Stratified economic exchange on networks

We investigate a model of stratified economic interactions between agents when the notion of spatial location is introduced. The agents are placed on a network with near-neighbor connections. Interactions between neighbors can occur only if the difference in their wealth is less than a threshold value that defines the width of the economic classes. By employing concepts from spatiotemporal dynamical systems, three types of patterns can be identified in the system as parameters are varied: laminar, intermittent and turbulent states. The transition from the laminar state to the turbulent state is characterized by the activity of the system, a quantity that measures the average exchange of wealth over long times. The degree of inequality in the wealth distribution for different parameter values is characterized by the Gini coefficient. High levels of activity are associated to low values of the Gini coefficient. It is found that the topological properties of the network have little effect on the activity of the system, but the Gini coefficient increases when the clustering coefficient of the network is increased.     

A multi-agent dynamic model based on different kinds of bequests

We investigate how wealth transfer that happens at the end of an agent’s life affects its final distribution based on a multi-agent dynamic model. We discuss two kinds of wealth transfers: to a single agent and to charities. The first kind of bequest is common in our realistic world and is always regarded by the public as unequal inheritance. The bequests to charities will be gathered and then equally redistributed among the survivors in our model. We find that when all the decedents choose the second kind of bequest, the final distribution is the Gibbs exponential function. When all the decedents choose the first kind of bequest, the result is condensation that a single individual accumulates all the available wealth. When an increasing portion of decedents choose the one-heir bequests, the exponential distribution evolves towards a power law shape (accompanied by deteriorating inequality). This shape firstly appears from the intermediate range of wealth and extends towards the top end of the simulated distribution, while the distribution remains exponential for high values of the wealth. At the same time, the Gini coefficient increases and the wealth accumulation becomes serious. At last, we analyze the source of the inequality. We find that not only unequal inheritances, but also equal division of the charity’s wealth can relatively contribute to an inequality of wealth distribution.     

Study of a market model with conservative exchanges on complex networks

Many models of market dynamics make use of the idea of conservative wealth exchanges among economic agents. A few years ago an exchange model using extremal dynamics was developed and a very interesting result was obtained: a self-generated minimum wealth or poverty line. On the other hand, the wealth distribution exhibited an exponential shape as a function of the square of the wealth. These results have been obtained both considering exchanges between nearest neighbors or in a mean field scheme. In the present paper we study the effect of distributing the agents on a complex network. We have considered archetypical complex networks: Erdös–Rényi random networks and scale-free networks. The presence of a poverty line with finite wealth is preserved but spatial correlations are important, particularly between the degree of the node and the wealth. We present a detailed study of the correlations, as well as the changes in the Gini coefficient, that measures the inequality, as a function of the type and average degree of the considered networks.     

Emergence of power-law in a market with mixed models

We investigate the problem of wealth distribution from the viewpoint of asset exchange. Robust nature of Pareto's law across economies, ideologies and nations suggests that this could be an outcome of trading strategies. However, the simple asset exchange models fail to reproduce this feature. A Yardsale (YS) model in which amount put on the bet is a fraction of minimum of the two players leads to condensation of wealth in hands of some agent while theft and fraud (TF) model in which the amount to be exchanged is a fraction of loser's wealth leads to an exponential distribution of wealth. We show that if we allow few agents to follow a different model than others, i.e., there are some agents following TF model while rest follow YS model, it leads to distribution with power-law tails. Similar effect is observed when one carries out transactions for a fraction of one's wealth using TF model and for the rest YS model is used. We also observe a power-law tail in wealth distribution if we allow the agents to follow either of the models with some probability.     

Laplace transform analysis of a multiplicative asset transfer model

We analyze a simple asset transfer model in which the transfer amount is a fixed fraction f of the giver’s wealth. The model is analyzed in a new way by Laplace transforming the master equation, solving it analytically and numerically for the steady-state distribution, and exploring the solutions for various values of f∈(0,1). The Laplace transform analysis is superior to agent-based simulations as it does not depend on the number of agents, enabling us to study entropy and inequality in regimes that are costly to address with simulations. We demonstrate that Boltzmann entropy is not a suitable (e.g. non-monotonic) measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model.     

Pareto and Boltzmann-Gibbs behaviors in a deterministic multi-agent system

A deterministic system of interacting agents is considered as a model for economic dynamics. The dynamics of the system is described by a coupled map lattice with nearest neighbor interactions. The evolution of each agent results from the competition between two factors: the agent’s own tendency to grow and the environmental influence that moderates this growth. Depending on the values of the parameters that control these factors, the system can display Pareto or Boltzmann-Gibbs statistical behaviors in its asymptotic dynamical regime. The regions where these behaviors appear are calculated on the space of parameters of the system. Other statistical properties, such as the mean wealth, the standard deviation, and the Gini coefficient characterizing the degree of equity in the wealth distribution are also calculated.     

Effects of introduction of new resources and fragmentation of existing resources on limiting wealth distribution in asset exchange models

Pareto law, which states that wealth distribution in societies has a power-law tail, has been the subject of intensive investigations in the statistical physics community. Several models have been employed to explain this behavior. However, most of the agent based models assume the conservation of number of agents and wealth. Both these assumptions are unrealistic. In this paper, we study the limiting wealth distribution when one or both of these assumptions are not valid. Given the universality of the law, we have tried to study the wealth distribution from the asset exchange models point of view. We consider models in which (a) new agents enter the market at a constant rate (b) richer agents fragment with higher probability introducing newer agents in the system (c) both fragmentation and entry of new agents is taking place. While models (a) and (c) do not conserve total wealth or number of agents, model (b) conserves total wealth. All these models lead to a power-law tail in the wealth distribution pointing to the possibility that more generalized asset exchange models could help us to explain the emergence of a power-law tail in wealth distribution.     

Evolution of the Distribution of Wealth in an Economic Environment Driven by Local Nash Equilibria

We present and analyze a model for the evolution of the wealth distribution within a heterogeneous economic environment. The model considers a system of rational agents interacting in a game theoretical framework, through fairly general assumptions on the cost function. This evolution drives the dynamic of the agents in both wealth and economic configuration variables. We consider a regime of scale separation where the large scale dynamics is given by a hydrodynamic closure with a Nash equilibrium serving as the local thermodynamic equilibrium. The result is a system of gas dynamics-type equations for the density and average wealth of the agents on large scales. We recover the inverse gamma distribution as an equilibrium in the particular case of quadratic cost functions which has been previously considered in the literature.     

Degree and wealth distribution in a network induced by wealth

A network induced by wealth is a social network model in which wealth induces individuals to participate as nodes, and every node in the network produces and accumulates wealth utilizing its links. More specifically, at every time step a new node is added to the network, and a link is created between one of the existing nodes and the new node. Innate wealth-producing ability is randomly assigned to every new node, and the node to be connected to the new node is chosen randomly, with odds proportional to the accumulated wealth of each existing node. Analyzing this network using the mean value and continuous flow approaches, we derive a relation between the conditional expectations of the degree and the accumulated wealth of each node. From this relation, we show that the degree distribution of the network induced by wealth is scale-free. We also show that the wealth distribution has a power-law tail and satisfies the 80/20 rule. We also show that, over the whole range, the cumulative wealth distribution exhibits the same topological characteristics as the wealth distributions of several networks based on the Bouchaud-Mèzard model, even though the mechanism for producing wealth is quite different in our model. Further, we show that the cumulative wealth distribution for the poor and middle class seems likely to follow by a log-normal distribution, while for the richest, the cumulative wealth distribution has a power-law behavior.     

Nestedness-Based Measurement of Evolutionarily Stable Equilibrium of Global Production System

A nested structure is a structural feature that is conducive to system stability formed by the coevolution of biological species in mutualistic ecosystems The coopetition relationship and value flow between industrial sectors in the global value chain are similar to the mutualistic ecosystem in nature. That is, the global economic system is always changing to form one dynamic equilibrium after another. In this paper, a nestedness-based analytical framework is used to define the generalist and specialist sectors for the purpose of analyzing the changes in the global supply pattern. We study why the global economic system can reach a stable equilibrium, what the role of different sectors play in the steady status, and how to enhance the stability of the global economic system. In detail, the domestic trade network, export trade network and import trade network of each country are extracted. Then, an econometric model is designed to analyze how the microstructure of the production system affects a country’s macroeconomic performance.     

Ideal-gas-like market models with savings: Quenched and annealed cases

We analyze the ideal-gas-like models of markets and review the different cases where a ‘savings’ factor changes the nature and shape of the distribution of wealth. These models can produce similar distribution of wealth as observed across varied economies. We present a more realistic model where the saving factor can vary over time (annealed savings) and yet produce Pareto distribution of wealth in certain cases. We discuss the relevance of such models in the context of wealth distribution, and address some recent issues in the context of these models.     

Crime and punishment: the economic burden of impunity

Crime is an economically relevant activity. It may represent a mechanism of wealth distribution but also a social and economic burden because of the interference with regular legal activities and the cost of the law enforcement system. Sometimes it may be less costly for the society to allow for some level of criminality. However, a drawback of such a policy is that it may lead to a high increase of criminal activity, that may become hard to reduce later on. Here we investigate the level of law enforcement required to keep crime within acceptable limits. A sharp phase transition is observed as a function of the probability of punishment. We also analyze other consequences of criminality as the growth of the economy, the inequality in the wealth distribution (the Gini coefficient) and other relevant quantities under different scenarios of criminal activity and probabilities of apprehension.     

Economic inequality and mobility in kinetic models for social sciences

Statistical evaluations of the economic mobility of a society are more difficult than measurements of the income distribution, because they require to follow the evolution of the individuals’ income for at least one or two generations. In micro-to-macro theoretical models of economic exchanges based on kinetic equations, the income distribution depends only on the asymptotic equilibrium solutions, while mobility estimates also involve the detailed structure of the transition probabilities of the model, and are thus an important tool for assessing its validity. Empirical data show a remarkably general negative correlation between economic inequality and mobility, whose explanation is still unclear. It is therefore particularly interesting to study this correlation in analytical models. In previous work we investigated the behavior of the Gini inequality index in kinetic models in dependence on several parameters which define the binary interactions and the taxation and redistribution processes: saving propensity, taxation rates gap, tax evasion rate, welfare means-testing etc. Here, we check the correlation of mobility with inequality by analyzing the mobility dependence from the same parameters. According to several numerical solutions, the correlation is confirmed to be negative.     

Effect of Savings on a Gas-Like Model Economy with Credit and Debt

In kinetic exchange models, agents make transactions based on well-established microscopic rules that give rise to macroscopic variables in analogy to statistical physics. These models have been applied to study processes such as income and wealth distribution, economic inequality sources, economic growth, etc., recovering well-known concepts in the economic literature. In this work, we apply ensemble formalism to a geometric agents model to study the effect of saving propensity in a system with money, credit, and debt. We calculate the partition function to obtain the total money of the system, with which we give an interpretation of the economic temperature in terms of the different payment methods available to the agents. We observe an interplay between the fraction of money that agents can save and their maximum debt. The system’s entropy increases as a function of the saved proportion, and increases even more when there is debt.     

Wealth condensation in a Barabasi-Albert network

We study the flow of money among agents in a Barabasi-Albert (BA) scale free network, where each network node represents an agent and money exchange interactions are established through links. The system allows money trade between two agents at a time, betting a fraction f of the poorer’s agent wealth. We also allow for the bet to be biased, giving the poorer agent a winning probability p. In the no network case there is a phase transition involving a relationship between p and f. In the networked case, we also found a condensation interface, however, this is not a complete condensation due to the presence of clusters in the network and its topology. As can be expected, the winner is always a well-connected agent, but we also found that the mean wealth decreases with the agents’ connectivity.     

The effect of asymmetric payoff mechanism on evolutionary networked prisoner’s dilemma game

In social and biological systems, there are obvious individual divergence and asymmetric payoff phenomenon due to the strength, power and influence differences. In this paper, we introduce an asymmetric payoff mechanism to evolutionary Prisoner’s Dilemma Game (PDG) on scale-free networks. The co-effects of individual diversity and asymmetric payoff mechanism on the evolution of cooperation and the wealth distribution under different updating rules are investigated. Numerical results show that the cooperation is highly promoted when the hub nodes are favored in the payoff matrix, which seems to harm the interest of the majority. But the inequality of social wealth distribution grows with the unbalanced payoff rule. However, when the node difference is eliminated in the learning strategy, the asymmetric payoff rule will not affect the cooperation level. Our work may sharpen the understanding of the cooperative behavior and wealth inequality in the society.     

Inequality reversal: Effects of the savings propensity and correlated returns

In the last decade, a large body of literature has been developed to explain the universal features of inequality in terms of income and wealth. By now, it is established that the distributions of income and wealth in various economies show a number of statistical regularities. There are several models to explain such static features of inequality in a unifying framework, and the kinetic exchange models in particular provide one such framework. Here we focus on the dynamic features of inequality. In the process of development and growth, inequality in an economy in terms of income and wealth follows a particular pattern of rising in the initial stage followed by an eventual fall. This inverted U-shaped curve is known as the Kuznets Curve. We examine the possibilities of such behavior of an economy in the context of a generalized kinetic exchange model. It is shown that under some specific conditions, our model economy indeed shows inequality reversal.     

A kinetic approach to some quasi-linear laws of macroeconomics

Some previous works have presented the data on wealth and income distributions in developed countries and have found that the great majority of population is described by an exponential distribution, which results in idea that the kinetic approach could be adequate to describe this empirical evidence. The aim of our paper is to extend this framework by developing a systematic kinetic approach of the socio-economic systems and to explain how linear laws, modelling correlations between macroeconomic variables, may arise in this context. Firstly we construct the Boltzmann kinetic equation for an idealised system composed by many individuals (workers, officers, business men, etc.), each of them getting a certain income and spending money for their needs. To each individual a certain time variable amount of money is associated - this meaning him/her phase space coordinate. In this way the exponential distribution of money in a closed economy is explicitly found. The extension of this result, including states near the equilibrium, give us the possibility to take into account the regular increase of the total amount of money, according to the modern economic theories. The Kubo-Green-Onsager linear response theory leads us to a set of linear equations between some macroeconomic variables. Finally, the validity of such laws is discussed in relation with the time reversal symmetry and is tested empirically using some macroeconomic time series. Received 25 February 2002 / Received in final form 11 July 2002 Published online 19 November 2002     

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.