Stability and direction of Hopf bifurcations of a cyclical growth model with two-time delays and one-delay dependent coefficients

Abstract

• We study the impact of two discrete-time delays on the basic Goodwin growth cycle model. • We study a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delay. • Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation. • We analyze the direction and stability of Hopf bifurcation by means of the first Lyapunov coefficient. This paper deals with the impact of two discrete-time delays on the basic Goodwin growth cycle model. The former concerns the existence of a finite time delay for building capital goods as suggested by Kalecki. The latter pertains to the wage lag hypothesis. This is because, taking the current change of the employment rate into account, workers and capitalists bargain new wage periodically. There are no examples in the literature on the Goodwin model of the use of both those lags in order to explore the GDP dynamics. From the analytical point-of-view, what we obtain is a delayed differential equation system with discrete-time delays and delay-dependent coefficients depending only on one of the time delays. Having chosen the time delays as bifurcation parameters, we study the stability-switching properties of the transcendental characteristic equation resulting from the stability analysis and the direction of the Hopf bifurcations. Although the system with no lag displays a stable focus, the introduction of the two lags preserves the stable solution only for particular combinations of parameters and length of the lags. In any other case, instability prevails and regular cycles or chaotic fluctuations emerge. Finally, we provide the analytical results with the necessary economic interpretations.