Uncovering a two-phase dynamics from a dollar exchange model with bank and debt

TL;DR

This paper analyzes a money exchange model with debt limits, revealing a two-phase mean-field dynamics and convergence to an asymmetric wealth distribution, with implications for wealth inequality.

Contribution

It introduces a formal mean-field limit for the model, uncovering a two-phase ODE dynamics and analyzing the role of banking and debt in wealth distribution.

Findings

Convergence to a two-sided geometric wealth distribution.

Identification of a two-phase dynamical process.

Numerical exploration of debt and bank effects on inequality.

Abstract

We investigate the unbiased model for money exchanges with collective debt limit: agents give at random time a dollar to one another as long as they have at least one dollar or they can borrow a dollar from a central bank if the bank is not empty. Surprisingly, this dynamics eventually leads to an asymmetric Laplace distribution of wealth (conjectured in [22] and shown formally in a recent work [18]). In this manuscript, we carry out a formal mean-field limit as the number of agents goes to infinity where we uncover a two-phase (ODE) dynamics. Convergence towards the unique equilibrium (two-sided geometric) distribution in the large time limit is also shown and the role played by the bank and debt (in terms of Gini index or wealth inequality) will be explored numerically as well.