Dynamic externalities: Comparing conditions for Hopf bifurcation under laissez-faire and planning

Consider an economy described by two states. The first state describes a pr ivate stock subject to a firm's (or a consumer's) control, while the second state captures market interactions and is exogenous data to the individual firm. Considering rational expectations, a market equilibrium can be deriv ed. This set-up is typical, in particular for the recently investigated new endogenous growth models. In contrast to the market outcome, planning atte mpts to internalise this externality. In both cases, the policies - either the optimal intertemporal policy of competitive firms exposed to this exter nality, or the social optimum - are characterised by a two-dimensional plan e. Thus, complex solutions in particular limit cycles are possible. This pa per compares the conditions of stability and, in particular, the conditions for limit cycles under these two different institutional set-ups, when the externality is or is not properly internalised. This comparison is first t heoretical and then applied to a deliberately simple economic example: firm s accumulate a capital stock (e.g., sewage treatment, energy saving technol ogies) involving convex investment costs and this stock lowers emissions (o r kinds of waste) that add to a stock of pollution (e.g. global warming, po llution of water and soil, etc.).