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The Generalized Fisher Index: the Generic Formulae of Siegel and of Shapley

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  • de Boer, P.M.C.

Abstract

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.

Suggested Citation

  • de Boer, P.M.C., 2011. "The Generalized Fisher Index: the Generic Formulae of Siegel and of Shapley," Econometric Institute Research Papers EI 2011-20, Erasmus University Rotterdam, Erasmus School of Economics (ESE), Econometric Institute.
    • Handle: RePEc:ems:eureir:23798
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    Keywords

    generalized Fisher index; generic formulae of Siegel and of Shapley; index decomposition analysis; input-output structural decomposition analysis;
    All these keywords.

    JEL classification:

    • C43 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - Index Numbers and Aggregation
    • Q43 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Energy - - - Energy and the Macroeconomy
    • Q51 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Environmental Economics - - - Valuation of Environmental Effects

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