Long-Run Cycles in a Growth Model with Natural Externalities

We develop an AK model augmented in two directions: first, by modeling nature as a productive externality, and, second, by considering a process of natural capital accumulation where a reproduction force is moderated by the negative impact of productive pollution. Differently from the existing literature, more focused on the planner's solution, we study here a market economy and the effects of its imperfections. We introduce two alternative natural accumulation processes: a power law, more usual in economics, and a logistic law, more popular in ecology. In both cases, we observe the occurrence of cycles of period two and of period a power of two, and chaos through a flip bifurcation and period-doubling bifurcations. In the case of the power law, dynamics are even richer because of the occurrence of limit cycles through a Neimark-Sacker bifurcation. Worth to note, in this case, cycles occur under a larger Total Factor Productivity, which amplifies the negative impact of pollution on natural regeneration. In the case of a logistic law, cycles take place when the saturation effect of nature has a larger negative impact on the reproduction process.