A note on Boiteux' surplus function and dual Pareto efficiency

The purpose of this note is to study first a notion of a surplus function that originates in the work of [Boiteux, M., 1951. Le Revenu Distribuable et les Pertes Économiques. Econometrica 112-133] and to rely upon this notion to study dual Pareto efficiency in an exchange economy. This function, which we call the Boiteux' surplus function, measures how many units of income an individual must be given to move from a reference utility level, to another utility level. We prove several properties of the Boiteux' surplus function, and study in particular its links with the expenditure and the indirect utility functions. With regard to dual Pareto efficiency and the Boiteux' surplus function our results are as follows. A feasible reference price-income pair is dual Pareto efficient if and only if it zero-maximizes the sum of individual Boiteux' surplus functions among all feasible price-income pairs. We use these results to give a new proof (for the case of an exchange economy with positive prices) of the characterization by Luenberger of dual Pareto optima.