The fundamental theorems of welfare economics

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 hyperplane theorem: The basic mathematical tools used by Arrow and Debreu to demonstrate the relationship between prices and Pareto optimality.

The first major theorem of welfare economics states that a competitive equilibrium is a Pareto-optimal production and distribution plan. The proof offered by Arrow is astonishingly brief; a line or two of mathematical argument replaces the tedious evaluation of vast arrays of marginal productivities and marginal rates of substitution. Moreover, convexity assumptions are required neither on the consumption nor production sides of the economy, though, of course, in the absence of such assumptions, the theorem is, in general, vacuous. The second major theorem – that a Pareto-optimal production and distribution plan can be supported by competitive prices – does require that consumer preferences satisfy a convexity assumption and that production sets be convex. Its proof is then a simple exercise in the application of Minkowski's separating hyperplane theorem.