Wealth Distributions in Models of Capital Exchange

TL;DR

This paper introduces a dynamical model of capital exchange between individuals, analyzing resulting wealth distributions under various exchange rules, including greedy and multiplicative processes, and explores extensions to spatial and growth scenarios.

Contribution

The paper presents a comprehensive analysis of wealth distributions in a new dynamical model of capital exchange, including analytical solutions and extensions to growth and spatial dimensions.

Findings

Fermi-like wealth distribution in greedy exchange models

Steady state in random multiplicative exchange

Power law distribution in greedy multiplicative exchange

Abstract

A dynamical model of capital exchange is introduced in which a specified amount of capital is exchanged between two individuals when they meet. The resulting time dependent wealth distributions are determined for a variety of exchange rules. For ``greedy'' exchange, an interaction between a rich and a poor individual results in the rich taking a specified amount of capital from the poor. When this amount is independent of the capitals of the two traders, a mean-field analysis yields a Fermi-like scaled wealth distribution in the long-time limit. This same distribution also arises in greedier exchange processes, where the interaction rate is an increasing function of the capital difference of the two traders. The wealth distribution in multiplicative processes, where the amount of capital exchanged is a finite fraction of the capital of one of the traders, are also discussed. For random multiplicative exchange, a steady state wealth distribution is reached, while in greedy multiplicative exchange a non-steady power law wealth distribution arises, in which the support of the distribution continuously increases. Finally, extensions of our results to arbitrary spatial dimension and to growth processes, where capital is created in an interaction, are presented.