A Model on Knowledge
and Endogenous Growth
Derek H. C. Chen* and Hiau Looi Kee**
The World Bank
Washington DC 20433
Abstract
This paper presents a model of endogenous growth in which the main engine of
economic development is knowledge. Using a twosector closed economy model that
comprises of a conventional goodsproducing sector and a research and development
sector, our model incorporates two key aspects of knowledge: technology and human
capital. Steadystate equilibrium conditions show that the growth rate of per capita
income hinges on the growth rate of human capital. While the growth rate of human
capital has been previously shown to affect the growth of the economy in transition
between steady states or balanced growth paths, this paper is the first to link the
growth rate of human capital to the steadystate growth rate of productivity and
output per worker. Furthermore, this result does not exhibit scale effects or policy
invariance, both of which have been longstanding concerns with the predictions of
endogenous growth models developed in the 1990s.
World Bank Policy Research Working Paper 3539, March 2005
The Policy Research Working Paper Series disseminates the findings of work in progress to encourage the
exchange of ideas about development issues. An objective of the series is to get the findings out quickly,
even if the presentations are less than fully polished. The papers carry the names of the authors and should
be cited accordingly. The findings, interpretations, and conclusions expressed in this paper are entirely
those of the authors. They do not necessarily represent the view of the World Bank, its Executive Directors,
or the countries they represent. Policy Research Working Papers are available online at
http://econ.worldbank.org.
* The Knowledge for Development Program, World Bank Institute. ** International Trade Team,
Development Research Group.
Chen and Kee: Knowledge and Endogenous Growth 2
Table of Contents
1. Introduction
2. Theoretical Model
2.1 Derivation of the SteadyState Condition for Capital Accumulation
2.2 Derivation of the SteadyState Condition for Technological Growth
2.3 Derivation of the SteadyState Solution
2.4 Effect of Growth of Human Capital on SteadyState Technological
Growth
2.5 Derivation of SteadyState Output Growth
3. Policy Exercises
3.1 An Increase in the Growth Rate of Human Capital
3.2 An Improvement in the ICT Infrastructure
4. Summary and Conclusion
References
Figures
Chen and Kee: Knowledge and Endogenous Growth 3
1. Introduction
Early endogenous growth models, such as Romer (1990), Grossman and Helpman
(1991), and Aghion and Howitt (1992), attempt to explain productivity growth by the
introduction of a research and development (R&D) sector, with human capital or skilled
labor as an input. These models show that the steadystate growth rate of output per
worker depends positively on the level of available resources for R&D in the economy,
such as the stock of human capital or endowment of skilled labor. Hence, an increase in
the average educational attainment of the labor force, for example, will lead to a
permanent increase in the longterm growth rate of per capita income. The literature
refers to these effects, where the level of resources influences the longterm growth rate,
as scale effects (see Jones, 1999).
Many researchers have pointed out that such scale effects fail to reflect reality.
The high growth rates of the East Asian Newly Industrializing Economies (NIEs), which
averaged 8 percent annually for the past three decades despite initial low levels of human
capital stocks, as compared to the 2 percent exhibited by the developed countries, provide
an excellent counter example.
Subsequent R&Dbased growth models have attempted to eliminate this
prediction. For example, Jones (1995), Kortum (1997) and Segerstrom (1998) develop
models in which the growth rate of per capita income depends only on parameters that
are usually taken as exogenous, such as the growth rate of population, and no longer
depends on the level of R&D resources in the economy. In such models, levels of human
capital and other R&D resources affect only the longrun level of per capita income, but
not the growth rate.
While the predictions of the latter class of models certainly do not preclude the
experience of the NIEs, they imply the growth rate of per capita income is, to a large
extent, invariant to government policies. Hence, policies such as tax incentives for R&D
or education subsidies, which are commonly accepted to promote technical progress and
hence longterm economic growth, will have no effect on the steadystate growth rate of
per capita income. In short, these predictions are at odds with conventional wisdom.
In this paper, we develop a theoretical model of endogenous growth in which
knowledge is the main engine of economic development. Similar to the models
mentioned above, apart from the conventional goodsproducing sector, an R&D sector is
introduced to endogenize the accumulation of knowledge via technical progress, which is
the underlying source of longterm growth in per capita income. As in Romer (1990), we
reemphasize the explicit role of human capital as an input into both sectors.
Chen and Kee: Knowledge and Endogenous Growth 4
We show that in our model with human capital as a factor of production, the
steadystate growth rate of per capita income depends positively on the growth rate of
human capital. While the growth rate of human capital has been previously shown to
affect the growth of the economy in transition between steady states or balanced growth
paths, this paper is the first to link the growth rate of human capital to the steadystate
growth rate of productivity and output per worker.
Furthermore, we note that scale effects, where the steadystate growth rate is
dependent on the level of R&D resources or the stock of human capital, are absent from
our model. In addition, given that the growth rate of human capital is a determinant of
the steadystate growth rate of per capita income, the prediction of this model provides a
channel through which government policies can influence the longterm growth rate.
This paper therefore provides an important contribution to the literature as it
demonstrates that relatively simple neoclassical models with human capital as a factor of
production are able to resolve the concerns of scale effects and policy invariance that
characterize the predictions of the two classes of models discussed above.1
The paper also proceeds to examine the steadystate effects of two scenarios that
could result from common developmentoriented policies. The first considers an
exogenous increase in the rate of human capital accumulation, while the second assumes
a onetime exogenous increase or improvement in the economy's information and
communications technology (ICT) infrastructure. Both exercises show that even one
time positive shocks can have longrun positive effects on technological growth, capital
accumulation and economic development.
This paper is laid out as follows. Section 2 derives the twosector model. Section
3 illustrates the steadystate effects of two policy shocks: an exogenous increase in the
growth rate of human capital and an exogenous improvement in the ICT infrastructure.
Section 4 presents a summary with the main conclusions.
1 To date, a number of other models have been developed using various methods to resolve problems of
scale effects and/or policy invariance in endogenous growth models. For example, see Aghion and Howitt
(1998), Dinopoulos and Thompson (1998), Howitt and Aghion (1998), Peretto (1998), Young (1998),
Segerstrom (2000), and LloydEllis and Roberts (2002).
Chen and Kee: Knowledge and Endogenous Growth 5
2. Theoretical Model
The following model draws on a model derived in Romer (1996).
We first assume that the economy comprises of two sectors: a goodsproducing
sector and an R&D sector. The former produces conventional output, while the latter
produces new technology, which adds to the existing level of technology. As mentioned
above, factors of production in the economy, namely, capital (K), labor (L) and human
capital (H) are allocated for use either in the goods or R&D sector.
Let
aK be the fraction of Kt used in the R&D sector
aH be the fraction of Ht used in the R&D sector
aL be the fraction of Lt used in the R&D sector
This implies that
(1  aK) is the fraction of Kt used in the goodsproducing sector
(1  aH) is the fraction of Ht used in the goodsproducing sector
(1  aL) is the fraction of Lt used in the goodsproducing sector
Note that the use of technology in one instance does not preclude the simultaneous use of
the same technology by another agent. In other words, technology has the characteristic
of being is nonrival, and hence the entire existing level of technology (A) is used in both
sectors2,3.
Output in time t is given by:
Yt = [(1 aK )Kt ] [(1 aH )Ht ] [At (1 aL )Lt ]1  , 0 < <1, 0 < <1, + <1
(1)
The level of innovation in the economy depends on the amount of capital, labor
and human capital devoted to the R&D sector and on the current level of technology. For
this model, we assume a generalized CobbDouglas production function for the R&D
sector.
2We therefore assume that there is zero excludability associated with technology in this economy.
3Although the acquisition of human capital by a worker involves learning, there is a clear conceptual
distinction between human capital and disembodied knowledge in the form of technology. Human capital
consists of the abilities, skills, knowledge and experience of particular workers. Thus, like conventional
economic goods, human capital is rival and excludable. For example, if an engineer's full effort is being
devoted to one activity, it precludes the use of his or her skills in another. In contrast, if an algorithm is
being used in one activity, that in no way makes its use in another more difficult or less productive.
(Romer 1996)
Chen and Kee: Knowledge and Endogenous Growth 6
A&t = B(aK Kt )a(aH Ht )b(aLLt )c At ,
B > 0, a 0, b 0, c 0, 0 (2)
As in the Solow model, the savings rate is exogenous and constant. Also, we set
depreciation to be zero for simplicity. This implies that the rate of capital accumulation
or investment will be given by:
K&t = sKYt, 0 sK 1 (3)
We treat population growth and human capital growth to be constant and exogenous.
L&t = nLt, n 0 (4)
H& t = mHt, m 0 (5)
where X&t Xt
t
Note that there are two state variables, Kt and At, and that in steady state, all growth rates
are constant.
2.1 Derivation of the SteadyState Condition for Capital Accumulation
To derive the steadystate condition for the capital (K) accumulation, we first substitute
(1) into (3) and we get,
K&t = sK [(1 aK )Kt ] [(1 aH )Ht ] [At (1 aL )Lt ]1

(6)
= sK (1 aK ) (1 aH ) (1 aL )1 Kt Ht At
1 1Lt
Let gKt be the growth rate of Kt, hence
gKt K&t = sK(1 aK )(1 aH ) (1 aL)1 Kt Ht At
1 1 1 Lt
Kt
Chen and Kee: Knowledge and Endogenous Growth 7
Letting cK sK (1 aK ) (1 aH ) (1 aL )1 , we get
gKt = cK Kt Ht At
1 1 1Lt
= cK Kt Ht At
1 1 1Lt Kt Kt

1
= cK Ht AKLt t t
Kt
Taking logs on both sides, we get
1
ln gKt = lncK t t
Ht AKLt
Kt
= ln cK + ln Ht
Kt + (1  )ln AtLt
Kt
Taking differential with respect to time, we get
g&Kt (lngKt ) = 1 Ht  1 At +1 Lt
t Ht t Kt t
1 Kt +(1  ) At t Lt t Kt t
1 Kt
= (gHt  gKt)+(1 )(gAt + gLt  gKt)
= (m gKt)+(1 )(gAt +n gKt)
For the steadystate condition, we set g&Kt = 0 resulting in
m  gK +(1  ) g*A + n gK = 0
( * ) ( * ) (7)
with rearranging, we get
gK* = 11  n 11  g*A +1 m (8)
+
Chen and Kee: Knowledge and Endogenous Growth 8
Recall it was assumed that 0 < <1, 0 < <1, + <1. Hence, the coefficient of gA
will be 0 < 1 
1 < 1. Figure 1 illustrates the phase diagram with the locus of
points where g&K = 0.
To see the dynamics around the locus g&K = 0, suppose that initially gK > gK , *
then from (7) we can see that g&K < 0 . This implies that gK will decrease until gK = gK .*
Graphically, this corresponds to all points that are above the g&K = 0 locus will have a
tendency to converge downwards to the g&K = 0 locus. Alternatively, suppose that
gK < gK , then from (7) we can see that g&K > 0. This implies that gK will increase until
*
gK = gK . Graphically, this corresponds to all points that are below the g&K = 0 locus will
*
have a tendency to converge upwards to the g&K = 0 locus.
2.2 Derivation of the SteadyState Condition for Technological Growth
To derive the steadystate condition for technological growth (A), recall from (2) that
A&t = B(aKKt )a(aHHt )b(aLLt )c At ,
B > 0, a 0, b 0, c 0
Let gAt be the growth rate of At, hence
gAt A&t = B(aKKt )a(aHHt )b(aLLt )c At
1
At
= BaKaHaLKt Ht Lt At
a b c a b c 1
Letting cA BaKaHaL , we get
a b c
gAt = cAKt Ht Lct At
a b 1
Taking logs, we get
ln gAt = ln cA + a ln Kt + b ln Ht + c ln Lt + ( 1)ln At
Chen and Kee: Knowledge and Endogenous Growth 9
Taking differential with respect to time, we get
g& At (ln gAt ) = a 1 Kt + b 1 Ht + c1 Lt 1 At
t Kt t Ht t Lt t + ( 1) At t
= ag Kt + bg Ht + cg Lt + ( 1)g At
= ag Kt + bm + cn + ( 1)g At
For the steadystate condition, we set g&At = 0 resulting in
agK + bm + cn + ( 1)g*A = 0
* (9)
Upon rearranging, we get
g*A = agK + bm + cn
*
(10a)
1
or equivalently,
gK* =  m  n +
b c
a a 1 g*A (10b)
a
It will be seen later that in order for the existence of a steadystate solution, we will need
to assume that 0 <1. This implies that the slope of the g&A = 0 locus will be strictly
positive. Figure 2 illustrates the phase diagram with the locus of points where g&A = 0,
assuming 0 <1.
To see the dynamics around the locus g&A = 0, suppose that initially gA > g*A ,
then from (9) we can see that g&A < 0 . This implies that gA will decrease until gA = g*A .
Graphically, this corresponds to all points that are to the right of the g&A = 0 locus will
have a tendency to converge leftwards to the g&A = 0 locus. Alternatively, suppose that
gA < g*A , then from (9) we can see that g&A > 0. This implies that gA will increase until
gA = g*A . Graphically, this corresponds to all points that are to the left of the g& A = 0
locus will have a tendency to converge rightwards to the g&A = 0 locus.
Chen and Kee: Knowledge and Endogenous Growth 10
2.3 Derivation of the SteadyState Solution
To summarize, we have 2 equations representing steadystate conditions with 2
unknowns:
From setting g&Kt = 0
gK* = 11  n 11  g*A +1 m
+ (8)
From setting g&At = 0
gK* =  m  n +
b c
a a 1 g*A (10b)
a
To find the intersection of the two steadystate conditions, we equate (8) and (10b)
11 n+11 g*A
+1 m =  m n+b c
a a 1 g*A
a
With rearranging, it can be shown that
g*A = 1
1a 11 1 a + m+
b 1  c
1 + n (11)
a
Note that the denominator of RHS term in equation (11) characterizes the nature of the
steadystate solution for this economy. More specifically, the existence of the steady
state solution hinges on the sign of 1 11 . We now examine the various
a
possible scenarios.
Chen and Kee: Knowledge and Endogenous Growth 11
Case 1: 1 11
a <
Firstly, note that because the g&A = 0 locus has a negative gK axis intercept, the
starting point of the g&A = 0 locus will always be vertically below that of the g&K = 0
locus. Consequently, in this case when the slope of the g&K = 0 locus is steeper than that
of the g&A = 0 locus, the two loci constantly diverge and hence there is no intersection.
Figure 3 illustrates the phase diagram that plots the two steadystate conditions
simultaneously in the gKgA space. It can be seen that regardless of where the economy's
initial point, it eventually enters the region between the two loci. Once this occurs, the
growth rates of both A and K, and hence the growth rate of output, increase continually.
Consequently, the economy exhibits everincreasing growth, and there is no tendency to
converge to a steadystate solution.
Case 1a: 1 11 ,
a < >1
This scenario is a special version of Case 1, where the slope of the g&K = 0 locus
is still steeper than that of the g&A = 0 locus, the only difference being that the g&A = 0
locus now exhibits a negative slope. The dynamics is identical to Case 1 in that the two
loci are constantly diverging leading to the nonexistence of an intersection between the
two loci and the economy does not converge to a steadystate solution (Figure 4).
Case 2: 1 11
a =
This case is again similar to that of Case 1. Here slope of the g&K = 0 locus is
equal to that of the g&A = 0 locus, thus the two loci are parallel and again nonintersecting.
Once again there is no tendency for the economy to converge to a steadystate solution
(Figure 5).
Case 3: 1 11
a >
The final possibility is when slope of the g&K = 0 locus is less steep than that of
the g&A = 0 locus, which is illustrated in Figure 6. It can be seen that in this case, the two
steadystate loci intersect at point E. In terms of the dynamics, it can be verified that
Chen and Kee: Knowledge and Endogenous Growth 12
regardless of the economy's initial point, there is a tendency for the economy to converge
to point E. This implies that point E is a steadystate solution for the economy.
In light of the above analysis of the various possible cases, it can be concluded
that for the existence of a steadystate solution, the following condition must hold:
1a 11 > 0 (12)
Given that 0 < 1 
1 <1, equation (12) also implies that <14.
2.4 Effect of Growth of Human Capital on SteadyState Technological
Growth
With reference to (11), it can be seen that the steadystate growth rate of technology will
be depend positively on the growth rate of human capital if the coefficient
1 a + b
1a 11 is positive. We note that the numerator 1 + is b
a
unambiguously positive. In addition, we have just shown that the denominator
1a 11 must be positive for a steadystate solution to exist. The ratio must
therefore be positive.
In summary, the model shows that an increase in the growth rate of human capital
leads to increases in the steadystate growth rate of technology.
4For this reason, the g& A = 0 locus has been plotted with a positive slope.
Chen and Kee: Knowledge and Endogenous Growth 13
2.5 Derivation of SteadyState Output Growth
We are ultimately interested in the steadystate growth rate of output or per capita output.
From (1) and upon rearranging, we get:
Yt = (1  aK ) (1  aH ) (1  aL )1  K H
1  1 
t t At Lt
Taking logs on both sides, we get
lnYt = ln(1 aK ) (1 aH ) (1 aL )1 +
lnKt + ln Ht + (1  )ln At +(1  )ln Lt
Taking differential with respect to time, we get
gYt (lnYt ) = 1 Kt + 1 Ht 1 At 1 Lt
t Kt t Ht t + (1  ) At t + (1  )Lt t
= gKt + gHt + (1  )gAt + (1  )gLt
= gKt + m + (1  )gAt + (1  )n
(13)
For steadystate output growth, we first substitute (8) into (13) and get:
gY
* = 1  n 11  g*A
1 + +1 
m + m +
(14)
(1  )g*A +(1  )n
With rearranging, (14) can be simplified to
1
gY* = [(1 ] (15)
1   )n + m + (1  )g*A
Chen and Kee: Knowledge and Endogenous Growth 14
Next, we substitute (11) into (15)
1
gY* = [(1  )n + m]
1  +
1 (1  )
1  + n
c
1a 11  1 a
 + m +
b 1 
1  a
Upon simplifying, we get
[ (1 )+ b(1  )]m + (1  )(1 + c)n
gY* = (1 )(1 ) a(1  ) (16)
Note that the condition for the existence of a steadystate solution (12) implies that the
denominator of the RHS term of (16) is positive:
1a 11 > 0 (17)
(1)(1)a(1  )
a(1) > 0
(1)(1)a(1  ) > 0
We are able to arrive at the expression immediately above because we know that
a(1) > 0 .
Thus, we see that coefficient of m is positive, implying that increases in the growth rate
of human capital leads to increases in the steadystate growth rate of output.
Similarly, in terms of steadystate per capita output growth:
[ (1 )+ b(1  )]m + (1  )(1 + c)n
gY  n =
*
(1 )(1 ) a(1  )  n
[ (1 )+ b(1  )]
= (1 )(1 ) a(1  ) 1
(1  )(1 + c)
(1 )(1 ) a(1  )m + n
(18)
Chen and Kee: Knowledge and Endogenous Growth 15
Thus increases in the growth rate of human capital also lead to increases in the steady
state growth rate of output per capita.
3. Policy Exercises
In this section, we consider the steadystate effects of two scenarios that could
result from common developmentoriented policies. The first exercise focuses on
education and considers an exogenous increase in the rate of human capital accumulation,
which could result from the country experiencing a onetime increase in the number of
new schools. In the second scenario, we examine the role of information and
communications technology (ICT) in economic development, by assuming an exogenous
onetime increase or improvement in the economy's ICT infrastructure. Both exercises
show that even onetime positive shocks can have longrun effects on technological
growth, capital accumulation and economic development.
3.1 An Increase in the Growth Rate of Human Capital Accumulation
Consider an exogenous increase in the growth rate of human capital accumulation
(from m0 to m1). With reference to Figure 7, it can be seen that this decreases the
intercept of the g&A = 0 locus, and consequently, resulting in a parallel downward shift
from (g&A = 0)0 to (g&A = 0)1 . At the same time, the increase in the growth rate of human
capital accumulation increases the intercept of the g&K = 0 locus, and consequently, leads
to an upward parallel shift from (g&K = 0)0 to (g&K = 0)1 . As a result, the steadystate
solution for the economy moved from E0 to E1, with the steadystate growth rate of
technology increasing from g*A to g*A , and the steadystate growth rate of capital
0 1
increasing from gK to gK .
*0 *1
Intuitively, the increase in the growth rate human capital first results in a larger
stock of human capital in the economy. Assuming that the share of human capital being
allocated to the R&D sector remains unchanged, the increase in human capital in the
R&D sector leads to more innovations and discoveries, which results in the increase in
the growth rate of technology, from g*A to g*A . The larger growth rate of the human
0 1
capital also simultaneously increases the amount of resources being used in the
conventional goodsproducing sector. In this sector, the larger growth rate of the human
capital stock, together with the more rapid level of technological growth (resulting from
the R&D sector), leads to an increase in the growth rate of output. Given that a fixed
Chen and Kee: Knowledge and Endogenous Growth 16
proportion of output is invested as new capital, the rate of capital accumulation also
increases with the increased rate of output production, which results in the increase from
gK to gK . To summarize, an exogenous increase in the growth rate of human capital in
*0 *1
our model increases the longterm growth rate of technology, capital and output.
3.2 An Improvement in the ICT Infrastructure
Information and communications technologies (ICTs) are the backbone of the
knowledge economy and in recent years have been recognized as an effective tool for
promoting economic growth and sustainable development. With relatively low usage
costs and the ability to overcome distance, ICTs have revolutionized the transfer of
information, knowledge and technology around the world.
ICT infrastructure in an economy refers to the accessibility, reliability and
efficiency of computers, phones, television and radio sets, and the various networks that
link them. The World Bank Group defines ICT to consist of hardware, software,
networks, and media for collection, storage, processing transmission, and presentation of
information in the form of voice, data, text, and images. They range from the telephone,
radio and television to the Internet (World Bank, 2003a and 2003b).
Over the past decade, there has been a series of studies that show that both ICT
production and ICT usage have contributed to economic growth5. One of the most
obvious benefits associated with ICT usage is the increased flow of information and
knowledge. Because ICTs allow information to be transmitted relatively inexpensively
and efficiently, ICT usage increases the flow of information, technology and knowledge,
and hence technologies can be acquired and adapted more easily leading to increased
innovation and productivity.
In light of the above, we argue that an improvement or an increase in the level of
the economy's ICT infrastructure increases the efficiency of which the existing level of
technology contributes to the production of innovation and discoveries. More
specifically, an improvement in the ICT infrastructure will increase . 6 Figure 8
illustrates the effect of an exogenous improvement in the ICT infrastructure on our two
5 See Chen and Dahlman (2004), Pilat and Lee (2001), Jorgenson and Stiroh (2000), Oliner and Sichel
(2000), Whelan (2000), and Schreyer (2000).
6ICT usage has also been cited to increase the rate of human capital accumulation because ICTs tend to
increase the access to existing knowledge and information (see World Bank, 2003a and 2003b). Given that
the case of an increase in the rate of human capital accumulation has been analyzed above, we will focus
here exclusively on the positive effect of ICTs on the contribution of existing technology to innovation
output.
Chen and Kee: Knowledge and Endogenous Growth 17
sector economy via an increase in the value of . As increases from to , the 0 1
slope of the g&A = 0 locus becomes more gentle leading to the clockwise pivot of the
locus from (g&A = 0)0 to (g&A = 0)1 . This results in the steadystate solution of the
economy to move from E0 to E1, and the steadystate growth rate of technology to
increase from g*A to g*A , and the steadystate growth rate of capital to increase from gK
0 1 *0
to gK .
*1
Intuitively, the increased flow of information and knowledge, resulting from the
improvement in the ICT infrastructure, allows innovation to be produced more
efficiently, holding constant the level of existing technology. This increase in efficiency
increases the steadystate growth rate of technology, resulting in the increase from g*A to 0
g*A . Subsequently, the increase in the steadystate growth rate of technology increases
1
the steadystate growth rate of output, which leads to the increase in the steadystate
growth rate of capital, resulting in the increase from gK to gK .
*0 *1 7
4. Summary and Conclusion
The first generation of endogenous growth models, such as Romer (1990), predict
that the longrun growth rate of an economy is proportionate to the level of resources
devoted to R&D, such as the stock of human capital. Such scale effects are not consistent
with the growth experience of the East Asia NIEs and many industrialized countries.
This issue of scale effects was addressed by the second generation of endogenous growth
models, such as Jones (1995). However, along with the scale effects, the latter class of
models also removed any channels via which government policies could influence the
longterm growth rate of per capita income.
In this paper, we have developed an endogenous growth model that relates the
longrun growth rate of an economy to the growth of human capital. This is consistent
with the growth experience of the East Asia NIEs, where education and schooling have
improved tremendously in a short period of time, even though the overall stock of human
capital is still lacking behind the industrialized countries. Furthermore, this model re
introduces the possibility of steadystate growth effects via government intervention by
affecting the growth rate of human capital.
7Note that the effect of the improvement in the ICT infrastructure differs from the increase in the growth
rate of human capital accumulation in that in the former only the locus moved, whereas in the latter, we
observed shifts in both of the steadystate loci. The key reason for this difference lies in the number of
sectors being initially affected by the shocks. The ICT improvement initially affects the R&D sector and
permeates the economy only through the R&D sector. In the human capital growth rate exercise, the
increase in the growth rate of human capital leads to a larger stock of human capital that simultaneously
enters both output and R&D sectors, resulting in the movement of both steadystate loci.
Chen and Kee: Knowledge and Endogenous Growth 18
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Chen and Kee: Knowledge and Endogenous Growth 21
Figure 1: Phase Diagram for the Dynamics of
the Growth Rate of Capital
gK
g&K < 0 g&K = 0
1 
1
11 n+1 m
g&K > 0
gA
Figure 2: Phase Diagram for the Dynamics
of the Growth Rate of Knowledge
gK Assume >1
g&A = 0
g&A > 0 1
a
g&A < 0
gA
 m n
b c
a a
Chen and Kee: Knowledge and Endogenous Growth 22
Figure 3: Phase Diagram for the Dynamics of the
Growth Rate of Capital and Knowledge
gK
Assume 1 1 
<
a 1 , <1
g&K = 0
1 
1
11 n
+1 m
g&A = 0
1 gA
 m n
b c a
a a
Figure 4: Phase Diagram for the Dynamics of the
Growth Rate of Capital and Knowledge
gK
Assume 1 1 
<
a 1 , 1
g&K = 0
1 
1
11 n
+1 m
 m n
b c gA
a a
1
a
g&A = 0
Chen and Kee: Knowledge and Endogenous Growth 23
Figure 5: Phase Diagram for the Dynamics of the
Growth Rate of Capital and Knowledge
gK
Assume 1 1 
=
a 1
g&K = 0
1 
1
11 n
+1 m g&A = 0
1
a
gA
 m n
b c
a a
Figure 6: Phase Diagram for the Dynamics of the
Growth Rate of Capital and Knowledge
gK
Assume 1 1 
> g&A = 0
a 1
g&K = 0
1 
gK* E 1
11 n+1 m
1
a
g*A gA
 m n
b c
a a
Chen and Kee: Knowledge and Endogenous Growth 24
Figure 7: An Increase in the Growth Rate of Human
Capital Accumulation
gK m1 > m0 (g&A = 0)0 (g&A = 0)1
(g&K = 0)1
gK
*1 E1
(g&K = 0)0
gK *0 E0
11 n m1
+1 
11 n
+1 m0
g*A
0 g*A
1 gA
 m0  n
b c
a a
 m1  n
b c
a a
Figure 8: An Improvement in the ICT Infrastructure
1 >0
gK (g&A = 0)0 (g&A = 0)1
1 0
a
gK *1 E1 g&K = 0
gK*0 E0 11
a
11 n+1 m 1 
1
g*A
0 g*A
1
gA
 m n
b c
a a