A Kaldor-Kalecki model of business cycles: Stability and limit cycles

Combining ideas proposed by Kaldor and Kalecki leads to a non-linear, time delayed, model for business cycle dynamics. In this paper, we analyse the stability and the local Hopf bifurcation properties of a Kaldor-Kalecki type model. In the analysis of such models, it is common to assume that the time delay continuously varies, and hence it is treated as a bifurcation parameter. However, this may not be a realistic assumption in various economic environments. We use an exogeneous, and non-dimensional, parameter as the bifurcation parameter to show that the underlying system undergoes a local Hopf bifurcation. Further, as this parameter gradually varies beyond the Hopf condition, we expect limit cycles to emerge from the stable equilibrium. We then, using Poincaré normal forms and the center manifold theory, outline the analysis to verify the type of the Hopf bifurcation and determine the stability of the limit cycles. The theoretical analysis is illustrated with some numerical examples.