Delayed Keen Model with Inflation

TL;DR

This work extends Keen-type macroeconomic models by introducing a time delay $\tau$ in the inflation feedback term $Z(\omega(t-\tau))$, resulting in a delayed three-variable system for wage share $\omega$, employment rate $\lambda$, and debt ratio $b$. By linearizing around an economically meaningful equilibrium and deriving the delayed characteristic equation, the authors prove that a Hopf bifurcation occurs when the delay crosses a critical value $\tau_0$, under explicit conditions on the model parameters. A concrete numerical example confirms a Hopf at $\tau_0 \approx 0.83$, identifying it as subcritical with unstable periodic solutions, and illustrates the transition from stability to oscillations. The results demonstrate that inflation delays can generate macroeconomic cycles in Keen-type models and provide a framework for further exploration of delayed feedback in macroeconomics with potential policy implications.

Abstract

Keen's model describes the dynamics between wage share, employment rate and debt ratio. In literature, the model was extended to represent the effects of inflation and also the speculative money flow. Based on the inflationary model, we take into account a time delay in the inflation term which stands for the period before the effects of inflation are seen. We observe that, the delayed system may experience a Hopf bifurcation and exhibit cyclic behavior around an equilibrium point, although the non-delayed model is stable under the same conditions.

Figures (3)

• Figure 1: $\tau=0$
• Figure 2: $\tau=\tau_0=0.82998$
• Figure 3: $\tau=0.85$

Theorems & Definitions (6)

• Lemma 1
• Proposition 1
• Lemma 2
• Corollary 1
• Lemma 3
• Theorem 1