On the volume-ranking of opportunity sets in economic environments

We provide a characterization of the volume-ranking of opportunity sets as defined on the set of all polyconvex sets, i.e. finite unions of convex, compact, Euclidean sets. In fact, such a domain is large enough to encompass most of the opportunity sets typically encountered in economic environments, including non-linear or even non-convex budget sets, and opportunity sets arising from production sets. Our result relies on a valuation-based volume-characterization theorem due to Klain and Rota (Introduction to Geometric Probability, Cambridge University Press, Cambridge, 1997) and helps to highlight some quite unusual conditions under which the volume-ranking can be justified as a freedom-ranking of opportunity sets. Therefore, it may also help to understand why the latter has been so conspicuously ignored in welfare analysis. Thanks are due to Larry Kranich, Massimo Marinacci, Uri Rothblum, Ernesto Screpanti, John Weymark, Yongsheng Xu, Claudio Zoli and two anonymous referees for helpful comments and suggestions.