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Book ChapterDOI

Towards Golden Rule of Capital Accumulation: A Genetic Algorithm Approach

16 Jun 2017-pp 481-491

TL;DR: 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%.

AbstractThe 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 attaining 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 0142% and standard deviation 0021%

Summary (2 min read)

1 Introduction

  • Economic growth depends on several factors; one of them is consumption.
  • Motivated by this, in the current work the authors consider Genetic Algorithm (GA) to maximize consumption.
  • Generally, the GA has numerous advantages including its flexibility to model the problem’s constraints, and its easy convergence to the optimal solution inspired by Darwinian principle [6].
  • The simulation results indicated that the GA is capable of capturing different features from the experimental nature of the subjects under consideration.
  • Next in section 3 the economical steady state is introduced and explained.

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.
  • At maximum consumption rate, economic growth will take place.
  • The aforesaid formalism is mathematically established using a basic Coubb-Douglaus production function [11].
  • There is a positive relationship between investment per worker and savings rate.

3 Economical Steady State

  • From the Figure 4 the authors depict how the economy approaches the steady level of capital.
  • Now, to examine whether at point ‘𝑀’ the economy reached steady state level of capital or not the authors take two points.
  • The first point is point ‘𝐴’which is below the steady state level.
  • At point ‘𝐴’it can be observed that the investment curve is steeper than the depreciation, so here investment is greater than the depreciation thus, if investment takes place it would enrich the capital stock which would lead to higher output.
  • If investment takes place it would shrink the capital stock as depreciation is far greater than the investment.

4 Golden Rule level of Capital

  • After achieving the steady state level of capital, there is a state where the authors maximize the consumption per worker that is generally known as the Golden rule level of capital [11, 18].
  • There are many factors which decide the growth of an economy.
  • 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].
  • The slope of this curve is 𝑑𝑦 𝑑𝑘 which is denoted by 𝑀𝑃𝑘.
  • Where, if one unit of capital is added, output will increase more than the depreciation.

5 Genetic Algorithm based methodology

  • The model is highly inspired from Darwinian principle of Natural Evolution, and involves a population which participates in finding the solution of a particular problem under consideration.
  • In the proposed work, the GA is applied to determine the optimal consumption value, where the GA proves its effectiveness for superior convergence toward global optimization compared to other global search algorithm.
  • The chromosome representation considered in the current study is as follows; Algorithm: Genetic Algorithm Start Generate random chromosomes’ population that represents solutions Evaluate the population-fitness Create new population using the following steps:.
  • Further, each of them is associated with a fitness value indicating the superiority of that particular solution.
  • The genetic operators such as crossover, mutation and several other versions of these two are popular [7-8].

6 Results & Discussion

  • The experiments are carried out using Intel Core i3 4GB Machine.
  • Depending on the difficulty of the problem being solved the initial size of population is set to 200 and maximum number of generation is set to 500.
  • Single point cross over strategy is adopted with a crossover probability of 0.15 which indicates that a significantly low probability would be sufficient to ensure the better quality population, while it is kept low in order to prevent the process from being trapped in local optima.
  • The objective function is already described and explained in section 4 (equation 7).
  • 𝛿 denotes depreciation rate and 𝑘∗ denotes amount of capital accumulated.

7 Experimental results

  • Table 2 reports the performance of Genetic Algorithm.
  • Azadeh, Ali, SeyedFaridGhaderi, B. PourvalikhanNokhandan, and Mohammad Sheikha- lishahi.

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© 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 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 problem’s constraints,
and its easy convergence to the optimal solution inspired by Darwinian princi-
ple [6]. Nicoară [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].
󰇛 󰇜
󰇛
󰇜
󰇛 󰇜
󰇛
󰇜
󰇛󰇜
Where, denotes capital per worker, denotes output per worker,  is con-
sumption per worker,  is investment per worker,  is total output,  de-
notes total labor and is total capital.
Suppose, a firm earns  and saves  fraction of. 󰇛 󰇜 fraction of 
goes to its consumption. Hence;

󰇛
 
󰇜
󰇛󰇜
Where,  is income per worker. And the relation between income per worker
and consumption is given by;
 󰇛󰇜
From (2) and (3) we get;
󰇛
 
󰇜

󰇛
󰇜
󰇛󰇜
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, 
󰇛
󰇜
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;
  󰇛󰇜
From equation (4) we get;

󰇛
󰇜

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 point. Now, to ex-
amine whether at point  the economy reached steady state level of capital or
not we take two points. The first point is point which is below the steady
state level. And the second point is point  which is above the steady state
level. At point it 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 and moves toward point
 while in case of the point  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 point.
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;
 󰇛󰇜
 

As, the difference between output and depreciation curve is zero (At point 
of Figure 4) Hence, 

󰇛
󰇜
󰇛󰇜



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Frequently Asked Questions (12)
Q1. What are the contributions mentioned in the paper "Towards golden rule of capital accumulation: a genetic algorithm approach" ?

The traditional Solow model based approach is well-studied and computationally complex. 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 %. 

In the current study a wellknown Darwinian principle inspired metaheuristic Genetic Algorithm is employed in maximizing the consumption (The Golden rule of capital accumulation) in terms of depreciation and capital accumulation. 

Initial condition to achieve the Golden rule level of capital is; marginal product of capital should be equals to depreciation rate. 

Maximization of consumption is traditionally done by employing a long time consuming and computationally complex Solow modelbased mathematical approach. 

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]. 

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. 

𝑀’.After achieving the steady state level of capital, there is a state where the authors maximize the consumption per worker that is generally known as the Golden rule level of capital [11, 18]. 

The optimum results indicated that in maximizing the consumption per worker the value of depreciation rate (𝛿) tends to the lower bound of the given range which indicates a lower depreciation rate. 

At point ‘𝐴’it can be observed that the investment curve is steeper than the depreciation, so here investment is greater than the depreciation thus, if investment takes place it would enrich the capital stock which would lead to higher output. 

The capital increases from the point ‘𝐴’and moves toward point ‘𝑀’ while in case of the point ‘𝐵’ the depreciation is steeper than the depreciation. 

The plots reveal that GA is extremely fast in maximizing the objective function and is better than traditional methods [18] because these involve calculation of mathematical differentiation of complex functions and is proved to be NP - Hard (exponential complexity)[17]. 

The solution generated by the proposed system is a pair of values consisting 𝛿 (depreciation rate) and 𝑘∗ (amount of capital accumulated) with in the given range of search.