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Symmetric Means and the Expected Utility Theorem

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Abstract

This paper shows how symmetric means can be useful in providing an axiomatic basis for making choices under uncertainty. In section 2 of the paper, symmetric means are defined by means of four axioms which are shown to be independent and they imply an interesting fifth axiom. In section 3, a consistency in aggregation axiom is added and is shown to imply that the symmetric mean must be quasilinear or additively separable. In the final section, the various axioms for a symmetric mean are used to establish a version of the Expected Utility Theorem.

Keywords

  • Preference Function
  • Axiomatic Characterization
  • Separability Axiom
  • Separability Assumption
  • Continuous Utility Function

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

This research was supported by a Strategic Grant from the Social Science and Humanities Research Council of Canada. Thanks are due Shelley Hey and Louise Hebert for typing a difficult manuscript, and to J. Aczél, W. Bossert, D. Donaldson, L.G. Epstein and F. Stehling for valuable comments and to C. Blackorby for valuable discussions. The material in this paper has been adapted from Diewert (1992).

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References

  • AczéL, J. (1966), Lectures on Functional Equations and their Applications, New York: Academic Press.

    Google Scholar 

  • Arrow, K.J. (1951), “Alternative Approaches to the Theory of Choice in Risk-Taking Situations,” Econometrica 19, 404–437.

    CrossRef  Google Scholar 

  • Arrow, K.J. (1953), “Le rôle des valeurs boursières pour la répartition la meilleure des risques.” In Econométrie, Paris: Centre National de la Recherche Scientifique, 41–48. Reprinted in English translation in 1964 as “The Role of Securities in the Optimal Allocation of Risk-Bearing,” The Review of Economic Studies 31, 91–96.

    Google Scholar 

  • AArrow, K.J. (1984), “Exposition of the Theory of Choice under Uncertainty.” In K.J. Arrow, Individual Choice under Certainty and Uncertainty, Cambridge, Mass.: Harvard University Press, 172–208. (An expanded version of a lecture originally given in 1963.)

    Google Scholar 

  • BErnoulli, D. (1738), “Specimen Theoriae Novae de Mensura Sortis,” Commentarii Academiae Scientiarum Imperialis Petropolitanae 5, 175–192. Reprinted in English translation by L. Sommer in 1954 as “Exposition of a New Theory on the Measurement of Risk,” Econometrica 22, 23–36.

    CrossRef  Google Scholar 

  • Blackorby, C., R. Davidson and D. Donaldson (1977), “A Homiletic Exposition of the Expected Utility Hypothesis,” Economica 44, 351–358.

    CrossRef  Google Scholar 

  • Blackorby, C. and D. Donaldson (1984), “Social Criteria for Evaluating Population Change,” Journal of Public Economics 25, 13–33.

    CrossRef  Google Scholar 

  • Blackorby, C., D. Donaldson and M. Auersperg (1981), “A New Procedure for the Measurement of Inequality Within and Among Population Subgroups,” Canadian Journal of Economics 14, 664–685.

    CrossRef  Google Scholar 

  • Blackorby, C. and D. Primont (1980), “Index Numbers and Consistency in Aggregation,” Journal of Economic Theory 22, 87–98.

    CrossRef  Google Scholar 

  • Bossert, W. and A. Pfingsten (1990), “Intermediate Inequality: Concepts, Indices and Welfare Implications,” Mathematical Social Sciences 19, 117–134.

    CrossRef  Google Scholar 

  • Chew, S.H. and L.G. Epstein (1989), “A Unifying Approach to Axiomatic Non-Expected Utility Theories,” Journal of Economic Theory 49, 207–240.

    CrossRef  Google Scholar 

  • Chew, S.H. and L.G. Epstein (1992), “A Unifying Approach to Axiomatic Non-Expected Utility Theories: Corrigenda,” Journal of Economic Theory, forthcoming.

    Google Scholar 

  • Debreu, G. (1959), Theory of Value, New York: John Wiley and Sons.

    Google Scholar 

  • Diewert, W.E. (1978), “Superlative Index Numbers and Consistency in Aggregation,” Econometrica 46, 883–900.

    CrossRef  Google Scholar 

  • Diewert, W.E. (1992), “Symmetric Means and Choice Under Uncertainty,” Chapter 14 in Essays in Index Number Theory, Vol. 1, W.E. Diewert and A.O. Nakamura, (eds.), Amsterdam: North-Holland.

    Google Scholar 

  • Eichhorn, W. (1976), “Fisher’s Tests Revisited,” Econometrica 44, 247–256.

    CrossRef  Google Scholar 

  • Eichhorn, W. (1978), Functional Equations in Economics, Reading, Mass.: Addison-Wesley.

    Google Scholar 

  • Eichhorn, W. and W. Gehrig (1982), “Measurement of Inequality in Economics.” In Modern Applied Mathematics: Optimization and Operations Research, B. Korte (ed.), Amsterdam: North-Holland, 657–693.

    Google Scholar 

  • Eichhorn, W. and J. Voeller (1976), Theory of the Price Index: Fisher’s Test Approach and Generalizations, Lecture Notes in Economics and Mathematical Systems, Vol. 140, Berlin: Springer-Verlag.

    Google Scholar 

  • Gorman, W.M. (1968) “The Structure of Utility Functions,” Review of Economic Studies 35, 367–390.

    CrossRef  Google Scholar 

  • Hardy, G.H., J.E. Littlewood, and G. Polya (1934), Inequalities, Cambridge: Cambridge University Press.

    Google Scholar 

  • Kolm, S.-C. (1969), “The Optimal Production of Social Justice.” In Public Economics, J. Margolis and H. Guitton (eds.), London: Macmillan, 145–200.

    Google Scholar 

  • Kolmogoroff, A. (1930), “Sur la notion de la moyenne,” Atti della Reale Accamedia Nazionale dei Lincei, Rendiconti 12, 388–391.

    Google Scholar 

  • Marshak, J. (1950), “Rational Behavior, Uncertain Prospects and Measurable Utility,” Econometrica 18, 111–141.

    CrossRef  Google Scholar 

  • Nagumo, M. (1930), “Über eine Klasse der Mittelwerte,” Japanese Journal of Mathematics 7, 71–79.

    Google Scholar 

  • Ramsey, F.P. (1926), “Truth and Probability.” Reprinted in The Foundations of Mathematics and other Logical Essays, R.B. Braithwaite (ed.), New York: The Humanities Press, 1950, 156–198.

    Google Scholar 

  • Samuelson, P.A. (1952), “Probability Utility and the Independence Axiom,” Econometrica 20, 670–678.

    CrossRef  Google Scholar 

  • Savage, L.J. (1954), The Foundations of Statistics, New York: John Wiley.

    Google Scholar 

  • Segal, U. (1992), “Additively Separable Representations on Non-Convex Sets,” Journal of Economic Theory 56, 89–99.

    CrossRef  Google Scholar 

  • Vartia, Y.O. (1974), Relative Changes and Economic Indices, Licensiate Thesis in Statistics, University of Helsinki, June.

    Google Scholar 

  • Vartia, Y.O. (1976), “Ideal Log-Change Index Numbers,” Scandinavian Journal of Statistics 3, 121–126.

    Google Scholar 

  • von Neumann, J. and O. Morgenstern (1947), Theory of Games and Economic Behavior, Second Edition, Princeton: Princeton University Press.

    Google Scholar 

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© 1993 Springer-Verlag Berlin · Heidelberg

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Diewert, W.E. (1993). Symmetric Means and the Expected Utility Theorem. In: Diewert, W.E., Spremann, K., Stehling, F. (eds) Mathematical Modelling in Economics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-78508-5_48

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  • DOI: https://doi.org/10.1007/978-3-642-78508-5_48

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-78510-8

  • Online ISBN: 978-3-642-78508-5

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