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An Analysis of Pseudo-Goodwin Cycles in a Wage-Led Minsky Model

Johannes Buchner

TL;DR

The paper formalizes pseudo-Goodwin cycles within a wage-led Minsky framework by casting the dynamics as a three-variable autonomous system in $(y,w,f)$ and by embedding a reserve-army wage mechanism. It develops a simplified Goodwin sub-model and extends it with Minsky-like fragility dynamics and distribution-function–driven wage feedback, culminating in a wage-led demand term $s y w$ that modulates stability. A Hopf bifurcation analysis shows that as $s$ crosses zero, the system transitions between stable and oscillatory dynamics, clarifying that observed cycles can be pseudo-periodic rather than true Goodwin cycles. The work highlights how combining wage shares, financial fragility, and distributional feedback yields a rich set of dynamical regimes, supported by simulations illustrating different behaviors across parameter ranges.

Abstract

The goal of these notes is to make the concept of "pseudo goodwin cycles" mathematically more precise. At first the title seems like a contradiction to have a wage-led model and still find goodwin cycles in it, but the point we try to make in the paper is that those are only `pseudo-goodwin' cycles, and not real goodwin cycles.

An Analysis of Pseudo-Goodwin Cycles in a Wage-Led Minsky Model

TL;DR

The paper formalizes pseudo-Goodwin cycles within a wage-led Minsky framework by casting the dynamics as a three-variable autonomous system in and by embedding a reserve-army wage mechanism. It develops a simplified Goodwin sub-model and extends it with Minsky-like fragility dynamics and distribution-function–driven wage feedback, culminating in a wage-led demand term that modulates stability. A Hopf bifurcation analysis shows that as crosses zero, the system transitions between stable and oscillatory dynamics, clarifying that observed cycles can be pseudo-periodic rather than true Goodwin cycles. The work highlights how combining wage shares, financial fragility, and distributional feedback yields a rich set of dynamical regimes, supported by simulations illustrating different behaviors across parameter ranges.

Abstract

The goal of these notes is to make the concept of "pseudo goodwin cycles" mathematically more precise. At first the title seems like a contradiction to have a wage-led model and still find goodwin cycles in it, but the point we try to make in the paper is that those are only `pseudo-goodwin' cycles, and not real goodwin cycles.

Paper Structure

This paper contains 15 sections, 1 theorem, 5 equations, 12 figures.

Table of Contents

  1. Acknowledgements
  2. Definitions
  3. Variables
  4. Differential equations
  5. A simplified version of the Goodwin model
  6. Reserve army effect
  7. Profit squeeze theory of accumulation
  8. A simplified version of the Goodwin model
  9. Minsky models
  10. A simple Minsky model
  11. Minsky cycle with a reserve army effect
  12. A Minsky model with distribution function and a wage-led effect in the demand function
  13. Next steps
  14. Hopf bifurcation in System \ref{['eq:Minsky_distribution_wage']} as $s$ crosses $0$
  15. Appendix

Key Result

Lemma 1

A pseudo-Goodwin cycle necessarily has at least one more variable that is neither enslaved to $y(t)$ nor enslaves $y(t)$.

Figures (12)

  • Figure 1: Sample orbit of the Goodwin model \ref{['eq:Goodwin_model']} for $r=c=1$ and initial condition $y(0)=0.6,w(0)=0.5$. The time series in panel (A) show that peaks in output $y(t)$ precede peaks in the wage rate $w(t)$. Panel (B) shows the vector field given by System \ref{['eq:Goodwin_model']} using blue arrows, and the orbit (shown in black) moves counterclockwise.
  • Figure 2: Relationship among variables in the Goodwin model [System \ref{['eq:Goodwin_model']}]. Blue edges denote positive feedbacks; red edges denote negative feedbacks. Edge labels are the terms in the ODEs in System \ref{['eq:Goodwin_model']}.
  • Figure 3: Sample orbit of the Minsky model \ref{['eq:Minsky_model']} for $p=1$ and initial condition $y(0)=0.6,f(0)=0.5$. The time series in panel (A) show that peaks in output $y(t)$ precede peaks in the wage rate $w(t)$. Panel (B) shows the vector field given by System \ref{['eq:Minsky_model']} using blue arrows, and the orbit (shown in black) moves counterclockwise.
  • Figure 4: Relationship among variables in the Minsky model [System \ref{['eq:Minsky_model']}]. Blue edges denote positive feedbacks; red edges denote negative feedbacks. Edge labels are the terms in the ODEs in System \ref{['eq:Minsky_model']}.
  • Figure 5: Sample orbit of the Minsky model with a reserve army effect [System \ref{['eq:Minsky_model']} augmented with Eq. \ref{['eq:wage_Minsky']}] for $p=r=c=1$ and $40$ time steps (top row) and the parameters in Stockhammer2014 ($p=2,r=5,c=3/2$, $200$ time steps, bottom row) and initial condition $y(0)=0.6,f(0)=0.5, w(0) = 0.4$. For the first set of parameters (top row), the wages $w(t)$ damp to zero, whereas for the second set of parameters (bottom row) the wage share $w(t)$ do not damp to zero (as in the example in Stockhammer2014).
  • ...and 7 more figures

Theorems & Definitions (11)

  • Definition 1
  • Remark 1
  • Definition 2: Reserve army effect
  • Definition 3: Profit squeeze
  • Definition 4: Goodwin cycle
  • Definition 5: Minsky financial fragility
  • Definition 6: Inverse relationship between output growth and financial fragility
  • Remark 2
  • Definition 7: Pseudo-Goodwin cycle
  • Lemma 1
  • ...and 1 more