Abstract
This paper develops several optimization principles relating the fundamental concepts of Pareto efficiency and competitive equilibria. The beginning point for this development is the introduction of a new function describing individual preferences, closely related to willingness-to-pay, termed the benefit function. An important property of the benefit function is that it can be summed across individuals to obtain a meaningful measure of total benefit relative to a given set of utility levels; and the optimization principles presented in the paper are based on maximization of this total benefit.
Specifically, it is shown that, under appropriate technical assumptions, a Pareto-efficient allocationX maximizes the total benefit relative to the utility levels it yields. Conversely, if an allocationX yields zero benefit and maximizes the total benefit function, then that allocation is Pareto efficient. The Lagrange multipliersp of the benefit maximization problem serve as prices; and the (X,p) pair satisfies a generalized saddle-point property termed a Lagrange equilibrium. This in turn is equivalent, under appropriate assumptions, to a competitive equilibrium.
There are natural duals to all of the results stated above. The dual optimization principle is based on a surplus function which is a function of prices. The surplus is the total income generated at pricesp, minus the total income required to obtain given utility levels. The dual optimization principle states that prices that are dual (or indirect) Pareto efficient minimize total surplus and render it zero. Conversely, a set of prices that minimizes total surplus and renders it zero is a dual Pareto efficient set of prices.
The results of the paper can be viewed as augmenting the first and second theorems of welfare economics (and their duals) to provide a family of results that relate the important economic concepts of Pareto efficiency, equilibrium, dual (or indirect) Pareto efficiency, total benefit, Lagrange equilibrium, and total surplus.
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The author wishes to thank Charles R. Bowman and Andrew J. Yates for several valuable suggestions and corrections.
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Luenberger, D.G. New optimality principles for economic efficiency and equilibrium. J Optim Theory Appl 75, 221–264 (1992). https://doi.org/10.1007/BF00941466
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DOI: https://doi.org/10.1007/BF00941466