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A representative individual from Arrovian aggregation of parametric individual utilities


  • Herzberg, Frederik


Abstract This article investigates the representative-agent hypothesis for a population which faces a collective choice from a given finite-dimensional space of alternatives. Each individual's preference ordering is assumed to admit a utility representation through an element of an exogenously given set of admissible utility functions. In addition, we assume that (i) the class of admissible utility functions allows for a smooth parametrization and only consists of strictly concave functions, (ii) the population is infinite, and (iii) the social welfare function satisfies Arrovian rationality axioms. We prove that there exists an admissible utility function r, called representative utility function, such that any alternative which maximizes r also maximizes the social welfare function. Given the structural similarities among the admissible utility functions (due to parametrization), we argue that the representative utility function can be interpreted as belonging to an - actual or invisible- individual. The existence proof for the representative utility function utilizes a special nonstandard model of the reals, viz. the ultrapower of the reals with respect to the ultrafilter of decisive coalitions; this construction explicitly determines the parameter vector of the representative utility function.

Suggested Citation

  • Herzberg, Frederik, 2010. "A representative individual from Arrovian aggregation of parametric individual utilities," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1115-1124, November.
  • Handle: RePEc:eee:mateco:v:46:y:2010:i:6:p:1115-1124

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    References listed on IDEAS

    1. Schmitz, Norbert, 1977. "A further note on arrow's impossibility theorem," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 189-196, August.
    2. Campbell, Donald E., 1990. "Intergenerational social choice without the Pareto principle," Journal of Economic Theory, Elsevier, vol. 50(2), pages 414-423, April.
    3. Anderson, Robert M., 1991. "Non-standard analysis with applications to economics," Handbook of Mathematical Economics,in: W. Hildenbrand & H. Sonnenschein (ed.), Handbook of Mathematical Economics, edition 1, volume 4, chapter 39, pages 2145-2208 Elsevier.
    4. Clark, Stephen A., 1992. "The representative agent model of probabilistic social choice," Mathematical Social Sciences, Elsevier, vol. 23(1), pages 45-66, February.
    5. Alan P. Kirman, 1992. "Whom or What Does the Representative Individual Represent?," Journal of Economic Perspectives, American Economic Association, vol. 6(2), pages 117-136, Spring.
    6. Grafe, F. & Grafe, J., 1983. "On arrow-type impossibility theorems with infinite individuals and infinite alternatives," Economics Letters, Elsevier, vol. 11(1-2), pages 75-79.
    7. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    8. James E. Hartley, 1996. "Retrospectives: The Origins of the Representative Agent," Journal of Economic Perspectives, American Economic Association, vol. 10(2), pages 169-177, Spring.
    9. Sen, Amartya, 1995. "Rationality and Social Choice," American Economic Review, American Economic Association, vol. 85(1), pages 1-24, March.
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    Cited by:

    1. Bedrosian, Geghard & Herzberg, Frederik, 2016. "Microeconomic foundations of representative agent models by means of ultraproducts," Center for Mathematical Economics Working Papers 514, Center for Mathematical Economics, Bielefeld University.
    2. repec:spr:etbull:v:1:y:2013:i:1:d:10.1007_s40505-013-0004-6 is not listed on IDEAS

    More about this item


    Representative individual Arrovian social choice Ultrafilter Ultraproduct Nonstandard analysis;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations


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