate of working human capital, the residual humancapital is (1 − uY − uA )h, which represents the portion of human capital that isbeing educated in order to increase human capital. Thus, the dynamics of humancapital of the household is assumed as follows:
Assumption H'(ḣ(t) = b 1−uY (t)−uA (t) h(t)−n h(t)+ - n h(t),b > 0,

0<- <1(≡ - ),
1+&

where b denotes the efficiency of education, and - captures the factor of population dynamics described below.
We assume that human capital is accumulated by human capital investment,
and the increment is linear for the investment (b(1 − uY − uA )h). We term thisas the Uzawa-type human capital accumulation function in this paper, as suchhuman capital was described by Uzawa (1965) and Lucas (1988). Furthermore,since we clearly distinguish between patented and embodied knowledge and ourmodel explicitly incorporates population growth, we can capture the propertiesof the knowledge possessed by human beings in our human capital dynamics.
The effects of population growth are assumed to cause two factors that influence
accumulation. First, because human capital is excludable, per capita value inthe household decreases with population growth (this is reflected by −n h). Thisdecrease is similar to that of (physical) capital holdings in the usual Ramsey modelwith population growth. Second, we introduce entry and retirement in the workforce. We assume that each agent enters the human capital (or labor) market witha constant endowment of human capital. We assume that this human capital isalso depreciated by factors such as aging or mortality and therefore also linearly
depends on the population growth rate. Thus, we can denote these factors as - n hin the dynamic equation. The restriction 0 < - < - (< 1) implies that human9capital accumulation is promoted by exogenous population growth, but that theeffects of such growth are not too large. Section 4 shows that the results of thegrowth rate for this assumption are consistent with those of empirical studies.
Aggregating h in the whole economy, we obtain the following equation:Ḣ(t) = bHH (t) + - n H(t),(14)

where H and HH are the aggregated stock of human capital and the educational
input of human capital respectively.
From the objective function and constraints, we obtain the optimal conditions
as follows: With regard to consumption, the usual Keynes-Ramsey rule is given
as
$
%
ċ(t)
Ẏ (t)+=+− n = r(t) − , − n,
(15)
c(t)
Y (t)
where we use (6) and C = cN.
From the optimal condition of human capital allocation, we obtain. (t)w(t) ≥ µ (t)b,with equality wheneveruH (t) > 0.

(16)

where . , w, and µ denote the shadow price of capital, wage rate of human capital,and shadow price of human capital, respectively.
The dynamic equation for µ is expressed as follows:,−µ̇ (t) . (t)w(t)=− b(1 − uY (t) − uA (t)) − (1 − - )n
µ (t)
µ (t)

(17)

If equality does not exist in condition (16), human capital investment is nil (i.e.,
uH = 0), and the conditions are summarized as follows:b < µ̃ (t)(18)*µ̃˙ (t) .̇ (t) ẇ(t) )=+− , − µ̃ (t) + (1 − - )n ,µ̃ (t) . (t) w(t)(19)

where µ̃ ≡ .µw .
Finally, the transversality conditions (TVC) are given as follows:
lim e−, t . (t)a(t) = 0,

t→)

and

10

lim e−, t µ (t)h(t) = 0.

t→)

(20)

3

Dynamics and the Steady States

In this section, we investigate the dynamics and steady states of the economy. We
denote the growth rate of the variable z by gz (namely, gz ≡ ż/z) and value z in
the SGP by z∗ . The model contains two growth engines–knowledge accumulation (R&D) and human capital accumulation–and we observe two types of steadystates: the Arnold regime (positive R&D and positive human capital accumulation) and the Jones regime (positive R&D and no human capital accumulation).
We obtain the following common equations among these regimes: (5) and (6)imply
gY (t) = gC (t) = gHY (t) + gA (t).

(21)

Using (15) and (21), we explain the growth rates as follows.

+ gc (t) = + (gY (t) − n) = + (gHY (t) + gA (t) − n) = r(t) − , − n.

(22)

In the case of positive human capital investment, uH > 0 holds. Therefore, uA +uY < 1. We can thus eliminate µ and obtain the optimal condition as a dynamic
equation about w as follows:gw (t) = r(t) − b − - n.

(23)Then, time differentiating (9), and uniting (11), we obtain the following dynamics:
gv (t) = gw (t) + (1 − & )gHA (t) − % gA (t) = r(t) −

$ (t)
.
v(t)

(24)

We can show that the steady state of the system is saddle stable (see AppendixB.1). Thus, in the main text, we focus on the steady state analysis.

3.1

Two Types of Long-run SGPs

In this section, we limit our attention to the case of the SGP and derive somefeatures of the model. The present model has two SGPs: one for the internal
solution case about human capital accumulation (with HH > 0) and the other forthe corner solution case about the same (with HH = 0).
First, the human capital of each allocation must grow at a constant rate(s);HH > 0 yields g∗HA = g∗HY = g∗HH = g∗H and hH = 0 yields g∗HA = g∗HY = g∗H = - n
and g∗HH = 0. From (8) and gHA = gH , the following condition is necessary for thesteady growth equilibrium:g∗A − % g∗A = & g∗HA .
11

(25)

Note here that Jones technology under international knowledge spillover affects
gA through the efficiency parameters (% and & ) and international knowledge growthrate, but by the level parameter a.
Further, (21) implies that in the steady state, the following equation holds:g∗ ≡ gC∗ = gY∗ = g∗H + g∗A .

(26)

By combining (25) and (26), we obtain the condition relating the growth rate ofhuman capital to the growth rate of GDP asg∗ = (1 + & )g∗H + % g∗A .

3.2

(27)

The Arnold Regime (A-Regime)

Since the Arnold (1998) model contains human capital accumulation derived by
the Uzawa-type educational function and R&D activities driven by Jones technology, we term the growth regime with these factors as the Arnold regime (Aregime).
When the condition hH > 0 holds, w = (1 − ! ) HYY and (23) make the following
equation hold in the steady state:AAAg∗ − g∗H = r∗ − b − - n,(28)

where ∗A indexes the steady state in the A-regime.
AA(5) and w = (1 − ! ) HYY imply gw = gA , and (22) implies r∗ = + (g∗ − n) −, − n. Substituting these two properties into (28) yields
'
(
,
A
1+
g∗H =
−(1 − + )n + b + - n − , + (1 − + )% g∗A .
(29)/where / ≡ 1 − (1 − + )(1 + & ). From (27) and (29), the growth rates of GDP and
per capita GDP are given as
"
#
'
(
1+&
% ∗
∗A
g =
−(1 − + )n + b + - n − , +
g ,
(30)
/
1+& A
'
(
,
A
1+
g∗y =
−n + (1 + & ) b + - n − , + % g∗A ,(30' )/
where y ≡ Y /N. Note that these can be equilibria if the conditions for the Aregime are satisfied.
Further, (30) to (29) show that each growth rate in the A-regime is a linear
combination of the factor of the population growth rate n, factor of contributionof the usual Uzawa type human capital accumulation b − , , and additive term - n,which is an added factor in the Uzawa-type human capital accumulation in Assumption H. Note that / > 0 under the assumption + > + , and that 1// functions
as a multiplier of growth factors such as n and b − , .
12

3.3

The Jones Regime (J-Regime)

The above discussion solves the steady state by assuming that h∗H > 0, which
implies gH > - n. However, this condition is not always satisfied. We term thegrowth regime with only R&D activities driven by Jones technology as the Jones
regime (J-regime).
Therefore, if the condition g∗H > - n is lacking, then the economy is stuck inthe equilibrium hH = 0; therefore, by uniting these properties and (28), we obtainthe steady-state human capital accumulation rate and knowledge growth rate inthis regime asJg∗H = - n,Jg∗A = & - n + % g∗A .

and

(31)

Therefore, under the condition HH = 0, combining (27) and (31) yields the aggregate and per capita GDP growth rates in this regime as follows:Jg∗ =(1 + & )- n + % g∗A ,

(32)

∗J

(32' )

gy =(1 + & )- n + % g∗A − n,(32) shows that the per capita GDP growth rate exhibits a semi-endogenous growthproperty: the growth rate is pinned down to the population growth rate and exogenous world knowledge growth rate.

4

Regime Determination

We derive regime determination in this section. Empirical studies on the relationship between population growth and the per capita GDP growth rate, such as
Kelley (1988), Kelley and Schmidt (1995), and Ahituv (2001), report a (weak)negative correlation between per capita income growth and population growth.
Thus, we assume here that " g∗y /" n < 0. This assumption and the steady-state percapita GDP growth rate g∗y yield the following condition:0 ≡ 1 − (1 + & )- > 0,

equivalently - <1(= - ).
1+&

Therefore, under the assumption that population growth does not have very large
effects on human capital accumulation, our model satisfies the property " g∗y /" n <0.
Since the A-regime is a case of an inner solution about human capital investment, we obtain the feasibility condition on human capital accumulation. (14)yields g∗H ∈ [- n, b + - n]. Thus, the lower bound condition gH > - n provides'(
Ag∗H > - n ⇔ b > , + (+ − 1) % g∗A − 0 n ,(33)13and the upper bound condition gH < b + - n provides
$
.
%
,
1
<
∗+
∗
gH < b + - n ⇔ b+ 0 n − % gA ,> 1+& 1−+
1>+ >+
for
.
+ >1

(34)

The correspondence between the TVC and (34) can be easily verified. Thus, forAg∗H to be the steady-state value, both (33) and (34) must hold.
For the J-regime to be in a steady state, (18) and (19) provide the following
conditions:Jb < µ̃ ∗

)

∗J

*

(35)

g. ∗J + gw∗J − , − µ̃ − (1 − - )n = 0,(36)Jwhere ∗J represents the index of the value in the J-regime. Substituting gY∗ andJg∗H derived in (32)-(31), and g. = −+ (gY − n) and gw = gY − gH into the above
conditions, we haveJ-regime⇔b < , + (+ − 1)(% g∗A − 0 n)

(37)

Thus, the conditions (33) and (37) imply that the equationb = , + (+ − 1)(% g∗A − 0 n)(≡ LB(+ ; g∗A ))divides the parameter domain into the A-regime and J-regime. ”LB” stems fromthe lower bound; that is, this line marks the lower bound of positive human capital
investment.
The domain of a regime and the regime switch caused by change of international spillover are depicted in Figure 1. A spillover has two types of effects–a∗∗∗higher growth rate in world knowledge g∗∗
A ) and that of a lower one gA (< gA ).
The main dividing line is LB, which is shifted by the change in g∗A . From Figure
1, we can determine the domain where the higher g∗A makes a country shift fromthe A-regime to the J-regime. Thus, we obtain the following lemma:
Lemma I If + is larger than 1, that is, the intertemporal elasticity of substitution
is small, the increase in the growth rate of world knowledge spillover results insome countries in the A-regime falling under the J-regime.

14∗
We assume that the knowledge growth rate increases from g∗A to g∗∗
A (> gA ),which makes a country shift from the A-regime to the J-regime. Therefore, thesteady state changes from an A-regime with the world knowledge growth rate g∗A
to a J-regime with the world knowledge growth rate g∗∗
A . We denote the per capita∗ ) and g∗J (g∗∗ ) respectively.
GDP growth rates of these two steady states as g∗A
(g
y
y
A
A
∗
Then, we check the change in the per capita growth rate between g∗Ay (gA ) and∗∗g∗Jy (gA ). Note that this situation is + > 1 from Lemma I; therefore, / > 1 fromthe definition.
∗∗J ∗∗The difference of gA∗
y (gA ) and gy (gA ) is shown as follows:∗∗∗A ∗1gy ≡ g∗Jy (gA ) − gy (gA )1= % (/g∗∗− g∗ ) −(+ − 1)(1 + & )0 n − (1 + & )(b − , ) .
A
23
4 1
23
4
23 A4 1
/ 1
+

−

(38)
(39)

±

Therefore, for example, if the population growth rate is very high, the regimechange from A to J decreases in the growth rate of the country.
Lemma II If 1gy < 0, which is realized in the case that, for example, the powerof world knowledge stock in the R&D function (% ) is smaller, the population
growth rate is higher, and education efficiency is higher than the subjective discount rate5 , the increase in the growth rate of international knowledge stock generates the regime switch from A to J, and decreases the county’s per capita GDP
growth rate.
From the above two lemmas, the proposition of this study is stated as follows:
Proposition If a country has a large intertemporal elasticity of substitution anda high population growth rate, an increase in the world knowledge growth rate
might cause the county to fall into low-growth traps without educational investment.

5

Conclusion

This study develops an endogenous growth model that incorporates variety expansion, human capital accumulation, and international knowledge spillover. We
5 Note that b − , > 0 is the condition for positive long-run growth in the Uzawa-Lucas model(see footnote 6 in Appendix A). Furthermore, if this condition is not satisfied, extremely highpopulation growth can lead to the same result.

15

established a relationship between the determination of the growth phase of acountry and its population growth rate and world knowledge growth rate, when
we assumed the non-unity CRRA parameter (Assumption U), human capital accumulation affected by population growth (Assumption H), and R&D function
with international knowledge spillover (Assumption A). The main finding of thisstudy is that in some domains, the increase in international knowledge spillover
might negatively affect a developing country’s growth rate.
Because the present study focuses on the negative effects of (positive) international knowledge spillover on developing countries, some important factors
are simplified. First, we assume that the only international factor is knowledge
spillover, and hence, some important factors, for example, good translation (international trade) and immigration, are ignored. Second, the present model is anorth-south (i.e., a vertical) one, and we focus on the south. Consequently, thetechnological growth rate, a main source of long-run growth in this study, is eventually equivalent to exogenous technological progress. Endogenizing the technological progress of the north country makes the model describe the dynamics of
the world economy, and relaxing the assumption of a small country with regard toworld knowledge accumulation makes the model analyze the cross–relationshipbetween two countries; in other words, the model becomes a north-north (i.e.,horizontal) one. Research along these lines is left for the future.

A

Separately and simultaneously in the Uzawa-Lucasmodel

The following is a simple explanation of separately and simultaneously in the
Uzawa-Lucas model. Suppose a typical setup of the Uzawa-Lucas model; thenthe optimal conditions are written asg∗w =gY∗ − g∗H = r − b,
+ gC∗ =+ gY∗ = r − , ,where g∗Z , w, Y , H, r, b, + , C, and , denote the growth rate of variable Z in
the steady states, wage rate of human capital, output, human capital stock, interest rate, efficiency of education, constant relative risk averse (CRRA) parameter,
consumption, and subjective discount rate, respectively. Therefore, a steady stateis given byg∗H = b − , + (1 − + )gY∗ .

(∗)

The equilibrium values of gY and gH are determined by combining this equation
and another relationship between gY and gH that is derived from final goods production.
16

The equation (∗) shows that in the case of the log-linear utility function (namely,+ = 1), gH is determined by the difference of b and , and not by gY . Further, inthe case of the non-log-linear utility function, it alternatively yields one conditionon the relationship between gY and gH . Alternatively, in the simple Uzawa-Lucasmodel, the condition g∗H = gY∗ is also derived in the steady state from the structureof the final goods production. In this case, the relationship between gY and gH isalso simply determined as6 gY∗ = g∗H = (1/+ )(b − , ) under the condition + )= 1.
Thus, in the case where + )= 1 and gY∗ )= g∗H , the determination of growth rates is
no longer simple, and at this point, we find some new implications.

B

Dynamic System and Steady States

The analysis is simplified by using variables that are constant in steady states,
we define new variables uY ≡ HY /H and uA ≡ HA /H. By using these notations,the human capital allocated to education sector can be written as HH = (1 − uY −uA )H.
From the discussions in Section 2, the model has two types of regimes: withand without education. In both cases, the economies follow the rules of common
dynamics (8), (24), (14), and (22). (8) and (14) are respectively noted as'(gA (t) = gA uA (t), 2 (t) = B(uA (t)2 (t))& ,(40)'(
gH (t) = gH uA (t), uH (t) = b(1 − uA (t) − uY (t)) + - n,(41)
where 2 ≡ H/(A1/& A−% /& ).
(21) and w in (2) yields$%
ẇ(t) Ẏ (t)u̇Y (t)Ȧ(t)=−+ gH (t) =w(t) Y (t)uY (t)A(t)Plugging (7), (10), and (42) into (24) yields
"
#
Ȧ(t)
u̇A (t)uY (t)
r(t) =+ (1 − & )+ gH (t) − % gA (t) + !
gA (t)A(t)uA (t)uA (t)(42)(43)

Differentiating 2 with respect to time, and substituting (41) and (40) into it
yields
"
#
'
( 1 '
( %
2̇ (t) = gH uY (t), uA (t) − gA uA (t), 2 (t) + gA (t) 2 (t)
(44)
&
&

6 From this equation, b − , > 0 is necessary for positive long-run growth in the Uzawa-Lucasmodel (see footnote 5 in Section 4).

17

B.1

The Case of the Arnold Regime

By adding (23) and (42), this case contains the following optimization condition:r(t) − b − - n =ẇ(t) Ȧ(t)=.
w(t) A(t)(45)

Eliminating r and w by uniting (43) and (45), we obtain the dynamics of uA in theinternal solution case:(uY (t) ''( b + - n + % gA (t) − ! uA (t) gA uA (t), 2 (t) u̇A (t) = −gH uY (t), uA (t) +u (t).

 A
1−&
(46)

Eliminating r and w by using (22) and (45), we haveu̇Y (t)Ẏ (t)= −gH (t) + b + - n − , + (1 − + ).
uY (t)Y (t)Substituting (21) into the above equation, and solving with respect to u̇Y , we havethe following dynamics of uY in the internal solution case asb−, -nu˙Y (t) =
+
(47)
+
+
.
(
'
( 1−+ '
gA 2 (t), uA (t) uY (t),−gH uY (t), uA (t) ++From (40) and (41), gA and gH are functions that depend on uA , uY , and 2 . Thus,
all three equations, (44), (46), and (47), are a function with variables uA , uY ,and 2 , and the dynamics of the system are completely represented by these three
dynamics.
The linearized system of q, ( , 2 , uY , and uA around a steady state is written as

B∗A
B∗A
B∗A 
" 2̇ B" 2̇ B" 2̇ B  "2 B∗A" uA B" uY B
2
−
2
2̇

B∗A
B∗A
B∗A 
A 

" u̇A B" u̇A B u˙A  = 
 " u̇A BB
  uA − u∗A  ," uA B" uY B
 "2

B∗A
B∗A
B∗A  uY − u∗Au˙YY" u̇Y B" u̇Y B" u̇Y B" uA B" uY B"2 B
where, by using

BA
A
g∗A
" gA BB∗= & ∗A ,
"2 B
2

BA
A
g∗A
" gA BB∗= & ∗A ," uA B
uA
18

A

g∗H =

1 ∗A % ∗g − gA ,& A&

Aand b(u∗A + uY∗ ) = b + n − g∗H , we can derive the following:

where

B∗A
B∗A
D
E
B
∗A
g
A
A
" 2̇ BB"2̇
B
B = −g∗A ,B = − b + A∗A 2 ∗ ,"2 B" uA B
uA
B∗A
BA
BA
A
" 2̇ BB" u̇A BB∗" u̇A BB∗∗A
∗A
= 31 u A ,= 32 u∗A ,B = −b 2 ,B
B" uY B
"2
" uA
BA
BA
A
" u̇A BB∗" u̇Y BB∗
1 − + g∗A ∗A∗A= 33 u A ,& ∗A uY ,=" uY B
"2 B
+
2
G
B∗A F
BA
A
A
" u̇Y BB" u̇Y BB∗
1 − + g∗A∗A
= b+
& ∗A uY ,= b uY∗ ,
B
B
" uA
+
" uY
uA
A

A

!& uY∗ g∗A
31 = −,
1 − & u∗AA 2 ∗AAA

32 = b + ! H

uY∗

A

u∗A

! g∗A
33 = b −.
1 − & u∗AA∗A
I2 gA ,

The characteristic equation is assumed to be given as
AAA

4(5 ) = −5 3 + Tr∗ 5 2 − BJ ∗ 5 + Det ∗ .
From (46), we obtainu∗! Y∗ g∗A = −(1 − & )g∗H + b + - n + % g∗A
uA

(48)

Considering this and (25), we can prove that the trace Tr∗+ is positive as follows:
Tr

∗A

B∗A
BA
BA
" 2̇ BB" u̇A BB∗" u̇Y BB∗
=
+
B +
"2 B
" uA B" uY BA= b(u∗A + uY∗ ) + b + - n − g∗H > 0,23
4
1 23 4 1>0>0A

where b + - n − g∗H > 0 is obtained from the property that the upper bound of gH
is b + - n.

19

The determinant Det ∗A is derived as follows:
F
G
∗A
∗A
g
A
A
A
A
∗
∗
∗
∗
∗A
∗ A 1 − + gA ∗ A
A
Det = −gA 32 uA b uY − b + ∗A 2 33 uA& ∗A uY
+
2
uA
F
G
F
G
A
∗
∗A
g
g
A
A
A
A
1
−
+
1
−
+
& A∗A + g∗A 33 uY∗ b +& A∗A+b 2 ∗ 31 u∗A b +
+
+
uA
uA
F
G
A
A
g∗
g∗ A
A
A
A
A
A 1−+
+ b + A∗A 2 ∗ 31 u∗A buY∗ + b2 ∗ 32 u∗A& A∗A uY∗
+
2
uA
#
"
A
A
A
1−+
=b u∗A uY∗ g∗A 1 −& (31 − 32 + 33 ) .
+From 31 , 32 , and 33 , it is easily derived as
DEAA! uY∗ ∗A 1 − &1& uY∗
31 − 32 + 33 = −g− ∗A ! ∗A < 0.
A + ∗A
1 − & u∗AA A
2
u∗A
uY
uA

(49)

(50)

Uniting + > 1+& & , which is equivalent to 1 − 1−+ + & > 0, and (50), we obtain theAAA

result of Det ∗ < 0. From the results of Tr∗ > 0 and Det ∗ < 0, we can conclude
that the currently studied system is saddle stable under the assumption of / > 0.

B.2

The Case of the Jones Regime

As we are primarily interested in the case of a negative trickle-down effect, we
assume + > 1 in this section.
This regime imposes the following condition:uA (t) + uY (t) = 1.

(51)(51) makes (14)Ḣ(t) = - nH(t),J

namely gH (t) = - n = g∗H(52)(52) states that the rate of human capital accumulation in this case is constant. The
value of the corner-solution steady state is denoted by ∗J .
From (40), (51), (52) and the definition of 2 , the dynamics of 2 are obtained
as
"
#
1
%
2̇ (t) = - n − gA (t) + gA (t) 2 (t)
(53)
&
&
20

(15) and (32) give)*r = , + n + + (1 + & )- n + % gA − n .

(54)

Substituting (40), (51), (52) and (54) into (43), we have the dynamics of uA as
follows:
H
I


A (t)
 34 + (1 + + )% gA (t) − 1 + ! 1−u

g
(t)
A
uA (t)u̇A (t) =u (t).
(55)

 A
1−&
)
*
where 34 ≡ , + n + (1 + + )& + + − 1 - n.
Thus, the system of this case comprises two dynamics: 2 , depicted by (53),and uA , depicted by (55).
We can obtain the values in a steady state as follows. Eliminating g∗A by using(53) and (55), we obtain u∗J
A as follows:
J

u∗A =

! (& - n + % g∗A ).
34 + (1 + + )% g∗A − (1 − ! )(& - n + % g∗A )
J

∗J
J∗
∗
If u∗JA ∈ (0, 1) is satisfied, uA can be an equilibrium, and uniting this uA , gA
∗J
∗J∗J
derived in (31) and (40) yields 2 . These (uA , 2 ) give the steady state. Thus,
the steady state of the Jones regime is also uniquely given if all feasible conditions
are satisfied.
We derive 2̇ = 0 and u˙A = 0 loci. (53) and (55) respectively yield 2̇ = 0 andu˙A = 0 loci as

1
&
∗
+
)
%
g
3
+
(1
+
4
IA  u−1u̇A = 0-loci : 2 = U(uA ) =  H
(56)
A ,
1−uA
B 1 + ! uA"#1
& - n + % g∗A & −1
2̇ = 0-loci : 2 = 6(uA ) =
uA .
(57)
B

We can easily show that U ' (·) < 0, 6' (·) < 0, and U(1) > 6(1). Furthermore, we
can derive
.
.
>
<
u̇A
= 0 ⇐⇒ 2
U(uA ).
(58)
<
>
.
.
>
<
2̇
= 0 ⇐⇒ 2
6(uA ).
(59)<>
From these properties, the existence condition for the steady state and its
uniqueness, we have the phase diagram of the J-regime as described in Figure
2, and we determine the saddle stability of this regime.
21

g∗A upb,g∗A up

•

•

+

+

1

Domain of
Positive Tricle down

Domain of
Negative Tricle down∗
For g∗∗A (> gA )For g∗A
A-Regime
J-Regime

Figure 1: Growth regimes2222̇ = 0(2 = 6(uA ))2 ∗J

u̇A = 0(2 = U(uA ))

•

U(1)
6(1)

u∗J
A

1

uA

Figure 2: Phase diagram of J-regime

References[1] Aghion, P. and Bolton, P., 1997. A Theory of Trickle-Down Growth andDevelopment, Review of Economic Studies, 64(2), 151-172.
[2] Ahituv, A., 2001. Be Fruitful or Multiply: On the Interplay between Fertility
and Economic Development, Journal of Population Economics, 14, 51-71.
[3] Arnold, L. G., 1998. Growth, Welfare, and Trade in an Integrated Model of
Human-Capital Accumulation and Research, Journal of Macroeconomics,
20(1), 81-105.
[4] Coe, D. T. and Helpman, E., 1995. International R&D Spillovers, European
Economic Review, 39, 859-887.
[5] Coe, D. T., Helpman, E., and Hoffmaister, A. W., 1997. North-South R&D
Spillovers, Economic Journal, 107, 134-149.
[6] Easterly, W., 1994. Economic Stagnation, Fixed Factors and Policy Thresholds, Journal of Monetary Economics, 33, 525-557.
[7] Grossman, G. M. and Helpman, E., 1991. Innovation and growth in the
global economy MIT Press (Cambridge, MA).
23

[8] Holod, D., and Reed III, R. R., 2009. Regional External Economies and Economic Growth under Asymmetry, Southern Economic Journal 75(4), 11231140.
[9] Jones, C. I., 1995a. Time Series Tests of Endogenous Growth Models, Quarterly Journal of Economics, 110, 495-525.
[10] Jones, C. I., 1995b. R&D-based Models of Economic Growth, Journal of
Political Economy, 103(4), 759-784.
[11] Kelley, A., 1988. Economic Consequences of Population Change in the
Third World, Journal of Economic Literature, 26(4), 1685-1728.
[12] Kelley, A. and Schmidt, R., 1995. Aggregate Population and Economic
Growth Correlations: The Role of the Components of Demographic Change,
Demography, 32, 543-555.
[13] Lucas Jr., R. E., 1988. On the Mechanism of Economic Development, Journal of Monetary Economics, 22(1), 3-42.
[14] Momota, A., 2009. A Population-Macroeconomic Growth Model for Currently Developing Countries, Journal of Economic Dynamics & Control,
33(2), 431-453.
[15] Quah, D. T., 1996. Empirics for Economic Growth and Convergence, European Economic Review, 40(6), 1353-1375.
[16] Quah, D. T., 1997. Empirics for Growth and Distribution: Polarization,
Stratification, and Convergence Clubs, Journal of Economic Growth, 2(1),27-59.
[17] Romer, P. M., 1990. Endogenous Technological Change, Journal of Political
Economy, 98(5), S71-S102.
[18] Solow R., 1957. Techonological Change and the Aggregate Production
Function, The Review of Economics and Statistics, 39(3), 312-320.
[19] Uzawa, H., 1965. Optimum Technical Change in An Aggregative Model of
Economic Growth, International Economic Review, 6, 18-31.

24