Equilibrium distributions and relaxation times in gas-like economic models: an analytical derivation

TL;DR

This paper presents an analytical method to derive the evolution equations of probability density functions in gas-like economic models, providing insights into their relaxation times and equilibrium states, applicable to various kinetic exchange rules.

Contribution

It introduces a step-by-step analytical technique to derive exact dynamical evolution equations and stationary solutions for kinetic wealth exchange models.

Findings

Derived explicit evolution equations for PDFs.

Calculated relaxation times for different models.

Identified conditions for analytical solutions.

Abstract

A step by step procedure to derive analytically the exact dynamical evolution equations of the probability density functions (PDF) of well known kinetic wealth exchange economic models is shown. This technique gives a dynamical insight into the evolution of the PDF, e.g., allowing the calculation of its relaxation times. Their equilibrium PDFs can also be calculated by finding its stationary solutions. This gives as a result an integro-differential equation, which can be solved analytically in some cases and numerically in others. This should provide some guidance into the type of probability density functions that can be derived from particular economic agent exchange rules, or for that matter, any other kinetic model of gases with particular collision physics.