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

Abstract

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.

A Simple Mechanism Causing Wealth Concentration

“…The observation has been also supported by analytic arguments. In contrast to kinetic exchange models e.g., [ 10 , 11 , 12 , 13 , 15 ], here, the wealth inequalities grow for a large range of initial conditions and this growth is not limited by a specific distribution with exponential tails. In contrast to other models with growing economy [ 14 ], no disproportionately better treatment of wealthier agents is required to fuel the growth of wealth inequalities.…”

“…This observation led to several mathematical models attempting to explain this phenomenon, i.e., so-called kinetic wealth exchange models that are based on microeconomic interactions between economic actors who exchange wealth between them over the trade cycle [ 9 ]. These include models introduced by Angle [ 10 ], Bennati [ 11 ], Chakraborti and Chakrabarti [ 12 ], Dragulescu and Yakovenko [ 13 ] and recently also the approaches by Vallejos et al [ 14 ] and Lim and Min [ 15 ], which share some common features with our approach.…”

“…In this section, we will briefly review the basic features of kinetic wealth exchange models following [ 9 , 14 , 15 ]. As usual, the economy is assumed to consist of N agents with wealth ( ).…”

“…Lim and Min [ 15 ] consider various kinetic exchange models in search for solidarity effects consider multiple variants of the model introducing heterogeneous savings parameter and wealth dependent trading rules with r being random variable uniformly distributed over and updated every transaction.…”

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

“…When studying the distribution of income and wealth with these models, different trends have been verified, such as the formation of classes [12,13]. Indeed, it has been seen that in the presence of labor unions, government policies to reduce inequality, or other solidarity factors of social protection, the corresponding wealth distribution changes to a bimodal distribution [14]. Although these distributions are rare and seem unrealistic, they have also been found when considering risky transactions and other stressful situations [15], such as an epidemic outbreak [16].…”