Wealth Exchange Models Research Articles

Wealth distribution on a dynamic complex network

We present an agent-based model that examines the microscopic exchange of wealth in a dynamic network to investigate the topological characteristics associated with economic inequality. The model consists of two processes: conservative wealth exchange between connected agents and the rewiring of connections, which depends on the wealth of the agents. The dynamics of wealth and connections are interrelated, as the network structure influences which agents interact with each other. We analyze the time evolution and asymptotic characteristics of the model for different values of a social protection factor (f), which favors the poorest agent in each wealth transaction. Our results show that for f=0, wealth and connections condense in a single agent, in accordance with mean-field models of wealth exchange. When f is low, agents from the middle and upper classes become favored, leading to the formation of network hubs. However, as f increases, the restriction of the network on exchanges results in an egalitarian society departing from the outcomes observed in the mean-field exchange models.

Emergence of Inequality in Income and Wealth Dynamics.

Increasing wealth inequality is a significant global issue that demands attention. While the distribution of wealth varies across countries based on their economic stages, there is a universal trend observed in the distribution function. Typically, regions with lower wealth values exhibit an exponential distribution, while regions with higher wealth values demonstrate a power-law distribution. In this review, we introduce measures that effectively capture wealth inequality and examine wealth distribution functions within the wealth exchange model. Drawing inspiration from the field of econophysics, wealth exchange resulting from economic activities is likened to a kinetic model, where molecules collide and exchange energy. Within this framework, two agents exchange a specific amount of wealth. As we delve into the analysis, we investigate the impact of various factors such as tax collection, debt allowance, and savings on the wealth distribution function when wealth is exchanged. These factors play a crucial role in shaping the dynamics of wealth distribution.

Islamic and capitalist economies: Comparison using econophysics models of wealth exchange and redistribution

Islamic and capitalist economies have several differences, the most fundamental being that the Islamic economy is characterized by the prohibition of interest (riba) and speculation (gharar) and the enforcement of Shariah-compliant profit-loss sharing (mudaraba, murabaha, salam, etc.) and wealth redistribution (waqf, sadaqah, and zakat). In this study, I apply new econophysics models of wealth exchange and redistribution to quantitatively compare these characteristics to those of capitalism and evaluate wealth distribution and disparity using a simulation. Specifically, regarding exchange, I propose a loan interest model representing finance capitalism and riba and a joint venture model representing shareholder capitalism and mudaraba of an Islamic profit-loss sharing system; regarding redistribution, I create a transfer model representing inheritance tax and waqf of an Islamic wealth redistribution system. As exchanges are repeated from an initial uniform distribution of wealth, wealth distribution approaches a power-law distribution more quickly for the loan interest than the joint venture model; and the Gini index, representing disparity, rapidly increases. The joint venture model's Gini index increases more slowly, but eventually, the wealth distribution in both models becomes a delta distribution, and the Gini index gradually approaches 1. Next, when both models are combined with the transfer model to redistribute wealth in every given period, the loan interest model has a larger Gini index than the joint venture model, but both converge to a Gini index of less than 1. These results quantitatively reveal that in the Islamic economy, disparity is restrained by prohibiting riba and promoting reciprocal exchange in mudaraba and redistribution through waqf. Comparing Islamic and capitalist economies provides insights into the benefits of economically embracing the ethical practice of mutual aid and suggests guidelines for an alternative to capitalism.

A poor agent and subsidy: an investigation through CCM model.

In this work, the dynamics of agents below a threshold line in some modified CCM type kinetic wealth exchange models are studied. These agents are eligible for subsidy as can be seen in any real economy. An interaction is prohibited if both of the interacting agents' wealth fall below the threshold line. A walk for such agents can be conceived in the abstract gain-loss space (GLS) and is macroscopically compared to a lazy walk. The effect of giving subsidy once to such agents is checked over, giving repeated subsidy from the point of view of the walk in GLS. It is seen that the walk has more positive drift if the subsidy is given once. The correlations and other interesting quantities are studied. This article is part of the theme issue 'Kinetic exchange models of societies and economies'.

Optimal control about multi-agent wealth exchange and decision-making competence

A kinetic model, describing the influence of decision-making competence in evolution of wealth for a multi-agent system with feedback control, is investigated. The control term is analyzed by the minimization of a quadratic cost functional, which aims to decrease the wealth gap between agents. The constant saving propensities and diffusion coefficients in the model are modified as functions of wealth and decision competence, respectively. In our micro binary wealth exchange model, an extra risk shared by all agent is discussed, which increases the inequality of wealth distribution. The main macro properties of the kinetic model and the marginal distributions of wealth and decision-making competence are considered through several examples. Our results illustrate that the disequilibrium of decision-making competence is one of the reasons to increase wealth inequality.

Wealth exchange models and machine learning: Finding optimal risk strategies in multiagent economic systems.

The Yard-Sale Model, a well known wealth exchange model whose observed macroscopic behavior hides many underlying aspects of particular complexity, was studied at the microscopic level. The performance of the agents during the successive transactions allows for the definition of successful or disadvantageous strategies according to the profit they achieve at the end of the process. Optimal strategies were found that maximize the individual wealth of each agent by performing their training through a genetic algorithm. The addition of different levels of rationality given by the amount of available information from their environment showed promising results, at both the microscopic and macroscopic levels. Remarkably, after the training process, the rational agents were able to determine when it would be convenient to interact with their opponents. Additionally, a region of parameters was found for which the distribution of wealth is a power law throughout the whole wealth range. As a general result, the incorporation of rational agents in this type of systems leads to greater inequality at the collective level.

Optimal risk in wealth exchange models: Agent dynamics from a microscopic perspective

In this work we present a new type of microscopic analysis for two well-known wealth exchange models that have been deeply studied at the macroscopic level for the past decades. This approach allowed us to study and classify the individual strategies carried out by the agents undergoing transactions. We analyze the role of their interaction parameter, the risk propensity, and find a critical risk such that agents with risk above that value always end up losing everything when the system approaches equilibrium. Moreover, we find that the wealth of the agents is maximum for a range of risk values that depend on particular characteristics of the model, such as the social protection factor. Our findings allow to determine a region of parameters for which the strategies of the economic agents are successful.

Non-conservative kinetic model of wealth exchange with saving of production

In this paper we propose a non-conservative kinetic model of wealth exchange with saving of production as an extension of the Chakraborti–Chakrabarti model of money exchange. Using microeconomic arguments, we achieve rules of interaction between economic agents that depend on two exogenous parameters, the exchange aversion of the agents (λ) and the saving of production (s), such that in the limit s = 0, these rules can be reduced to the ones of the Chakraborti–Chakrabarti model. The non-conservative dynamics are approached analytically through a mean field approximation and the Boltzmann kinetic equation. Both approximations allow us to compute a theoretical rate of exponential growth (g) and to fit the emergent distributions of wealth to a gamma probability density function, in such a way that g, the fit parameters and the Gini index can be expressed analytically in terms of λ and s. In general, the emergent distributions do not reach a stationary state, however it is possible to study the emergence of self-similar distributions that hold the gamma pattern and maximize the Shannon entropy. With the purpose of addressing labor income, we explore additionally the effect of salary income in the model by defining a two-class structure where population is separated into workers and producers. This assumption leads to an emergent rate of economic growth ge. The macroeconomic implications of this model are studied by means of the wealth/income ratio, which can be predicted as s∕ge, in accordance with the Solow model of economic growth. The results in this paper allow to tie some of the important facts of the modern economic speech, as well as the microeconomic theory, with some methods and ideas developed in the context of non-conservative exchange models.

Wealth distribution under the spread of infectious diseases.

We develop a mathematical framework to study the economic impact of infectious diseases by integrating epidemiological dynamics with a kinetic model of wealth exchange. The multiagent description leads to the study of the evolution over time of a system of kinetic equations for the wealth densities of susceptible, infectious, and recovered individuals, whose proportions are driven by a classical compartmental model in epidemiology. Explicit calculations show that the spread of the disease seriously affects the distribution of wealth, which, unlike the situation in the absence of epidemics, can converge toward a stationary state with a bimodal form. Furthermore, simulations confirm the ability of the model to describe different phenomenon characteristics of economic trends in situations compromised by the rapid spread of an epidemic, such as the unequal impact on the various wealth classes and the risk of a shrinking middle class.

Analysis of Solidarity Effect for Entropy, Pareto, and Gini Indices on Two-Class Society Using Kinetic Wealth Exchange Model.

It is well known that two different underlying dynamics lead to different patterns of income/wealth distribution such as the Boltzmann–Gibbs form for the lower end and the Pareto-like power-law form for the higher-end. The Boltzmann–Gibbs distribution is naturally derived from maximizing the entropy of random interactions among agents, whereas the Pareto distribution requires a rational approach of economics dependent on the wealth level. More interestingly, the Pareto regime is very dynamic, whereas the Boltzmann–Gibbs regime is stable over time. Also, there are some cases in which the distributions of income/wealth are bimodal or polymodal. In order to incorporate the dynamic aspects of the Pareto regime and the polymodal forms of income/wealth distribution into one stochastic model, we present a modified agent-based model based on classical kinetic wealth exchange models. First, we adopt a simple two-class society consisting of the rich and the poor where the agents in the same class engage in random exchanges while the agents in the different classes perform a wealth-dependent winner-takes-all trading. This modification leads the system to an extreme polarized society with preserving the Pareto exponent. Second, we incorporate a solidarity formation among agents belonging to the lower class in our model, in order to confront a super-rich agent. This modification leads the system to a drastic bimodal distribution of wealth with a varying Pareto exponent over varying the solidarity parameter, that is, the Pareto-regime becomes narrower and the Pareto exponent gets larger as the solidarity parameter increases. We argue that the solidarity formation is the key ingredient in the varying Pareto exponent and the polymodal distribution. Lastly, we take two approaches to evaluate the level of inequality of wealth such as Gini coefficients and the entropy measure. According to the numerical results, the increasing solidarity parameter leads to a decreasing Gini coefficient not linearly but nonlinearly, whereas the entropy measure is robust over varying solidarity parameters, implying that there is a trade-off between the intermediate party and the high end.

Effect of segregation on inequality in kinetic models of wealth exchange

Empirical distributions of wealth and income can be reproduced using simplified agent-based models of economic interactions, analogous to microscopic collisions of gas particles. Building upon these models of freely interacting agents, we explore the effect of a segregated economic network in which interactions are restricted to those between agents of similar wealth. Agents on a 2D lattice undergo kinetic exchanges with their nearest neighbours, while continuously switching places to minimize local wealth differences. A spatial concentration of wealth leads to a steady state with increased global inequality and a magnified distinction between local and global measures of combatting poverty. Individual saving propensity proves ineffective in the segregated economy, while redistributive taxation transcends the spatial inhomogeneity and greatly reduces inequality. Adding fluctuations to the segregation dynamics, we observe a sharp phase transition to lower inequality at a critical temperature, accompanied by a sudden change in the distribution of the wealthy elite.

Statistics of Correlations and Fluctuations in a Stochastic Model of Wealth Exchange.

In our recently proposed stochastic version of discretized kinetic theory, the exchange of wealth in a society is modelled through a large system of Langevin equations. The deterministic part of the equations is based on non-linear transition probabilities between income classes. The noise terms can be additive, multiplicative or mixed, both with white or Ornstein–Uhlenbeck spectrum. The most important measured correlations are those between Gini inequality index G and social mobility M, between total income and G, and between M and total income. We describe numerical results concerning these correlations and a quantity which gives average stochastic deviations from the equilibrium solutions in dependence on the noise amplitude.

Kinetic Wealth Exchange Models in Economics and Sociology

Kinetic Wealth Exchange Models in Economics and Sociology

Inequality measures in kinetic exchange models of wealth distributions

In this paper, we study the inequality indices for some models of wealth exchange. We calculated Gini index and newly introduced k-index and compare the results with reported empirical data available for different countries. We have found lower and upper bounds for the indices and discuss the efficiencies of the models. Some exact analytical calculations are given for a few cases. We also exactly compute the quantities for Gamma and double Gamma distributions.

Inequality Measures in Kinetic Exchange Models of Wealth Distributions

In this paper, we study the inequality indices for some models of wealth exchange. We calculated Gini index and newly introduced k-index and compare the results with reported empirical data available for different countries. We have found lower and upper bounds for the indices and discuss the efficiency of the models. Some exact analytical calculations are given for a few cases. We also exactly compute the quantities for Gamma and double Gamma distributions.

Evolution of consumption distribution and model of wealth distribution in China between 1995 and 2012

We study the evolution of the distribution of consumption of individuals in the majority population in China during the period 1995–2012 and find that its probability density functions (PDFs) obey the rule Pc(x)=K(x−μ)e−(x−μ)22σ2. We also find (i) that the PDFs and the individual income distribution appear to be identical, (ii) that the peaks of the PDFs of the individual consumption distribution are consistently on the low side of the PDFs of the income distribution, and (iii) that the average of the marginal propensity to consume (MPC) is large, MPC¯=0.77, indicating that in the majority population individual consumption is low and strongly dependent on income. The long right tail of the PDFs of consumption indicates that few people in China are participating in the high consumption economy, and that consumption inequality is high. After comparing the PDFs of consumption with the PDFs of income we obtain the PDFs of residual wealth during the period 1995–2012, which exhibit a Gaussian distribution. We use an agent-based kinetic wealth-exchange model (KWEM) to simulate this evolutional process and find that this Gaussian distribution indicates a strong propensity to save rather than spend. This may be due to an anticipation of such large potential outlays as housing, education, and health care in the context of an inadequate welfare support system.

An approach to trading valuable points in a virtual wealth exchange model for closed virtual communities

Today, different types of virtual communities connect billions of people around the world via the internet, particularly given expanding cellular networks and the growth of smart mobile devices. For community operators, one major problem is increasing users' economic activity in virtual worlds. As such, connecting various virtual economic systems in one all-encompassing system can be an alternative posing inexhaustible potential for e-commerce. Scholars have proposed the reliable virtual wealth trading model for exchanging virtual wealth across closed virtual communities. However, the model lacks a quantity concept for virtual products that could obstruct the model's applicability. In this paper, we propose a virtual wallet concept to resolve the model's quantity deficit. Since communities can be either regular or lean, we also offer two corresponding versions of wallet processes. Ultimately, this article provides two complete implementations to serve as reference designs, as well as explains the usability of the approach.

Emergence of communities in a coevolutive model of wealth exchange

We present a model in which we investigate the structure and evolution of a random network that connects agents capable of exchanging wealth. Economic interactions between neighbors can occur only if the difference between their wealth is less than a threshold value that defines the width of the economic classes. If the interchange of wealth cannot be done, agents are reconnected with another randomly selected agent, allowing the network to evolve in time. On each interaction there is a probability of favoring the poorer agent, simulating the action of the government. We measure the Gini index, having real world values attached to reality. Besides the network structure showed a very close connection with the economic dynamic of the system.

Clausius inequality and [formula omitted]-theorems for some models of random wealth exchange

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution equation that describes random wealth exchanges. In such kinetic models economical agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy. It appears useful to reformulate the problem and represent the dynamics as a combination of two processes. The first is a linear transformation of a two-particle distribution function during the act of exchange while the second one corresponds to new random pairing of agents and plays a role of some kind of feedback control. This representation leads to a Clausius-type inequality which suggests a new interpretation of the exchange process as an irreversible relaxation due to a contact with a reservoir of a special type. Only in some special cases when equilibrium distribution is exactly a gamma distribution, this inequality results in the H-theorem with monotonically growing ‘entropy’ functional which differs from the Boltzmann entropy by an additional term. But for arbitrary exchange rule the evolution has some features of relaxation to a non-equilibrium steady state and it is still unclear if any general H-theorem could exist.

Kinetics of wealth and the Pareto law.

An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy change due to interactions with other molecules. We pursue this analogy and derive a Boltzmann equation for the time evolution of the wealth distribution of a population of agents for the so-called Yard-Sale Model of wealth exchange. We examine the solutions to this equation by a combination of analytical and numerical methods and investigate its long-time limit. We study an important limit of this equation for small transaction sizes and derive a partial integrodifferential equation governing the evolution of the wealth distribution in a closed economy. We then describe how this model can be extended to include features such as inflation, production, and taxation. In particular, we show that the model with taxation exhibits the basic features of the Pareto law, namely, a lower cutoff to the wealth density at small values of wealth, and approximate power-law behavior at large values of wealth.