我们回顾了Angle [J. Angle，Social Forces 65（1986）293; J.Math.Sociol.26（2002）217]，Bennati [E. Bennati，Rivista Internazionale di Scienze Economiche Commerciali 35（1988）735]，Chakraborti and Chakrabarti [A.查克拉博蒂湾K.查克拉巴蒂J. B 17（2000）167]，以及Dragulescu和Yakovenko [A. Dragulescu，V. M.雅科文科，欧洲J. B 17（2000）723]。提出了均衡财富分布的解析拟合形式。研究了异质性对财富分布的影响，讨论了财富分布中厚尾现象的出现和向均衡态的松弛。一个统一的重新考虑的模型建议。
英文摘要：
We review the basic kinetic wealth-exchange models of Angle [J. Angle, Social Forces 65 (1986) 293; J. Math. Sociol. 26 (2002) 217], Bennati [E. Bennati, Rivista Internazionale di Scienze Economiche e Commerciali 35 (1988) 735], Chakraborti and Chakrabarti [A. Chakraborti, B. K. Chakrabarti, Eur. Phys. J. B 17 (2000) 167], and of Dragulescu and Yakovenko [A. Dragulescu, V. M. Yakovenko, Eur. Phys. J. B 17 (2000) 723]. Analytical fitting forms for the equilibrium wealth distributions are proposed. The influence of heterogeneity is investigated, the appearance of the fat tail in the wealth distribution and the relaxation to equilibrium are discussed. A unified reformulation of the models considered is suggested.
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.
We propose a novel kinetic exchange model differing from previous ones in two main aspects. First, the basic dynamics is modified in order to represent economies where immediate wealth exchanges are carried out, instead of reshufflings or uni-directional movements of wealth. Such dynamics produces wealth distributions that describe more faithfully real data at small values of wealth. Secondly, a general probabilistic trading criterion is introduced, so that two economic units can decide independently whether to trade or not depending on their profit. It is found that the type of the equilibriu…
We propose in this work a kinetic wealth-exchange model of economic growth by introducing saving as a non consumed fraction of production. In this new model, which starts also from microeconomic arguments, it is found that economic transactions between pairs of agents leads the system to a macroscopic behavior where total wealth is not conserved and it is possible to have an economic growth which is assumed as the increasing of total production in time. This last macroeconomic result, that we find both numerically through a Monte Carlo based simulation method and analytically in the framework …
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…
The paper provides a survey of results related to the "$κ$-generalized distribution", a statistical model for the size distribution of income and wealth. Topics include, among others, discussion of basic analytical properties, interrelations with other statistical distributions as well as aspects that are of special interest in the income distribution field, such as the Gini index and the Lorenz curve. An extension of the basic model that is most able to accommodate the special features of wealth data is also reviewed. The survey of empirical applications given in this paper shows the $κ$-gene…
We introduce a class of new one-dimensional linear Fokker--Planck type equations describing the evolution in time of the wealth in a multi-agent society. The equations are obtained, via a standard limiting procedure, by introducing an economically relevant variant to the kinetic model introduced in 2005 by Cordier, Pareschi and Toscani according to previous studies by Bouchaud and Mézard. The steady state of wealth predicted by these new Fokker--Planck equations remains unchanged with respect to the steady state of the original Fokker--Planck equation. However, unlike the original equation, it…
The spreading of 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 modelling, 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 dist…
It has been pointed out by Patriarca et al. (2005) that the power-law tailed equilibrium distribution in heterogeneous kinetic exchange models with a distributed saving parameter can be resolved as a mixture of Gamma distributions corresponding to particular subsets of agents. Here, we propose a new four-parameter statistical distribution which is a $κ$-deformation of the Generalized Gamma distribution with a power-law tail, based on the deformed exponential and logarithm functions introduced by Kaniadakis(2001). We found that this new distribution is also an extension to the $κ$-Generalized d…
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 gl…
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 proced…
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 develop a mathematical framework to study the economic impact of infectious diseases by integrating epidemiological dynamics with a kinetic model of wealth exchange. The multi-agent description leads to study 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 conver…
The lattice gas automaton (LGA) is proposed for a closed economic market of agents with heterogeneous saving interests. There are two procedures in the standard LGA, i.e., "propagation" + "transaction". If the propagation step is removed and the transaction is conducted among all agents, the LGA reduces to a more simplified kinetic model. In addition, two dealing rules are imposed on the transaction phase. Under Rule I, the trading volume depends on the average saving propensities of an arbitrary pair of agents in trade. Under Rule II, the exchange is governed by a stochastic parameter between…
Relations Among Different Inequality Measures in Complex Systems: From Kinetic Exchange to Earthquake Models
Indian Institute of Science Education and Research（印度科学教育研究所）；S. N. Bose National Centre for Basic Sciences（S.N. 玻色基础科学国家中心）；Indian Statistical Institute（印度统计研究所）；Saha Institute of Nuclear Physics（萨哈核物理研究所）
In this article, we briefly review the different aspects and applications of kinetic exchange models in economics and sociology. Our main aim is to show in what manner the kinetic exchange models for closed economic systems were inspired by the kinetic theory of gas molecules. The simple yet powerful framework of kinetic theory, first proposed in 1738, led to the successful development of statistical physics of gases towards the end of the 19th century. This framework was successfully adapted to modeling of wealth distributions in the early 2000's. In later times, it was applied to other areas…
Social and economic inequality is a plague of the XXI Century. It is continuously widening, as the wealth of a relatively small group increases and, therefore, the rest of the world shares a shrinking fraction of resources. This situation has been predicted and denounced by economists and econophysicists. The latter ones have widely used models of market dynamics which consider that wealth distribution is the result of wealth exchanges among economic agents. A simple analogy relates the wealth in a society with the kinetic energy of the molecules in a gas, and the trade between agents to the e…
In the manuscript, we are interested in using kinetic theory to better understand the time evolution of wealth distribution and their large scale behavior such as the evolution of inequality (e.g. Gini index). We investigate three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics. At the particle level, one agent is picked randomly based on its wealth and one of its dollar is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques […
Limiting risk to reduce inequality: insights from the Yard-Sale model
Centro Atómico Bariloche (CNEA)（巴里洛切原子中心）；CONICET（阿根廷国家科学研究委员会）；Instituto Balseiro, Universidad Nacional de Cuyo（门多萨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）
We consider a nonlinear kinetic equation of Boltzmann type which takes into account the influence of conviction during the formation of opinion in a system of agents which interact through the binary exchanges introduced in [G. Toscani, Commun. Math. Sci. 4, 481 (2006)]. The original exchange mechanism, which is based on the human tendency to compromise and change of opinion through self-thinking, is here modified in the parameters of the compromise and diffusion terms, which now are assumed to depend on the personal degree of conviction. The numerical simulations show that the presence of con…
We report the numerical results for the steady state income or wealth distribution $P(m)$ and the resulting inequality measures (Gini $g$ and Kolkata $k$ indices) in the kinetic exchange models of market dynamics. We study the variations of $P(m)$ and of the indices $g$ and $k$ with the saving propensity $λ$ of the agents, with two different kinds of trade (kinetic exchange) dynamics. In the first case, the exchange occurs between randomly chosen pairs of agents and in the next, one of the agents in the chosen pair is the poorest of all and the other agent is randomly picked up from the rest o…
