Technology and Investment, 2013, 4, 24-26
Published Online Febr uary 2013 (
Copyright © 2013 SciRes. TI
A Localization of Solow Growth Model with Labor Growth
Pattern in China
Wang Wanxin, Guo Zequn
Faculty of Business and Economics, The University of Hong Kong, Hong Kong, China
Department of Automation, Tsinghua University, Beijing, China
Received 2012
This paper investigates the Solow Growth Model on a country-specific level by applying the demographic growth pat-
tern in China to it. To localize the neoclassic model, China population growth estimation function based on the Verhulst
Population Model is introduced to transform the population growth rate from a constant to a function, altering the orig-
inal model assumption. By inserting the population growth function into Solow's work, an economy growth phase dia-
gra m for China is obtained. MATLAB programming is used to depict the diagram in a three-dimensional space and to
show that the set of optimal cap ital-labor ratio values lies in the intersecting line of two planes rather than in the inter-
secting point of two curves in the ori ginal model setting. An neoclassical aggregate feasible growth path for China's
economy can be depicted based on a chosen optimal value. The dynamic equilibrium in this case should not be unique;
instead, capital-labor ratio together with population growth situation at a certain time point should be jointly taken into
consideration to solve t he optimization problem in the country's long term economy development.
Keywords: Solow Growth Model; China Population Growth; Optimal Economy Developmen t
1. Introduction
Researchers have been investigating on generalization of
the neoclassical Solow Growth Model [1] in different
directions by applying certain patterns or constraints to
different model factors. The standard neoclassic Solow
Growth Model has an essential assumption that the la-
bor(population) growth in the economy is constant. Ac-
cinelli and Brida [2], B ucci and Guerrini [3], Ferrara and
Guerr i ni [4] implemented several population (labor)
models, for example, Ramsey population model and
Uzawa-Lucase model [5] to Solow's work. However,
researches have not explored enough in terms of revising
the model to a regional level. China is a country worth
attention in terms of its demographical situation, with the
world's largest quantity and relatively high growing rate.
Solow's assumption about a constant population growth
does not suit in the case of China. By inserting a function
simulating China's population growth to the original
model set, a new phase diagram describing the country's
economy growth process can be depicted. The paper
helps study the dynamic equilibrium, namely the coun-
try's optimal growth pattern, in a more accurate stage,
with both the values of population growth and capi-
tal-labor ratio taken in to consideration.
2. The neoclassical Solow Growth Model
Consider the most general case: at time t, an economy
produces product called national product Yt with an ag-
gregate production function F
having two factors,
namely, labor (population) Lt and Kt capital. F is twice
differentiable and indicates the diminishing return of
each factor. By assuming population growths in a con-
stant continuous rate r we can get the neoclassical ag-
gregate feasible growt h path kt=f(k t)=(r+ s
)kt–ct, where
is the depreciation rate of capital, s is the propensity to
save. Also, the Solow's path (ksolow,csolow), which is called
the optimal balanced growth path, could be attained
through the formula:
()( )ksf krsk
tt t
= −+
. (1)
3. The localization to China Population
Growth Pattern
According to Luo's [6] modeling work of China's popu-
lation since year 2003, a function Pt, which simulates
annual demographical situation in China, is derived by
revising the Verhulst Population Model [8] with adjust-
ing to country-specific indexes and using Binary Bi-
nomial regression. The function is formulated as:
Copyright © 2013 SciRes. TI
0.0014( 2003)
t0.0014( 2003)
1 1.3279
, 2003t. (2)
The graph of Equation (2) is shown in Figure 1. And we
can get the growth rate of China's population r'
c by:
0.0014( 2003)
0.0014( 2003)
1 1.3279
= −
. (3)
The graph of Equation (3) is shown in Fig ure 2. We can
definitely see that the population growth rate in China is
not constant and we can also get that over
lim 0r
, which is expected in the neoclassical
model. In order to fit the discrete population growth rate
to the continuous case in Solow's model, we transform
ln(1 )rr
= +
, therefore satisfying:
Pe r
P Pe
= =
. (4)
With a certain population growth pattern localized to
China, we should get a revised growth model for China.
The revised optimal path should be described as:
Figure 1. China Population Simulation.
Figure 2. China Population Growth Rate (Simulated).
. (5)
We present the determination process of optimal capi-
tal-labor ratio, namely ksolow by depicting a phase diagram
(Figure 3) in three-dimensional space. By implementing
China's population growth model, we finally generalize
the neoclassical Solow Growth model to a coun-
try-specific level.
The three curve in the original model phase diagram turn
to three planes. The intersecting line reflecting the sit ua-
tion described in formula (1) is shown in the diagram,
pointing out a possible set of optimal capital-labor ratio
values of Chinas economy. The dynamic equilibrium of
the economy should be decided given consideration to
both population growth and capital-labor ratio situation
at a certain time point. To be specific, given a certain
year tm, corresponding population growth rate rcm and
capital-labor ratio
at the point can be obtained, and a
short term optimal economy growth path towards the
long term dynamic equilibrium point for the country can
be depicted, with . Since the population growth rate va-
ries continuously, the optimal path will turn slightly,
changing its slope possibly from km to km+1. Over a long
span of time, it should be expected that under Solow's
model settings with China population growth pattern
inserted, the country's economy develops to its long term
optimum following a path not straight, but with certain
radians. For other countries with diverse population
growth patterns, the same methodology is appropriate to
be adopted to describe economy development paths in a
country-specific level.
4. Acknowledgments
Sincere gratitude is extended to the following people
who gave me lot of insights during the research process:
Professor J. Wu, and Professor S. Ching, who directed
me to the field of mathematical economics and gave me
inspirations and advice.
Figure 3. Economy Phase Diagram (China)
All my peer students, who made great company during
my whole research period.
Copyright © 2013 SciRes. TI
[1] R.M. Solow, "A Contribution to The Theory of Economic
Growth," The Quarterly Journal of Economics, Vol. 70,
No. 1, 1956, pp. 65-94.
[2] E. Accinelli and J. G. Brida, "Population Growth and the
Solow-Swan Model", International Journal of Ecological
Economics & Statistics, Vol. 8, No. S07, 2007, pp. 54-63.
[3] A. Bucci and L. Guerrini, "Transitional Dynamics in the
Solow-Swan Growth Model with AK Technology and
Logistic Population Change, B.E. Journal of Macroeco-
nomics, Vol. No. 9, 2009, pp. 1-16.
[4] M. Ferrara and L. Guerrini, "The Ramsey model with
logistic population growth and Benthamite felicity func-
tion revisited", WSEAS Transactions on Mathematics,
Vol. 8, 2009, pp. 41-50.
[5] M. Ferrara and L. Guerrini, "A Note on the Uzawa-Lucas
Model with Unskilled Labor", Applied Science, Vol. 12,
2008, pp.90-95.
[6] L. Xiang 罗翔, "The Formation and Application of China
Population Growth Estimation Model" 中国人口增长预
测模型的建立与应用, Journal of Henan Institute of
Science and Technology
, Vol 36, No.
3, 2008.
[7] L. Guerrini, "The Solow-Swan Model with a Bounded
Population Growth Rate", Journal of Mathematical Eco-
nomics Vol. 42, 2006, pp.
[8] L. Guerrini, A Closed-form Solution to the Ramsey Mod-
el with Logistic Population Growth, Economic Modeling,
Vol. 27, 2010, pp.1178-1182.