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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.

This paper revisits the derivation and properties of the Allen-Uzawa and Morishima elasticities. Using a Swiss dataset, this paper empirically estimates various elasticities both in a dual and primal framework using a production theory open economy model and tests for linear homogenous technology. In addition to reporting elasticity at the mean, the standard practice in the literature, this paper also calculates nonparametric distribution of various elasticities. The paper aims to discuss these issues.

To assess the effect of price change on input, the paper estimates a translog cost function and to assess the effect of quantity change on price, the paper estimates the translog distance function using the data on Swiss economy. The paper estimates Allen-Uzawa and Morishima elasticity both under homogenous and non-homogenous technology using the Swiss dataset of one aggregate gross output and four inputs (resident labor, non-resident labor, imports, and capital) over 1950-1986. Elasticities are reported and compared at the mean as well as explored by looking at the range and nonparametric distribution.

This paper shows that constant returns to scale are easily rejected in this dataset and that the elasticities, both qualitatively and quantitatively, are very different under homogenous and non-homogenous technology. These elasticities can switch from complements to substitutes or vice versa when one moves away from the mean of the sample. The equality of the nonparametric elasticity distributions under homogenous vs non-homogenous technology is rejected in all cases except one.

This paper gives a clear derivation and interpretation of different elasticities as well as demonstrates using a dataset how to systematically go about empirically estimating these elasticities in a dual and primal framework. It shows that linear homogenous technology can be easily rejected and the elasticities, both quantitatively and qualitatively, are very different under homogenous and non-homogenous technology. This paper is also very valuable because it shows that the standard practice of reporting elasticity at the mean might not be adequate and there is a possibility that these elasticities can switch from complements to substitutes or vice versa when one moves away from the mean of the sample.