Convex Input and Output Projections of Nonconvex Production Possibility Sets

In this paper we characterize the smallest production possibility set that contains a specified set of (input, output) combinations. In accordance with neoclassical production economics, this possibility set has convex projections into the input and output spaces (convex isoquants), and it satisfies the assumption of free disposability. We obtain it by means of a possibly infinite recursion which builds the possibility set as an ever larger union of convex sets. We remark on the nature of the approximations obtained by truncating the recursion, and we obtain a necessary and sufficient condition, checkable in one iteration for the recursion to stop in the next. For the case in which the recursion stops, we provide a succinct characterization of the dominance relations among the constituent sets produced by the procedure. Finally, we present examples of both finite and infinite cases. The example for the finite case illustrates the construction of the possibility set along with its associated production and consumption sets.