### Discussion Paper

## Abstract

A long literature on inter-distributional inequality (IDI) has developed statistical tools for measuring the extent of inequality between two groups (e.g. men versus women). Firstly, I introduce the property of group-specific disadvantage focus (GDF). Indices satisfying this property are only sensitive to inequalities that are disadvantageous to one specific group. Then the paper reviews some of the most prominents IDI indices proposed in the last four decades. The assessment focuses on whether these indices satisfy GDF and, if not, how they react to inequalities that are disadvantageous to different groups. I also discuss whether these indices are informative, or not, regarding other interesting features related to IDI comparisons, e.g. distributional equality, absence of distributional overlap and presence of first-order stochastic dominance. Finally, I propose amendments to several of these indices in order to render them in fulfillment of GDF and more informative on the mentioned distributional features.

Paper submitted to the special issue

The Measurement of Inequality and Well-Being: New Perspectives

## Comments and Questions

See attached file

Dear Editor,

I have received the referee report for my paper. Firstly, I would like to thank the referee for having reviewed the paper and for the very helpful comments and recommendations in the report. In my understanding of the report, the referee asks me to improve the paper ...[more]

... by providing: 1) a more appealing motivation of all the required properties and a rigurous definition of them (i.e. not just of GDF); and 2) a guide on how to choose among the propose indices.

The referee also highlighted four minor points.

I agree with the referee that implementing all these proposals would improve the paper substantially and I will be happy to undertake them if the editor asks me to submit a new version.

See attached file

Dear Editor,

I have received the second referee report on my paper. Firstly, I would like to thank the referee for having reviewed the paper and for the very helpful comments and recommendations in the report. The referee makes three points:

1) The GDF axiom conflates two different ...[more]

... properties which should be stated separately (in two different axioms). These properties are a) an index should account only for advantages of a group against another one; and b) assessing these advantages at each quantile or percentile. I think this is the referee´s main concern and I am very grateful to him/her for drawing this point to my attention. I think the referee is right in making this distinction between focusing on a group´s specific (dis)advantage, on one hand, and choosing how to measure the latter (e.g. by comparing quantiles), on the other. If the editor gives me the green light for a resubmission I will be happy to make this distinction and refine the axioms accordingly. My view is that while the second property (measuring at the quantile level) may require independent motivation from the first one, as the referee rightly asserts, I think it is easier to handle than the first property, since, as the referee says, the latter can be motivated in several ways, i.e. there is an a priori ambiguity as to what focusing on one group´s advantage may mean. This in itself is an interesting issue, whose discussion, in my view, has not yet been exhausted in the literature.

2) I thank the referee for correcting me for the wrong combination of the words "cumulative" and "density". As for expression (1), I would have expected that it should not be controversial once one visualizes its meaning. For continuous variables, the expression can be derived using integration by parts. (I am happy to provide my own derivation if necessary). I am not sure exactly as to what the referee wants me to notice regarding ordinal variables, but I can see that actually equation 1 does not hold for ordinal variables because, even though the left-hand side has a straightforward summation equivalent, its right hand side depends on the actual scaling of the variable, which is arbitrary in the case of ordinal variables. I am happy to remove the references to ordinal variables from that discussion. On the other hand, expression (1) is still useful because definition 2, i.e the dual of 1, enables the use of the GDF property in the case of indices that map from cumulative distributions (e.g. Garstwith´s). The case of ordinal variables would require special attention (probably beyond the scope of the paper).

3) Ommitted references: I am very grateful to the referee for pointing these. I will be happy to incorporate them in a revised version. I was aware of the orange Hancock and Morris textbook, but forgot that, indeed, it offers several indices of relative distributions. As for Le Breton et al. (2011, forthcoming JET), it looks like an updated version of the 2008 working paper by the same authors that I cited in the paper. I will be happy to update this reference (and the content related to it).