Testing for optimality in the absence of convexity

The modern treatment of the fundamental theorems of welfare economics was developed by two of the masters of our trade: Kenneth J. Arrow (1951) and Gerard Debreu (1951). The contrast between their presentation and that offered by a distinguished predecessor — Abba Lerner — is a striking illustration of the emergence of a new line of argumentation in economic theory. The index to Lerner’s book, The economics of control (1944), contains not a single reference to convex sets nor to the separating hyper-plane theorem: The basic mathematical tools used by Arrow and Debreu to demonstrate the relationship between prices and Pareto optimality.