Vindication of a "Common Mistake" in Welfare Economics

"Such a solution to the Pareto-efficiency problem will not necessarily satisfy the necessary conditions for a solution to the maximization problem, so those conditions are not necessary for Pareto efficiency" (p. 863). Professor Dorfman's criticism is unfounded, and those (unnamed) authors whom he reproves have made, at worst, a semantic error. Let x = (x1,. . ., xm) be a vector characterizing social states, and let U'(x), i = 1, 2, . . ., n, be differentiable utility functions representing the preferences of the agents over the states. Suppose there are constraints on the states given by the differentiable functions Ft(x) > 0 t = 1, 2,..., T.1 The work of S. Smale (1974b),2 in a more general mathematical setting,3 implies that a necessary condition for x* to be Pareto optimal, subject to the constraints, is that there exist nonnegative numbers w1,. . ., wn; All ... , AT, not all zero, such that