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A representative individual from Arrovian aggregation of parametric individual utilities


  • Herzberg, Frederik

    (Center for Mathematical Economics, Bielefeld University)


This article investigates the representative-agent hypothesis for an infinite population which has to make a social choice from a given finite-dimensional space of alternatives. It is assumed that some class of admissible strictly concave utility functions is exogenously given and that each individual's preference ordering can be represented cardinally through some admissible utility function. In addition, we assume that (i) the class of admissible utility functions allows for a smooth parametrization, and (ii) 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. The proof utilizes a special nonstandard model of the reals, viz. the ultraproduct 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, 2011. "A representative individual from Arrovian aggregation of parametric individual utilities," Center for Mathematical Economics Working Papers 411, Center for Mathematical Economics, Bielefeld University.
  • Handle: RePEc:bie:wpaper:411

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    References listed on IDEAS

    1. 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.
    2. Clark, Stephen A., 1992. "The representative agent model of probabilistic social choice," Mathematical Social Sciences, Elsevier, vol. 23(1), pages 45-66, February.
    3. 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.
    4. 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.
    5. Schmitz, Norbert, 1977. "A further note on arrow's impossibility theorem," Journal of Mathematical Economics, Elsevier, vol. 4(2), pages 189-196, August.
    6. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
    7. Campbell, Donald E., 1990. "Intergenerational social choice without the Pareto principle," Journal of Economic Theory, Elsevier, vol. 50(2), pages 414-423, April.
    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


    Ultraproduct; Nonstandard analysis; Ultrafilter; Arrovian social choice; Representative individual;

    JEL classification:

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

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