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Additive representation of separable preferences over infinite products

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  • Pivato, Marcus

Abstract

Let X be a set of states, and let I be an infinite indexing set. Our first main result states that any separable, permutation-invariant preference order (>) on X^I admits an additive representation. That is: there exists a linearly ordered abelian group A and a `utility function' u:X-->A such that, for any x,y in X^I which differ in only finitely many coordinates, we have x>y if and only if the sum of [u(x_i)-u(y_i)] over all i in I is positive. Our second result states: If (>) also satisfies a weak continuity condition, then, for any x,y in X^I, we have x>y if and only if the `hypersum' of [u(x_i)-u(y_i)] over all i in I is positive. The `hypersum' is an infinite summation operator defined using methods from nonstandard analysis. Like an integration operator or series summation operator, it allows us to define the sum of an infinite set of values. However, unlike these operations, the hypersum does not depend on some form of convergence (recall: A has no topology) ---it is always well-defined. Also, unlike an integral, the hypersum does not depend upon a sigma-algebra or measure on the indexing set I. The hypersum takes values in a linearly ordered abelian group A*, which is an ultrapower extension of A. These results are applicable to infinite-horizon intertemporal choice, choice under uncertainty, and variable-population social choice.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 28262.

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Date of creation: 19 Jan 2011
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Handle: RePEc:pra:mprapa:28262

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Keywords: additive; separable; intertemporal choice; intergenerational choice; risk; uncertainty; variable-population social choice; generalized utilitarian; nonstandard analysis; hyperreal; linearly ordered abelian group; non-Archimedean utility; lexicographical utility;

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References

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  1. Lauwers, Luc & Van Liedekerke, Luc, 1995. "Ultraproducts and aggregation," Journal of Mathematical Economics, Elsevier, vol. 24(3), pages 217-237.
  2. Blackorby,Charles & Bossert,Walter & Donaldson,David J., 2005. "Population Issues in Social Choice Theory, Welfare Economics, and Ethics," Cambridge Books, Cambridge University Press, number 9780521825511, November.
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  12. Kaushik Basu & Tapan Mitra, 2003. "Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian," Econometrica, Econometric Society, vol. 71(5), pages 1557-1563, 09.
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Citations

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Cited by:
  1. Pivato, Marcus, 2013. "Multiutility representations for incomplete difference preorders," Mathematical Social Sciences, Elsevier, vol. 66(3), pages 196-220.
  2. Pivato, Marcus, 2011. "Social choice with approximate interpersonal comparison of welfare gains," MPRA Paper 32252, University Library of Munich, Germany.
  3. Pivato, Marcus, 2011. "Variable-population voting rules," MPRA Paper 31896, University Library of Munich, Germany.

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