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

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.

💡 Research Summary

The paper introduces a novel extension of the classic Bennati‑Dragulescu‑Yakovenko (BDY) wealth‑exchange model by imposing both an upper wealth ceiling (b) and a lower wealth floor (a) on agents. In the original BDY model, at random exponential times a randomly chosen agent who possesses at least one dollar gives one dollar to another randomly chosen agent, preserving the total wealth N µ. The authors modify this mechanism so that an agent i can give a dollar only if his wealth S_i exceeds the floor a, and the receiving agent j can accept only if his wealth S_j is below the ceiling b. The parameters satisfy 0 ≤ a < µ < b, where µ is the fixed average wealth per agent.

Mean‑field limit. By letting the number of agents N tend to infinity, the empirical distribution of wealth converges (in the sense of propagation of chaos) to a deterministic probability mass function p(t) = (p₀(t), p₁(t), …). The evolution of p(t) is governed by a coupled infinite system of nonlinear ordinary differential equations \