Derivation of wealth distributions from biased exchange of money

TL;DR

This paper uses kinetic theory to analyze how different biased exchange dynamics influence wealth distribution and inequality, revealing complex behaviors like dispersive waves and wealth concentration.

Contribution

It introduces a kinetic model for wealth exchange with unbiased, poor-biased, and rich-biased dynamics, and proves convergence to equilibrium or complex dispersive behaviors.

Findings

Unbiased and poor-biased dynamics converge to equilibrium wealth distributions.

Rich-biased dynamics generate dispersive waves leading to maximum inequality.

Dispersive waves accumulate wealth, causing Gini index to approach 1.

Abstract

In the manuscript, we are interested in using kinetic theory to better understand the time evolution of wealth distribution and their large scale behavior such as the evolution of inequality (e.g. Gini index). We investigate three type of dynamics denoted unbiased, poor-biased and rich-biased dynamics. At the particle level, one agent is picked randomly based on its wealth and one of its dollar is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individual approaches infinity using both coupling techniques [48] and martingale-based approach [36]. Equipped with the limit equation, we identify and prove the convergence to specific equilibrium for both the unbiased and poor-biased dynamics. In the rich-biased dynamics however, we observe a more complex behavior where a dispersive wave emerges. Although the dispersive wave is vanishing in time, its also accumulates all the wealth leading to a Gini approaching 1 (its maximum value). We characterize numerically the behavior of dispersive wave but further analytic investigation is needed to derive such dispersive wave directly from the dynamics.