Unification theorem for the two basic dualities of homothetic demand theory.
When income elasticities of demand are all unity, every dollar being spent in the same proportions at all levels of income, a homogeneous-first-degree, concave utility function exists to serve as an unequivocal measure of real output. Dual to it, and with identical concavity and homogeneity properties, is the minimized-cost-per-unit-of-output, a function of prices. Distinct from this production dual is the indirect-utility dual, representing, except for algebraic sign, the maximized level of utility attainable as a function of prices relative to income. These basic alternative dualities are shown to be related by a unifying theorem: The logarithm of either of the pair of production-dual functions has for its indirect-utility dual the logarithm of the other function. What is shown to be the same thing, the indirect-utility dual of the output function is, except for sign, the reciprocal of the output's production-dual.
Samuelson
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Department of Economics, Massachusetts Institute of Technology, Cambridge, Mass. 02139.