A Duality Theory

The theorems of welfare economics describe relationships linking equilibrium allocations and Pareto optima. The duality theory that is set out in this chapter can be viewed as a mathematical extension of these theorems. We shall show that there is a duality relationship between general equilibrium theory on one side and the theory of Pareto efficient allocations compatible with a priori given budget constraints (also named the B.C.P.E. allocations, where B.C.P.E. stands for budget constrained Pareto efficient) on the other. We shall make use of this duality relationship in two different directions. Firstly, duality provides a powerful means of establishing and proving properties of B.C.P.E. allocations by straightforward translation of the corresponding properties and proofs of general equilibrium theory. We shall illustrate this use of duality in several examples, one of them dealing with the existence of B.C.P.E. allocations. Secondly, this duality relationship possesses the remarkable, although a priori unexpected, property of providing a mathematical formulation of general equilibrium theory that lends itself more readily to the study of questions related to the number of equilibria or to singular economies than the original formulation through envelope theory. This potency of the dual formulation stands out most evidently in the several examples investigated in the last part of this chapter.