The chapter reviews and extends the theory of exact and superlative index numbers. Exact index numbers are empirical index number formula that are equal to an underlying…
The chapter reviews and extends the theory of exact and superlative index numbers. Exact index numbers are empirical index number formula that are equal to an underlying theoretical index, provided that the consumer has preferences that can be represented by certain functional forms. These exact indexes can be used to measure changes in a consumer's cost of living or welfare. Two cases are considered: the case of homothetic preferences and the case of nonhomothetic preferences. In the homothetic case, exact index numbers are obtained for square root quadratic preferences, quadratic mean of order r preferences, and normalized quadratic preferences. In the nonhomothetic case, exact indexes are obtained for various translog preferences.
- exact index numbers
- superlative index numbers
- flexible functional forms
- Fisher ideal index
- normalized quadratic preferences
- mean of order r indexes
- homothetic preferences
- nonhomothetic preferences
- cost of living indexes
- the measurement of welfare change
- translog functional form
- duality theory
- Allen quantity index
A concise introduction to the normalized quadratic expenditure or cost function is provided so that the interested reader will have the necessary information to understand…
A concise introduction to the normalized quadratic expenditure or cost function is provided so that the interested reader will have the necessary information to understand and use this functional form. The normalized quadratic is an attractive functional form for use in empirical applications as correct curvature can be imposed in a parsimonious way without losing the desirable property of flexibility. We believe it is unique in this regard. Topics covered include the problem of cardinalizing utility, the modeling of nonhomothetic preferences, the use of spline functions to achieve greater flexibility, and the use of a “semiflexible” approach to make it feasible to estimate systems of equations with a large number of commodities.
Recent years have been marked by an extensive empirical testing of profit, cost, expenditure, and demand equations based on duality theory and the development of specific functional forms as approximations compatible with any neoclassical production or utility function. In estimating systems of demand equations, the approach taken is almost always to assume consumers maximize utility of current period consumption subject to a budget constraint (or combined time‐budget constraint). There are many times that consumers face other constraints on their purchases. Many of these cases are associated with price discrimination and other market imperfections that result in several products being sold in packages (or, equivalently, several characteristics being contained in one product). This constraint on the proportions in which commodities must be purchased presents some significant problems for using an indirect utility function, particularly if one is interested in testing if the proportions are equal to or differ from what consumers would desire. One example of this problem is whether merged charities allocate funds in accordance with donor's preferences. Franklin Fisher showed, using a Stone‐Geary utility function, that there were reasonable cases where a merged charity could increase contributions by allocating funds differently than donors prefer.