Analyzing money distributions in `ideal gas' models of markets

TL;DR

This paper studies ideal gas-like market models, proposing a new money distribution fit, confirming a power law tail with Pareto index 1, and analyzing the dynamics of money distribution and differences.

Contribution

It introduces a new fit for money distribution in fixed saving markets and analytically confirms the Pareto index in markets with random saving factors.

Findings

Money distribution follows a power law with Pareto index 1.

New fitting function for money distribution in fixed saving markets.

Derived master equations and solutions for the evolution of money distribution.

Abstract

We analyze an ideal gas like models of a trading market. We propose a new fit for the money distribution in the fixed or uniform saving market. For the marketwith quenched random saving factors for its agents we show that the steady state income ($m$) distribution $P(m)$ in the model has a power law tail with Pareto index $\nu$ exactly equal to unity, confirming the earlier numerical studies on this model. We analyze the distribution of mutual money difference and also develop a master equation for the time development of $P(m)$. Precise solutions are then obtained in some special cases.