On Revealed Preference and Indivisibilities

Affiliation(s)

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.

Department of Economics and Related Studies, University of York, York, UK.

Research Institute for Mathematical Sciences, Kyoto University, Kyoto, Japan.

Department of Economics and Related Studies, University of York, York, UK.

ABSTRACT

We consider a practical market model in which all commodities are inherently indivisible and thus are traded in integer quantities, or consumption choices are available only in discrete quantities. We ask whether a finite set of price-quantity observations satisfying the Generalized Axiom of Revealed Preference (GARP) is consistent with utility maximization. Due to the absence of perfect divisibility and continuity, the existing argument and also familiar assumptions such as non-satiation cannot be used in the current discrete model. We develop a new approach to deal with this problem and establish a discrete analogue of Afrita’s celebrated theorem. We also introduce a new concept called tight budget demand set which is a natural refinement of the standard notion of demand set and plays a crucial role in the current analysis. Exploring network structure and a new and easy-to-use variant of GARP, we propose an elementary, simple, combinatorial and constructive proof for our result.

We consider a practical market model in which all commodities are inherently indivisible and thus are traded in integer quantities, or consumption choices are available only in discrete quantities. We ask whether a finite set of price-quantity observations satisfying the Generalized Axiom of Revealed Preference (GARP) is consistent with utility maximization. Due to the absence of perfect divisibility and continuity, the existing argument and also familiar assumptions such as non-satiation cannot be used in the current discrete model. We develop a new approach to deal with this problem and establish a discrete analogue of Afrita’s celebrated theorem. We also introduce a new concept called tight budget demand set which is a natural refinement of the standard notion of demand set and plays a crucial role in the current analysis. Exploring network structure and a new and easy-to-use variant of GARP, we propose an elementary, simple, combinatorial and constructive proof for our result.

Cite this paper

S. Fujishige and Z. Yang, "On Revealed Preference and Indivisibilities,"*Modern Economy*, Vol. 3 No. 6, 2012, pp. 752-758. doi: 10.4236/me.2012.36096.

S. Fujishige and Z. Yang, "On Revealed Preference and Indivisibilities,"

References

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[2] R. W. Blundell, M. Browning and I. Crawford, “Nonparametric Engel Curves and Revealed Preference,” Econometrica, Vol. 71, No. 1, 2003, pp. 205-240. doi:10.1111/1468-0262.00394

[3] M. Browning and P. A. Chiappori, “Efficient Intrahouse- hold Allocations: A General Characterization and Empirical Tests,” Econometrica, Vol. 66, No. 6, 1998, pp. 1241-1278. doi:10.2307/2999616

[4] L. Cherchye, B. De Rock and F. Vermeulen, “The Collective Model of Household Consumption: A Non-Parametric Characterization,” Econometrica, Vol. 75, No. 2, 2007, pp. 553-574. doi:10.1111/j.1468-0262.2006.00757.x

[5] P. A. Chiappori, “Rational Household Labor Supply,” Econometrica, Vol. 56, No. 1, 1988, pp. 63-89. doi:10.2307/1911842

[6] E. Diewert, “Afriat and Revealed Preference Theory,” Review of Economic Studies, Vol. 40, No. 3, 1973, pp. 419-426. doi:10.2307/2296461

[7] A. Fostel, H. Scarf and M. J. Todd, “Two New Proofs of Afriat’s Theorem,” Economic Theory, Vol. 24, No. 1, 2004, pp. 211-219. doi:10.1007/s00199-003-0438-4

[8] C. P. Teo and R. V. Vohra, “Afriat’s Theorem and Negative Cycles,” Preprint, 2003.

[9] H. R. Varian, “The Non-Parametric Approach to Demand Analysis,” Econometrica, Vol. 50, No. 4, 1982, pp. 945- 974. doi:10.2307/1912771

[10] H. R. Varian, “Microeconomic Analysis,” 3rd Edition, W. W. Norton, New York, 1992.

[11] H. R. Varian, “Revealed Preference,” In: M. Szenberg, L. Ramrattan and A. A. Gottesman, Eds., Samuelsonian Economics and the 21st Century, Oxford University Press, Oxford, 2006.

[12] K. J. Arrow and F. H. Hahn, “General Competitive Ana- lysis,” Holden-Day, San Francisco, 1971.

[13] G. Debreu, “Theory of Value,” Yale University Press, New Haven, 1959.

[14] A. Kelso and V. P. Crawford, “Job Matching, Coalition Formation, and Gross Substitutes,” Econometrica, Vol. 50, No. 6, 1982, pp. 1483-1504. doi:10.2307/1913392

[15] T. C. Koopmans and M. Beckmann, “Assignment Problems and the Location of Economic Activities,” Econometrica, Vol. 25, No. 1, 1957, pp. 53-76. doi:10.2307/1907742

[16] A. Lerner, “The Economics of Control,” Macmillan, New York, 1944.

[17] P. Milgrom, “Package Auctions and Package Exchanges,” Econometrica, Vol. 75, No. 4, 2007, pp. 935-966. doi:10.1111/j.1468-0262.2007.00778.x

[18] H. Scarf, “The Allocation of Resources in the Presence of Indivisibilities,” Journal of Economic Perspectives, Vol. 8, No. 4, 1994, pp. 111-128. doi:10.1257/jep.8.4.111

[19] L. Shapley and H. Scarf, “On Cores and Indivisibilities,” Journal of Mathematical Economics, Vol. 1, No. 1, 1974, pp. 23-37. doi:10.1016/0304-4068(74)90033-0

[20] N. Sun and Z. Yang, “Equilibria and Indivisibilities: Gross Substitutes and Complements,” Econometrica, Vol. 74, No. 5, 2006, pp. 1385-1402. doi:10.1111/j.1468-0262.2006.00708.x

[21] H. Houthakker, “Revealed Preference and the Utility Function,” Economica, Vol. 17, No. 66, 1950, pp. 159-174. doi:10.2307/2549382

[22] P. A. Samuelson, “Consumption Theory in Terms of Revealed Preference,” Economica, Vol. 15, No. 60, 1948, pp. 243-253. doi:10.2307/2549561

[23] S. Fujishige, “Submodular Functions and Optimization,” 2nd Edition, Elsevier, Amsterdam, 2005.

[1] S. N. Afriat, “The Construction of a Utility Function from Expenditure Data,” International Economic Review, Vol. 8, No. 1, 1967, pp. 67-77. doi:10.2307/2525382

[2] R. W. Blundell, M. Browning and I. Crawford, “Nonparametric Engel Curves and Revealed Preference,” Econometrica, Vol. 71, No. 1, 2003, pp. 205-240. doi:10.1111/1468-0262.00394

[3] M. Browning and P. A. Chiappori, “Efficient Intrahouse- hold Allocations: A General Characterization and Empirical Tests,” Econometrica, Vol. 66, No. 6, 1998, pp. 1241-1278. doi:10.2307/2999616

[4] L. Cherchye, B. De Rock and F. Vermeulen, “The Collective Model of Household Consumption: A Non-Parametric Characterization,” Econometrica, Vol. 75, No. 2, 2007, pp. 553-574. doi:10.1111/j.1468-0262.2006.00757.x

[5] P. A. Chiappori, “Rational Household Labor Supply,” Econometrica, Vol. 56, No. 1, 1988, pp. 63-89. doi:10.2307/1911842

[6] E. Diewert, “Afriat and Revealed Preference Theory,” Review of Economic Studies, Vol. 40, No. 3, 1973, pp. 419-426. doi:10.2307/2296461

[7] A. Fostel, H. Scarf and M. J. Todd, “Two New Proofs of Afriat’s Theorem,” Economic Theory, Vol. 24, No. 1, 2004, pp. 211-219. doi:10.1007/s00199-003-0438-4

[8] C. P. Teo and R. V. Vohra, “Afriat’s Theorem and Negative Cycles,” Preprint, 2003.

[9] H. R. Varian, “The Non-Parametric Approach to Demand Analysis,” Econometrica, Vol. 50, No. 4, 1982, pp. 945- 974. doi:10.2307/1912771

[10] H. R. Varian, “Microeconomic Analysis,” 3rd Edition, W. W. Norton, New York, 1992.

[11] H. R. Varian, “Revealed Preference,” In: M. Szenberg, L. Ramrattan and A. A. Gottesman, Eds., Samuelsonian Economics and the 21st Century, Oxford University Press, Oxford, 2006.

[12] K. J. Arrow and F. H. Hahn, “General Competitive Ana- lysis,” Holden-Day, San Francisco, 1971.

[13] G. Debreu, “Theory of Value,” Yale University Press, New Haven, 1959.

[14] A. Kelso and V. P. Crawford, “Job Matching, Coalition Formation, and Gross Substitutes,” Econometrica, Vol. 50, No. 6, 1982, pp. 1483-1504. doi:10.2307/1913392

[15] T. C. Koopmans and M. Beckmann, “Assignment Problems and the Location of Economic Activities,” Econometrica, Vol. 25, No. 1, 1957, pp. 53-76. doi:10.2307/1907742

[16] A. Lerner, “The Economics of Control,” Macmillan, New York, 1944.

[17] P. Milgrom, “Package Auctions and Package Exchanges,” Econometrica, Vol. 75, No. 4, 2007, pp. 935-966. doi:10.1111/j.1468-0262.2007.00778.x

[18] H. Scarf, “The Allocation of Resources in the Presence of Indivisibilities,” Journal of Economic Perspectives, Vol. 8, No. 4, 1994, pp. 111-128. doi:10.1257/jep.8.4.111

[19] L. Shapley and H. Scarf, “On Cores and Indivisibilities,” Journal of Mathematical Economics, Vol. 1, No. 1, 1974, pp. 23-37. doi:10.1016/0304-4068(74)90033-0

[20] N. Sun and Z. Yang, “Equilibria and Indivisibilities: Gross Substitutes and Complements,” Econometrica, Vol. 74, No. 5, 2006, pp. 1385-1402. doi:10.1111/j.1468-0262.2006.00708.x

[21] H. Houthakker, “Revealed Preference and the Utility Function,” Economica, Vol. 17, No. 66, 1950, pp. 159-174. doi:10.2307/2549382

[22] P. A. Samuelson, “Consumption Theory in Terms of Revealed Preference,” Economica, Vol. 15, No. 60, 1948, pp. 243-253. doi:10.2307/2549561

[23] S. Fujishige, “Submodular Functions and Optimization,” 2nd Edition, Elsevier, Amsterdam, 2005.