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Essential data, budget sets and rationalization


  • Françoise Forges
  • Vincent Iehlé



According to a minimalist version of Afriat’s theorem, a consumer behaves as a utility maximizer if and only if a feasibility matrix associated with his choices is cyclically consistent. An “essential experiment” consists of observed consumption bundles $$(x_{1}, \ldots , x_{n})$$ and a feasibility matrix $$\varvec{\alpha }$$ . Starting with a standard experiment, in which the economist has access to precise budget sets, we show that the necessary and sufficient condition for the existence of a utility function rationalizing the experiment, namely, the cyclical consistency of the associated feasibility matrix, is equivalent to the existence, for any budget sets compatible with the deduced essential experiment, of a utility function rationalizing them (and typically depending on them). In other words, the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate. Starting with an essential experiment $$(x_{1}, \ldots , x_{n}; \varvec{\alpha }$$ ) only, we show that the cyclical consistency of $$\varvec{\alpha }$$ , together with a further consistency condition involving both $$(x_{1}, \ldots , x_{n})$$ and $$\varvec{\alpha }$$ , guarantees the existence of a budget representation and that the essential experiment is rationalizable almost robustly, in the sense that there exists a single utility function which rationalizes at once almost all budget sets which are compatible with $$(x_{1}, \ldots , x_{n}; \varvec{\alpha }$$ ). The conditions are also trivially necessary. Copyright Springer-Verlag 2013

Suggested Citation

  • Françoise Forges & Vincent Iehlé, 2013. "Essential data, budget sets and rationalization," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 449-461, November.
  • Handle: RePEc:spr:joecth:v:54:y:2013:i:3:p:449-461 DOI: 10.1007/s00199-012-0716-0

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    References listed on IDEAS

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    2. A. Fostel & H. Scarf & M. Todd, 2004. "Two new proofs of Afriat’s theorem," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 24(1), pages 211-219, July.
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    Cited by:

    1. Alfred Galichon & John Quah, 2013. "Symposium on revealed preference analysis," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 419-423, November.
    2. Forges, Françoise & Iehlé, Vincent, 2014. "Afriat’s theorem for indivisible goods," Journal of Mathematical Economics, Elsevier, vol. 54(C), pages 1-6.
    3. Hans Keiding & Mich Tvede, 2013. "Revealed smooth nontransitive preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 463-484, November.

    More about this item


    Afriat’s theorem; Budget sets; Cyclical consistency; Rational choice; Revealed preference; D11; C81;

    JEL classification:

    • D11 - Microeconomics - - Household Behavior - - - Consumer Economics: Theory
    • C81 - Mathematical and Quantitative Methods - - Data Collection and Data Estimation Methodology; Computer Programs - - - Methodology for Collecting, Estimating, and Organizing Microeconomic Data; Data Access


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