A statistical model with a standard Gamma distribution

TL;DR

This paper demonstrates that a simple interacting units model reaches an equilibrium distribution that fits a standard Gamma distribution, linking economic trading and physical energy exchange models.

Contribution

It shows that a basic interacting units model naturally results in a Gamma distribution at equilibrium, unifying economic and physical interpretations.

Findings

Equilibrium distribution fits a standard Gamma distribution.

Model can be interpreted as a closed economy or a physical gas.

Numerical simulations confirm the theoretical distribution.

Abstract

We study a statistical model consisting of $N$ basic units which interact with each other by exchanging a physical entity, according to a given microscopic random law, depending on a parameter $\lambda$. We focus on the equilibrium or stationary distribution of the entity exchanged and verify through numerical fitting of the simulation data that the final form of the equilibrium distribution is that of a standard Gamma distribution. The model can be interpreted as a simple closed economy in which economic agents trade money and a saving criterion is fixed by the saving propensity $\lambda$. Alternatively, from the nature of the equilibrium distribution, we show that the model can also be interpreted as a perfect gas at an effective temperature $T(\lambda)$, where particles exchange energy in a space with an effective dimension $D(\lambda)$.