Banking loan duopoly dynamics with a macroprudential instrument: Bifurcations and virtual fixed points

We study the dynamics of a banking loan duopoly under a macroprudential policy based on the loan-to-deposit-ratio reserve requirement (LDR-RR). Unlike previous duopoly contributions focusing on banks with a capital adequacy ratio above the incentive threshold, we consider the case in which the capital ratio remains below that threshold, so that the regulatory mechanism becomes two-sided and reserve requirement penalties may arise both when the loan-to-deposit ratio falls below the lower bound and when it exceeds the upper bound. Starting from a simplified balance-sheet framework, we derive a two-dimensional piecewise-smooth map describing the evolution of the two banks’ loan volumes and partition the phase space into nine regions. For each subsystem, we determine the fixed points, distinguish between real and virtual equilibria, and derive existence and local stability conditions. The analysis shows that only three diagonal equilibria may become locally asymptotically stable, while the others are either unstable or always virtual. Instability may arise through both flip (period-doubling) and border-collision bifurcations, depending on the adjustment speed and on the LDR-RR parameters. Numerical simulations support the analytical results and reveal low-period cycles, multistability, and chaotic dynamics. Overall, the results highlight the crucial role played by macroprudential thresholds and penalty coefficients in shaping the stability of aggregate bank lending.