A Minimal Probabilistic Minsky Model: 3D Continuous-Jump Dynamics

This paper proposes a formalization of Hyman P. Minsky’s theory of financial instability. The model includes private-sector borrowing, capacity utilization, and the stock of private-sector debt. The model is based on self-reinforcing borrowing and output dynamics that repeatedly come to a sudden stop, with discontinuous downward jumps in the three variables. The paper treats as endogenous the instantaneous probability of a jump and the size distribution of jump vectors. Formally, the model comprises three ordinary differential equations and a compound Poisson process, with jumps drawn from a heavy-tailed stable distribution. The paper shows it can be stated in three equations in the jump differentials and the usual differentials. A section sketches a nonlinear mechanism that can bound the system. The paper analyzes the dynamics of a simplified version of the main model and a more-SFC model with feedbacks from debt to borrowing and capacity utilization via debt-service effects. The paper reports (1) eigenvalues for the linear parts of both the simplified analytical model and a numerical example of the more-SFC model, (2) a phase diagram for the analytical model, and, (3) analytical stability conditions for the more-SFC model. The model replicates the upward instability and abrupt crises of Minsky’s theory.