ยฉ Springer-Verlag Berlin Heidelberg 2011
Towards Golden Rule of Capital Accumulation: A
Genetic Algorithm Approach
Sankhadeep Chatterjee
1
, Rhitaban Nag
2
, Soumya Sen
3
, Amitrajit Sarkar
4
1
Department of Computer Science and Engineering, University of Calcutta, Kolkata,
India, E-mail: chatterjeesankhadeep.cu@gmail.com
2
Department of Economics, Raja Peary Mohan College, Uttarpara, Hooghly, India,
E-mail: nag.rhitaban@gmail.com
3
A.K.Choudhury School of Information Technology, University of Calcutta, Kolkata,
India, E-mail โ soumyasen1@acm.org
4
Department of Computing, Ara Institute of Canterbury, Christchurch, New Zealand,
E-mail: sarkara@cpit.ac.nz
Abstract. The current study deals with maximizing consumption per worker in
connection with the economic growth of society. The traditional Solow model
based approach is well-studied and computationally complex. The present work
proposes a Genetic Algorithm (GA) based consumption maximization in attain-
ing the Golden rule. An objective function derived from traditional Solow model
based on depreciation rate and amount of accumulated capital is utilized. The
current study considered a constant output per worker to incorporate a constant
efficiency level of labor. Different ranges of Depreciation rate and accumulated
capital are tested to check the stability of the proposed GA based optimization
process. The mean error and standard deviation in optimization process is utilized
as a performance metric. The experimental results suggested that GA is very fast
and is able to produce economically significant result with an average mean error
0.142% and standard deviation 0.021%.
Keywords: Accumulated capital, Depreciation rate, Golden rule, Genetic Algo-
rithm, Metaheuristic, Solow model
1 Introduction
Economic growth of society directly impacts economy of a country. Economic
growth depends on several factors; one of them is consumption. Thus, con-
sumption maximization is an imperative task to ensure ever growing economic
stability. Solow model [11] deals with the growth of the economy in terms of
basic production, investment and depreciation. The steady level of capital plays
a vital role in the same. Maximization of consumption is traditionally done by
employing a long time consuming and computationally complex Solow model
based mathematical approach. Motivated by this, in the current work we con-
sider Genetic Algorithm (GA) to maximize consumption. Typically, the eco-
nomic problems can be framed as an optimization problem, such as cost mini-
mization, and revenue maximization. Thus, efficient optimization methods are
required to deliver accurate results. Consequently, several optimization algo-
rithms can be involved to solve the required maximum profit level, such meth-
ods include the genetic algorithm (GA), particle swarm optimization (PSO),
firefly algorithm, and cuckoo search algorithm (CS). Several studies have es-
tablished the ingenuity and accuracy of GA [1-4]. Generally, the GA has nu-
merous advantages including its flexibility to model the problemโs constraints,
and its easy convergence to the optimal solution inspired by Darwinian princi-
ple [6]. Nicoarฤ [12] revealed about the GA relevance compared to the tradi-
tional methods for manufacturing structure optimization. Geisendorf [13] em-
ployed the GA to solve Resource Economic problem using two different as-
sumptions to calculate the optimal extraction rate in order to achieve optimal
benefits. Arifovic [14] solved decision rules of future production and sales by
employing the GA in a competitive cobweb model in a market of single prod-
uct. The simulation results indicated that the GA is capable of capturing differ-
ent features from the experimental nature of the subjects under consideration.
Riechmann [15] established that the GA can be connected with the evolutionary
game theory. Hommeset al. [16] reveled that GA can converge to a series of
near Nash equilibrium solutions, where the heterogeneous agent behavior has
been modeled using GA.
The rest of the work is arranged as follows; first in section 2 the basic economic
background is introduced and mathematically explained. Next in section 3 the
economical steady state is introduced and explained. In section 4 Golden rule
of capital based on the Solow model is formulated. The appropriateness of
choosing the objective function is mathematically established. Section 5 intro-
duces the GA based proposed method. Section 6 reported the experimental set
up of GA and finally, Section 7 reveals experimental results. It discusses the
economic significance of the obtained results as well.
2 Basic Economic Background
The Solow model deals with growth of economy. It includes the living standard
of every citizen currently living in the economy. Their living standard depends
on various kinds of determinant; one of them is their income. The consumption
increases with increasing earning of citizens thereby raising the overall con-
sumption of economy. These results in potential growth in economic systems
as the consumption rate of citizens have increased inside that economy. Conse-
quently, consumption turns out to be a major indicator of growth. At maximum
consumption rate, economic growth will take place. The golden rule of capital
accumulation is a tool which has used to maximize the consumption. Golden
rule actually indicates steady state with maximum consumption. The aforesaid
formalism is mathematically established using a basic Coubb-Douglaus pro-
duction function [11].
๎ดณ ๏ ๎ต๓ฐ๎ดฎ๎ก ๎ดญ๓ฐ
๎ดณ
๎ดฎ
๏ ๎ต๓ฐ
๎ดฎ
๎ดฎ
๎ก
๎ดญ
๎ดฎ
๓ฐ
๎ต ๏ ๎ต๓ฐ๎ฅณ๎ก ๎ต๓ฐ
๎ต ๏ ๎ต
๓ฐ
๎ต
๓ฐ
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅณ๓ฐ
Where,๎ฎ๎ต๎ฏ denotes capital per worker,๎ฎ๎ต๎ฏ denotes output per worker, ๎ฎ๎ดฟ๎ฏ is con-
sumption per worker, ๎ฎ๎ต
๎ฏ is investment per worker, ๎ฎ๎ดณ๎ฏ is total output, ๎ฎ๎ดฎ๎ฏ de-
notes total labor and๎ฎ๎ดญ๎ฏ is total capital.
Suppose, a firm earns ๎ฎ๎ต๎ฏ and saves ๎ฎ๎ต๎ฏ fraction of๎๎ฎ๎ต๎ฏ. ๓ฐ๎ฅณ๎ ๏ ๎๎ต๓ฐ fraction of ๎ฎ๎ต๎ฏ
goes to its consumption. Hence;
๎ดฟ๎ ๏ ๎
๓ฐ
๎ฅณ๎๎ ๎๎ต
๓ฐ
๎ฏ ๎๎ต๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅด๓ฐ
Where, ๎ฎ๎ต๎ฏ is income per worker. And the relation between income per worker
and consumption is given by;
๎ต ๏ ๎ดฟ ๏
๎ต
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅต๓ฐ
From (2) and (3) we get;
๎ต ๏
๓ฐ
๎ฅณ๎๎ ๎๎ต
๓ฐ
๎ฏ ๎๎ต ๏
๎ต
๎ต
๏ ๎ต ๎ฏ ๎ต
๎ต
๏ ๎ต ๎ฏ ๎ต
๓ฐ
๎ต
๓ฐ
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅถ๓ฐ
Investment per worker depends on the savings rate of the firm (s). There is a
positive relationship between investment per worker and savings rate. As sav-
ing rate rises, investment per worker also rises. An increasing savings rate shifts
the investment per worker vs. capital per worker curve upward as depicted in
Figure 2.
Fig. 1.Capital per worker vs. output per
worker curve
Fig. 2.Combining output per worker and in-
vestment per worker curve. Where, ๎ต๎ต
๓ฐ
๎ต
๓ฐ
de-
notes investment per worker curve
Fig. 3.Relationship between capital and
total depreciation
Fig. 4.Combining Figure 1 and Figure 3. Depict-
ing economic steady state
The relationship between change in capital per worker and depreciation is
given as;
๎ฆฟ๎ต ๏ ๎ต
๏ ๎๎ท ๎ฏ ๎ต๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅท๓ฐ
From equation (4) we get;
๎ฆฟ๎ต ๏ ๎ต ๎ฏ ๎ต
๓ฐ
๎ต
๓ฐ
๏ ๎๎ท ๎ฏ ๎ต
Figure 3 depicts that depreciation is increased with increased capital per
worker.
3 Economical Steady State
From the Figure 4 we depict how the economy approaches the steady level of
capital. In Figure 4 we have shown the steady level is at point๎ฎ๎ดฏ๎ฏ. Now, to ex-
amine whether at point ๎ฎ๎ดฏ๎ฏ the economy reached steady state level of capital or
not we take two points. The first point is point ๎ฎ๎ดฃ๎ฏwhich is below the steady
state level. And the second point is point ๎ฎ๎ดค๎ฏ which is above the steady state
level. At point ๎ฎ๎ดฃ๎ฏit can be observed that the investment curve is steeper than
the depreciation, so here investment is greater than the depreciation thus, if in-
vestment takes place it would enrich the capital stock which would lead to
higher output. The capital increases from the point ๎ฎ๎ดฃ๎ฏand moves toward point
๎ฎ๎ดฏ๎ฏ while in case of the point ๎ฎ๎ดค๎ฏ the depreciation is steeper than the deprecia-
tion.If investment takes place it would shrink the capital stock as depreciation
is far greater than the investment. Thus, the level of capital will move down-
ward to the point๎ฎ๎ดฏ๎ฏ.
4 Golden Rule level of Capital
After achieving the steady state level of capital, there is a state where we max-
imize the consumption per worker that is generally known as the Golden rule
level of capital [11, 18]. The Golden rule addresses the question of presence of
any growth in an economy. There are many factors which decide the growth of
an economy. The current study focuses on the consumption. Maximization of
consumption is an imperative task in order to achieve economic growth. The
Golden rule of capital indicates a state (a unique value of depreciation rate and
amount of capital accumulated) that ensures maximum consumption level,
thereby assures a strong economic growth [18].
From equation 3 and 5 we get;
๎ดฟ ๏ ๎ต ๏ ๎ต
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅธ๓ฐ
๎ฆฟ๎ต ๏ ๎ต
๏ ๎๎ท ๎ฏ ๎ต
๎ต
๏ ๎๎ท ๎ฏ ๎ต
As, the difference between output and depreciation curve is zero (At point ๎ฎ๎ดฏ๎ฏ
of Figure 4) Hence, ๎ฆฟ๎ต ๏ ๎ฅฒ
๎ดฟ ๏ ๎ต
๓ฐ
๎ต
๎ฏ
๓ฐ
๏ ๎ท ๎ฏ ๎ต
๎ฏ
๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๎๓ฐ๎ฅน๓ฐ
