On the Study of Wealth Distribution with Non-Maxwellian Collision Kernels and Variable Trading Propensity

Abstract

A class of dynamic equations containing a non-Maxwellian collision kernel is used to investigate the distribution of wealth. A trading rule, in which the trading propensity $\gamma$ of agents is a function of wealth w (namely, $\gamma =\gamma \left(w\right)$), is considered. Two different trading propensity functions are discussed. One is that $\gamma \left(w\right)$ increases with wealth. The other is that $\gamma \left(w\right)$ decreases with the increase in wealth. In a single transaction, when the transaction tendency increases with the increase in wealth, the rich invest more in transactions. The gap between the rich and the poor in society is reduced under suitable conditions. Through numerical simulation, we conclude that an escalation in market risk intensifies the inequality in wealth distribution.

1. Introduction

The methods of statistical mechanics have been utilized to investigate various social problems [1,2,3,4]. Bassetti and Toscani [5] employ statistical mechanics to analyze the evolution of populations, demonstrating that the solution of the corresponding kinetic model tends toward equilibrium over time. Bisi [6] proposes a kinetic model to describe the trade transactions between agents in different countries. Trading depends on the agent’s trading tendency and random effects. On the kinetic level, considering the continuous trading restriction, a Fokker–Planck equation is derived in [6]. Based on the method of statistical mechanics, Maldarella and Pareschi [7] introduce a Boltzmann model for interactions in a multi-agent market. In [7], the asymptotic state of investment and price distribution is studied by using the kinetic model, and it is determined that the price distribution converges to lognormal distribution. Düring et al. [8] discuss the optimal control strategies for the kinetic models of wealth distribution. The results in [8] show that control has improved wealth inequality. Slanina [9] studies the economic behavior of agents by using the principle of statistical mechanics, arguing that agent interactions lead to wealth redistribution (see [10,11,12,13]).

Pareto [14] observed that wealth in Western societies follows a power-law tail, now termed the Pareto law. For the wealth density function $f_{\infty }\left(w\right)$ at steady state, if there is a constant $\alpha \in [1,+\infty )$ such that

$$f_{\infty }\left(w\right)\propto w^{-(\alpha +1)}asw\to \infty .$$

We say that $f_{\infty }\left(w\right)$ satisfies the Pareto law, and $\alpha$ denotes the Pareto index. The closer the Pareto index is to 1, the more inequality there is in the distribution of social wealth [15].

When building a kinetic model to study the distribution of wealth, the most important thing is to determine the interaction rules of wealth. Saving effect and randomness are two important characteristics that describe the rules of wealth interaction. Agents often save part of their wealth, a behavior known as the saving effect. Randomness reflects the uncertainty in wealth due to market risks. Chatterjee et al. [16] obtain the wealth distribution curve with a power law tail when agents have different saving tendencies. Cordier et al. [17] use a kinetic model containing binary trading rules to explain the formation of Pareto tails. Pareschi and Toscani [18] consider knowledge in interaction rules, obtaining a kinetic model that describes the impact of knowledge on wealth evolution. Bisi [19] examines the saving tendency that changes with wealth in trading rules and compares the Pareto index under two different trading tendencies. It is found that as the trading tendency function increases with wealth, the Pareto index also increases, showing that the trading strategy of the rich basically determines the distribution of wealth in society.

The collision kernel is often utilized to describe the interaction frequency of wealth. The traditional Maxwell collision kernel shows that agents with minimal or zero wealth also participate in market transactions. Under the economic background, everyone wants to trade with an agent who has a certain level of wealth. Thus, in the trading process, we employ a non-Maxwellian collision kernel, excluding transactions with zero or a very small amount of wealth. Ballante et al. [20] employ collision kernel $\kappa {\left(vw\right)}^{\delta }$ to investigate the influence of uncertain factors in interaction rules on saving tendency. Del-Mul [21] extends the collision kernel $\kappa {\left(vw\right)}^{\delta }$ and discusses other forms of non-Maxwellian collision kernels. Zhou et al. [22] investigate the distribution of wealth by adding a linear collision kernel into the Boltzmann equation. The Fokker–Planck equation is obtained and the steady-state solution is gained in [22]. Furioli et al. [23] utilize the non-Maxwellian collision kernel $\kappa {\left(vw\right)}^{\delta }$ (constant $\kappa >0$, $\delta \in (0,1]$, w and v stand for wealth of agents before the transaction) to analyze the wealth distribution in society.

Comparing with the works in [23], we state the main contribution of this work.

(I)

In [23], the trading tendency within trading rules is assumed to be constant. In this work, we propose a trading rule in which the agent’s trading tendency $\gamma =\gamma \left(w\right)$ depends on wealth w. This generalization captures realistic behavior of agents which depends on risk–reward dynamics.

(II)

We analyze two different trading tendency functions $\gamma \left(w\right)$: (i) $\gamma \left(w\right)$ is a decreasing function about w; (ii) $\gamma \left(w\right)$ is an increasing function about w. In a single transaction, when the trading tendency increases with the increase in wealth, the rich invest more in the transaction, and we find the Pareto index increases. This indicates that the trading strategies employed by the rich significantly affect the distribution of wealth among individuals, while the investments of the poor redistribute a small amount of wealth to other individuals.

The structure of this paper is as follows. In the second section of this paper, we introduce a kinetic model with the non-Maxwellian collision kernel $\kappa {\left(vw\right)}^{\delta }$. In the third section, we discuss the properties of uniformly bounded moments. In the fourth section, we obtain the Fokker–Planck equation and its the steady-state solution by using the continuous trading restriction. In the fifth section, we discuss two different trading tendency functions. Finally, we analyze the influence of market risk on the steady-state wealth density.

2. Kinetic Modeling of Trading Activity

Kinetic modeling involves micro-interaction, which is similar to binary interaction in rarefied gas dynamics. Trade refers to the exchange of wealth between agents following specific interaction rules. At any time $t\ge 0$, the state of an agent is characterized by wealth $w\ge 0$. The state of the agent system is represented by the density function $g=g(t,w)$, in which the wealth $w\in R_{+}$ and $t\ge 0$. Usually, for an interval $D\subseteq R_{+}$, the integral

$${\int }_{D}g(t,w)dw$$

denotes the quantity of agents depicted by wealth $w\in D$ when $t\ge 0$. Assuming that the distribution function is standardized to 1, namely, at any time $t\ge 0$, we have

$${\int }_{R_{+}}g(t,w)dw=1.$$

In the kinetic models in [15,24], the trading propensity parameters in the interaction rules are the same for the whole system. In fact, when the wealth of agents is different, their trading propensity also changes. For example, when a rich person realizes that high risk corresponds to high income, he puts most of the money into trading. At this time, his propensity to save is low. However, some wealthy agents have a high propensity to save and keep almost all their wealth. For the poor, there are also two kinds of trading situations. One is to store almost all their wealth, and the other is to risk almost all their wealth, expecting their wealth to increase rapidly. Because the agent is affected by the transaction, the amount of wealth is constantly updated according to the trading rules in the transaction. We assume that the trading rule is

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& v^{*}=(1-\gamma \left(v\right))v+\gamma \left(w\right)w+{\eta }_{1}v,\hfill \\ & w^{*}=(1-\gamma \left(w\right))w+\gamma \left(v\right)v+{\eta }_{2}w.\hfill \end{array}\right.\end{array}$$

(1)

In which the first item in (1) denotes the surplus wealth of agents who enter the trading market with $\gamma \left(v\right)v$ wealth. The second item represents the wealth received by agents in the trading market. The last item represents the risk of trading, where $\gamma \left(w\right)$ and $\gamma \left(v\right)$ represent trading propensity, which are functions dependent on wealth w and v. Trading propensity indicates the fraction of wealth which an agent invests in a single transaction. ${\gamma }_0$ is a constant and $0<\gamma \left(w\right)\le {\gamma }_0<1$, $0<\gamma \left(v\right)\le {\gamma }_0<1$. $w^{*}$ and $v^{*}$ indicate the agent’s wealth after the transaction. ${\eta }_1$ and ${\eta }_2$ are independent and identically distributed random variable with mean of zero and a variance of $\sigma$. The variance $\sigma$ quantifies market risk, capturing the uncertainty in wealth redistribution due to stochastic factors such as price fluctuations or external shocks. In order to ensure that the wealth after the transaction $(v^{*},w^{*})$ is not negative, assume that ${\eta }_1$ and ${\eta }_2$ belong to the interval $(-1+{\gamma }_0,1-{\gamma }_0)$.

Using the ideas in [2,3], for all smooth functions $\phi \left(w\right)$ with a supported set in $R_{+}$, using the trading rule Equation (1) and taking the expectation over random variables ${\eta }_1$, ${\eta }_2$, we derive that the agent’s wealth density $g(t,w)$ obeys the following Boltzmann Equation (2). This equation represents a balance law for the wealth distribution

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\int }_{R_{+}}g(t,w)\phi \left(w\right)dw={\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\int }_{R_{+}^2}K(w,v)\left(\phi \left(w^{*}\right)-\phi \left(w\right)\right)g(t,w)h(t,v)dwdv\right].\end{array}$$

(2)

In Equation (2), we set $v\in R_{+}$ to stand for the wealth in the market and its distribution is represented by $h(v,t)$. The non-Maxwellian collision kernel $K(w,v)=\kappa {\left(vw\right)}^{\delta }$ is chosen to exclude transactions involving agents with negligible wealth ($w\approx 0$ or $v\approx 0$). This is consistent with the economic behavior observed in the real world. The ${\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[·\right]$ represents the expectation of the random variable ${\eta }_1$ and ${\eta }_2$.

Utilizing the non-Maxwellian collision kernel $K(w,v)=\kappa {\left(vw\right)}^{\delta }$, in which $\kappa >0$, and $\delta \in (0,1]$ are constants, v and w stand for wealth of two agents before transaction, and Equation (2) becomes

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}{\int }_{R_{+}}g(t,w)\phi \left(w\right)dw={\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\int }_{R_{+}^2}\kappa {\left(vw\right)}^{\delta }\left(\phi \left(w^{*}\right)-\phi \left(w\right)\right)g(t,w)h(t,v)dwdv\right].\end{array}$$

(3)

3. Uniform Bounded Moment

We assume that

$$\begin{array}{c}\hfill M_j={\int }_{R_{+}}{v}^{j}h(t,v)dv<+\infty ,1\le j\le 4.\end{array}$$

If the initial function $g(0,w)$ is a probability density function, then $g(t,w)$ is still the probability density at any later time. We need to study the mean value and variance of the density $g(t,w)$. In the following content, let $m_j\left(t\right)$ represent the j-order moment of $g(t,w)$

$$\begin{array}{c}\hfill m_j\left(t\right)={\int }_{R_{+}}{w}^{j}g(t,w)dw,1\le j\le 3.\end{array}$$

Selecting $\phi \left(w\right)=w$ in Equation (3) and noticing Equation (1) yields

$$\begin{array}{c}\hfill {\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[w^{*}-w\right]=\gamma \left(v\right)v-\gamma \left(w\right)w,\end{array}$$

from which we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{m}_1\left(t\right)\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}{\left(vw\right)}^{\delta }(\gamma \left(v\right)v-\gamma \left(w\right)w)g(t,w)h(t,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}(\gamma \left(v\right)v^{1+\delta }{w}^{\delta }-\gamma \left(w\right)v^{\delta }{w}^{1+\delta })g(t,w)h(t,v)dwdv\hfill \\ \hfill \le & \kappa {\int }_{R_{+}^2}\gamma \left(v\right)v^{1+\delta }{w}^{\delta }g(t,w)h(t,v)dwdv.\hfill \end{array}$$

(4)

Using the Jensen’s inequality, we deduce that

$$\begin{array}{cc}\hfill {\int }_{R_{+}}{w}^{\delta }g(t,w)dw\le {\left[{\int }_{R_{+}}wg\left(t,w\right)dw\right]}^{\delta }& =m_1^{\delta }\left(t\right).\hfill \end{array}$$

Thus, inequality Equation (4) becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{m}_1\left(t\right)\hfill \\ \hfill \le & \kappa {\int }_{R_{+}^2}(\gamma \left(v\right)v^{1+\delta }{w}^{\delta }g(t,w)h(t,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}}\left[{\int }_{R_{+}}\gamma \left(v\right)v^{1+\delta }h(t,v)dv\right]w^{\delta }g(t,w)dw\hfill \\ \hfill \le & \kappa \left[{\int }_{R_{+}}\gamma \left(v\right)v^{1+\delta }h(t,v)dv\right]m_1^{\delta }\left(t\right).\hfill \end{array}$$

(5)

In inequality Equation (5), we set

$${\int }_{R_{+}}\gamma \left(v\right)v^{1+\delta }h(t,v)dv=N_1$$

and obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{m}_1\left(t\right)\le & \kappa N_1{m}_1^{\delta }\left(t\right).\hfill \end{array}$$

(6)

From inequality Equation (6), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}& m_1\left(t\right)\le \sqrt[1-\delta ]{\left(\kappa N_{1}t+m_1\left(0\right)\right)(1-\delta )},\hfill & \hfill 0<\delta <1,\\ & m_1\left(t\right)\le m_1\left(0\right)e^{\kappa N_{1}t},\hfill & \hfill \delta =1.\end{array}\right.\end{array}$$

(7)

System Equation (7) shows that the mean is uniformly bounded. Similarly, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{m}_2\left(t\right)=& \kappa {\int }_{R_{+}^2}{\left(vw\right)}^{\delta }{\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}\right)}^2-w^2\right]g(t,w)h(t,v)dwdv,\hfill \end{array}$$

where ${\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}\right)}^2-w^2\right]$ satisfies

$$\begin{array}{c}\hfill {\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}\right)}^2-w^2\right]=\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)w^2+2\gamma \left(v\right)(1-\gamma \left(w\right))vw+{\gamma }^2\left(v\right)v^2.\end{array}$$

Thus, the second moment satisfies

$$\begin{array}{cc}\hfill {\displaystyle \frac{d}{dt}}{m}_2\left(t\right)=& \kappa {\int }_{R_{+}^2}{\left(vw\right)}^{\delta }[\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)w^2\hfill \\ \hfill & +2\gamma \left(v\right)(1-\gamma \left(w\right))vw+{\gamma }^2\left(v\right)v^2]g(t,w)h(t,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}[\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)v^{\delta }{w}^{2+\delta }\hfill \\ \hfill & +2\gamma \left(v\right)(1-\gamma \left(w\right)){\left(vw\right)}^{1+\delta }+{\gamma }^2\left(v\right)v^{2+\delta }{w}^{\delta }]g(t,w)h(t,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)v^{\delta }{w}^{2+\delta }g(t,w)h(t,v)dwdv\hfill \\ \hfill & +\kappa {\int }_{R_{+}^2}2\gamma \left(v\right)(1-\gamma \left(w\right)){\left(vw\right)}^{1+\delta }g(t,w)h(t,v)dwdv\hfill \\ \hfill & +\kappa {\int }_{R_{+}^2}{\gamma }^2\left(v\right)v^{2+\delta }{w}^{\delta }g(t,w)h(t,v)dwdv.\hfill \end{array}$$

(8)

In addition, if $\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)<0$, we know that the coefficient $w^{2+\delta }$ in Equation (8) is negative. Utilizing the Jensen’s inequality, we derive that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{dt}}{m}_2\left(t\right)\hfill \\ \hfill \le & \kappa \left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)M_{\delta }{m}_2^{1+{\displaystyle \frac{\delta }{2}}}\left(t\right)+\kappa 2\gamma \left(v\right)(1-\gamma \left(w\right))M_{1+\delta }{m}_2^{{\displaystyle \frac{1}{2}}+{\displaystyle \frac{\delta }{2}}}\left(t\right)\hfill \\ \hfill & +\kappa {\gamma }^2\left(v\right)M_{2+\delta }{m}_2^{{\displaystyle \frac{\delta }{2}}}\left(t\right)\hfill \\ \hfill =& \kappa m_2^{{\displaystyle \frac{\delta }{2}}}\left(t\right)[\left(\sigma +{\gamma }^2\left(w\right)-2\gamma \left(w\right)\right)M_{\delta }{m}_2\left(t\right)+2\gamma \left(v\right)(1-\gamma \left(w\right))M_{1+\delta }{m}_2^{{\displaystyle \frac{1}{2}}}\left(t\right)\hfill \\ \hfill & +{\gamma }^2\left(v\right)M_{2+\delta }].\hfill \end{array}$$

(9)

The right side of Equation (9) is a quadratic equation with one variable in $m_2^{{\displaystyle \frac{1}{2}}}\left(t\right)$ and the coefficient of the quadratic term is negative. Let

$$\begin{array}{cc}\hfill K& ={\left\{{\displaystyle \frac{{\lambda }_1+\sqrt{{\lambda }_1^2+{\lambda }_2{\gamma }^2\left(v\right)M_{\delta }{M}_{2+\delta }}}{{\lambda }_2{M}_{\delta }}}\right\}}^2,\hfill \end{array}$$

(10)

where

$$\begin{array}{cc}\hfill & {\lambda }_1=\gamma \left(v\right)(1-\gamma \left(w\right))M_{1+\delta },\hfill \\ \hfill & {\lambda }_2=2\gamma \left(w\right)-\sigma -{\gamma }^2\left(w\right).\hfill \end{array}$$

From Equations (9) and (10), we have

$$\begin{array}{c}\hfill m_2\left(t\right)\le max\left\{m_2\left(0\right);K\right\}.\end{array}$$

(11)

Inequality Equation (11) states that the second moment $m_2\left(t\right)$ is uniformly bounded.

4. The Fokker–Planck Equation

Employing Equation (1), we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& w^{*}-w=\gamma \left(w\right)w-\gamma \left(v\right)v+{\eta }_{2}v,\hfill \\ & {\left(w^{*}-w\right)}^2={\left(\gamma \left(w\right)w-\gamma \left(v\right)v+{\eta }_{2}v\right)}^2.\hfill \end{array}\right.\end{array}$$

(12)

Applying Equation (12), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[w^{*}-w\right]=\gamma \left(v\right)v-\gamma \left(w\right)w,\hfill \\ & {\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}-w\right)}^2\right]={\gamma }^2\left(w\right)w^2+{\gamma }^2\left(v\right)v^2-2\gamma \left(w\right)\gamma \left(v\right)wv+\sigma w^2.\hfill \end{array}\right.\end{array}$$

(13)

Using the Taylor series expansion on $\phi \left(w^{*}\right)$ at w yields

$$\begin{array}{c}\hfill \phi \left(w^{*}\right)-\phi \left(w\right)={\displaystyle \frac{{\phi }^{\prime }\left(w\right)}{1!}}(w^{*}-w)+{\displaystyle \frac{{\phi }^{″}\left(w\right)}{2!}}{\left(w^{*}-w\right)}^2+{\displaystyle \frac{1}{6}}{\phi }^{‴}\left(\tilde{w}\right){\left(w^{*}-w\right)}^3,\end{array}$$

(14)

where $\tilde{w}=\theta w^{*}+(1-\theta )w$, $\theta \in (0,1)$. We discuss the asymptotic behavior of the solution for Equation (2) through continuous transaction restriction. We rewrite the trading propensity $\gamma \left(v\right)={\gamma }_0\tilde{\gamma }\left(v\right)$, $\gamma \left(w\right)={\gamma }_0\tilde{\gamma }\left(w\right)$ and assume that the interaction Equation (1) produces a small average change in wealth through scaling

$$\begin{array}{c}\hfill \gamma \left(v\right)=\epsilon {\gamma }_0\tilde{\gamma }\left(v\right),\gamma \left(w\right)=\epsilon {\gamma }_0\tilde{\gamma }\left(w\right),\sigma =\epsilon \sigma {\gamma }_0,\end{array}$$

(15)

where positive $\epsilon \ll 1$ is a small parameter and $\tilde{\gamma }\left(v\right)$ is the dimensionless trading propensity function normalized by the maximum propensity ${\gamma }_0$. From Equations (13)–(15), we have

$$\begin{array}{c}\hfill {\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[\phi \left(w^{*}\right)-\phi \left(w\right)\right]=\epsilon {\gamma }_0\left({\phi }^{\prime }\left(w\right)\left(\tilde{\gamma }\left(v\right)v-\tilde{\gamma }\left(w\right)w\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma w^2\right)+R_{\epsilon }\left(w\right),\end{array}$$

(16)

where

$$\begin{array}{c}\hfill R_{\epsilon }\left(w\right)={\displaystyle \frac{1}{2}}{\epsilon }^2{\gamma }_0^2\left[{\tilde{\gamma }}^2\left(w\right)w^2+{\tilde{\gamma }}^2\left(v\right)v^2-2\tilde{\gamma }\left(w\right)\tilde{\gamma }\left(v\right)wv\right]+{\displaystyle \frac{1}{6}}{\phi }^{‴}\left(\tilde{w}\right){\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}-w\right)}^3\right]\end{array}$$

represents the remainder of the Taylor formula.

Letting $\tau =\epsilon {\gamma }_{0}t$ and $g(t,w)$=$g(\tau ,w)$, combining Equations (3) and (16) and scaling in time give rise to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\tau }}{\int }_{R_{+}}\phi \left(w\right)g(\tau ,w)dw\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}{\left(vw\right)}^{\delta }\left({\phi }^{\prime }\left(w\right)\left(\tilde{\gamma }\left(v\right)v-\tilde{\gamma }\left(w\right)w\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma w^2\right)g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill & +{\displaystyle \frac{\kappa }{\epsilon {\gamma }_0}}{\int }_{R_{+}^2}{\left(vw\right)}^{\delta }{R}_{ϵ}\left(w\right)g(\tau ,w)h(\tau ,v)dwdv.\hfill \end{array}$$

(17)

When $\epsilon \to 0$, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\kappa }{\epsilon {\gamma }_0}}{\int }_{R_{+}^2}{\left(vw\right)}^{\delta }{R}_{ϵ}\left(w\right)g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill =& {\displaystyle \frac{\kappa }{\epsilon {\gamma }_0}}{\int }_{R_{+}^2}{\left(vw\right)}^{\delta }\{{\displaystyle \frac{1}{2}}{\epsilon }^2{\gamma }_0^2\left[{\tilde{\gamma }}^2\left(w\right)w^2+{\tilde{\gamma }}^2\left(v\right)v^2-2\tilde{\gamma }\left(w\right)\tilde{\gamma }\left(v\right)wv\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{6}}{\phi }^{‴}\left(\tilde{w}\right){\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}-w\right)}^3\right]\}g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill =& {\displaystyle \frac{1}{2}}\kappa \epsilon {\gamma }_0{\int }_{R_{+}^2}{\left(vw\right)}^{\delta }\{\left[{\tilde{\gamma }}^2\left(w\right)w^2+{\tilde{\gamma }}^2\left(v\right)v^2-2\tilde{\gamma }\left(w\right)\tilde{\gamma }\left(v\right)wv\right]\hfill \\ \hfill & +{\displaystyle \frac{1}{6}}{\phi }^{‴}\left(\tilde{w}\right){\mathbb{E}}_{{\eta }_1,{\eta }_2}\left[{\left(w^{*}-w\right)}^3\right]\}g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill =& 0.\hfill \end{array}$$

When the scale Equation (15) is used, the weak form of dynamic model Equation (3) is approximately the weak form of the Fokker–Planck equation. As a result, Equation (17) becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\tau }}{\int }_{R_{+}}\phi \left(w\right)g(\tau ,w)dw\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}{\left(vw\right)}^{\delta }\left[{\phi }^{\prime }\left(w\right)\left(\tilde{\gamma }\left(v\right)v-\tilde{\gamma }\left(w\right)w\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma w^2\right]g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}^2}\left[{\phi }^{\prime }\left(w\right)\left(\tilde{\gamma }\left(v\right)v^{1+\delta }{w}^{\delta }-\tilde{\gamma }\left(w\right)v^{\delta }{w}^{1+\delta }\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma v^{\delta }{w}^{2+\delta }\right]g(\tau ,w)h(\tau ,v)dwdv\hfill \\ \hfill =& \kappa {\int }_{R_{+}}\left[{\phi }^{\prime }\left(w\right)\left({\tilde{N}}_1{w}^{\delta }-\tilde{\gamma }\left(w\right)M_{\delta }{w}^{1+\delta }\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma M_{\delta }{w}^{2+\delta }\right]g(\tau ,w)dw,\hfill \end{array}$$

(18)

where

$${\int }_{R_{+}}\tilde{\gamma }\left(v\right)v^{1+\delta }h(\tau ,v)dv={\tilde{N}}_1.$$

When $\gamma \left(w\right)={\gamma }_0$, namely $\tilde{\gamma }\left(w\right)=1$ is constant. Integrating the right side of Equation (18) by parts, we obtain the Fokker–Planck equation

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial g}{\partial \tau }}=& \kappa {\displaystyle \frac{\sigma M_{\delta }}{2}}{\displaystyle \frac{{\partial }^2\left(w^{2+\delta }g\right)}{\partial w^2}}-\kappa {\displaystyle \frac{\partial \left[\left({\tilde{N}}_1{w}^{\delta }-M_{\delta }{w}^{1+\delta }\right)g\right]}{\partial w}}.\hfill \end{array}$$

(19)

Based on the works in [25], when $t>0$, the standard boundary condition to ensure the conservation of mass is given by

$$\begin{array}{cc}\hfill \{& \kappa {\displaystyle \frac{\sigma M_{\delta }}{2}}{\displaystyle \frac{\partial \left(w^{2+\delta }g\right)}{\partial w}}-\kappa \left({\tilde{N}}_1{w}^{\delta }-M_{\delta }{w}^{1+\delta }\right)g\}{|}_{w=0,+\infty }=0.\hfill \end{array}$$

When $t\to \infty$, solving Equation (19), the steady-state solution $g_{\infty }\left(w\right)$ satisfies

$$\begin{array}{c}\hfill g_{\infty }\left(w\right)={\displaystyle \frac{{\left(\frac{2{\tilde{N}}_1}{\sigma M_{\delta }}\right)}^{1+\delta +\frac{2}{\sigma }}}{\Gamma (1+\delta +\frac{2}{\sigma })}}{\displaystyle \frac{exp\left(-\frac{2{\tilde{N}}_1}{\sigma M_{\delta }w}\right)}{w^{2+\delta +\frac{2}{\sigma }}}}.\end{array}$$

(20)

From Equation (20), we know that the Pareto index is ${\alpha }_1=1+\delta +{\displaystyle \frac{2}{\sigma }}$.

Figure 1 illustrates the variation in the steady-state probability density $g_{\infty }\left(w\right)$ endowed with different $\sigma$ values, when $\delta =0.5$. As the market risk $\sigma$ increases, the peak of wealth density curve $g_{\infty }\left(w\right)$ shifts to the left and the tail becomes thicker. This indicates that the wealth of middle-class and low-class agents is deteriorating, while wealth is highly concentrated among a few people. Consequently, the increase in market risk intensifies the inequality of wealth.

It is very important to understand the significance and influence of the factor $\delta$ in the steady-state solution. Figure 2 describes the behavior of the steady-state solution $g_{\infty }\left(w\right)$ when $\delta$ is different. Figure 2 shows that with the increase in $\delta$, the central part of the solution becomes wider and the maximum value of the peak value becomes lower. In economics, this shows that with the increase in $\delta$, the middle class becomes richer and richer, and the average wealth of steady-state solutions increases. At this time, the distribution of wealth in society is also more equal.

Besides, according to Equation (20) and assuming $\delta +{\displaystyle \frac{2}{\sigma }}>1$, the variance value is given by

$$\begin{array}{cc}\hfill Var\left(g_{\infty }\right)& ={\displaystyle \frac{{\left(\frac{2{\tilde{N}}_1}{\sigma M_{\delta }}\right)}^2}{(\delta +\frac{2}{\sigma })(\delta +\frac{2}{\sigma }-1)}}-1.\hfill \end{array}$$

When the condition $\delta +{\displaystyle \frac{2}{\sigma }}>1$ holds. With a fixed value of $\sigma$ under the action of non-Maxwellian collision kernel $K(w,v)=\kappa {\left(vw\right)}^{\delta }$, we observe that when $\delta$ increases, the variance $Var\left(g_{\infty }\right)$ decreases. It means that with the increase in $\delta$, the wealth gap between individuals or groups diminishes and the distribution of wealth becomes fair.

5. Kinetic Model with Two Different Trading Tendency Functions

In this section, we discuss $\gamma \left(w\right)$ in two cases: (i) $\gamma \left(w\right)$ is a decreasing function about w; (ii) $\gamma \left(w\right)$ is an increasing function about w.

5.1. $\gamma \left(w\right)$ Is a Decreasing Function About w

We consider that the trading tendency satisfies

$$\begin{array}{cc}\hfill \gamma \left(w\right)& ={\gamma }_0{\displaystyle \frac{1}{4w+1}},\hfill \end{array}$$

(21)

which describes a market in which the proportion of wealth invested by an agent in a single transaction is reduced. To be precise, for wealth w ranging from $(0,+\infty )$, the function $\gamma \left(w\right)$ is reduced from ${\gamma }_0$ to 0. This situation describes a group where the wealthy invest only a small part of their wealth, while the poor invest nearly all their money in transactions in order to improve their current situation.

From Equation (21), we derive $\tilde{\gamma }\left(w\right)={\displaystyle \frac{1}{4w+1}}$, Equation (18) becomes

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\tau }}{\int }_{R_{+}}\phi \left(w\right)g(\tau ,w)dw\hfill \\ \hfill =& \kappa {\int }_{R_{+}}\left[{\phi }^{\prime }\left(w\right)\left({\tilde{N}}_1{w}^{\delta }-{\displaystyle \frac{1}{4w+1}}{M}_{\delta }{w}^{1+\delta }\right)+{\displaystyle \frac{1}{2}}{\phi }^{″}\left(w\right)\sigma M_{\delta }{w}^{2+\delta }\right]g(\tau ,w)dw.\hfill \end{array}$$

(22)

From Equation (22), we obtain the Fokker–Planck equation

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial g}{\partial \tau }}=& \kappa {\displaystyle \frac{\sigma M_{\delta }}{2}}{\displaystyle \frac{{\partial }^2\left(w^{2+\delta }g\right)}{\partial w^2}}-\kappa {\displaystyle \frac{\partial \left[\left({\tilde{N}}_1{w}^{\delta }-\frac{1}{4w+1}{M}_{\delta }{w}^{1+\delta }\right)g\right]}{\partial w}}.\hfill \end{array}$$

(23)

The steady-state solution of the Fokker–Planck Equation (23) is

$$\begin{array}{c}\hfill g_{1,\infty }\left(w\right)=C_1{(4w+1)}^{{\displaystyle \frac{2}{\sigma }}}{\displaystyle \frac{exp\left(-\frac{2{\tilde{N}}_1}{\sigma M_{\delta }w}\right)}{w^{2+\delta +\frac{2}{\sigma }}}},\end{array}$$

(24)

where $C_1$ satisfies ${\int }_{R_{+}}{g}_{1,\infty }\left(w\right)dw=1$. Therefore, when $\gamma \left(w\right)={\gamma }_0{\displaystyle \frac{1}{\xi w+1}}$$(\xi >1)$ is selected, the Pareto index is $\widetilde{{\alpha }_1}=1+\delta$, which is smaller than the Pareto index ${\alpha }_1=1+\delta +{\displaystyle \frac{2}{\sigma }}$. It illustrates that in a single transaction, when trading propensity $\gamma \left(w\right)$ is a decreasing function of wealth w, the rich invest a small amount of money and retain the majority of their wealth. In this situation, the Pareto index decreases and the distribution of wealth becomes even more unequal.

Figure 3 shows the change in the steady-state probability density $g_{1,\infty }\left(w\right)$ with different $\sigma$. When the trading tendency function is a decreasing function, the increase of market risk $\sigma$ makes the wealth density curve shift to the left and the tail becomes thicker, which reflects the phenomenon of wealth concentration. This indicates that the increase in market risk has widened the gap between the rich and the poor in society.

5.2. $\gamma \left(w\right)$ Is an Increasing Function About w

We consider a case in which the trading tendency $\gamma \left(w\right)$ increases with wealth. It describes a market in which the proportion of wealth invested by agents in a single transaction increases relative to wealth. This situation describes a group, where the poor invest only a small part of their wealth, while the rich invest a lot of money in transactions. We choose

$$\begin{array}{cc}\hfill \gamma \left(w\right)& ={\gamma }_0\left(1-{\displaystyle \frac{1}{w^4+\beta }}\right),\hfill \end{array}$$

where $\beta >1$. For $0\le w<+\infty$ we have ${\gamma }_0(1-{\displaystyle \frac{1}{\beta }})<\gamma \left(w\right)\le {\gamma }_0$. When $\tilde{\gamma }\left(w\right)=\left(1-{\displaystyle \frac{1}{w^4+\beta }}\right)$, using Equation (18) and the similar derivation of Equation (23), we derive the Fokker–Planck equation

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial g}{\partial \tau }}=& \kappa {\displaystyle \frac{\sigma M_{\delta }}{2}}{\displaystyle \frac{{\partial }^2\left(w^{2+\delta }g\right)}{\partial w^2}}-\kappa {\displaystyle \frac{\partial \left[\left({\tilde{N}}_1{w}^{\delta }-\left(1-\frac{1}{w^4+\beta }\right)M_{\delta }{w}^{1+\delta }\right)g\right]}{\partial w}}.\hfill \end{array}$$

(25)

The steady-state solution of the Fokker–Planck Equation (25) is

$$\begin{array}{cc}\hfill g_{2,\infty }\left(w\right)& =C_2{(w^4+\beta )}^{-{\displaystyle \frac{1}{2\sigma \beta }}}{w}^{{\displaystyle \frac{2}{\sigma \beta }}-2-{\displaystyle \frac{2}{\sigma }}-\delta }{e}^{-{\displaystyle \frac{2{\tilde{N}}_1}{M_{\delta }\sigma w}}}.\hfill \end{array}$$

The Pareto index is $\widetilde{{\alpha }_2}=1+\delta +{\displaystyle \frac{2}{\sigma }}$, which is consistent with the Pareto index ${\alpha }_1=1+\delta +{\displaystyle \frac{2}{\sigma }}$, and the constant $C_2$ satisfies ${\int }_{R_{+}}{g}_{2,\infty }\left(w\right)dw=1$. In a single transaction, when the trading tendency increases with the increase in wealth, the rich invest more wealth in trading, the redistributive wealth increases, and the Pareto index increases. This indicates that the trading strategies employed by the rich significantly affect the distribution of wealth among individuals. The investments of the poor redistribute a small amount of wealth to other individuals. Thus, their trading tendencies do not substantially impact the long-term dynamics of wealth distribution.

Figure 4 shows the changes in the steady-state probability densities $g_{2,\infty }\left(w\right)$ under different $\sigma$. Similarly, as the trading tendency function is an increasing function, the peak of the steady-state probability density also shifts to the left with the increase in market risk, and the thin tail of the wealth distribution curve becomes a thick tail, meaning that a minority group or individual occupies a dominant position in wealth distribution. This shows that the increase of market risk has aggravated the gap between the rich and the poor in society.

6. Conclusions

We introduce a class of kinetic equations containing non-Maxwellian collision kernel to depict the distribution of wealth within a multi-agent society. We consider a trading rule in which the agent’s trading tendency $\gamma$ is a function of wealth w. We further investigate two different trading tendency functions. In a single transaction, when the trading tendency decreases with the increase in wealth, it indicates that the rich primarily keep their wealth instead of trading. In this case, the Pareto index drops, and the wealth distribution in society becomes more and more unequal. In a single transaction, when the trading tendency increases with the increase in wealth, the rich invest more wealth in the transaction. At this time, the redistributive wealth in society increases, thus increasing the Pareto index. This demonstrates that the trading strategy of the wealthy fundamentally determines the formation of the Pareto tail in a proper asymptotic state. Numerical analysis illustrates that the rise of market risk intensifies the gap between the rich and the poor in society.