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Revealed Preference and the Number of Commodities

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Author Info
Jan Heufer ()

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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.

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Publisher Info
Paper provided by Rheinisch-Westfälisches Institut für Wirtschaftsforschung, Ruhr-Universität Bochum, Universität Dortmund, Universität Duisburg-Essen in its series Ruhr Economic Papers with number 0036.

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Length: 20 pages
Date of creation: Dec 2007
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Handle: RePEc:rwi:repape:0036

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Related research
Keywords: Homotheticity nonparametric tests preference cycles revealed preference SARP WARP

Find related papers by JEL classification:
C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: General - - - Semiparametric and Nonparametric Methods
C60 - Mathematical and Quantitative Methods - - Mathematical Methods and Programming - - - General
D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
D12 - Microeconomics - - Household Behavior - - - Consumer Economics: Empirical Analysis

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References listed on IDEAS
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  1. Knoblauch, Vicki, 1992. "A Tight Upper Bound on the Money Metric Utility Function," American Economic Review, American Economic Association, vol. 82(3), pages 660-63, June. [Downloadable!] (restricted)
  2. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-73, July. [Downloadable!] (restricted)
  3. 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 971-78, September. [Downloadable!] (restricted)
  4. 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. [Downloadable!] (restricted)
  5. Shafer, Wayne J., 1977. "Revealed preference cycles and the slutsky matrix," Journal of Economic Theory, Elsevier, vol. 16(2), pages 293-309, December. [Downloadable!] (restricted)
  6. John, Reinhard, 1997. "A Simple Cycle Preserving Extension of a Demand Function," Journal of Economic Theory, Elsevier, vol. 72(2), pages 442-445, February. [Downloadable!] (restricted)
  7. Samiran Banerjee & James Murphy, 2006. "A simplified test for preference rationality of two-commodity choice," Experimental Economics, Springer, vol. 9(1), pages 67-75, April. [Downloadable!] (restricted)
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