Abstract
The wealth substitution rate, which describes the substitution relationship between agents’ investment in wealth, is introduced into the collision kernel of the Boltzmann equation to study wealth distribution. Using the continuous trading limit, the Fokker–Planck equation is derived and the steady-state solution is obtained. The results show that the inequality of wealth distribution decreases as the wealth substitution rate increases under certain assumptions. The wealth distribution has a bimodal shape if the wealth substitution rate does not equal one.
Highlights
Since Pareto [1] discovered that wealth distribution in stable economy presents power law characteristics, the research on wealth distribution in multiagent society has been widely concerned by scholars, including economists, mathematicians, and physicists
During and Toscani [11] assume that there are two distinct types of agents, some of which tend to have higher saving propensity and some tend to have a lower saving propensity. e results show that wealth distribution presents a bimodal shape. e binary transaction in [8] contains knowledge, which affects saving propensity and risks. e results illustrate that knowledge is one of the reasons for the Pareto tail formation
Formulating the linear Boltzmann equation and seeking the asymptotic Fokker–Planck equation, we obtain the steady-state solution. e results show that the inequality of wealth distribution becomes small as the wealth substitution rate increases if (β/α) ≠ 1, and the wealth substitution rate has no effect on the inequality of wealth distribution when (β/α) 1
Summary
Since Pareto [1] discovered that wealth distribution in stable economy presents power law characteristics, the research on wealth distribution in multiagent society has been widely concerned by scholars, including economists, mathematicians, and physicists. The wealth distribution model with binary transactions analyzes the Pareto tail formation, many scholars use the Maxwell-type collision kernel (constant collision kernel) when constructing a kinetic model. If K(v, w) is constant, it is the Maxwell-type collision kernel [6,7,8, 11, 13, 14], which enables agents with zero wealth or extremely small wealth in the market to participate in trading. E linear collision kernel (1) considers the rationality of agents with a certain amount of wealth participating in transactions This selection brings some difficulties to our subsequent discussions. We obtain that the influence of saving propensity and market risk on wealth distribution is consistent with the Maxwell-type collision kernel.
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