An $H$ theorem for Boltzmann's equation for the Yard-Sale Model of asset exchange

TL;DR

This paper proves that the Gini coefficient acts as an $H$ function for the Yard-Sale Model of asset exchange, confirming that wealth concentrates over time without redistribution, aligning with previous numerical conjectures.

Contribution

It establishes a rigorous $H$ theorem for the Yard-Sale Model, demonstrating the Gini coefficient's role as an $H$ function for the Boltzmann and Fokker-Planck equations.

Findings

Gini coefficient is an $H$ function for the model.

Wealth concentrates into oligarchy over time.

Theoretical proof confirms previous numerical conjectures.

Abstract

In recent work, Boltzmann and Fokker-Planck equations were derived for the "Yard-Sale Model" of asset exchange. For the version of the model without redistribution, it was conjectured, based on numerical evidence, that the time-asymptotic state of the model was oligarchy -- complete concentration of wealth by a single individual. In this work, we prove that conjecture by demonstrating that the Gini coefficient, a measure of inequality commonly used by economists, is an $H$ function of both the Boltzmann and Fokker-Planck equations for the model.