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Revealed Preference and the Number of Commodities

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  • Heufer, Jan

Abstract

This work consists of two parts: First, it is shown that for a two-dimensional commodity space any homothetic utility function that rationalizes each pair of observations in a set of consumption data also rationalizes the entire set of observations. The result is stated as a pairwise version of Varian's Homothetic Axiom of Revealed Preference and is used to provide a simplified nonparametric test of homotheticity. In the second part a unifying proof technique is presented to show that the Weak Axiom of Revealed Preference (WARP) implies the Strong Axiom of Revealed Preference (SARP) for two commodities yet not for more commodities. It also shows that preference cycles can be of arbitrary length.While these results are already known, the proof here generalizes and unifies the existing ones insofar as it gives necessary and sufficient conditions for preference cycles to exist. It is then shown that in two dimensions the necessary condition cannot be fulfilled, whereas in more than two dimensions the sufficient conditions can always be met. The proof admits an intuitive understanding of the reason by giving a geometric interpretion of preference cycles as paths on indifference surfaces.

Suggested Citation

  • Heufer, Jan, 2007. "Revealed Preference and the Number of Commodities," Ruhr Economic Papers 36, RWI - Leibniz-Institut für Wirtschaftsforschung, Ruhr-University Bochum, TU Dortmund University, University of Duisburg-Essen.
    • Handle: RePEc:zbw:rwirep:36
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    References listed on IDEAS

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    1. Liu, Pak-Wai & Wong, Kam-Chau, 2000. "Revealed homothetic preference and technology," Journal of Mathematical Economics, Elsevier, vol. 34(3), pages 287-314, November.
    2. Knoblauch, Vicki, 1993. "Recovering homothetic preferences," Economics Letters, Elsevier, vol. 43(1), pages 41-45.
    3. Hugh Rose, 1958. "Consistency of Preference: The Two-Commodity Case," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 25(2), pages 124-125.
    4. Shafer, Wayne J., 1977. "Revealed preference cycles and the slutsky matrix," Journal of Economic Theory, Elsevier, vol. 16(2), pages 293-309, December.
    5. Samiran Banerjee & James Murphy, 2009. "A simplified test for preference rationality of two-commodity choice," Experimental Economics, Springer;Economic Science Association, vol. 12(2), pages 252-252, June.
    6. Kihlstrom, Richard E & Mas-Colell, Andreu & Sonnenschein, Hugo, 1976. "The Demand Theory of the Weak Axiom of Revealed Preference," Econometrica, Econometric Society, vol. 44(5), pages 971-978, September.
    7. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-973, July.
    8. Knoblauch, Vicki, 1992. "A Tight Upper Bound on the Money Metric Utility Function," American Economic Review, American Economic Association, vol. 82(3), pages 660-663, June.
    9. Peters Hans & Wakker Peter, 1994. "WARP Does Not Imply SARP for More Than Two Commodities," Journal of Economic Theory, Elsevier, vol. 62(1), pages 152-160, February.
    10. John, Reinhard, 1997. "A Simple Cycle Preserving Extension of a Demand Function," Journal of Economic Theory, Elsevier, vol. 72(2), pages 442-445, February.
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    Cited by:

    1. repec:zbw:rwirep:0070 is not listed on IDEAS
    2. Jan Heufer, 2008. "Stochastic Revealed Preference and Rationalizability," Ruhr Economic Papers 0070, Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen.
    3. Heufer, Jan, 2008. "Stochastic Revealed Preference and Rationalizability," Ruhr Economic Papers 70, RWI - Leibniz-Institut für Wirtschaftsforschung, Ruhr-University Bochum, TU Dortmund University, University of Duisburg-Essen.

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    More about this item

    Keywords

    Homotheticity; nonparametric tests; preference cycles; revealed preference; SARP; WARP;
    All these keywords.

    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General
    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • D12 - Microeconomics - - Household Behavior - - - Consumer Economics: Empirical Analysis

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