Master equation for a kinetic model of trading market and its analytic solution

TL;DR

This paper develops a master equation for a kinetic trading market model with random savings, deriving an exact steady-state income distribution with a Pareto tail of index one, confirming previous numerical findings.

Contribution

It introduces an analytical master equation approach to a kinetic trading market model, providing exact solutions for the steady-state income distribution.

Findings

Steady state income distribution has a Pareto tail with index one.

Analytical solutions confirm earlier numerical results.

Model incorporates quenched random savings factors.

Abstract

We analyze an ideal gas like model of a trading market with quenched random saving factors for its agents and 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. The analysis starts with the development of a master equation for the time development of $P(m)$. Precise solutions are then obtained in some special cases.