Laplace transform analysis of a multiplicative asset transfer model

TL;DR

This paper introduces a Laplace transform approach to analyze a multiplicative asset transfer model, providing analytical solutions for the steady-state distribution and insights into entropy and inequality without relying on agent-based simulations.

Contribution

It presents a novel Laplace transform method for solving the master equation of a multiplicative asset transfer model, enabling efficient analysis of steady states and disorder measures.

Findings

Laplace transform yields analytical steady-state distributions.

Boltzmann entropy is unsuitable for this system.

An equivalent asymmetric stochastic process is proposed.

Abstract

We analyze a simple asset transfer model in which the transfer amount is a fixed fraction $f$ of the giver's wealth. The model is analyzed in a new way by Laplace transforming the master equation, solving it analytically and numerically for the steady-state distribution, and exploring the solutions for various values of $f\in(0,1)$. The Laplace transform analysis is superior to agent-based simulations as it does not depend on the number of agents, enabling us to study entropy and inequality in regimes that are costly to address with simulations. We demonstrate that Boltzmann entropy is not a suitable (e.g. non-monotonic) measure of disorder in a multiplicative asset transfer system and suggest an asymmetric stochastic process that is equivalent to the asset transfer model.