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Public Choice

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06-03-2018

Social welfare with net utilities

Authors: Jon X. Eguia, Dimitrios Xefteris

Published in: Public Choice | Issue 1-2/2019

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Abstract

We consider a society facing a binary choice, in an environment in which differences in utility are comparable across individuals. In such an environment, net utility is the difference between the utility that an individual attains from one alternative, and the utility she attains from the other alternative. A social welfare ordering is a preference relation over net utility profiles. We show that a social welfare ordering satisfies a collection of standard normative axioms if and only if it is representable by a collective utility function defined by the sums of a given power of net individual utilities.

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Note, however, that our use of the term is not exactly the same as Moulin’s. Moulin defines a social welfare ordering as a preference over the set of utility profiles. Our social welfare ordering is a preference over the set of net utility profiles.

We use the symbol \(\lnot\) to denote the negation of a logical statement.

We use Moulin’s (1988) Theorem 2.6.b, a version of Roberts’ (1980) Theorem 6. Roberts (1980) imposes his conditions on social welfare functionals; Moulin (1988) reinterprets the axioms to apply them to social welfare orderings, an approach we follow. Roberts (1980) credits previous literature, citing Arrow (1965) and Hicks (1965) for the mathematical insight behind his Theorem 6. The first proof we are aware of is in Katzner (1970) in the context of consumer theory. The functions characterized by these these theorems are often called Bergson functions, in reference to Bergson (née Burk) (1936).

This implication derives from the Debreu-Gorman separability theorem (Debreu 1960; Gorman 1968). See as well Blackorby et al. (1998).

Arrow, K. J. ([1951] 1963). Social choice and individual values, 2nd ed. Cowles Foundation No. 12. New Haven: Yale University Press.

Arrow, K. J. (1965). Aspects of the theory of risk-bearing. Helsinki: Yrjö Jahnssonin Säätiö.

Blackorby, C., Primont, D., & Russell, R. (1998). Separability: A survey. In S. Barberà, P. Hammond, & Seidl, C. (Eds.), Handbook of utility theory I (pp. 49–92). Springer.

Burk, A. (1936). Real income, expenditure proportionality, and Frisch’s ‘New Methods of Measuring Marginal Utility’. Review of Economic Studies, 4(1), 33–52.CrossRef

d’Aspremont, C., & Gevers, L. (1977). Equity and the informational basis of collective choice. Review of Economic Studies, 44(2), 199–209.CrossRef

Debreu, G. (1954). Representation of a preference ordering by a numerical function. In R. Thrall, C. Coombs, & R. Davis (Eds.), Decision processes. New York: Wiley.

Debreu, G. (1960). Topological methods in cardinal utility theory. In K. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in the social sciences (pp. 16–26). Redwood: Stanford University Press.

Gorman, W. M. (1968). The structure of utility functions. Review of Economic Studies, 35(4), 367–390.CrossRef

Hicks, J. R. (1965). Capital and growth. Oxford: Oxford University Press.

Katzner, D. W. (1970). Static demand theory. New York: Macmillan.

Maskin, E. (1978). A theorem on utilitarianism. Review of Economic Studies, 45(1), 93–96.CrossRef

May, K. O. (1952). A set of independent necessary and sufficient conditions for simple majority decision. Econometrica, 20(4), 680–684.CrossRef

Moulin, H. (1988). Axioms of cooperative decision making. Cambridge: Cambridge University of Press.CrossRef

Roberts, K. W. (1980). Interpersonal comparability and social choice theory. Review of Economic Studies, 47(2), 421–439.CrossRef

Sen, A. K. (1966). A possibility theorem on majority decisions. Econometrica, 34(2), 491–499.CrossRef

Sen, A. K. (1970). Interpersonal aggregation and partial comparability. Econometrica, 38, 393–409.CrossRef

Sen, A. K. (1977). On weights and measures: Informational constraints in social welfare analysis. Econometrica, 45(7), 1539–1572.CrossRef

Title: Social welfare with net utilities
Authors: Jon X. Eguia
Dimitrios Xefteris
Publication date: 06-03-2018
Publisher: Springer US
Published in: Public Choice / Issue 1-2/2019
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-018-0527-3

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