Dynamic externalities:Comparing conditions for Hopf bifurcationunder laissez‐faire and planning

Consider an economy described by two states. The first state describes a private stocksubject to a firm's (or a consumer's) control, while the second state captures market interactionsand is exogenous data to the individual firm. Considering rational expectations, amarket equilibrium can be derived. This set‐up is typical, in particular for the recentlyinvestigated new endogenous growth models. In contrast to the market outcome, planningattempts to internalise this externality. In both cases, the policies ‐ either the optimal intertemporalpolicy of competitive firms exposed to this externality, or the social optimum ‐are characterised by a two‐dimensional plane. Thus, complex solutions in particular limitcycles are possible. This paper compares the conditions of stability and, in particular, theconditions for limit cycles under these two different institutional set‐ups, when the externalityis or is not properly internalised. This comparison is first theoretical and then applied to adeliberately simple economic example: firms accumulate a capital stock (e.g., sewage treatment,energy saving technologies) involving convex investment costs and this stock lowersemissions (or kinds of waste) that add to a stock of pollution (e.g. global warming, pollutionof water and soil, etc.).

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