Index-number tests and the common-scaling social cost-of-living index

For a change in prices, the common-scaling social cost-of-living index is the equal scaling of every individual’s expenditure level needed to restore the level of social welfare to its pre-change value. This index does not, in general, satisfy two standard index-number tests. The reversal test requires the index value for the reverse change to be the reciprocal of the original index. And the circular test requires the product of index values for successive price changes to be equal to the index value for the whole change. We show that both tests are satisfied if and only if the Bergson-Samuelson indirect social-welfare function is homothetic in prices, a condition which does not require individual preferences to be homothetic. If individual preferences are homothetic, however, stronger conditions on the Bergson-Samuelson indirect must be satisfied. Given these results, we ask whether the restrictions are empirically reasonable and find, in the case that individual preferences are not homothetic, that the restrictions make little difference to estimates of the index. 1 Thanks to Charlie Blackorby and Walter Bossert for comments and assistance with proofs, and to the Social Sciences and Humanities Research Council for research support. Recently, Thomas Crossley and Krishna Pendakur [2009] have proposed a social cost-of-living index that is a departure from the standard ones. For a price change, their common-scaling social cost-of-living index is the equal scaling of every individual’s expenditure level needed to restore the pre-change level of social welfare. Using a social-welfare function, the index takes account of a social attitude towards inequality of well-being when assessing the social cost of price changes. Like the common-scaling index, the other general ethical index, due to Pollak [1980, 1981], uses a social-welfare function. The value of Pollak’s index depends on a reference level of social welfare and is equal to the ratio of total expenditure needed to achieve that level after the change to total expenditure needed before the change. Blackorby and Donaldson [1983] investigate conditions under which the index is independent of the reference level of social welfare, given that the social-welfare function is additively separable. The commonly used plutocratic index is the share-weighted value of individual costof-living indexes. It is equal to the ratio of the total cost of keeping each person at his or her pre-change utility level to total expenditures before the change. Given appropriate social-welfare and individual utility functions, the common-scaling index is the plutocratic index (see Section 2). The arithmetic mean of individual indexes is also used, but has no desirable ethical properties. The Pollak and common-scaling indexes are generalizations of individual cost-of-living indexes, which can be defined as the ratio of expenditures needed to achieve a reference level of individual utility, parallel to Pollak’s index, or as the scaling of individual expenditure need to preserve the pre-change utility level, parallel to the common-scaling index. Because cost-of-living adjustments are typically made with across-the-board percentage changes, the common-scaling index may be more attractive. Suppose that an economy consists of n single adults, where n is a positive integer, with m goods, m ≥ 2. Individual i has a continuous indirect utility function V i:R ++ → R which is increasing in expenditure, weakly decreasing in prices, and homogeneous of degree zero. The social-welfare function W :V 1(R ++ )× . . .×V n(R ++ )→ R is assumed to be continuous, increasing (Pareto-inclusive), and symmetric (anonymous). The Bergson-Samuelson indirect social-welfare function is B:R ++ → R is given by B(p, x) = W ( V (p, x1), . . . , V (p, xn) ) , (1) where p = (p1, . . . , pm) is the (common) price vector and x = (x1, . . . , xn) is the vector of individual expenditures. B is homogeneous of degree zero. Because W is increasing in individual utilities which are, in turn, increasing in individual expenditures, B is increasing in x. Suppose prices change from pb to pa (b for ‘before,’ a for ‘after’), and x gives the vector of individual expenditures corresponding to pb (the ‘before’ expenditure vector). The common-scaling social cost-of-living index I:R ++ → R is written I(pa, pb, x) and defined by I(p, p, x) = μ⇔ W ( V (p, x1), . . . , V (p, xn) ) = W ( V (p, μx1), . . . , V (p, μxn) ) , (2)