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In this work, we investigate a biased dollar exchange model with collectivedebt limit, in which agents picked at random (with a rate depending on theamount of dollars they have) give at random time a dollar to another agentbeing picked uniformly at random, as long as they have at least one dollar intheir pockets or they can borrow a dollar from a central bank if the bank isnot empty. This dynamics enjoys a mean-field type interaction and partiallyextends the recent work \cite{cao_uncovering_2022} on a related model. Weperform a formal mean-field analysis as the number of agents grows to infinityand as a by-product we discover a two-phase (ODE) dynamics behind theunderlying stochastic $N$-agents dynamics. Numerical experiments on thetwo-phase (ODE) dynamics are also conducted where we observe the convergencetowards its unique equilibrium in the large time limit.

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