On the non-existence of a marginal cost pricing equilibrium and the Ioffe normal cone

Abstract

In his recent paper, Khan (1987) proved a version of the second fundamental theorem of welfare economics with the Ioffe normal cone as a formalization of marginal rates of substitution. His introduction of the Ioffe normal cone was motivated by the fact that, in the absence of free disposal, the Clarke normal cone may be too "big" and the result based on it, by Khan and Vohra (1987) and Cornet (1986), is not sharp enough to be satisfactory. An interesting property of the Ioffe normal cone is that it is not necessarily convex. Khan showed that the lack of convexity is of no consequence for the second welfare theorem, but left open the question as to whether the convexity property is essential for the existence of a marginal cost pricing equilibrium as originally formulated by Guesnerie (1975); see also Beato and Mas-Colell (1985) and Brown et al. (1986). In this note we answer this question in the negative.