Uncovering a Two-Phase Dynamics from a Dollar Exchange Model with Bank and Debt

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 dynamic eventually leads to an asymmetric Laplace distribution of wealth (conjectured in [N. Xi, N. Ding, and Y. Wang, Phys. A, 357 (2005), pp. 543--555] and shown formally in a recent work [N. Lanchier and S. Reed, J. Stat. Phys., 176 (2019), pp. 1115-1137]). 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 dynamic. Con-vergence 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.