MULTIPLICATIVE DECOMPOSITION AND INDEX NUMBER THEORY: AN EMPIRICAL APPLICATION OF THE SATO–VARTIA DECOMPOSITION

Abstract

In de Boer (2008), additive decompositions of aggregate changes in a variable into its factors were considered. We proposed using the ‘ideal’ Montgomery decomposition, developed in index number theory as an alternative to the commonly used methods in structural decomposition analysis, and applied it to the example analyzed by Dietzenbacher and Los (1998) (D&L). In this paper we consider multiplicative decompositions and show that the method proposed by D&L of taking the geometric mean of all elementary decompositions is ‘ideal’. However, it requires the computation of an ever-increasing number of decompositions when the number of factors increases. As an alternative, we propose using the Sato–Vartia decomposition, which is also ‘ideal’, but requires the computation of only one decomposition. Application to the example of D&L reveals that the two methods yield results that are very close to each other.

Keywords:

• Multiplicative decomposition
• Index number theory
• Structural decomposition analysis
• Siegel-Shapley decomposition

Acknowledgements

The author is indebted to Bert Balk for his suggestion to apply the Sato–Vartia indices to structural decomposition analysis and to Bart Los for putting the data sets at his disposal. He wishes to thank an anonymous referee and an editor for their comments that largely improved the paper.

Notes

¹ De Boer (2009) showed that, by collecting duplicates, the computation of r! permutations can be reduced to the computation of combinations, which, in the case of 10 factors, amounts to 512 combinations.

² The main properties of these logarithmic means are (see Balk, 2003): is continuous; ; and . According to Balk (2003) the logarithmic mean was introduced in the economics literature by Törnqvist in 1935 in an unpublished memo of the Bank of Finland (see Törnqvist et al., 1985). Lorenzen (1990) provided a proof of the fact that .

³ In de Boer (2008) it is argued that a change in stocks is not an appropriate final demand category and he splits a stock change over all other items of the row according to the pertinent shares in total output.

⁴ Equation 21 is the generalization of Gini (1937) of the Fisher index to three factors. Siegel (1945) has generalized the Fisher index to an arbitrary number of factors (r) and supplies a.o. this result for r = 3. It is also equivalent to Equation 7 given in the article by Ang et al. (2004). They make use of the very complicated formula of the n-factor Shapley value (Shapley, 1953).