Econometrica: May 1979, Volume 47, Issue 3

Expenditure Functions, Local Duality, and Second Order Approximations<579:EFLDAS>2.0.CO;2-N
p. 579-602

Charles Blackorby, W. E. Diewert

This paper provides a complete set of local duality results for a utility maximizing consumer (or single output cost minimizing firm). Given a continuous local expenditure function defined on a compact, convex set of positive prices we establish the existence of continuous local direct, indirect utility and distance functions. This procedure avoids troublesome continuity problems at the boundary of R^N+. In addition it is shown that if two utility functions are second order approximations at some point, then their respective expenditure, distance, and indirect utility functions are also second-order approximations to each other at some point. This latter result provides additional impetus for using duality theory and substantial justification for the use of "flexible" functional forms which can provide second-order differential approximations to any twice continuously differentiable function at a point.

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