Derivation of wealth distributions from biased exchange of money

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 types of dynamics denoted unbiased, poor-biased and rich-biased exchange models. At the individual level, one agent is picked randomly based on its wealth and one of its dollars is redistributed among the population. Proving the so-called propagation of chaos, we identify the limit of each dynamics as the number of individuals approaches infinity using both coupling techniques [55] and a martingale-based approach [45]. 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, it 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.