On the Volume-Ranking of Opportunity Sets in Economic Environments

We provide a characterization of the volume-ranking of oppor- tunity sets as de ned on the set of all polyconvex sets, i.e. nite 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 justi ed 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.