We investigate the effect of applying nonlinear redistributive taxes to the yard-sale dynamics of assets. An amount of money is collected from each individual (tax) and distributed back equally. We consider (i) a piecewise linear tax, exempting those with wealth below a threshold $w_0$, and taxing the excess wealth otherwise, and (ii) a power-law tax with exponent $α>0$, which allows embracing regressive, proportional and progressive rules. The distribution of wealth obtained from numerical simulations of the agent-based dynamics is compared with the solution of its associated Fokker-Planck equation for the probability density function $P(w,t)$ of wealth $w$ at time $t$, in good agreement. Based on these solutions, we analyze how the different rules modify the distribution of wealth across the population, quantifying the level of inequality through the Gini coefficient. We note that the introduction of an exemption threshold does not always diminish inequality, depending on the implementation details. Moreover, nonlinearity brings new stylized facts in the distribution of wealth compared to the linear case, e.g., negative skewness, bimodality, indicating stratification, or a flat shape meaning equality populated wealth layers.
A Fokker-Planck approach to a stochastic multiplicative wealth model with taxation and redistribution
Universidade Federal de Viçosa (UFV)（维索萨联邦大学）；Ibitipoca Institute of Physics (IbitiPhys)（伊比蒂波卡物理研究所）；Pontifícia Universidade Católica do Rio de Janeiro (PUC–Rio)（里约热内卢天主教大学）
In this work, we modify the affine wealth model of wealth distributions to examine the effects of nonconstant redistribution on the very wealthy. Previous studies of this model, restricted to flat redistribution schemes, have demonstrated the presence of a phase transition to a partially wealth-condensed state, or "partial oligarchy," at the critical value of an order parameter. These studies have also indicated the presence of an exponential tail in wealth distribution precisely at criticality. Away from criticality, the tail was observed to be Gaussian. In this work, we generalize the flat r…
Given the wealth inequality worldwide, there is an urgent need to identify the mode of wealth exchange through which it arises. To address the research gap regarding models that combine equivalent exchange and redistribution, this study compares an equivalent market exchange with redistribution based on power centers and a nonequivalent exchange with mutual aid using the Polanyi, Graeber, and Karatani modes of exchange. Two new exchange models based on multi-agent interactions are reconstructed following an econophysics approach for evaluating the Gini index (inequality) and total exchange (ec…
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 p…
In earlier work, we derived a nonlinear, nonlocal Fokker-Planck equation for the Yard-Sale Model of asset exchange. In the absence of redistribution, we showed that the Gini coefficient is a Lyapunov functional for this model, tending to one in the time-asymptotic limit, corresponding to maximal inequality. When a one-parameter model of redistribution is introduced, we showed that the model admits a steady state similar to Pareto's Law. In this work, we analyze the form of this distribution in greater detail, both analytically and numerically. We find that, while Pareto's Law is approximately …
Diffusion-Induced Stabilization of Nonreciprocal Wealth Dynamics
Econophysics Lab, Institut Louis Bachelier（经济物理学实验室，路易·巴舍利耶研究所）；LadHyX UMR CNRS 7646, École Polytechnique（拉迪克斯实验室（CNRS联合研究单位7646），巴黎综合理工学院）；Capital Fund Management（资本基金管理公司）
In our multi-agent model agents generate wealth from repeated interactions for which a prisoner's dilemma payoff matrix is assumed. Their gains are taxed by a government at a rate $α$. The resulting budget is spent to cover administrative costs and to pay a bonus to cooperative agents, which can be identified correctly only with a probability $p$. Agents decide at each time step to choose either cooperation or defection based on different information. In the local scenario, they compare their potential gains from both strategies. In the global scenario, they compare the gains of the cooperativ…
Phase Transitions in Economic Inequality:Taxation and Extremal Replacement Dynamics
Centro Atómico Bariloche (CNEA) and CONICET（巴里洛切原子中心（CNEA）和国家科学研究委员会）；Instituto Balseiro, Universidad Nacional de Cuyo（乌斯帕尔塔国立大学巴尔塞罗研究所）；Universidade Federal do Rio Grande do Sul（南里奥格兰德联邦大学）；Instituto Nacional de Ciência e Tecnologia de Sistemas Complexos, CBPF（复杂系统国家科技研究所，巴西粒子物理中心）
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 Bo…
Fokker-Planck Description of Wealth Dynamics and the Origin of Pareto's Law
所谓的财富分配的“庭院销售模型”假定，财富在经济主体之间转移是交易的结果，交易的规模与较不富裕的主体的财富成正比。在最近的工作中[B]。Boghosian，“Kinetics of Wealth and the Pareto Law”，{ \it Phys. Rev. E} { \bf 89}（2014）042804]，表明这导致了支配财富分配的福克-普朗克方程。随着财富再分配机制的增加，进一步表明，该模型导致了在形式上与帕累托定律非常相似的固定财富分配。本文给出了该Fokker-Planck方程的一种更为简单的推导方法。
Understanding the statistical dynamics of growth and inequality is a fundamental challenge to ecology and society. Recent analyses of wealth and income dynamics in contemporary societies show that economic inequality is very dynamic and that individuals experience substantially different growth rates over time. However, despite a fast growing body of evidence for the importance of fluctuations, we still lack a general statistical theory for understanding the dynamical effects of heterogeneneous growth across a population. Here we derive the statistical dynamics of correlated growth rates in he…
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 …
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 for its redistribution. Here we analyze how the effect of a social protection policy, which favors agents of lower wealth during the exchange, influences stability and some relevant economic indicators of the system. On the other hand, we study how periods of interruption of these policies prod…
Based on interactions between individuals and others and references to social norms, this study reveals the impact of heterogeneity in time preference on wealth distribution and inequality. We present a novel approach that connects the interactions between microeconomic agents that generate heterogeneity to the dynamic equations for capital and consumption in macroeconomic models. Using this approach, we estimate the impact of changes in the discount rate due to microeconomic interactions on capital, consumption and utility and the degree of inequality. The results show that intercomparisons w…
A mean-field like stochastic evolution equation with growth and reset terms (LGGR model) is used to model wealth distribution in modern societies. The stationary solution of the model leads to an analytical form for the density function that is successful in describing the observed data for all wealth categories. In the limit of high wealth values the proposed density function has the accepted Tsallis-Pareto shape. Our results are in agreement with the predictions of an earlier approach based on a mean-field like wealth exchange process.
We study a system of $N$ agents, whose wealth grows linearly, under the effect of stochastic resetting and interacting via a tax-like dynamics -- all agents donate a part of their wealth, which is, in turn, redistributed equally among all others. This mimics a socio-economic scenario where people have fixed incomes, suffer individual economic setbacks, and pay taxes to the state. The system always reaches a stationary state, which shows a trivial exponential wealth distribution in the absence of tax dynamics. The introduction of the tax dynamics leads to several interesting features in the sta…
We study an agent-based model of evolution of wealth distribution in a macro-economic system. The evolution is driven by multiplicative stochastic fluctuations governed by the law of proportionate growth and interactions between agents. We are mainly interested in interactions increasing wealth inequality that is in a local implementation of the accumulated advantage principle. Such interactions destabilise the system. They are confronted in the model with a global regulatory mechanism which reduces wealth inequality. There are different scenarios emerging as a net effect of these two competin…
Stochastic multiplicative dynamics characterize many complex natural phenomena such as selection and mutation in evolving populations, and the generation and distribution of wealth within social systems. Population heterogeneity in stochastic growth rates has been shown to be the critical driver of diversity dynamics and of the emergence of wealth inequality over long time scales. However, we still lack a general statistical framework that systematically explains the origins of these heterogeneities from the adaptation of agents to their environment. In this paper, we derive population growth …
The recent book by T. Piketty (Capital in the Twenty-First Century) promoted the important issue of wealth inequality. In the last twenty years, physicists and mathematicians developed models to derive the wealth distribution using discrete and continuous stochastic processes (random exchange models) as well as related Boltzmann-type kinetic equations. In this literature, the usual concept of equilibrium in Economics is either replaced or completed by statistical equilibrium. In order to illustrate this activity with a concrete example, we present a stylised random exchange model for the distr…
We review and classify stochastic processes without detailed balance condition. We obtain stationary distributions and investigate their stability in terms of generalized entropic divergences beyond the Kullback-Leibler formula. A simple stochastic model with local growth rates and direct resetting to the ground state is investigated and applied to various networks, scientific citations and Facebook popularity, hadronic yields in high energy particle reactions, income and wealth distributions, biodiversity and settlement size distribution.
