Detrimental Externalities and Non-Convexity of the Production Set

Abstract

This paper undertakes to show that detrimental externalities tend to induce non-convexity of the social production possibility set. In particular we show that if externalities are sufficiently strong, convexity conditions must break down. It is not our objective here to review in any detail the difficulties caused by non-convexity. Some of these consequences have long been recognized and are widely known.2 However, until the recent appearance of papers by Starett [8], Portes [5], Kolm [3] and Baumol [2], it was apparently not recognized that externalities themselves are a source of non-convexity. These more recent writings suggest more than one connection between the two phenomena. However, one particularly straightforward relationship seems to have received little or no attention. With sufficiently strong interactive effects non-convexity follows from the simple fact that if either of two mutually interfering activities is operated at zero level the other suffers no hindrance. The goal of this paper is to explore this phenomenon and show very clearly how it is that sufficiently severe detrimental externalities of the form described and non-convexity necessarily go together. In the first three sections we show both with the aid of illustrative examples and more general analysis that detrimental externalities of sufficient magnitude must always produce non-convexity in the production possibility set for two activities: one generating the externality and one affected by it. In Section IV we show that the problem is reduced but not generally eliminated by the possibility of spatial separation of offender and offended. Achievement of the right spatial separation turns out not always to be a simple matter, however. Section V contains some speculations about the way in which the number of local peaks in