A stylized model for wealth distribution

TL;DR

This paper introduces a stylized stochastic model for wealth distribution, connecting discrete and continuous approaches and illustrating how finite models relate to Boltzmann-like kinetic equations.

Contribution

It presents a new simplified wealth exchange model, analyzing its equilibrium and linking discrete Markov chains to continuous kinetic equations.

Findings

Existence of equilibrium distribution in the discrete model

Connection established between Markov chain and Boltzmann-like equations

Framework demonstrates transition from finite to continuous wealth models

Abstract

The recent book by T. Piketty (Capital in the Twenty-First Century) promoted the important issue of wealth inequality. In the last twenty years, physicists and mathematicians developed models to derive the wealth distribution using discrete and continuous stochastic processes (random exchange models) as well as related Boltzmann-type kinetic equations. In this literature, the usual concept of equilibrium in Economics is either replaced or completed by statistical equilibrium. In order to illustrate this activity with a concrete example, we present a stylised random exchange model for the distribution of wealth. We first discuss a fully discrete version (a Markov chain with finite state space). We then study its discrete-time continuous-state-space version and we prove the existence of the equilibrium distribution. Finally, we discuss the connection of these models with Boltzmann-like kinetic equations for the marginal distribution of wealth. This paper shows in practice how it is possible to start from a finitary description and connect it to continuous models following Boltzmann's original research program.