A biased dollar exchange model involving bank and debt with discontinuous equilibrium

In this work, we investigate a biased dollar exchange model with collective debt limit, in which agents picked at random (with a rate depending on the amount of dollars they have) give at random time a dollar to another agent being picked uniformly at random, as long as they have at least one dollar in their pockets or they can borrow a dollar from a central bank if the bank is not empty. This dynamics enjoys a mean-field type interaction and partially extends the recent work \cite{cao_uncovering_2022} on a related model. We perform a formal mean-field analysis as the number of agents grows to infinity and as a by-product we discover a two-phase (ODE) dynamics behind the underlying stochastic $N$-agents dynamics. Numerical experiments on the two-phase (ODE) dynamics are also conducted where we observe the convergence towards its unique equilibrium in the large time limit.