Affiliations: [a] Department of Economics, University of British
Columbia, Vancouver, B.C., Canada V6T 1Z1. Tel.: +1 604 822 2544; Fax: +1 604
822 5915; E-mail: [email protected]
Correspondence:
[*]
University of British Columbia and NBER. The
author thanks the Social Sciences and Humanities Research Council of Canada for
financial support and Bert Balk and Rolf Färe for helpful comments
Abstract: It was shown in 1976 that a difference in a quadratic function of N variables
evaluated at two points is exactly equal to the sum of the arithmetic average of
the first order partial derivatives of the function evaluated at the two points
times the differences in the independent variables. In the present paper, this
result is generalized and the resulting generalized quadratic approximation
lemma is used to establish all of the superlative index number formulae that
were derived in Diewert [4]. In addition, some new exact decompositions of the
percentage change in the Fisher and Walsh superlative indexes into N components
are derived. Each component in this decomposition represents the contribution of
a change in a single independent variable to the overall percentage change in
the index. Finally, these components are given economic interpretations.
