  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.
