# Towards Golden Rule of Capital Accumulation: A Genetic Algorithm Approach

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%.

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 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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##### Frequently Asked Questions (12)

###### Q2. What is the significance of the results of the current study?

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.

###### Q3. what is the condition to achieve the Golden rule level of capital?

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

###### Q4. How is the growth of society influenced by the consumption of the country?

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

###### Q5. What is the definition of the Golden rule of capital?

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].

###### Q6. How did Hommeset al. find the GA?

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.

###### Q7. What is the level of capital at which the economy reached steady state level?

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].

###### Q8. What is the significance of the results?

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.

###### Q9. What is the point where the economy reached steady state level of capital?

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.

###### Q10. What is the level of capital at the point ‘M’?

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.

###### Q11. What is the difference between the two plots?

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].

###### Q12. What is the solution generated by the proposed system?

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.