# Essential data, budget sets and rationalization

## Author Info

• 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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## Bibliographic Info

Article provided by Springer 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

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## Related research

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

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Find related papers by 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

## References

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1. Herbert E. Scarf & Ana Fostel & Michael J. Todd, 2004. "Two New Proofs of Afriat's Theorem," Yale School of Management Working Papers ysm377, Yale School of Management.
2. Francoise Forges & Enrico Minelli, 2006. "Afriat's Theorem for General Budget Sets," Working Papers ubs0609, University of Brescia, Department of Economics.
3. Donald Brown & Caterina Calsamiglia, 2007. "The Nonparametric Approach to Applied Welfare Analysis," Economic Theory, Springer, vol. 31(1), pages 183-188, April.
4. Varian, Hal R, 1982. "The Nonparametric Approach to Demand Analysis," Econometrica, Econometric Society, vol. 50(4), pages 945-73, July.
5. Shapley, Lloyd & Scarf, Herbert, 1974. "On cores and indivisibility," Journal of Mathematical Economics, Elsevier, vol. 1(1), pages 23-37, March.
6. Christopher Chambers & Federico Echenique, 2009. "Profit maximization and supermodular technology," Economic Theory, Springer, vol. 40(2), pages 173-183, August.
7. Yatchew, Adonis John, 1985. "A note on non-parametric tests of consumer behaviour," Economics Letters, Elsevier, vol. 18(1), pages 45-48.
8. Andrés Carvajal, 2010. "The testable implications of competitive equilibrium in economies with externalities," Economic Theory, Springer, vol. 45(1), pages 349-378, October.
9. Matzkin, Rosa L, 1991. "Axioms of Revealed Preference for Nonlinear Choice Sets," Econometrica, Econometric Society, vol. 59(6), pages 1779-86, November.
10. 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.
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## Citations

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Cited by:
1. Alfred Galichon & John Quah, 2013. "Symposium on revealed preference analysis," Economic Theory, Springer, vol. 54(3), pages 419-423, November.
2. Hans Keiding & Mich Tvede, 2013. "Revealed smooth nontransitive preferences," Economic Theory, Springer, vol. 54(3), pages 463-484, November.
3. Francoise Forges & Vincent Iehlé, 2013. "Afriat's Theorem for Indivisible Goods," CESifo Working Paper Series 4498, CESifo Group Munich.

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