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In the manuscript, we are interested in using kinetic theory to betterunderstand the time evolution of wealth distribution and their large scalebehavior such as the evolution of inequality (e.g. Gini index). We investigatethree 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 oneof its dollar is redistributed among the population. Proving the so-calledpropagation of chaos, we identify the limit of each dynamics as the number ofindividual approaches infinity using both coupling techniques [48] andmartingale-based approach [36]. Equipped with the limit equation, we identifyand prove the convergence to specific equilibrium for both the unbiased andpoor-biased dynamics. In the rich-biased dynamics however, we observe a morecomplex behavior where a dispersive wave emerges. Although the dispersive waveis vanishing in time, its also accumulates all the wealth leading to a Giniapproaching 1 (its maximum value). We characterize numerically the behavior ofdispersive wave but further analytic investigation is needed to derive suchdispersive wave directly from the dynamics.

Derivation of wealth distributions from biased exchange of money