Separability, Externalities, and Competitive Equilibrium

Abstract

The characterization of external effects as "separable" has played an important role in the development of the theory of externalities. The separable case appears particularly well behaved when procedures for achieving an optimum allocation of resources in the presence of externalities are examined. Three examples can illustrate the range of conclusions which have been reached concerning separable externalities. Davis and Whinston [1962] find that separability assures the existence of a certain kind of equilibrium in bargaining between firms which create externalities, and that equilibrium does not exist without separability. Kneese and Bower [1968] argue that, with separability, the computation of Pigovian taxes to remedy externalities is particularly simple. Marchand and Russell [1974] demonstrate that certain liability rules regarding external effects lead to Pareto optimal outcomes if and only if externalities are separable. In these and other articles, an externality is defined as separable if the cost function of an affected firm has a specific form, stated in Definition 1 of this paper. With few exceptions, explorations of the implications of separability have assumed that equilibria and optima can be characterized in terms of classical first-order conditions of profit-maximization. Examination of the class of production functions which are compatible with separable externalities reveals, however, that separable externalities cause a distinctive non-convexity when the possibility that a firm will shut down rather than accept negative profits is introduced. Since numerous policies for dealing, for example, with environmental damage have been based on theoretical investigations of externalities, a defect in those investigations can have serious consequences. In this paper, we characterize the class of production functions which generate separable externalities. These results are used to show that all production functions in this class contain a nonconvex part. Some of the consequences of this non-convexity for market structure in the presence of separable externalities are examined. Finally, examples are given which suggest some conditions under which a competitive equilibrium may exist in the presence of externalities and some conditions under which it may not.

Files

Additional details