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# Essential data, budget sets and rationalization

## Author Info

Listed author(s):
• Françoise Forges
• Vincent Iehlé

()

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## 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 $$(x_{1}, \ldots , x_{n})$$ 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

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File URL: http://hdl.handle.net/10.1007/s00199-012-0716-0
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## Bibliographic Info

Article provided by Springer & Society for the Advancement of Economic Theory (SAET) in its journal Economic Theory.

Volume (Year): 54 (2013)
Issue (Month): 3 (November)
Pages: 449-461

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 Handle: RePEc:spr:joecth:v:54:y:2013:i:3:p:449-461 DOI: 10.1007/s00199-012-0716-0 Contact details of provider: Web page: http://www.springer.com Order Information: Web: http://www.springer.com/economics/economic+theory/journal/199/PS2

## References

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1. Christopher Chambers & Federico Echenique, 2009. "Profit maximization and supermodular technology," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(2), pages 173-183, August.
2. Donald J. Brown & Caterina Calsamiglia, 2005. "The Nonparametric Approach to Applied Welfare Analysis," Cowles Foundation Discussion Papers 1507, Cowles Foundation for Research in Economics, Yale University.
3. Matzkin, Rosa L, 1991. "Axioms of Revealed Preference for Nonlinear Choice Sets," Econometrica, Econometric Society, vol. 59(6), pages 1779-1786, November.
4. 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, 07.
5. Andrés Carvajal, 2010. "The testable implications of competitive equilibrium in economies with externalities," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 45(1), pages 349-378, October.
6. Forges, Françoise & Minelli, Enrico, 2009. "Afriat's theorem for general budget sets," Journal of Economic Theory, Elsevier, vol. 144(1), pages 135-145, January.
7. Yatchew, Adonis John, 1985. "A note on non-parametric tests of consumer behaviour," Economics Letters, Elsevier, vol. 18(1), pages 45-48.
8. repec:dau:papers:123456789/4099 is not listed on IDEAS
9. 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.
10. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-973, July.
11. 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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