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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.

Suggested Citation

  • William Barnett & Ki-Hong Choi, 2006. "Operational identification of the complete class of superlative index numbers: an application of Galois theory," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 200604, University of Kansas, Department of Economics.
  • Handle: RePEc:kan:wpaper:200604
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    References listed on IDEAS

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    1. Blackorby, Charles & Davidson, Russell & Schworm, William, 1991. "Implicit separability: Characterisation and implications for consumer demands," Journal of Economic Theory, Elsevier, vol. 55(2), pages 364-399, December.
    2. Lau, Lawrence J, 1979. "On Exact Index Numbers," The Review of Economics and Statistics, MIT Press, vol. 61(1), pages 73-82, February.
    3. Diewert, W E, 1992. "Exact and Superlative Welfare Change Indicators," Economic Inquiry, Western Economic Association International, vol. 30(4), pages 562-582, October.
    4. Theil, Henri, 1973. "A New Index Number Formula," The Review of Economics and Statistics, MIT Press, vol. 55(4), pages 498-502, November.
    5. Diewert, W Erwin, 1978. "Superlative Index Numbers and Consistency in Aggregation," Econometrica, Econometric Society, vol. 46(4), pages 883-900, July.
    6. Barnett, William A. & Choi, Ki-Hong & Sinclair, Tara M., 2003. "The Differential Approach to Superlative Index Number Theory," Journal of Agricultural and Applied Economics, Southern Agricultural Economics Association, vol. 35.
    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 566-593, September.
    8. Hulten, Charles R, 1973. "Divisia Index Numbers," Econometrica, Econometric Society, vol. 41(6), pages 1017-1025, November.
    9. Diewert, W. E., 1976. "Exact and superlative index numbers," Journal of Econometrics, Elsevier, vol. 4(2), pages 115-145, May.
    10. Sato, Kazuo, 1976. "The Ideal Log-Change Index Number," The Review of Economics and Statistics, MIT Press, vol. 58(2), pages 223-228, May.
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    Cited by:

    1. Thomas von Brasch & Diana-Cristina Iancu & Arvid Raknerud, 2018. "Productivity growth, firm turnover and new varieties," Discussion Papers 872, Statistics Norway, Research Department.
    2. 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.
    3. James J. Heckman & Apostolos Serletis, "undated". "Introduction to Internally Consistent Modeling, Aggregation, Inference, and Policy," Working Papers 2014-73, Department of Economics, University of Calgary, revised 29 Sep 2014.

    More about this item

    Keywords

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

    JEL classification:

    • C8 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs
    • E01 - Macroeconomics and Monetary Economics - - General - - - Measurement and Data on National Income and Product Accounts and Wealth; Environmental Accounts
    • D - Microeconomics

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