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Essential Data, Budget Sets and Rationalization

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  • Francoise Forges

    (LEDa - Laboratoire d'Economie de Dauphine - Université Paris IX - Paris Dauphine, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - CNRS : UMR7534 - Université Paris IX - Paris Dauphine)

  • Vincent Iehlé

    (LEDa - Laboratoire d'Economie de Dauphine - Université Paris IX - Paris Dauphine, CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - CNRS : UMR7534 - Université Paris IX - Paris Dauphine)

Abstract

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,..., x_n) and a feasibility matrix \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,..., x_n; alpha) only, we show that the cyclical consistency of alpha, together with a further consistency condition involving both (x_1,..., x_n) and 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,..., x_n; alpha). The conditions are also trivially necessary.

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Paper provided by HAL in its series Post-Print with number halshs-00727806.

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Date of creation: 01 Dec 2013
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Publication status: Published, Economic Theory, 2013, 54, 3, 449-461
Handle: RePEc:hal:journl:halshs-00727806

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Keywords: Afriat's theorem; budget sets; cyclical consistency; rational choice; revealed preference;

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  1. Yatchew, Adonis John, 1985. "A note on non-parametric tests of consumer behaviour," Economics Letters, Elsevier, vol. 18(1), pages 45-48.
  2. Teo Chung Piaw & Rakesh V. Vohra, 2003. "Afrait's Theorem and Negative Cycles," Discussion Papers 1377, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  3. Donald Brown & Caterina Calsamiglia, 2007. "The Nonparametric Approach to Applied Welfare Analysis," Economic Theory, Springer, vol. 31(1), pages 183-188, April.
  4. M.J. Todd & A. Fostel & H.E. Scarf, 2004. "Two New Proofs of Afriat's Theorem," Econometric Society 2004 North American Summer Meetings 632, Econometric Society.
  5. Francoise Forges & Enrico Minelli, 2006. "Afriat’s Theorem for General Budget Sets," CESifo Working Paper Series 1703, CESifo Group Munich.
  6. Andrés Carvajal, 2010. "The testable implications of competitive equilibrium in economies with externalities," Economic Theory, Springer, vol. 45(1), pages 349-378, October.
  7. Matzkin, Rosa L, 1991. "Axioms of Revealed Preference for Nonlinear Choice Sets," Econometrica, Econometric Society, vol. 59(6), pages 1779-86, November.
  8. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-73, July.
  9. Christopher Chambers & Federico Echenique, 2009. "Profit maximization and supermodular technology," Economic Theory, Springer, vol. 40(2), pages 173-183, August.
  10. Shapley, Lloyd & Scarf, Herbert, 1974. "On cores and indivisibility," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 23-37, March.
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Cited by:
  1. Francoise Forges & Vincent Iehlé, 2013. "Afriat's theorem for indivisible goods," Working Papers halshs-00870052, HAL.
  2. Alfred Galichon & John Quah, 2013. "Symposium on revealed preference analysis," Economic Theory, Springer, vol. 54(3), pages 419-423, November.
  3. Hans Keiding & Mich Tvede, 2013. "Revealed smooth nontransitive preferences," Economic Theory, Springer, vol. 54(3), pages 463-484, November.

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