Gibbs versus non-Gibbs distributions in money dynamics

TL;DR

This paper analyzes a simple closed economy model showing that the distribution of money among agents follows a Gibbs distribution when agents save nothing, and a non-Gibbs distribution when they do, with explicit formulas provided.

Contribution

It derives the explicit analytical form of the equilibrium money distribution for nonzero saving propensity, extending previous knowledge beyond the zero-saving case.

Findings

Gibbs distribution observed at zero saving propensity

Non-Gibbs distribution derived for nonzero saving propensity

Equilibrium distribution can be expressed as a Poisson distribution

Abstract

We review a simple model of closed economy, where the economic agents make money transactions and a saving criterion is present. We observe the Gibbs distribution for zero saving propensity, and non-Gibbs distributions otherwise. While the exact solution in the case of zero saving propensity is already known to be given by the Gibbs distribution, here we provide the explicit analytical form of the equilibrium distribution for the general case of nonzero saving propensity. We verify it through comparison with numerical data and show that it can be cast in the form of a Poisson distribution.