The Importance of Parameter Constancy for Endogenous Growth with Externality

The Importance of Parameter Constancy for Endogenous Growth with Externality
Kelly Yiyu Lin*
BMO Financial Group, U.S.A.
email:yiyu@msn.com
Wenti Du**
Akita International University, Japan
email:wenti.du@gmail.com
Abstract
The economic model of endogenous growth has been commonly discussed. It has been
specified by econometric models by Robert Barro (1986, 1990, and 1994) and Xavier Sala-
i-Martin (2003) but it is challenging to keep parameter constancy in the model. This paper
demonstrates how to find the stable growth rate converging to the steady state and the
optimal capital level at the steady state with parameter constancy. This paper also finds
the economy would converge to a stable steady state when the co-integration holds
between annual growth rate of GDP per capita and GDP per capita. We take an empirical
study of selected five countries (Indonesia, India, US, France and Japan) from 1960 to 2016
and specify econometric models of endogenous growth with externality and to test the
convergence.
JEL Codes: O11, O47
Keywords: externality, production function, endogenous growth, convergence, parameter
constancy
1

1. Introduction
Technology generates positive externalities and contributes to long-term growth in an
economy. The economic model of endogenous growth has been commonly discussed, for
example, see Romer (1986, 1990, 1994), Benhabib and Jovanovic (1989), Mankiw (1995);
however, it is challenging to keep parameter constancy. In this paper, we take an
empirical study of selected countries to specify an econometric model of endogenous
growth with externality and to test the convergence. Barro and Sala-i-Martin (2003)
propose the concepts of β convergence and σ convergence to specify the econometric
models for the rate of convergence. However, parameter constancy is the challenge in
their model. In order to find the stable growth rate converging to the steady state and
the optimal capital level at the steady state, this paper applies impulse indicators to
keep parameter constancy in the face of exogenous shocks.
The new technology generates one-time shock that affects economies. We assume that
such impact of new technology is exogenous and constant, which could be viewed as an
externality in an economy. Also, technology innovation contributes to creating more
physical capital and human capital in an economy. In the face of exogenous shocks, the
significant impulse indicators of the model demonstrate the current economy is
converging to a stable steady state if the co-integration relationship holds between
endogenous variables; otherwise, the economy will diverge to a saddle point (unstable
steady state). Our findings show that technology innovation has a different impact on
developed countries (France, Japan, and the U.S.) and developing countries (India and
Indonesia).
The effect of new technology on developed countries is insignificant, while the impact of
new technology on developing countries is shown to be significant. Moreover, our results
suggest that the rates of convergence in developed countries are lower than those in
developing countries. In particular, among our sample countries, the Indian economy
diverges to an unstable steady state, also known as “saddle point” because the co-
integration relationship does not hold between annual growth rate of GDP per capita and
GDP per capita. This paper is organized as follows. Section 2 analyzes economies in
Indonesia, India, U.S., France and the Japan. Section 3 explains the data we use and
specify the econometric model we adopt. Section 4 presents the empirical results. Section
5 discusses the AK model and the case in Indonesia. Section 6 concludes our research.
2

2. An Overview of the relationship between GDP per capita and Human Capital
After World War II, the analysis of economic growth became a topic of high priority
because of decolonization in Asia. The neo-classical, Solow growth theory (1956) was
embraced by economists, which explains long-run economic growth by focusing on
physical capital accumulation. However, economic growth could not only rely on physical
capital. While the ratio of the growth of physical capital to the growth of human capital
exceeds the equilibrium ratio, the rate of economic growth declines. Thus, human capital
is the crucial variable for economic growth. The innovation of technology is the critical
shock for economic growth. To obtain the best available technology, the economy needs
skilled labor force, which leads to thinking about human capital in economic growth. The
advances in endogenous growth theory from Paul Romer (1986) have been the ability to
formalize this idea. This section provides the economic analysis to precede the research of
the endogenous growth model for Indonesia, India, U.S., France and Japan.
The main topic of endogenous growth is human capital accumulation. Human capital
could produce output in production, and human capital could produce human capital
endogenously. GDP per capita is highly co-integrated with human capital index and also
the index of physical capital, so GDP per capita can apply as the aggregate impacts of
physical capital and human capital in modeling. The human capital and physical capital in
developed countries are higher than they are in developing countries. However, the Inada
rule indicates the poor economies have small capital stocks, and the marginal product of
capital should be high. Thus, capital should have the incentive to flow from rich countries
to emerging countries. The poor economies have the tendency to grow more rapidly than
rich economies. Human capital accumulation of endogenous growth is caused by the
shock of technology. Technology is kind of positive externality of the economy with
governmental intervention, and the one-time shock of technology can create the
permanent change of the economy. This paper will focus on the endogenous growth with
externality and test for the convergence of selected countries.
3

Graph 2.1 GDP per Capita and Human Capital Index
As shown in Graph 2.1, GDP per capita is co-integrated with Human Capital Index for
Indonesia, India, U.S., France and Japan. Because GDP per capita for the selected four
4

countries are highly co-integrated with human capital indexes of all selected countries,
GDP per capita could be the proxy of human capital.
Graph 2.2 Human Capitals Index
Moreover, during the period both GDP per Capita and human capitals of developed
countries (Japan, U.S, and France) are higher than those of developing countries (India and
Indonesia) as shown in Graph 2.2.
5

Graph 2.3 Annual Growths of GDP per Capita and Population
6

However, as illustrated in Graph 2.3, the annual growth rate of GDP per capita increases
when population growth rate decreases for all selected countries from 1961 to 2016. In
general, human capital increases when population growth decreases for developed
countries (Indonesia, U.S, France and Japan), while India is an exception (Graph 2.4). In the
case of India, the human capital increases along with the population growth rate before
1983, but afterward it increases with the decline of population growth rate. Thus, human
capital does not have a constant negative relationship with population growth rate.
Graph 2.4 the Relationship between Human Capital and Population Growth Rate
7

Graph 2.5 Results of Convergence Tests
8

Results from conditional convergence test suggest that the long-run growth of Japan, U.S.,
and France converge to stable steady states (Graph 2.5). However, there are divergences
for long-run growth of India, in particular; the economy of India would diverge to an
unstable steady state (saddle point). The intercept shift of annual growth of GDP per
capita will be explained in detail in the model specification in Section three. The result of
conditional convergence test of Indonesia is close to AK model. This paper will discuss the
case of Indonesia in section five.
3. Data and Model Specification
3.1 Data
The sample size of data is annual data from 1961 to 2016. The data of annual growth of
GDP per capita comes from the World Bank database. The data of GDP per capita comes
from Federal Reserve's Bank. Constant GDP per capita is selected for gross domestic
product divided by midyear population. GDP is the sum of gross value added by all
resident producers in the economy plus any product taxes and minus any subsidies not
included in the value of the products.
3.2 Model Specification
Romer (1986) introduces the externality into the production function. The institution of
Romer model indicates that firms increase capital (physical capital and human capital) for
new technology, and this creates an externality. The externality is the impact of the
invention of technology. People believe the new technology will promote the production
significantly, so government encourages firms to invest more physical capital in new
technology and skill training to increase the human capital to operate new technology.
The technology generates positive externality to capital, so the externality to each firm is
determined by GDP per capita (Romer model with Lucas modification). Because GDP per
capita is highly co-integrated with human capital and physical capital, GDP per capita
should include the impacts of physical capital and human capital.
The invention of new technology creates the impacts of the one-time shock of the
economy (externality) and causes the permanent change of the economy. From 1961 to
2016, the innovate technology is "computer”. The invention of the computer changes our
lives permanently. Government establishes computer courses in schools and also
encourages firms to invest more in physical capital and human capital to promote the
industries. In economic theory section, we discuss how the impacts of one-time shocks
could cause the permanent change of the economy with externalities. In the model
9

specification, we will introduce the economic model first and then specify the
econometric models to test for convergence to find the optimal level of capital per capita
at the steady state and the rate of convergence back to steady state for selected
countries.
3.3 Economic Model
The theory of endogenous growth demonstrates the concerns of convergence and the
perfect competition in aggregate level. Graph 2.5 reflects the sample data (1961 to 2016)
supports the convergence for the economies of Japan, U.S and France, and the sample
data also supports the divergence for the economies of India. The invention of technology
would generate the one-time shock impacts to cause the permanent change of the
economy. The effects of new technology are exogenous and later change peoples'
economic activities permanently, so we assume technology is exogenous and constant in
the model. To investigate the impacts (size) of externality on the economy, we will discuss
the model in following three cases.
The production function of Romer model is defined as follows:
Y = AKα
L1−α
Ҟη
Y: production
A: constant technology
Ҟ: externality
𝝶: size of the externality
𝝶 increases -> externality is more significant and important
𝝶=0, production, Y, is Cobb-Douglas function
For Romer, Ҟ=K investment increases -> K increases
For Lucas, Ҟ= ƙ = K/L -> Romer model with Lucas modification
Y = AKα
L1−α
Ҟη
= AKα
L1−α
Kη
L−η
= AKα+η
L1−α−η
Y = AKα+η
L1−α−η
y = Y/L , ƙ= K/L -> y= Aƙα+η
Ќ = s·Y - δ·K
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ƙ = K/L -> ln ƙ = ln K - ln L
ќ/ƙ = Ќ/K - Ľ/L
Ľ/L = n
n: population growth; population growth is exogenous
ќ/ƙ = (s·Y - δ·K )/K - n -> ќ = sy - (n+δ)·ƙ = sδ)·ƙ = s Aƙα+η
- (n+δ)·ƙ = sδ)·ƙ
ɣƙ = ќ/ƙ = s Aƙα+η−1
- (n+δ)·ƙ = sδ) -> the growth rate of ƙ, ɣƙ, depends on (α+δ)·ƙ = s 𝝶-1)
𝝶 > 0, ɣƙ >0, 0< s<1, s: saving rate; saving rate is exogenous
(α +δ)·ƙ = s 𝝶-1) decides if the model has a stable steady state, unstable steady state or no steady
state (AK model). The following three cases demonstrate the endogenous models with
different size of externalities.
Case 1: (α+δ)·ƙ = s 𝝶-1) <0 -> the impacts of externalities are not significant
-> Y is Neo-Classical production function
(the market is competitive)
-> there is a positive stable steady state
Graph 3.1 Convergence of the Endogenous Growth Model.
ɣƙ = ќ/ƙ = s Aƙ
α+η−1
- (n+δ)·ƙ = sδ) = 0
ƙ
ss
=[
sA
n+δ
]
1/(1−α−η)
11

Case 2: (α+δ)·ƙ = s 𝝶-1)>0 -> the impacts of externalities are very significant
-> Y is not Neo-Classical production function
(the market is not competitive)
-> there is a unique, positive unstable steady state
(saddle point)
Graph 3.2 Divergence of the Endogenous Growth Model.
ɣƙ = ќ/ƙ = s Aƙ
α+η−1
- (n+δ)·ƙ = sδ) = 0
ƙ
ss
=[
sA
n+δ
]
1/(1−α−η)
Case 3: (α+δ)·ƙ = s 𝝶-1)=0 -> the impacts of externalities are significant, stable and
consistent (𝝶 needs to be a specific value, 𝝶>0)
-> Y is not Neo-Classical production function
-> the market is perfectly competitive (constant return to scale holds)
-> AK model
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-> there is no steady state
ɣƙ = sA - (n+δ)·ƙ = sδ)
Graph 3.3 AK model of the Endogenous Growth Model.
The AK-type endogenous growth models, (named for the production function Y = AK),
including those of Romer (1986, 1987), Lucas (1988), and Rebelo (1991), suggest that
investment in the broadly defined capital has a positive long-run effect on growth. One
primary concern about the AK model is that it is based on a simple one-sector linear
production function without transitional dynamics. The diminishing returns of physical
capital trade off the increasing return of human capital. An essential property of the AK
model is the global absence of diminishing returns of capital (physical and human capital).
The AK model implies that the broadly defined rate of investment exerts a positive effect
on the long-run rate of growth. Because the constant return to scale holds in AK model,
AK growth model satisfies perfect competitive models.
In the endogenous growth model, the accumulation of human capital encounters the
diminishing returns. AK model (Y=AK) occurs when the increasing return of human capital
trades off the diminishing return of physical capital. AK model exhibits constant returns to
scale in the accumulated factors of production, so the market is competitive; this model
can generate perpetual (permanent) growth without the assumption of exogenous shifts
in the production function.
Case 1 and case 2 indicates the steady state exists when the annual growth of capital per
capita is equal to zero (ɣƙ =0). Graph 2.5, Graph 3.1 and Graph3.2 indicate there are
intercept shifts in Y-axis of scatter plots when the annual growth of capital per capita is
equal to zero (ɣƙ =0). In economic theory, the intercept shift is equal to population growth
13

rate plus the depreciation rate of capital. The steady-state level of capital per capita (ƙss
=
[
sA
n+δ
]
1/(1−α−η)
) is equal to the function of all exogenous parameters (saving rate, constant
technology, population growth rate and depreciation rate of capital). The economic model
of endogenous growth is easy to demonstrate in the discussion. However, how to specify
the econometric model of endogenous growth for application is the major topic of this
paper.
3.4 Econometric Model
Econometric modeling in this paper would demonstrate how to find the capital per capita
at steady state and the rate of convergence for selected countries. Graph 2.5 indicates
there are intercept shifts in Y-axis when the annual growth rates of capital per capita for
selected countries are equal to zero (ɣƙ =0). In an economic model, the intercept shifts in
Y-axis is equal to population growth rate plus depreciation rate of capital. For India and
Indonesia, population growth rates decline dynamically within the sample size (1961 to
2016). For developed countries (Japan, U.S, and France), population growth rates keep as
low constant rates. Depreciation rate should include the aggregate impacts of physical
capital and human capital. However, it is not difficult to find the precise depreciation rates
of all selected countries. Population growth rates and depreciation rates change
dynamically. We are not able to calculate the capital per capita at the steady state and
find out the rate of convergence by using algebra.
In econometric modeling, the intercept shifts in Y-axis come from the aggregate impact of
exogenous parameters such as population growth rate and depreciation rate of capital. In
econometric modeling, the shifts of intercept on Y-axis of scatter plots could be
incorporated in the constant term of the regression model if the aggregate impact of
exogenous shock generates recursive activities with endogenous variables. If the
aggregate effect of the exogenous shock generates a constant impact on annual growth
rate per capita, we should transform the dependent variable as the annual growth rate of
GDP per capita minus average of the annual growth rate of GDP per capita. There are two
steps for econometric modeling of endogenous growth. In the first step, we have to
specify the models for all selected countries by using OLS methodology to find the GDP
per capita at the steady state. We also have to do Granger co-integration test for all
models. After we obtain the capital per capita at the steady state, we would pursue the
second step to find the rates of convergence.
3.4.1 Step One
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We regress the annual growth rates of GDP per capita on the GDP per capita by using OLS
methodology and then test the co-integration relationship for all models. Graph 2.4
demonstrates all selected countries experienced the impacts of exogenous shocks from
the invention of new technology, governmental policy changes and financial crises within
the sample period. Although the innovation technology is one time shock and generates
consistent externality of the economy. The economy still experiences other shocks
dynamically. We have to impose the impulse indicators to capture the hit-and-run impacts
from the exogenous shocks and stabilize the parameters. Hendry (2005) states impulse
indicators simplify the dummy variables and stabilize parameters in the face of exogenous
shocks to avoid the misspecification of the model.
ɣƙ = ќ/ƙ = s Aƙα+η−1
- (n+δ)·ƙ = sδ) --------Equation 3.1
ɣƙ = C +δ)·ƙ = s βƙ +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation 3.2
ɣƙ = C +δ)·ƙ = s βƙ +δ)·ƙ = s ϵ -------Equation 3.3
ɣ= ɣƙ - Ʀ = C +δ)·ƙ = s βƙ +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation 3.4
ɣ=ɣƙ - Ʀ = C +δ)·ƙ = s βƙ +δ)·ƙ = s ϵ --------Equation3.5
ɣƙ: the annual growth rate of capital per capita
C: constant term (intercept)
ɸ: impulse indicators to capture hit-and-run impacts of exogenous shocks
Ԑ: error term of the estimated equation of annual growth rate of GDP per capita; Ԑ has to
be white noise
ϵ: error term of co-integration equation; ϵ has to be stationary
Ʀ: an average of the annual growth rate of GDP per capita
The economic model demonstrates the annual growth rate of capital per capita (ɣƙ) in
Equation 3.1, and then we specify the econometric model as Equation 3.2 that shows the
econometric model to estimate the endogenous growth model for Japan, India, and, U.S.
The intercept of Equation 3.2 incorporates the shift of the annual growth rate of capital
per capita in Graph 2.5.
Equation 3.2 provides stable and efficient parameters with white noise error terms to
demonstrate the economic interaction between the annual growth rate of GDP per capita
and GDP per capita. The significant impulse indicators mean the state of the economy is
15

deviating from the steady state in the face of impacts of exogenous shocks. Equation 3.3
demonstrates the co-integration equation (steady state) between the annual growth rate
of GDP per capita and GDP per capita. The definition of co-integration is that the
endogenous variables are not necessary to be stationary but the first differences of
endogenous variables have to be stationary, and the error term of the co-integration
equation has to be stationary. We applied Augmented Dickey-Fuller test to test for the
stationarity of endogenous variables, and the results are demonstrated in (Table4.1, Table
4.2, Table4.3, Table4.4, and Table4.5). The co-integration of endogenous variables
indicates the existence of the stable steady state of endogenous variables. However,
without the impulse indicators, the parameter of co-integration equation is not stable and
efficient enough to interpret the economic interaction of endogenous variables. The
significant impulse indicators of Equation 3.2 and Equation 3.4 demonstrate the current
economy is converging to steady state with stable growth rate if co-integration
relationship holds between endogenous variables; otherwise, the economy will diverge to
a saddle point (unstable steady state).
Equation 3.4 demonstrates the econometric model to estimate the endogenous growth
model for France. Equation 3.5 demonstrates the co-integration equation of endogenous
growth model of France. The aggregate impacts of exogenous parameters of France
approach a fixed constant. Thus, the constant term of an econometric model is not able to
incorporate the aggregate effects of exogenous parameters. Graph 2.5 demonstrates the
intercept shift of annual growth of GDP per capita of France is close to the average annual
growth of GDP per capita of France. The transformed annual growth rate of France should
be the annual growth rate of GDP per capita minus the average of the annual growth rate
of GDP per capita of France (2.14%).
The regression results of Equation 3.2 and Equation 3.4 demonstrate the economic
interaction between the annual growth rate of GDP per capita and GDP per capita. ɣƙ is
equal to zero at steady state so we can obtain the optimal GDP per capita at steady state
based on the regression results. We will demonstrate the results in Section 4 (Empirical
Results). As long as we obtain the optimal GDP per capita at steady state, we can pursue
to the second step to specify the econometric model for the rate of convergence for the
selected countries.
3.4.2 Step Two
We specify the econometric model of the rate of convergence based on the paper of
Mankiw (1995). The model is specified as follows:
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ќ = λ· (ƙ - ƙss
) +δ)·ƙ = s e-------Equation3.6
ќ: the first difference of GDP per capita; ƙss
: GDP per capita at steady state
e: the error term of estimation equation of rate of convergence; e has to be white noise
We have obtained the optimal GDP per capita at steady state in the first step, and then we
could specify a model for the rate of convergence as Equation 3.6. Equation 3.6
demonstrates the econometric model to estimate the rates of convergence for all selected
countries. λ is the convergence rate of GDP per capita. λ measures how quickly the
economy approach to the steady state.
4. Empirical Results
The previous section demonstrates the framework of econometric models for endogenous
growth. However, this section displays all results of the econometric models and explains
the results in detail. According to the framework of econometric models in Section 3, we
demonstrate the empirical results in two steps. The first step would describe how to
obtain the optimal GDP per capita at steady state for all well-specified models and also
demonstrate the importance of co-integration tests for all models. The second step would
apply the optimal GDP per capita at steady states to specify the models for rates of
convergence.
4.1 The First Step
The first step would demonstrate how to obtain the optimal GDP per capita at steady
states, and we also found the results of co-integration tests in relation to convergence,
which means the results of co-integration tests would support the models that converge
to a stable steady state. If the co-integration relation does not hold between endogenous
variables, the model should diverge to an unstable steady state (saddle point).
ɣJapan = 4.51394 -0.00015671·JAPANGDPPERCAPITA +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation4.1
ɣIndia = -1.97814 +δ)·ƙ = s 0.00241· INDIAGDPPERCAPITA +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation4.2
ɣUS = 2.50773 - 0.00007282 · USGDPPERCAPITA +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation4.3
ɣFRANCE = 3.4563 - 0.00011781· FRANCEGDPPERCAPITA +δ)·ƙ = s ɸ +δ)·ƙ = s Ԑ -------Equation4.4
ɣFRANCE = annual growth rate of GDP per capita of France - GDP per capita of France
(2.14%)
ɣFRANCE: the transformed annual growth rate of GDP per capita of France
17

ɸ: impulse indicators
The above equations demonstrate stable and efficient parameters to interpret economic
interactions between endogenous variables at steady states for selected countries,
although the current economic states are approaching to steady states. The error terms of
Equation4.1, Equation4.2, Equation4.3, and Equation4.4 are white noise, and the detail
regression results are demonstrated elaborately in Appendix. The results demonstrate
negative relationships between annual growth rates of GDP per capita and GDP per capita
for Japan, US, and France, which means the economies of Japan, US and France would
converge to stable steady states. The regression results show the positive relationship
between Indian annual growth rate of GDP per capita and GDP per capita, which indicates
that the economy of India would diverge to an unstable steady state (saddle point). The
regression results are consistent with the scatter plot results of Graph 2.5.
At steady states, the expected values of annual growth rates of GDP per capita of all
selected countries are equal to zero without impulse indicators (ɸ). Thus, we obtain the
following results with stable and efficient parameters:
0 = 4.51394 - 0.00015671·JAPANGDPPERCAPITAss
-> JAPANGDPPERCAPITAss
= 28804.42
0 = -1.97814 +δ)·ƙ = s 0.00241· INDIAGDPPERCAPITAss
-> INDIAGDPPERCAPITAss
= 820.8
0 = 2.50773 - 0.00007282 · USGDPPERCAPITAss
-> USGDPPERCAPITAss
= 34437.38
0 = 2.50773 - 0.00007282· FRANCEGDPPERCAPITAss
-> FRANCEGDPPERCAPITAss
=29337.92
4.1.1 Co-integration Results
The definition of co-integration states the endogenous variables are not necessary to be
stationary1
(Table 4.2, Table 4.3). But the first differences of endogenous variables2
(Table
1
Table 4.2 demonstrates annual growth rate of Japan is almost stationary; annual growth rates of GDP per
capita for other selected countries are stationary. Table 4.3 demonstrates GDP per capita for selected
countries are not stationary (P-value > 5%) by using Augmented Dickey-Fuller test.
18

4.4, Table 4.5) are stationary, and the error terms of co-integration equations are
stationary (Table 4.1). The co-integration equations are demonstrated as followings:
ɣJapan = 9.37562 - 0.00019393·JAPANGDPPERCAPITA +δ)·ƙ = s e-------Equation4.5
ɣIndia = 0.6191 +δ)·ƙ = s 0.00401· INDIAGDPPERCAPITA +δ)·ƙ = se-------Equation4.6
ɣUS = 4.06853 - 0.00005759· USGDPPERCAPITA +δ)·ƙ = s e -------Equation4.7
ɣFRANCE = 4.3147 - 0.0001423· FRANCEGDPPERCAPITA +δ)·ƙ = se-------Equation4.8
ɣFRANCE = annual growth of GDP per capita of France - GDP per capita of France (2.14%)
e: the error terms of co-integration equations
Table 4.1 Augmented Dickey-Fuller Tests for Residuals
1
2
3
4
5 <.0001 <.0001 0.9999 <.0001
P-Value
Augmented Dickey-FullerUnit Root Tests for the Residuals ofCo-integration Equation (Ho: Nonstationary )
<.0001 0.0001 0.0001 <.0001
<.0001 0.0001 0.0001 <.0001
<.0001 <.0001 <.0001 <.0001
<.0001 <.0001 0.0001 <.0001
Lags
Japan (Zero Mean) India (Zero Mean) US (Zero Mean) France (Zero Mean)
Table 4.2 Augmented Dickey-Fuller Tests for the Annual Growth Rates of GDP per Capita
1
2
3
4
5
P-Value
AugmentedDickey-FullerUnitRootTests forAnnualGrowthRate PerCapita( Ho: Nonstationary)
0.2202 0.4773 0.0004 0.0001
0.3451 0.2071 0.0004 0.0001
0.1479 0.0235 0.0005 <.0001
0.0517 0.0167 0.0005 <.0001
0.0042 0.0005 0.0005 <.0001
Lags
Japan(Single Mean) India(Single Mean) US(Single Mean) France (Trend)
Table 4.3 Augmented Dickey-Fuller Tests for GDP per Capita
2
Table 4.5 demonstrates the first differences of GDP per capita for Japan, U.S., and France are stationary but
not India.
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1
2
3
4
5
P-Value
AugmentedDickey-FullerUnit RootTests forGDP PerCapita( Ho: Nonstationary)
0.8726 0.9999 0.0036 0.9741
0.9191 0.9999 0.1415 0.9399
0.9413 0.9999 0.1586 0.9276
0.9536 0.9999 0.1126 0.9622
0.9378 0.9999 0.0903 0.9042
Lags
Japan(Trend) India (Trend) US(Trend) France (Trend)
Table 4.4 Augmented Dickey-Fuller Tests for the First Difference of the Annual Growth
Rates of GDP per Capita
1
2
3
4
5
P-Value
Augmented Dickey-Fuller Unit Root Tests forthe First Difference ofAnnual Growth Rate PerCapita (Ho: Nonstationary )
0.99990.9999 0.9999 0.9999
0.9999
0.9999 0.9999 0.9999 0.9999
0.9999 0.9999 0.9999
0.0001
0.9999 0.0001 0.0001 0.0001
0.0001 0.0001 0.0001
Lags
Japan (Zero Mean) India (Zero Mean) US (Zero Mean) France (Zero Mean)
Table 4.5 Augmented Dickey-Fuller Tests for the First Difference of GDP per Capita
1
2
3
4
5 0.014
P-Value
AugmentedDickey-FullerUnitRootTests forthe FirstDifference ofGDPPerCapita(Ho:Nonstationary)
0.0087 0.9972 0.0001
0.0004
0.003 0.9962 0.0001 0.0004
0.0004 0.9933 0.0001
0.0005
0.0005 0.9816 0.0005 0.0005
0.0005 0.6007 0.0005
Lags
Japan(Single Mean) India(Single Mean) US(Single Mean) France (Single Mean)
The results of co-integration test demonstrate the co-integration relationships hold for the
endogenous models of Japan, US, and France; however, the co-integration relationship
does not hold for the endogenous model of India. The co-integration relationship of
endogenous variables ensures the models would converge to stable steady states. The co-
integration relationship does not hold for the model of India. The model of India diverges
to an unstable steady state (saddle point). Thus, the co-integration ensures the stability of
endogenous variables.
Although the co-integration ensures the stability of endogenous variables to approach to a
stable steady state, the parameters of co-integration equation are not stable and efficient
because the error terms of co-integration equations are not white noise. The parameters
20

of co-integration equations are not able to interpret economic interactions of endogenous
variables at steady states. Thus, we have to apply the parameters of Equarion4.1,
Equation4.2, Equation4.3, and Equation4.4 to interpret the economic activities at steady
states and then obtain the optimal GDP per capita at steady states for all selected
countries.
4.2 The Second Step
Equation 3.6 demonstrates the framework of the rate of convergence3
. After we obtain
the optimal value of GDP per capita at steady state, we can estimate the rate of
convergence as followings:
DJAPANGDPPERCAPITA = λ · (JAPAGDPPERCAPITA - JAPANGDPPERCAPITAss
) +δ)·ƙ = s e---
Equation4.9
DINDIAGDPPERCAPITA =λ· (INDIAGDPPERCAPITA - INDIAGDPPERCAPITAss
) +δ)·ƙ = s e---
Equation4.10
DUSGDPPERCAPITA = λ· (USGDPPERCAPITA - USGDPPERCAPITAss
) +δ)·ƙ = s e-------
Equation4.11
DFGDPPERCAPITA =λ· (FRANCEGDPPERCAPITA-FRANCEGDPPERCAPITAss
) +δ)·ƙ = s e----
Equation4.12
Note: JAPANGDPPERCAPITAss
=28804.42, INDIAGDPPERCAPITAss
=820.8,
USGDPPERCAPITAss
=34437.38, FRANCEGDPPERCAPITAss
=29337.92
DJAPANGDPPERCAPITA: the first difference of GDP per capita of Japan
DINDIAGDPPERCAPITA: the first difference of GDP per capita of India
DUSGDPPERCAPITA: the first difference of GDP per capita of US
DFGDPPERCAPITA: the first difference of GDP per capita of France
Table4.6 Regression Results for GDP per Capita at the Steady States and the Rate of
Convergence for Selected Countries
3
Table4.6 demonstrates the regression results for GDP per capita at steady states and the rates of
convergence for all selected countries.
21

COUNTRY JAPAN INDIA US FRANCE
GDP at Steady State 28804.42 820.8 34437.38 29337.92
The Rate of Convergence -1.54% 5.95% -0.57% -2.69%
Graph 2.5 provides the simple scatter plots between annual growth rates of GDP per
capita and GDP per capita, which provides the outline to judge the convergence or
divergence of the economies. The empirical results provide the overall results of the test
of convergence, which include how to obtain the optimal GDP per capita ate steady states,
judge if the models would converge to a stable steady state or diverge to a saddle point,
and estimate the rate of convergence for all selected countries. The economies of Japan,
US, and France converge to stable steady states. The impacts of the invention of new
technology are insignificant. The markets of these countries are perfectly competitive. The
rate of convergence of these countries is lower than the rate of convergence of
developing countries. The economies of these countries are consistent with Case 1 of
economic model in Section 3. The economy of India would diverge to an unstable steady
state (saddle point), and the impacts of the invention of technology would be significant.
The market of Indian is a monopoly with market power. The rate of convergence of India
is higher than the rate of convergence of other developed countries (Japan, US, and
France). The economy of India is consistent with Case 2 of the economic model in Section
3.
5. A Brief Discussion of the AK Model
Economists have been discussing the existence of AK model of endogenous growth for
several decades. The AK model could occur when the diminishing return rate of physical
capital trade off the increasing return rate of human capital. The accumulation of human
capital encounters the diminishing returns. The market is perfectly competitive because
AK model exhibits constant returns to scale in the accumulated factors of production; this
model can generate perpetual (permanent) growth without the assumption of exogenous
shifts in the production function. We will discuss AK model of endogenous growth by using
the economy of Indonesia.
Graph 5.1 shows the economy of Indonesia would barely diverge to an unstable steady
state, while Graph 5.2 illustrates that the economy of Indonesia has no steady state in the
22

aggregate level of the annual growth rate of GDP in relation to GDP per capita. The
aggregate level of the annual growth rate of GDP in relation to GDP per capita would
demonstrate the economy of Indonesia would approach to AK model, which can generate
perpetual (permanent) growth without the assumption of exogenous shifts in the
production function. Because there is no steady state in AK models, we are not able to
apply econometric model to obtain the GDP at the steady state and the rate of
convergence to steady state.
Graph 5.1 Test for Convergence of Indonesia (1)
Graph 5.2 Test for Convergence of Indonesia (2)
Looking at the annual growth of Indonesia’s GDP, we notice that the global financial crisis
(2008) did not affect Indonesia severely (Graph 5.3). The annual growth of aggregate GDP
of Indonesia and annual growth of GDP per capita have the same movement. The impacts
of Asian financial crisis (1997) hit the economy of Indonesia seriously.
23

Graph 5.3 Annual Growth of GDP of Indonesia
We discuss AK model in this section. We believe the AK model and Indonesia should be a
topic for further research papers. Indonesia and other Asian countries suffer from Asian
financial crisis severely. However, Indonesia got better recovery than other countries. AK
model can generate perpetual (permanent) growth without the assumption of exogenous
shifts in the production function, which might assist Indonesia to get out of the recession
from Asian financial crisis efficiently. The market of Indonesia should turn from monopoly
to perfect competition. However, there is no transitional dynamic and steady state for AK
model at current economic state for Indonesia. As long as Indonesia accumulate sufficient
physical capital and human capital, GDP per capita of Indonesia would obtain significantly
improved, the impacts of externality would not be significant. The economy of Indonesia
might converge to stable steady state as well as the neo-classical model.
6. Conclusion
In the history, there are industrial revolutions with the new invention of technologies.
New technology changes our lives and then becomes part of our lives. We could say new
technology generates one-time shock impacts of the economy and we assume the effects
24

of new technology are exogenous and constant, which are externalities of the economy.
The new technology generates more physical capital and human capital in the economy.
The new technology creates different impacts for developed countries and developing
countries. For developed countries, the physical capital and human capital of developed
are a lot higher than developing countries. Developing countries have small capital stocks.
Although developed countries have higher human capital stocks, the increasing return of
human capital of developed nations is not as high as emerging ones. The marginal
production of the capital of developing countries is higher than developed countries,
which makes capital flows from developed countries to developing countries. Thus, the
growth rates of developing countries are higher than developed countries.
Romer (1986) proposed new endogenous growth model with externality to demonstrate
the impacts of new technology on the economy. For developed countries (Japan, US, and
France), the effects of externalities are insignificant; the economies are close to neo-
classical economies and converge to stable steady states. The markets are perfectly
competitive. For developing countries, the impacts of externalities are significant. The
economies diverge to an unstable steady state (saddle point) with imperfect completive
markets, or the economies could have no steady state with the perfectly competitive
market (AK model). If the increasing return of human capital dominates the decreasing
return of physical capital, the economy of developing country will diverge to a saddle
point. If the increasing returns of human capital trade-off the decreasing return of
physical capital, the economy of developing country would approach AK model. This paper
confirms when the co-integration relation holds between the annual growth rate of GDP
per capita and GDP per capita; the economy would converge to a steady state; if it does
not hold, the economy will diverge to a saddle point. This paper demonstrates the results
of convergence test including how to obtain GDP per capita and convergence rates at
steady states for selected countries in detail and also discuss the economy of Indonesia
could be the case of AK model. We accomplish our work to dedicate the contribution of
economists to the endogenous growth model.
25

Appendix:
India
Step 1: Find the optimal GDP per capita at the steady state of India.
Dependent variable is annual growth rate of GDP per capita of India.
Root MSE 0.64386 R-Square 0.9638
Dependent Mean 3.3379 Adj R-Sq 0.9602
CoeffVar 19.28934
Parameter Standard
Estimate Error Standard
Error
Intercept 1 -1.97814 0.27052 -7.31 <.0001 0.20896 -9.47 <.0001 0
INDIAGDPPERCAPITA 1 0.00241 0.00021631 11.15 <.0001 0.00018742 12.87 <.0001 1.09086
IDUM1 1 3.73002 0.29652 12.58 <.0001 0.21669 17.21 <.0001 1.29906
IDUM2 1 -1.55698 0.18408 -8.46 <.0001 0.16105 -9.67 <.0001 1.12099
IDUM3 1 2.73219 0.18883 14.47 <.0001 0.17058 16.02 <.0001 1.19806
IDUM4 1 -5.18208 0.49409 -10.49 <.0001 0.3357 -15.44 <.0001 1.1357
Variable t Value Pr > |t|
Parameter Estimates for Model of India
Heteroscedasticity Consistent
DF t Value Pr > |t|
Variance
Inflation
The steady state INDIAGDPPERCAPITA= (1.97814/0.00241)= 820.8
Durbin-Watson D 1.932
Number of Observations 56
1st Order Autocorrelation 0.015
Parameter Symbol Estimate
Mean Mu 0
Std Dev Sigma 0.613896
Test
Kolmogorov-Smirnov D 0.09513858 Pr > D >0.150
Cramer-von Mises W-Sq 0.06131651 Pr > W-Sq >0.250
Anderson-Darling A-Sq 0.40615653 Pr > A-Sq >0.250
p ValueStatistic
Parameters for Normal Distribution
Goodness-of-Fit Tests for Normal Distribution
Step 2: Find the stable growth rate of GDP per capita of India
26

CoeffVar 15.89832
Parameter Standard
Error
Intercept 1 20.91507 1.46528 14.27 <.0001 1.55237 13.47 <.0001 0
INDIASS 1 0.0595 0.00159 37.5 <.0001 0.00167 35.69 <.0001 1.19146
RIDUM1 1 15.07383 1.71336 8.8 <.0001 1.87688 8.03 <.0001 1.36819
RIDUM2 1 -8.7521 1.27478 -6.87 <.0001 1.18793 -7.37 <.0001 1.09205
RIDUM3 1 10.32725 1.51605 6.81 <.0001 1.09638 9.42 <.0001 1.56945
RIDUM4 1 21.56937 2.26909 9.51 <.0001 2.41325 8.94 <.0001 1.36726
Parameter Estimates for Model of India
Variance
InflationVariable DF t Value Pr > |t|
t Value Pr > |t|
The stable growth rate diverging to a saddle point (unstable steady state) of India is
0.0595.
Mean Mu 0
Test
Statistic p Value
US:
Step 1: Find the optimal GDP percapita at the steady state of US.
Dependent variable is annual growth rate of GDP per capita of US.
CoeffVar 28.41235
27

Parameter Standard
Error
Intercept 1 2.50773 0.30731 8.16 <.0001 0.4441 5.65 <.0001 0
USGDPPERCAPITA 1 -0.00007282 7.5E-06 -9.72 <.0001 7.88E-06 -9.24 <.0001 1.08544
UDUM1 1 3.14008 0.21141 14.85 <.0001 0.26328 11.93 <.0001 1.17727
UDUM2 1 -1.76926 0.17445 -10.14 <.0001 0.1846 -9.58 <.0001 1.22916
UDUM3 1 2.30261 0.20665 11.14 <.0001 0.15063 15.29 <.0001 1.39756
Variable DF t Value Pr > |t|
t Value Pr > |t|
Parameter Estimates for Model of US
Variance
Inflation
The steady state USGDPPERCAPITA= (2.50773/0.00007282) =34437.37984
1st Order Autocorrelation -0.026
Mean Mu 0
Test
Statistic p Value
Step 2: Find the stable growth rate of GDP per capita of US
CoeffVar 16.98732
Parameter Standard
Error
Intercept 1 8.9137 44.93412 0.2 0.8436 49.00046 0.18 0.8564 0
USSS 1 -0.00572 0.0014 -4.09 0.0002 0.00132 -4.32 <.0001 1.02649
RUDUM1 1 843.08977 44.24133 19.06 <.0001 42.2722 19.94 <.0001 1.1416
RUDUM2 1 -394.9506 30.44639 -12.97 <.0001 34.11445 -11.58 <.0001 1.05833
RUDUM3 1 587.84322 33.06683 17.78 <.0001 36.55981 16.08 <.0001 1.12961
RUDUM4 1 -1324.58677 80.92743 -16.37 <.0001 24.4364 -54.21 <.0001 1.07688
Parameter Estimates for Model of US
Variance
Inflation
t Value Pr > |t|
The stable growth rate converging to a stable steady state of US is 0.00572.
28

Mean Mu 0
Test
Statistic p Value
France:
Step 1: Find the optimal GDP percapita at the steady state of France.
Dependent variable is annual growth rate of GDP per capita of France.
Dependent Mean -0.00459 Adj R-Sq 0.9559
CoeffVar -8677.73562
Parameter Standard
Error
Intercept 1 3.4563 0.20401 16.94 <.0001 0.17738 19.49 <.0001 0
FRANCEGDPPERCAPITA 1 -0.00011781 0.00000627 -18.8 <.0001 0.00000541 -21.77 <.0001 1.08382
FDUM1 1 1.82828 0.11031 16.57 <.0001 0.10407 17.57 <.0001 1.04967
FDUM2 1 -0.99143 0.12043 -8.23 <.0001 0.09472 -10.47 <.0001 1.25112
FDUM3 1 -2.72724 0.20186 -13.51 <.0001 0.2833 -9.63 <.0001 1.16718
Variance
Inflation
ParameterEstimates forModel of France
Variable DF tValue Pr>|t|
HeteroscedasticityConsistent
tValue Pr>|t|
The steady state FRANCEGDPPERCAPITA= (3.4563/0.00011781)=29337.91698
Mean Mu 0
Test
Statistic p Value
Step 2: Find the stable growth rate of GDP per capita of France
29

CoeffVar 20.91544
Parameter Standard
Error
Intercept 1 446.15411 25.24861 17.67 <.0001 21.81685 20.45 <.0001 0
FRANCESS 1 -0.02693 0.00178 -15.09 <.0001 0.00178 -15.14 <.0001 1.11228
RFDUM1 1 421.67458 33.50593 12.59 <.0001 30.24962 13.94 <.0001 1.31475
RFDUM2 1 -218.67875 30.16153 -7.25 <.0001 23.57055 -9.28 <.0001 1.06539
RFDUM3 1 386.15133 44.73396 8.63 <.0001 38.85144 9.94 <.0001 1.28338
RFDUM4 1 -1015.97424 66.93262 -15.18 <.0001 144.3399 -7.04 <.0001 1.08263
Parameter Estimates for Model of France
Variance
Inflation
t Value Pr > |t|
The stable growth rate converging to a stable steady state of France is 0.02693.
1st Order Autocorrelation 0
Mean Mu 0
Test
Statistic p Value
Japan:
Step 1: Find the optimal GDP percapita of Japan at the steady state.
Dependent variable is annual growth rate of GDP per capita of Japan.
CoeffVar 25.20954
30

Intercept 1 4.51394 0.44526 10.14 <.0001 0.55259 8.17 <.0001 0
JGDPPERCAPITA 1 -0.00015671 0.0000098 -15.99 <.0001 0.00000826 -18.97 <.0001 1.19319
JDUM1 1 4.44399 0.33213 13.38 <.0001 0.4576 9.71 <.0001 1.06331
JDUM2 1 -2.49876 0.23595 -10.59 <.0001 0.22325 -11.19 <.0001 1.21253
JDUM3 1 5.01378 0.29817 16.82 <.0001 0.29856 16.79 <.0001 1.39664
t Value Pr >|t|
Variance
Inflation
Standard ERROR
Standard ERROR
Variable t Value Pr >|t|Parameter ERROR
Parameter Estimates for Model of Japan
DF
The steady state JGDPPERCAPITA= (4.51394/0.00015671)=28804.4158
Mean Mu 0
Test
Shapiro-Wilk W 0.988035 Pr < W 0.8513
Tests for Normality
Statistic p Value
Step 2: Find the stable growth rate of GDP per capita of Japan
CoeffVar 26.476
Parameter Standard
Error
Intercept 1 -12.23152 80.81876 -0.15 0.8803 80.31201 -0.15 0.8796 0
JAPANSS 1 -0.01536 0.00219 -7.01 <.0001 0.00181 -8.48 <.0001 1.04418
RJDUM1 1 984.55057 81.58071 12.07 <.0001 79.77229 12.34 <.0001 1.20167
RJDUM2 1 -551.97999 50.8594 -10.85 <.0001 50.03882 -11.03 <.0001 1.04806
RJDUM3 1 952.16883 60.33682 15.78 <.0001 59.27475 16.06 <.0001 1.06814
RJDUM4 1 -1663.35318 201.397 -8.26 <.0001 81.10363 -20.51 <.0001 1.17698
Parameter Estimates for Model of Japan
Variance
Inflation
t Value Pr > |t|
The stable growth rate converging to a stable steady state of Japan is 0.01536.
31

Mean Mu 0
Test
Statistic p Value
Compliance with Ethical Standards:
Conflict of Interest:
I _____KELLY YIYU LIN_______________- as corresponding author, certify that all authors
contributed significantly to the manuscript and no other authors were involved and were not part
of the team that created this manuscript. All used data are correct and results are based on
authors’ calculations. Author(s) is/are solely responsible for the entire content and results of the
paper published in the journal.
I certify that no part of this manuscript has been published before or is not under consideration for
publication elsewhere.
I certified that this manuscript was used as working paper only.
Title of the
manuscript
The Importance of Parameter Constancy for Endogenous
Growth with Externality
Authors’ name / affiliation / official email address Percentage
of
contributio
n
1. KELLY YIYU LIN 70%
2.WENTI DU 30%
32

Ethical Approval: This article does not contain any study
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