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Currently reading David Adler's text on distributional weights in benefit-cost analysis (https://scholarship.law.duke.edu/faculty_scholarship/3110/), I cannot manage to fully grasp the definition of the anonymity axiom.

The isoelastic/Atkinson social welfare function (SWF) allows for an inequality aversion parameter; greater weight is put on changes affecting worse-off individuals. Adler states that this function satisfies the anonymity/impartiality axiom, defined here as "meaning indifference between any given utility vectora and all rearrangements of component numbers". I understand this, broadly, as meaning that the levels of utility, and not the persons enjoying those levels, are what counts.

I can't manage to reconcile the fact that the "anonymity/impartial SWFs focus only on the patern of well-being, and not on the identities of the people who end up at particular well-being levels" (page 267 of the text) with the use of an inequality aversion parameter, which, by definition, treats individuals differently. Would this not mean focusing "on the identities of the people who end up at particular well-being levels"?

What am I missing ?

Any links towards readings on this subject would be much appreciated as well.

Many thanks.

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  • $\begingroup$ Is it possible if you can cite the specific quote you are having difficulty understanding? $\endgroup$
    – Bensstats
    Mar 4, 2018 at 18:29
  • $\begingroup$ "Anonymity/impartial SWFs focus only on the patern of well-being, and not on the identities of the people who end up at particular well-being levels" (page 267, first paragraph); which I cannot reconcile with the fact that the isoelastic SWF puts more weight on changes affecting less well off individuals. I must be missing something but haven't managed to find any documentation that helps... Many thanks $\endgroup$
    – Ploit88
    Mar 4, 2018 at 18:33
  • $\begingroup$ Is it possible if you can include that in your edited question? Sorry for the bother. $\endgroup$
    – Bensstats
    Mar 4, 2018 at 18:34
  • $\begingroup$ No worries, thanks for pointing out that it needed to be clarified. $\endgroup$
    – Ploit88
    Mar 4, 2018 at 18:38

2 Answers 2

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Anonymous/impartial SWFs focus only on the pattern of well-being, and not the identities of the people who end up at particular well-being levels.

Identities here simply means names. When applying an impartial SWF, one only looks at the profile of utilities, not who those utilities are associated with. Consider the following two cases: \begin{array}{ccc} &U_{\text{President}}&U_{\text{CEO}}&U_{\text{Farmer}} \\\hline \text{Case 1}& 100 & 80 & 10\\ \text{Case 2}& 10 & 100 & 80 \end{array} The patterns of the distribution of well-being/utility in the two cases are essentially the same, and yet they differ in who is better/worse off. An impartial SWF would generate the same outcome in both cases, since it only focuses on the pattern (i.e. the utility numbers) of the well-being distribution and not on who (i.e. the identity) has which level of utility. In particular, the fact that the President is the happiest person in Case 1 but the most miserable in Case 2 has no impact whatsoever on the outcome of the impartial SWF.

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Compare a) A social planner who prefers the gains to go to one group instead of another, and b) A social planner who equally values a gain regardless of who got it.

In terms of "inequality aversion", consider the specific mathematical definitions. The Gini index, for example, does not correct for whether the identities of the individuals are Mrs. Smith or Mr. Lee.

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