Double-Hopf bifurcation in an extended Goodwin model with Mechanization, Independent Investment, and Disequilibrium: Toward a Marxian-Keynesian Synthesis

This paper proposes an extended Goodwin model that synthesizes Marxian and Keynesian dynamics into a unified four-dimensional framework. The model integrates endogenous technical change via mechanization, investment behavior driven by effective demand, and goods market disequilibrium. We develop two three-dimensional closures–a Classical-Marxian and a Keynesian-Kaleckian formulation–each capable of generating persistent endogenous cycles through Hopf bifurcations. These are then combined into a Marxian-Keynesian (MK) system, which exhibits complex dynamics including quasi-periodicity and, under specific parameter values, a double-Hopf bifurcation. This result, to our knowledge not previously identified in extended Goodwin models, points to the potential for interacting oscillatory modes and long-run fluctuations even with relatively simple behavioral rules. Numerical simulations suggest that the MK synthesis captures rich endogenous fluctuations without relying on exogenous shocks and may exhibit chaotic dynamics under future extensions. These findings lay the groundwork for a more comprehensive Mars-Keynes-Schumpeter synthesis of capital instability, as suggested in the conclusion section.