Cycles in continuous-time economic models (with applications to Goodwin's and Solow's models)

This paper offers a unified perspective of the analytical detection of Hopf bifurcation, which is a crucial tool in dynamic economic modelling. We clarify the relations between stability theorems and the notions of Simple and General Hopf Bifurcations. A Lienard-Chipart-type theorem for detecting bifurcations, which appears of considerable usefulness in applications, is proved. Subsequently we show how to use the notions of "stability boundary" and "bifurcation boundary", providing a new, surprisingly straightforward, tool for detecting bifurcations in economics. An economic illustration is given by two models with time delay: a Solow-type demo-economic model and a Kaleckian extension of the Lotka-Volterra-Goodwin model.