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

Consider an economy described by two states. The first state describes a private 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 derived. This set-up is typical, in particular for the recently investigated new endogenous growth models. In contrast to the market outcome, planning attempts to internalise this externality. In both cases, the policies – either the optimal inter-temporal policy of competitive firms exposed to this externality, or the social optimum – are characterised by a two-dimensional plane. Thus, complex solutions in particular limit cycles are possible. This paper 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 theoretical and then applied to a deliberately simple economic example: firms accumulate a capital stock (e.g., sewage treatment, energy saving technologies) involving convex investment costs and this stock lowers emissions (or kinds of waste) that add to a stock of pollution (e.g. global warming, pollution of water and soil, etc.).