Wealth Distribution Model Research Articles

A wealth distribution model with a non-Maxwellian collision kernel

Abstract A non-Maxwellian collision kernel is employed to study the evolution of wealth distribution in a multi-agent society. The collision kernel divides agents into two different groups under certain conditions. Applying the kinetic theory of rarefied gases, we construct a two-group kinetic model for the evolution of wealth distribution. Under the continuous trading limit, the Fokker–Planck equation is derived and its steady-state solution is obtained. For the non-Maxwellian collision kernel, we find a suitable redistribution operator to match the taxation. Our results illustrate that taxation and redistribution have the property to change the Pareto index.

WE economy: Potential of mutual aid distribution based on moral responsibility and risk vulnerability.

Reducing wealth inequality is a global challenge that requires the transformation of the economic systems that produce inequality. The economic system comprises: (1) gifts and reciprocity, (2) power and redistribution, (3) market exchange, and (4) mutual aid without reciprocal obligations. Current inequality stems from a capitalist economy consisting of (2) and (3). To sublimate (1), the human economy, to (4), the concept of a "mixbiotic society" has been proposed in the philosophical realm. In this society, free and diverse individuals mix, recognize their respective "fundamental incapability," and sublimate them into "WE" solidarity. Moreover, the economy must have a moral responsibility as a co-adventurer and consider its vulnerability to risk. This study focuses on two factors of mind perception-moral responsibility and risk vulnerability-and proposes a novel wealth distribution model between the two agents following an econophysical approach, whereas the conventional model dealt with redistribution through taxes and institutions. Three models are developed: a joint-venture model in which profit/losses are distributed based on their factors, a redistribution model in which wealth stocks are redistributed periodically based on their factors in the joint-venture model, and a "WE economy" model in which profit/losses are distributed based on the ratio of each other's factors. A simulation comparison reveals that WE economies are effective in reducing inequality, resilient in normalizing wealth distribution as advantages, and susceptible to free riders as disadvantages. However, this disadvantage can be compensated for by fostering fellowship and using joint ventures. This study presents the effectiveness of moral responsibility and risk vulnerability, complementarity between the WE economy and joint economy, and the direction of the economy in reducing inequality. Future challenges include developing an advanced model based on real economic analysis and economic psychology and promoting its fieldwork for worker coops and platform cooperatives to realize a desirable mixbiotic society.

Secular stagnation: a Classical–Marxian view

We study a model of secular stagnation, income and wealth distribution, and employment in the Classical–Marxian (CM) tradition, with the purpose of drawing a contrast with established Neoclassical accounts of the topic (Piketty 2014, Gordon 2015). In these explanations, which assume full employment of labor at all times, an exogenous reduction in the growth rate g increases the difference with the endogenous rate of return to capital r. The capital–income ratio rises and, if the elasticity of substitution is above one, the wage share falls. Our explanation does not presuppose full employment, and features a crucial tension between profit-driven capital accumulation and wage-driven labor-augmenting technical change: both these features are defining for CM economics and have been emphasized in recent heterodox macro literature. Institutional or technological shocks that lower the wage share initially foster capital accumulation – which is profit-driven – and increase wealth inequality. However, the effect on long-run growth is negative, because a reduction in the wage share lessens the incentives by firms to introduce labor-saving innovation, which is wage-driven. The capital–income ratio must rise in order to restore balanced growth and stabilize employment in the long run; and the increase in wealth inequality is permanent. The ultimate effect on long-run employment depends on the relative strength of the response of technical change versus real wage growth to labor market institutions: we identify a simple condition that delivers either a wage-led or a profit-led long-run employment regime. We then test the model using time-series data for the US (1960–2019): impulse responses from vector error-correction model (VECM) estimators lend support to the main predictions of our model, and point to the employment–population ratio being wage-led.

Variation of Gini and Kolkata indices with saving propensity in the Kinetic Exchange model of wealth distribution: An analytical study

We study analytically the change in the wealth (x) distribution P(x) against saving propensity λ in a closed economy, using the Kinetic theory. We estimate the Gini (g) and Kolkata (k) indices by deriving (using P(x)) the Lorenz function L(f), giving the cumulative fraction L of wealth possessed by fraction f of the people ordered in ascending order of wealth. First, using the exact result for P(x) when λ=0 we derive L(f), and from there the index values g and k. We then proceed with an approximate gamma distribution form of P(x) for non-zero values of λ. Then we derive the results for g and k at λ=0.25 and as λ→1. We note that for λ→1 the wealth distribution P(x) becomes a Dirac δ-function. Using this and assuming that form for larger values of λ we proceed for an approximate estimate for P(x) centered around the most probable wealth (a function of λ). We utilize this approximate form to evaluate L(f), and using this along with the known analytical expression for g, we derive an analytical expression for k(λ). These analytical results for g and k at different λ are compared with numerical (Monte Carlo) results from the study of the Chakraborti–Chakrabarti model. Next we derive analytically a relation between g and k. From the analytical expressions of g and k, we proceed for a thermodynamic mapping to show that the former corresponds to entropy and the latter corresponds to the inverse temperature.

Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models.

The spread of the COVID-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modeling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated, resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming, on the one hand, that individuals in different compartments act differently in the economic process and, on the other hand, that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society.

The Study of a Wealth Distribution Model with a Linear Collision Kernel

The wealth substitution rate, which describes the substitution relationship between agents’ investment in wealth, is introduced into the collision kernel of the Boltzmann equation to study wealth distribution. Using the continuous trading limit, the Fokker–Planck equation is derived and the steady-state solution is obtained. The results show that the inequality of wealth distribution decreases as the wealth substitution rate increases under certain assumptions. The wealth distribution has a bimodal shape if the wealth substitution rate does not equal one.

Kinetic model of wealth distribution by trading stocks with geometric brownian motion

In this paper, we consider the wealth distribution obtained by trading (buying–selling) stocks whose prices follow the geometric Brownian motion (GBM), when both number of the ticker symbol of the stock and maximum number of the traded stock are limited to unity. The binary exchange of the cash and stock between two agents is expressed with the Boltzmann-type kinetic equation. The distribution function of the number of the agents with the specific number of the stock or specific amount of the cash can be demonstrated, theoretically, when the price of the stock is constant. The distribution function of the number of the agents with the specific amount of the total asset can be approximated by [Formula: see text]-distribution, when the price of the stock follows the GBM. Finally, the rule in the binary-exchange-game approximates the distribution function of the number of the agents with the specific amount of the total asset to the Feller–Pareto-like distribution at the high wealth tail.

Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach

Abstract We recast the Aiyagari–Bewley–Huggett model of income and wealth distribution in continuous time. This workhorse model—as well as heterogeneous agent models more generally—then boils down to a system of partial differential equations, a fact we take advantage of to make two types of contributions. First, a number of new theoretical results: (1) an analytic characterization of the consumption and saving behaviour of the poor, particularly their marginal propensities to consume; (2) a closed-form solution for the wealth distribution in a special case with two income types; (3) a proof that there is a unique stationary equilibrium if the intertemporal elasticity of substitution is weakly greater than one. Second, we develop a simple, efficient and portable algorithm for numerically solving for equilibria in a wide class of heterogeneous agent models, including—but not limited to—the Aiyagari–Bewley–Huggett model.

Classical Political Economy and Secular Stagnation

This paper presents a model of secular stagnation, income and wealth distribution, and employment in the Classical Political Economy tradition, that can be contrasted with the accounts by Piketty (2014) and Gordon (2015). In these explanations, an exogenous reduction in the growth rate g --because of declining fertility or the exhaustion of path-breaking scientific discoveries--increases the difference with the rate of return to capital r. The capital-income ratio rises, and if the elasticity of substitution is above one, the wage share falls. Both Piketty and Gordon assume full employment at all times. In our explanation, which does not presuppose full employment, the key tension is between profit-driven capital accumulation and wage-driven labor-augmenting technical change: both are defining for Classical Political Economy, and have been emphasized in recent heterodox macro literature. Labor-crushing institutional or technological shocks initially foster capital accumulation -which is profit-driven-- and increase wealth inequality. However, the effect on long-run growth is negative, because of the reduced incentives by firms to introduce labor-saving innovation, which is wage-driven. The capital/income ratio must rise in order to restore balanced growth in the long run; and the increase in wealth inequality is permanent. The ultimate effect on long-run employment depends on the strength of the response of labor-augmenting technical change vs. the response of real wage growth to labor market institutions: accordingly, long-run employment can either be wage-led or profit-led. We then test the model using time-series data for the US (1990-2019): the test offers support to the main predictions of our model, and to the employment-population ratio being wage-led.

Corporate Waqf: a sustainable model of Islamic wealth creation and distribution

Waqf has the potential to function as a mechanism towards improving the welfare of the ummah . Even though waqf has been long practised in all Muslim countries, each country has different methods of implementing it. Issues pertaining to waqf have become the most talked about agenda among Muslim scholars, administrators, and academics, particularly, the deficiencies of the conventional model and the lack of knowledge among practitioners, which has led to the mismanagement of waqf assets. The need to expand the conventional Waqf concept into business practices in achieving Islamic transformation agenda and wealth creation is, therefore, crucial. The aim of this study is to explore a holistic corporate waqf model to be used by organisations to address social issues and to create wealth for the ummah . A Tawhidic concept premised on the Quran and Sunnah was used to underpin the development of the proposed corporate waqf model. Content analyses of existing academic literature on Waqf , followed by preliminary semi-structured interviews, were conducted to provide a framework for the development of the corporate waqf model. The findings from this study would be of interest to academics, regulators, and practitioners in our quest to improve the existing waqf model of income distribution to maximize wealth for the ummah through a more efficient and effective wealth management system of governance and accountability.

Corporate Waqf: A Sustainable Model of Islamic Wealth Creation and Distribution

The aim of this study is to explore an enhanced corporate waqf model to be used by organisations so that they can address and help solve social issues, and create wealth for the ummah, specifically in Malaysia. A Tawhidic concept premised on the Quran and Sunnah was used to underpin the development of the proposed corporate waqf model. Content analyses of existing academic literature on Waqf, followed by preliminary semi-structured interviews on groups of Chief Executive Officers, SIRC officers, Islamic Scholars and Academics, were conducted. The key challenges faced by waqf organisations are as follows: i) the development of waqf properties is not economical; and ii) the current waqf model does not support the wealth creation, wealth enhancement and wealth distribution to the ummah. These weaknesses have motivated this study to propose a new corporate waqf model; one that encompasses an enhanced governance structure and management setting. This new model treats waqf as an effective mechanism that can strengthen the economic growth of Malaysia.

Can Zakat Charity Help Reduce Economic Inequality?

Purpose – This paper suggests an agent based model of economic behavior based on which it argues that the Islamic charity of Zakat can be rationalized as it significantly reduces wealth inequalities. 
 Design/methodology/approach – An Agent Based Model (ABM) of wealth distribution (Wilensky, 1998) is configured to introduce Zakat in the model. Simulations were run in the model with ordered and random distribution of Zakat charity and wealth distribution pattern compared with the standard simulation without Zakat.
 Findings – The impact of this very small charity on the part of the rich (2.5% of net wealth) showed significant impact on wealth distribution pattern as it changed the power law distribution into normal distribution of wealth.
 Practical implications – The results emphasize non-conventional handling of the economic problem of poverty and wealth signifying the importance of faith based policing
 Originality/value – The paper contributes an agent based simulation which is a relatively newer technique to study human behavior and is commensurate to complexity theory. Its use is relatively rare in Islamic Economics and the paper is the first to propose a decentralzied model simulation of Islamic charity of Zakat.

A simple and efficient kinetic model for wealth distribution with saving propensity effect: Based on lattice gas automaton

The dynamics of wealth distribution plays a critical role in the economic market, hence an understanding of its nonequilibrium statistical mechanics is of great importance to human society. For this aim, a simple and efficient one-dimensional (1D) lattice gas automaton (LGA) is presented for wealth distribution of agents with or without saving propensity. The LGA comprises two stages, i.e., random propagation and economic transaction. During the former phase, an agent either remains motionless or travels to one of its neighboring empty sites with a certain probability. In the subsequent procedure, an economic transaction takes place between a pair of neighboring agents randomly. It requires at least 4 neighbors to present correct simulation results. The LGA reduces to the simplest model with only random economic transaction if all agents are neighbors and no empty sites exist. The 1D-LGA has a higher computational efficiency than the 2D-LGA and the famous Chakraborti–Chakrabarti economic model. Finally, the LGA is validated with two benchmarks, i.e., the wealth distributions of individual agents and dual-earner families. With the increasing saving fraction, both the Gini coefficient and Kolkata index (for individual agents or two-earner families) reduce, while the deviation degree (defined to measure the difference between the probability distributions with and without saving propensities) increases. It is demonstrated that the wealth distribution is changed significantly by the saving propensity which alleviates wealth inequality.

Parameter Estimation and Measurement of Social Inequality in a Kinetic Model for Wealth Distribution

This paper deals with the modeling of wealth distribution considering a society with non-constant population and non-conservative wealth trades. The modeling approach is based on the kinetic theory of active particles, where individuals are distinguished by a scalar variable (the activity) which expresses their social state. A qualitative analysis of the model focusing on asymptotic behaviors and measurement of inequality through the Gini coefficient is presented. Finally, some specific case-studies are proposed in order to carry out numerical experiments to validate our model, characterize societies and investigate emerging behaviors.

Two-Phase Exponential Model of Wealth Distribution

Abstract The topic of wealth and money distribution attracts great attention of economists, as well as researchers from other scientific fields, such as statistical physics and econophysics. An increasing number of models and simulations are being created in order to understand the process of wealth distribution and reaching the steady state of the distribution system. Also, the number of papers dealing with analysis and determining the distribution proportion is constantly growing, and, unlike the previous years, when the Pareto principle was “80-20”, today that principle could be “90-10”and even “90-20”. In this paper we present an agent-based simulation model derived from econophysics that describes the dynamics of wealth distribution. Two models of exponential function are tested: a one-phase model that uses the Newton’s law of cooling and a two-phase exponential function model. We found that exponential decreasing function adequately described the dynamics of wealth distribution, especially in the models without the possibility of borrowing money, and the validity of the Pareto principle “80-20” in these models could be confirmed.

The Investigation of a Wealth Distribution Model on Isolated Discrete Time Domains

A wealth distribution model on isolated discrete time domains, which allows the wealth to exchange at irregular time intervals, is used to describe the effect of agent’s trading behavior on wealth distribution. We assume that the agents have different degrees of risk aversion. The hyperbolic absolute risk aversion (HARA) utility function is employed to describe the degrees of risk aversion of agents, including decreasing relative risk aversion (DRRA), increasing relative risk aversion (IRRA), and constant relative risk aversion (CRRA). The effect of agent’s expectation on wealth distribution is taken into account in our wealth distribution model, in which the agents are allowed to adopt certain trading strategies to maximize their utility and improve their wealth status. The Euler equation and transversality condition for the model on isolated discrete time domains are given to prove the existence of the optimal solution of the model. The optimal solution of the wealth distribution model is obtained by using the method of solving the rational expectation model on isolated discrete time domains. A numerical example is given to highlight the advantages of the wealth distribution model.

Wealth distribution models with regulations: Dynamics and equilibria

Simple agent based exchange models are a commonplace in the study of wealth distribution in an artificial economy. Generally, in a system that is composed of many agents characterized by their wealth and risk-aversion factor, two agents are selected sequentially and randomly to exchange wealth, allowing its redistribution. Here we analyze how the effect of social protection policies, which favor agents of lower wealth during the exchange, influences the stability and some relevant economic indicators of the system. On the other hand, we study how periods of interruption of these policies produce, in the short and long term, changes in the system. In all cases, a steady state is reached, but with varying relaxation times. We conclude that regulations may improve economic mobility and reduce inequality. Moreover, our results indicate that the removal of social protection entails a high cost associated with the hysteresis of the distribution of wealth. Economic inequalities increase during the period without social protection and decrease when they are restored. But recovery to the initial equilibrium may take longer than it took to disturb it. In some extreme cases, inequality is irreversible.

The Pareto Distribution of World’s GDP

The problem of wealth distribution has gathered the attention of researchers for many years. In their work, the researchers are mainly engaged in the issue of distribution of wealth between individuals by analyzing empirical results at the country level, or specific lists that particular organization form like the Forbes list. Research are also increasingly directed toward the analysis of new models such as Boltzmann Gibbs or application of Gama function that describes this distribution. An interesting issue is the analysis of the distribution of wealth among the countries themselves. In these works, the value of GDP is used as the wealth that country has. In this article, the author dealt with the analysis of the distribution of GDP between countries at the global level. Analysis were performed using the Pareto distribution model of wealth distribution and GINI coefficient based on the data of the value of GDP for countries from IMF estimation. The analysis was conducted for the period from 1980 to the present, as well as analysis of data provided by IMF estimates for the value of GDP by 2022. The goal is to determine the degree of uneven distribution between the countries themselves in the world, analyzing the dynamics of change in the degree of unevenness and an analysis of the degree of unevenness in the future based on forecasts of the IMF on the values of countries GDP. The author also wanted to test if Pareto's 80/20 rule applies when it comes to the distribution of GDP at world level.
 
 
 Keywords: GDP, distribution of wealth, IMF, Pareto distribution, GINI coefficient

The Affine Wealth Model: An agent-based model of asset exchange that allows for negative-wealth agents and its empirical validation

We present a stochastic, agent-based, binary-transaction Asset-Exchange Model (AEM) for wealth distribution that allows for agents with negative wealth. This model retains certain features of prior AEMs such as redistribution and wealth-attained advantage, but it also allows for shifts as well as scalings of the agent density function. We derive the Fokker–Planck equation describing its time evolution and we describe its numerical solution, including a methodology for solving the inverse problem of finding the model parameters that best match empirical data. Using this methodology, we compare the steady-state solutions of the Fokker–Planck equation with data from, inter alia, the United States Survey of Consumer Finances over a time period of 27 years. In doing so, we demonstrate agreement with empirical data with an average error less than 0.16% over this time period. We present the model parameters for the US wealth distribution data as a function of time under the assumption that the distribution responds to their variation adiabatically. We argue that the time series of model parameters thus obtained provides a valuable new diagnostic tool for analyzing wealth inequality.

Kinetic models for optimal control of wealth inequalities

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a finite time horizon approximation, or model predictive control, of the corresponding control problem for the microscopic agents’ dynamic and results in an alternative theoretical approach to the taxation and redistribution policy at a global level. It is shown that in general the control is able to modify the Pareto index of the stationary solution of the corresponding Boltzmann kinetic equation, and that this modification can be exactly quantified. Connections between previous Fokker–Planck based models for taxation-redistribution policies and the present approach are also discussed.