A Fokker-Planck approach to a stochastic multiplicative wealth model with taxation and redistribution

We develop a Fokker-Planck description of the dynamics of wealth distribution in a stochastic multiplicative economic growth model with taxation and redistribution, as introduced by P.M.C. de Oliveira. Extending the original formulation, our theoretical framework includes general redistribution protocols, encompassing a broad class of state-dependent transfer mechanisms. As a particular case, we investigate a two-state protocol designed to emulate conditional cash transfer programs. Analytical expressions for the stationary wealth distributions are derived, revealing how the interplay between multiplicative noise, taxation, and redistribution shapes the emergence of inequality. The theoretical results are corroborated by agent-based simulations. To quantify and compare the impact of the different protocols, we employ the Gini index as a measure of inequality. Our analysis highlights how specific nonuniform redistribution schemes can significantly mitigate wealth disparities.