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Inequality aversion and separability in social risk evaluation

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Abstract

This paper examines how to satisfy “independence of the utilities of the dead” (Blackorby et al. in Econometrica 63:1303–1320, 1995; Bommier and Zuber in Soc Choice Welf 31:415–434, 2008) in the class of “expected equally distributed equivalent” social orderings (Fleurbaey in J Polit Econ 118:649–680, 2010) and inquires into the possibility to keep some aversion to inequality in this context. It is shown that the social welfare function must either be utilitarian or take a special multiplicative form. The multiplicative form is compatible with any degree of inequality aversion, but only under some constraints on the range of individual utilities.

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Notes

  1. On the interpretation of the implications of Harsanyi’s theorem, see Weymark (1991) and Broome (1991). For a defense of Paretian (“ex ante”) criteria that evaluate the distribution of individual expected utilities with some inequality aversion, see, e.g., Diamond (1967) and Epstein and Segal (1992). For a defense of rational (“ex post”) criteria that compute the expected value of an inequality averse social welfare function, see, e.g., Adler and Sanchirico (2006) and Fleurbaey (2010).

  2. The equally distributed equivalent (Atkinson 1970) of a given distribution of utility is the level of utility that, if enjoyed uniformly by all individuals, would yield the same social welfare as the contemplated distribution.

  3. Certain apparent violations of dominance seem rational (Grant 1995). If a parent would rather flip a coin to allocate a sweet between two children than give it to one child without flipping a coin, this seems to violate dominance because the final distribution of sweets is the same anyway. But this behavior is compatible with dominance if, as is natural, one incorporates the fairness of the procedure in the description of the final consequences.

  4. Independence of the utilities of the dead is an important principle in intergenerational ethics. In the certainty case, it corresponds to Postulate 3b of Koopmans (1960) as discussed by Asheim Mitra and Tungodden (2012) who defend recursive social welfare objectives defined by this Postulate and a stationarity condition. These principles, however, apply to situations with an infinite number of generations. This infinite population case raises specific issues as discussed by Lauwers (2012). We do not address these issues in the present paper.

  5. The reader can in fact check that in the proof of Proposition 1, only Independence of the utilities of the dead is actually used.

  6. In the case \(i=1\), the equation is \(\hat{e}(U_{s}^{1},U_{s}^{2},\ldots ,U_{s}^{n})=a_{1}(U_{s}^{1})+b_{1}(U_{s}^{1})\hat{e}(u^{*},U_{s}^{2},\ldots ,U_{s}^{n})\).

  7. For a comparison of various multidimensional versions of the Pigou-Dalton principle, see Diez et al. (2007).

  8. It is permissible to let \(\varepsilon \rightarrow +\infty \) only if \(\inf X\ge 0\).

  9. Risk equity and catastrophe avoidance are two principles introduced in Keeney (1980). The former is the principle that, when individuals face independent risks of a specific damage (accident), inequalities in their probabilities of damage are undesirable. The latter principle seeks to minimize the risk of having a large number of fatalities. Keeney showed that the two principles are antinomic, because the best way to avoid a catastrophe is to concentrate the risk on a few (sacrificed) individuals. In an intergenerational setting with uncertain existence of future generations, Bommier and Zuber (2008) show that risk equity (resp., catastrophe avoidance) induces a low (resp., high) social discount rate.

  10. We thank the referee for suggesting this conclusion.

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Correspondence to Stéphane Zuber.

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We would like to thank François Maniquet and a referee for helpful remarks and comments, as well as the audience at the 2012 SCW congress. This research has been supported by the Chair on Welfare Economics and Social Justice at the Institute for Global Studies (FMSH—Paris) and the Franco-Swedish Program on Economics and Philosophy (FMSH—Risksbankens Jubileumsfond).

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Fleurbaey, M., Zuber, S. Inequality aversion and separability in social risk evaluation. Econ Theory 54, 675–692 (2013). https://doi.org/10.1007/s00199-012-0730-2

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