A Goodwin type cyclical growth model with two-time delays.

• Goodwin's model with delayed investment function. • Cyclical growth model with time delays and delay dependent coefficients. • Existence of Hopf bifurcations analyzed by choosing time delays as bifurcation parameters. • Chaotic dynamics. In this paper, we reconsider the Goodwin 1967 growth-cycle model, where the antagonistic relationship between wages and profits is assimilated to the prey-predator conflict modeled by Volterra in 1931. Here we propose an extension of Goodwin's basic model by adding two important elements of the business cycle theory: (i) a finite time delay between investment orders and deliveries of finished capital goods, as theorized by Kalecki (1935); (ii) a delayed reaction of real wages to the unemployment levels, as suggested by Chiarella (1990). Both these delays preserve the two-dimensionality of the original model, but it becomes a delayed differential equation system, with two discrete time delays and one-delay dependent parameters. The qualitative study of the system shows that without lags the economic meaningful equilibrium is structurally stable. Nevertheless, as soon the time delays become positive, that equilibrium loses its stability and, according to the combinations of parameters and length of the lags, either periodic or non-periodic (chaotic) fluctuations arise. Numerical simulations supporting the economic analysis show that, in the very long run, a "strange attractor" depicts the dynamic behavior of the system. [ABSTRACT FROM AUTHOR]