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At What Price?: Conceptualizing and Measuring Cost-of-Living and Price Indexes
when s = 1. This cost function represents homothetic preference, and the corresponding utility function is the constant elasticity of substitution (CES) utility function introduced into the economics literature by Arrow, Chenery, Minhas, and Solow (1961). The parameter s is the elasticity of substitution; when s = 0, the unit cost function defined by (47) is linear in prices and hence corresponds to a fixed-coefficients utility function with zero substitutability between all commodities. When s = 1, equation (48), the corresponding utility function is a Cobb-Douglas function. When s tends to infinity, the corresponding utility function approaches a linear utility function which exhibits infinite substitutability between all commodities. Even within the class of homothetic preferences, the CES cost function defined by (47) and (48) is not a fully flexible functional form (unless the number of commodities is two), but it is more flexible than the zero substitutability utility function that is exact for the Laspeyres and Paasche price indexes.
The base period cost-of-living index associated with (47) takes the form
Note that (49) is itself a CES function of the price relatives; in the mathematical literature, it is also known as the mean of order 1 - s. When s takes the value zero, (38) is the Laspeyres index; the Laspeyres is only a COLI when the consumer is unable (or unwilling) to substitute between goods, always consuming them in fixed proportions. As s tends to unity, (38) tends to the base period expenditure share weighted geometric mean. Provided not all the price relatives are the same, the CES index (49) is monotonically decreasing as the elasticity of substitution increases from 0 to infinity. If some consumers have an extreme aversion to substitution so that their elasticity of substitution is 0, then as relative prices change from period 0 to t, they will face a higher cost of living than consumers who substitute toward commodities that have decreased in relative price. Hence, if the elasticity of substitution s is positive and prices in period t are not proportional to prices in period 0, the Laspeyres price index, , will always b strictly greater than the corresponding CES price index, .
The CES cost-of-living index was first derived from CES preferences by Lloyd (1975), though it was Moulton (1996) who noted its usefulness for statistical agencies. In order to evaluate (50), the only requirements are information on the base period expenditure shares , the price relatives , and an estimate of the elasticity of substitution s. The first two requirements are met by the standard information that statistical agencies use to evaluate the Laspeyres price index. Hence, if the statistical agency is somehow able to estimate the elasticity of