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Operational identification of the complete class of superlative index numbers: an application of Galois theory

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  • William Barnett

    (Department of Economics, The University of Kansas)

  • Ki-Hong Choi

    (National Pension Research Institute)

Abstract

We provide an operational identification of the complete class of superlative index numbers to track the exact aggregator functions of economic aggregation theory. If an index number is linearly homogeneous and a second order approximation in a formal manner that we define, we prove the index to be in the superlative index number class of nonparametric functions. Our definition is mathematically equivalent to Diewert¡¯s most general definition. But when operationalized in practice, our definition permits use of the full class, while Diewert¡¯s definition, in practice, spans only a strict subset of the general class. The relationship between the general class and that strict subset is a consequence of Galois theory. Only a very small number of elements of the general class have been found by Diewert¡¯s method, despite the fact that the general class contains an infinite number of functions. We illustrate our operational, general approach by proving for the first time that a particular family of nonparametric functions, including the Sato-Vartia index, is within the superlative index number class.

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Bibliographic Info

Paper provided by University of Kansas, Department of Economics in its series WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS with number 200604.

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Length: 22 pages
Date of creation: Feb 2006
Date of revision:
Handle: RePEc:kan:wpaper:200604

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Keywords: Exact index numbers; superlative index number class; Divisia line integrals; aggregator function space; Galois theory.;

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  1. William A. Barnett & Ke- Hong Choi & Tara M. Sinclair, 2001. "The Differential Approach to Superlative Index Number Theory," Econometrics 0111002, EconWPA, revised 28 Dec 2001.
  2. Diewert, W. E., 1976. "Exact and superlative index numbers," Journal of Econometrics, Elsevier, vol. 4(2), pages 115-145, May.
  3. Blackorby, C. & Davidson, R. & Schworm, W., 1990. "Implicit Separability: Characterisation And Implications For Consumer Demands," G.R.E.Q.A.M. 90a16, Universite Aix-Marseille III.
  4. Hulten, Charles R, 1973. "Divisia Index Numbers," Econometrica, Econometric Society, vol. 41(6), pages 1017-25, November.
  5. Lau, Lawrence J, 1979. "On Exact Index Numbers," The Review of Economics and Statistics, MIT Press, vol. 61(1), pages 73-82, February.
  6. Diewert, W Erwin, 1978. "Superlative Index Numbers and Consistency in Aggregation," Econometrica, Econometric Society, vol. 46(4), pages 883-900, July.
  7. Samuelson, Paul A & Swamy, S, 1974. "Invariant Economic Index Numbers and Canonical Duality: Survey and Synthesis," American Economic Review, American Economic Association, vol. 64(4), pages 566-93, September.
  8. Theil, Henri, 1973. "A New Index Number Formula," The Review of Economics and Statistics, MIT Press, vol. 55(4), pages 498-502, November.
  9. Sato, Kazuo, 1976. "The Ideal Log-Change Index Number," The Review of Economics and Statistics, MIT Press, vol. 58(2), pages 223-28, May.
  10. Diewert, W E, 1992. "Exact and Superlative Welfare Change Indicators," Economic Inquiry, Western Economic Association International, vol. 30(4), pages 562-82, October.
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Cited by:
  1. Hennessy, David A. & Lapan, Harvey E., 2009. "Harmonic symmetries of imperfect competition on circular city," Journal of Mathematical Economics, Elsevier, vol. 45(1-2), pages 124-146, January.

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