Lim Gyuchang, Min Seungsik
Department of Natural Science, Korea Naval Academy, Changwon 51704, Korea.
Entropy (Basel). 2020 Mar 28;22(4):386. doi: 10.3390/e22040386.
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.
众所周知，两种不同的潜在动态会导致不同的收入/财富分配模式，比如低端的玻尔兹曼-吉布斯形式和高端的帕累托型幂律形式。玻尔兹曼-吉布斯分布自然地源于使主体间随机相互作用的熵最大化，而帕累托分布则需要一种依赖于财富水平的经济学理性方法。更有趣的是，帕累托机制非常动态，而玻尔兹曼-吉布斯机制随时间是稳定的。此外，在某些情况下，收入/财富分配是双峰或多峰的。为了将帕累托机制的动态方面以及收入/财富分配的多峰形式纳入一个随机模型，我们基于经典的动态财富交换模型提出了一个改进的基于主体的模型。首先，我们采用一个由富人和穷人组成的简单两阶层社会，同一阶层的主体进行随机交换，而不同阶层的主体进行依赖于财富的胜者全得交易。这种修改使系统走向一个极端两极分化的社会，同时保留帕累托指数。其次，我们在模型中纳入了属于下层阶级的主体之间的团结形成，以对抗超级富有的主体。这种修改使系统产生财富的剧烈双峰分布，随着团结参数的变化，帕累托指数也会变化，也就是说，随着团结参数增加，帕累托机制变得更窄，帕累托指数变得更大。我们认为团结形成是帕累托指数变化和多峰分布的关键因素。最后，我们采用两种方法来评估财富不平等程度，比如基尼系数和熵度量。根据数值结果，团结参数的增加导致基尼系数非线性而非线性地下降，而熵度量在团结参数变化时是稳健的，这意味着中间阶层和高端阶层之间存在权衡。
