A generalized Goodwin business cycle model in random environment

This work develops a generalization of the Goodwin-Lotka-Volterra models. The traditional model describes the evolution of the employment rate and the labor share. Motivated by recent developments on business cycles and Goodwin-type models, we consider a generalized Goodwin business cycle model with noise. We aim at investigating the impact of telegraph noise and white noise on the evolution of trajectories and the energy on trajectories of the model. It is proved that with the presence of the noises, all trajectories exit any compact subset of the open unit square D=(0,1)×(0,1) and the energy along trajectories can be arbitrarily large. Moreover, under certain additional conditions, the boundary of D is the stable limit cycle of the perturbed model by white noise. Noise impact on the period of business cycles is also discussed. Numerical examples are provided to demonstrate our findings.