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Endogenous Cycles in a Keen--Goodwin Model with Minsky Debt

Roberto Albarrán-García, Martha Alvarez-Ramírez, Carlos García-Azpeitia

Abstract

We analyze a three-dimensional Keen--Goodwin model that couples wage--employment dynamics with Minsky-style private debt. At zero real interest the interior equilibrium is nonhyperbolic and organized by a two-dimensional center manifold foliated by neutral Goodwin cycles. Introducing a small positive interest rate unfolds this degeneracy: we derive an explicit Hopf condition, prove persistence of the center manifold as a normally hyperbolic attracting surface, and obtain first-order amplitude and frequency corrections for the emergent limit cycle via phase--amplitude reduction. Numerical simulations support the asymptotic predictions and demonstrate how interest rates determine and modulate endogenous business cycles.

Endogenous Cycles in a Keen--Goodwin Model with Minsky Debt

Abstract

We analyze a three-dimensional Keen--Goodwin model that couples wage--employment dynamics with Minsky-style private debt. At zero real interest the interior equilibrium is nonhyperbolic and organized by a two-dimensional center manifold foliated by neutral Goodwin cycles. Introducing a small positive interest rate unfolds this degeneracy: we derive an explicit Hopf condition, prove persistence of the center manifold as a normally hyperbolic attracting surface, and obtain first-order amplitude and frequency corrections for the emergent limit cycle via phase--amplitude reduction. Numerical simulations support the asymptotic predictions and demonstrate how interest rates determine and modulate endogenous business cycles.

Paper Structure

This paper contains 18 sections, 8 theorems, 108 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Model
  3. Interior equilibrium$$
  4. Boundary equilibrium
  5. Infinite-debt equilibrium
  6. Center manifold with $r=0$
  7. The integrability for $r=0$
  8. Hopf bifurcation at the good equilibrium
  9. Normally stable invariant manifold for small $r$
  10. Phase-amplitude reduction for small interest rate
  11. Decomposition of the planar vector field
  12. Phase-amplitude coordinates
  13. Adjoint modes and phase-amplitude reduction
  14. Invariant graph of $d$ over the family
  15. Numerical experiments
  16. ...and 3 more sections

Key Result

Proposition 1

The Jacobian matrix evaluated at the interior equilibrium $(\omega _{0},\lambda _{0},d_{0})$ takes the form where The characteristic polynomial of eq:J_good is In particular, since $K_{0}K_{1}>0$ and $\alpha +\beta >0$, all eigenvalues have negative real parts if and only if $rK_{2}<0.$ Therefore, the good equilibrium is locally asymptotically stable if and only if $rK_{2}<0$, and it loses stab

Figures (3)

  • Figure 1: Amplitude of the periodic orbits in $\omega$ as a function of $\eta(\kappa_2)$. Each point corresponds to a periodic orbit. The red and green markers indicate the two representative periodic orbits used for comparison.
  • Figure 2: Comparison of two representative periodic orbits in the phase plane $(\omega,\lambda)$. The red orbit corresponds to $\kappa_2 \approx 13.09$ and the green orbit to $\kappa_2 \approx 12.49$.
  • Figure 3: Comparison of the two representative periodic orbits (red and green) in the full state space $(\omega, \lambda, d)$, illustrating how the debt variable embeds the planar dynamics into a three-dimensional invariant manifold.

Theorems & Definitions (13)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 4
  • proof
  • Proposition 5
  • proof
  • Proposition 7: Hopf bifurcation at the good equilibrium
  • proof
  • Theorem 8: Persistence of a normally hyperbolic invariant manifold for small $r$
  • ...and 3 more