Directed Random Market: the equilibrium distribution

TL;DR

This paper derives the explicit equilibrium wealth distribution for the Directed Random Market process, showing it is a Gamma distribution with shape 1/2, and proves convergence of the process to this equilibrium.

Contribution

It provides the first explicit expression for the equilibrium distribution of the Directed Random Market and proves convergence of the wealth distribution process.

Findings

Equilibrium wealth distribution is a Gamma distribution with shape 1/2.

The wealth distribution process converges to the equilibrium distribution.

Explicit formula for the equilibrium distribution is derived.

Abstract

We find the explicit expression for the equilibrium wealth distribution of the Directed Random Market process, recently introduced by Mart\'inez-Mart\'inez and L\'opez-Ruiz, which turns out to be a Gamma distribution with shape parameter $\frac{1}{2}$. We also prove the convergence of the discrete-time process describing the evolution of the distribution of wealth to the equilibrium distribution.