The Mordukhovich Normal Cone and the Foundations of Welfare Economics

The statement that Pareto optimal allocations require the equalization of marginal rates of substitution, or in an economy with public goods, require the equalization of the aggregate of the marginal rates in consumption to those in production, is formalized through the use of the Mordukhovich normal cone. Since this cone is strictly contained, in general, in the Clarke normal cone, the results generalize earlier work of Khan and Vohra, Quinzii, Yun, and Cornet. The results are an application of Mordukhovich's 1980 theorem on necessary conditions for optimality in constrained optimization problems involving functions that are not necessarily differentiable or quasi‐concave. As such, the results suggest a distinction between the mathematical programming approach to the “second welfare theorem,” as in the work of Hicks, Lange, and Samuelson, and that based on the separation of sets, as pioneered by Arrow and Debreu.