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. For example, 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 sepalability. 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.
We will argue in this paper that whenever an externality affecting a firm is separable, the production set of that firm is not convex in a neighborhood of zero output. The proposition is established by redefining separability in a manner which allows for the fact that in the long run a firm will shut down rather than accept negative profits. These definitions yield the theorem that separability implies a non-convexity of the production function, which may result in a discontinuous supply correspondence
