Breakdown of the mean-field approximation in a wealth distribution model

TL;DR

This paper critically examines a wealth distribution model, revealing that the mean-field approximation's stationary power-law distribution only holds approximately over finite times and is invalid for finite or heterogeneous agent populations.

Contribution

It demonstrates the limitations of the mean-field approximation in a wealth distribution model, especially for finite and heterogeneous agent systems.

Findings

Mean-field approximation is only valid for finite times.

No true stationary wealth distribution exists for finite agents.

Heterogeneity exacerbates the limitations of the approximation.

Abstract

One of the key socioeconomic phenomena to explain is the distribution of wealth. Bouchaud and M\'ezard have proposed an interesting model of economy [Bouchaud and M\'ezard (2000)] based on trade and investments of agents. In the mean-field approximation, the model produces a stationary wealth distribution with a power-law tail. In this paper we examine characteristic time scales of the model and show that for any finite number of agents, the validity of the mean-field result is time-limited and the model in fact has no stationary wealth distribution. Further analysis suggests that for heterogeneous agents, the limitations are even stronger. We conclude with general implications of the presented results.