Coupling cycle mechanisms: Minsky debt cycles and the multiplier-accelerator

While there exists a substantial literature on different business cycle mechanisms, there is little literature on economies with more than one business cycle mechanism operating and the relation of stability of these subsystems with the stability of the aggregate system. We construct a model where a multiplier-accelerator subsystem in output-investment space (a real cycle) and a Minskyian subsystem in investment-debt space (a financial cycle) can generate stable/unstable cycles in 2D in isolation. We then derive a theorem showing that if two independent cycle mechanisms that generate stable closed orbits in 2D share a self-destabilizing common variable and the true representation of the system is a fully-coupled 3D system where a weighted average of the common variable is in effect, then the 3D system will generate locally stable closed orbits in 3D if and only if the subsystems have the same frequencies and/or the self-destabilizing effects of the common variable evaluated at the fixed point are equal in both subsystems. Our results indicate that in the presence of multiple cycle mechanisms which share common variables in an economy, the stability of the aggregate economy crucially depends on the frequencies of these sub-cycle mechanisms.