Particle system approach to wealth redistribution

TL;DR

This paper investigates a stochastic particle system modeling wealth exchange among agents, revealing divergence of moments in non-conservative cases and stability with convergence properties in conservative economies, connecting microscopic dynamics to kinetic equations.

Contribution

It provides a detailed analysis of the particle system's behavior, including divergence of moments and stability results, bridging microscopic stochastic models with kinetic wealth distribution equations.

Findings

Non-conservative exchanges lead to diverging moments over time.

Conservative economies exhibit stability, moment propagation, and exponential convergence.

Explicit rate of propagation of chaos is established as N^{-1/3}.

Abstract

We study a stochastic $N$-particle system representing economic agents in a population randomly exchanging their money, which is associated to a class of one-dimensional kinetic equations modelling the evolution of the distribution of wealth in a simple market economy, introduced by Matthes and Toscani \cite{matthes-toscani2008}. We show that, unless the economic exchanges satisfy some exact conservation condition, the $p$-moments of the particles diverge with time for all $p>1$, and converge to 0 for $0<p<1$. This establishes a qualitative difference with the kinetic equation, whose solution is known to have bounded $p$-moments, for all $p$ smaller than the Pareto index of the equilibrium distribution. On the other hand, the case of strictly conservative economies is fully treated: using probabilistic coupling techniques, we obtain stability results for the particle system, such as propagation of moments, exponential equilibration, and uniform (in time) propagation of chaos with explicit rate of order $N^{-1/3}$.