In the framework of a multiplicative decomposition Ang et al. (2004) proposed to use in index decomposition analysis (IDA) a generalized Fisher approach. They based their formulae for the decomposition of an aggregate change in a variable in three or four factors on the generic formula that Shapley (1953) derived for his value of n-person games and mention that Siegel (1945) gave their formulae using a different route. De Boer (2009a) proved that input-output structural decomposition analysis (SDA) is equivalent to the generalized Fisher approach for the decompositions in two, three and four factors. Using Siegel’s formula, he provided tables from which the decomposition of an aggregate change in a variable in five or six factors can easily be derived. In this paper we give the complicated generic formulae of Siegel and of Shapley and show how to implement them in case of four factors. The formulae are used to derive the tables for decompositions in seven and eight factors.

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Keywords generalized Fisher index, generic formulae of Siegel and of Shapley, index decomposition analysis, input-output structural decomposition analysis
JEL Index Numbers and Aggregation (jel C43), Energy and the Macroeconomy (jel Q43), Valuation of Environmental Effects (jel Q51)
Publisher Erasmus School of Economics (ESE)
Persistent URL
de Boer, P.M.C. (2011). The Generalized Fisher Index: the Generic Formulae of Siegel and of Shapley (No. EI 2011-20). Report / Econometric Institute, Erasmus University Rotterdam (pp. 1–11). Erasmus School of Economics (ESE). Retrieved from