Stability and limit cycles in one-dimensional dynamic optimisations of competitive agents with a market externality

This paper considers low dimensional (more precisely, one state variable) dynamic optimisation problems of competitive agents. These individual decisions lead to a dynamic externality for the evolution of the system. However, the impact of an individual and competitive agent is negligible and thus each agent considers this evolution as exogenous data. This leads, assuming rational expectations (perfect foresight due to the deterministic set up), to motions in the three dimensional space of state, costate and externality. Considering the fact that such externalities are widespread, e.g., R&D in the literature on new growth theory, pollution in environmental economics, etc., the incorporation of such externalities due to competitive markets is important, yet this incorporation may alter the stability of the system. Indeed, complex policies such as stable limit cycles are sustainable in such a low-dimensional economy, even for a separable and strictly concave model.