Wealth exchange under ceiling and flooring constraints: a modified Bennati-Dragulescu-Yakovenko model

TL;DR

This work extends the Bennati-Dragulescu-Yakovenko (BDY) model by imposing wealth floor $a$ and wealth ceiling $b$, and studies the resulting dynamics through a mean-field limit. The authors derive a coupled ODE system for the wealth-distribution ${\bf p}(t)$ with densities $p_n(t)$, establish the conservation of total mass and mean, and identify a unique equilibrium ${\bf p}^*$ that is a truncated geometric distribution on $[a,b]$. They construct a generalized entropy-based Lyapunov functional ${\widetilde{\mathcal H}}_{ab}[{\bf p}]$ with positive weights to prove exponential convergence of ${\bf p}(t)$ to ${\bf p}^*$ in $\ell^p$ spaces, and analyze the impact of $a$ and $b$ on inequality via the Gini index. Numerically, increasing the floor $a$ tends to reduce inequality while increasing the ceiling $b$ can increase it; explicit expressions in the $b\to\infty$ limit corroborate the monotonicity, suggesting policy-like effects of wealth constraints on distributional outcomes.

Abstract

We investigate the classical Bennati-Dragulescu-Yakovenko (BDY) dollar exchange model introduced in \cite{dragulescu_statistical_2000} where the effects of wealth ceiling and wealth flooring are explored. In our model, $N$ identical economical agents involved in the BDY game are also subjected to certain policies issued by a (artificial) government, which prevent agents whose wealth exceeds some prescribed threshold value (denoted by $b \in \mathbb N_+$) from receiving money and which prohibit agents whose wealth falls below certain threshold value (denoted by $a \in \mathbb N$) from giving out their money. We derive a mean-field system of coupled nonlinear ordinary differential equations (ODEs) governing the evolution of the distribution of money as the number of agents $N$ tends to infinity and study the large time behavior of the resulting ODE system. The impact of a wealth cap and a wealth floor on economic inequality (measured by the Gini index) will also be explored numerically.

Figures (6)

• Figure 1: In the proposed binary exchange dynamics, an agent with less than $a$ dollars ( poor agent) represented on the left, can no longer give. Similarly, an agent with more than $b$ dollars ( rich agent) represented on the right, cannot receive. The middle class consists of agents whose wealth lies between $a$ and $b$ (represented in the middle), they can both give and receive money.
• Figure 2: Illustration of the wealth exchange dynamics in the mean-field limit \ref{['eq:law_limit']}-\ref{['eq:two-classes']} as $N \to \infty$. The population is described through a probability mass function ${\bf p}=(p_0,p_1,\dots)$ whose evolution is prescribed by the rates $λ_g$ and $λ_r$, which represent the proportion of givers and receivers, respectively.
• Figure 3: Left: Empirical wealth distributions obtained from particle simulations at time $t = 1$ for different $N$-agents, alongside the mean field ODE solution (solid line). Right: Error between particle distribution for different $N$-agents and ODE solution at $t = 1$ (with the plot being in the log-log scale).
• Figure 4: Geometric-type Equilibrium distribution ${\bf p}^*$eq:equi for different values of the mean value $\mu$. A) Corresponds to the case where the root resides within $(0,1)$ and the resulting distribution is decreasing within $[a,b]$. B) Corresponds to the case where the root is exactly at $x = 1$ and the distribution is (nearly) uniform within $[a,b]$. C) Corresponds to the case where the root lies within $(1,\infty)$ and the distribution is increasing within $[a,b]$.
• Figure 5: Left: Comparison between the numerical solution ${\bf p}(t)$ of the mean-field model \ref{['eq:law_limit']}-\ref{['eq:two-classes']} and the geometric-type equilibrium ${\bf p}^*$ for $0\leq t\leq 5$. Right: Exponential decay of the distance between ${\bf p}(t)$ toward the equilibrium ${\bf p}^*$.

Theorems & Definitions (11)

• Theorem 1
• Lemma 2.1
• Proposition 2.2
• Proposition 3.1
• Theorem 2: Dissipation of the generalized entropy
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Proposition 3.5
• Lemma A.1