The Kinetics of Wealth and the Origin of the Pareto Law

TL;DR

This paper models wealth distribution dynamics using kinetic theory analogies, deriving equations that explain the emergence of Pareto law features like wealth cutoff and power-law tail.

Contribution

It introduces a Boltzmann equation for wealth exchange, extending kinetic models to include taxation and other economic factors, explaining Pareto law characteristics.

Findings

Derivation of a Boltzmann equation for wealth distribution.

Model extension to include inflation, production, and taxation.

Explanation of Pareto law features such as wealth cutoff and power-law tail.

Abstract

An important class of economic models involve agents whose wealth changes due to transactions with other agents. Several authors have pointed out an analogy with kinetic theory, which describes molecules whose momentum and energy changes due to interactions with other molecules. We pursue this analogy and derive a Boltzmann equation for the time evolution of the wealth distribution of a population of agents for the so-called Yard-Sale Model of wealth exchange. We examine the solutions to this equation by a combination of analytical and numerical methods, and investigate its long-time limit. We study an important limit of this equation for small transaction sizes, and derive a partial integrodifferential equation governing the evolution of the wealth distribution in a closed economy. We then describe how this model may be extended to include features such as inflation, production and taxation. In particular, we show that the model with taxation is capable of explaining the basic features of the Pareto law, namely a lower cutoff to the wealth density at small values of wealth, and approximate power-law behavior at large values of wealth.