Review Reports

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I would like to thank the editors for the opportunity to review this manuscript. The authors have made an interesting contribution to the multivariate inequality literature.
However, I have some concerns about the precise presentation and (lack of) discussion of the mathematical motivation of the proposed indices. Furthermore, I am not convinced that Lemma 1 states a proper decomposition of the proposed Gini-like index into single (=marginal) inequality parts and an in-between part driven by pure dependence within the data. It seems to be rather an approximation derived by some kind of Taylor expansion, and it remains unclear how large the error in the approximation is (it may be so large that possible insights derived from the structure of the decomposition are worthless compared to the approximation error). The authors use the paper by Abul Naga and Geoffard (2006) to somehow vaguely motivate the decomposition by analogy. However, looking at that paper, it is not immediately clear that we have formal parallels here.

Some listed comments (not ordered)

- Either put more effort into a rigorous proof, derivation and analysis of the decomposition, or just remove this part from the paper.
- Definition 1 and 2: The motivation is too short here. Really write down why these are Gini-like indices, so make the link to Gini. The notation is hard to follow and the parts should be explained in more detail. Also explain the structure of the indices. How do these indices measure dependence, what are the implications?
- Please indent more formulas so that the reader can easily grasp them.
- 'Smoothing' a distribution by replacing different values with the same value does not sound like 'smoothing' anything. You could replace the word 'smoothing' throughout the paper.
- You could elaborate more on the differences between inequality and inequality of opportunity, and for both give the reader more references and ideas on recent multivariate approaches, and perhaps even compare your empirical results.
- While income is an absolute and objective number, health on a self-reported 5-point scale is not. So you might run into some fixed effects across countries and opportunity groups. At least comment on this in the paper. Maybe the indices can account for the effect by some mean normalisation, but I am not sure. Using total wealth as the second dimension might be less problematic, but of course less innovative.
- A summary/conclusion is missing.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Main Comments:

1. The article lacks specified the aim of the research, and research hypothesis(es) or research question(s).

2.  The introduction section should contain research background, research gap, aim of the research, and research hypothesis(es) or research question(s). It should then be followed by the literature review section. At present, it is a bit scarce (just 20 references).

3. Some of references are wrongly cited, for example here: This is consistent with some other research (Heckman and Landersø (2022)). It should read: This is consistent with some other research (Heckman and Landersø, 2022).

4. Please describe every abbreviation that is used. For example, the Authors use the IOP abbreviation (firstly on page 4). Further we can guess that it refers to “inequality of opportunity”, but it should be explicitly explained first time when it is used.

5. On page 5 the Authors write that they use  (𝑋𝜇)𝑗ℎ and (𝑋𝜇)𝑗. I think that the latter should read (𝑋𝜇)𝑗ℎ. In some places the latter term is written with and in some places without h in the superscript.

6. When the Authors present the proof for equation (3), they write that they get (4) (page 8). However, (4) refers to the MLD index. Probably the Authors meant that they get (3).

7. The MLD index of multivariate inequality of opportunity, as presented by equation (2), has the factor (2ℎ − 1). However, when the Authors present it for two outcomes (equation 4, page 8), this factor is missing. Please explain.

8. When the Authors describe the results of analysis for the Gini index (Figure 1), they write about positions of Sweden, Denmark and Norway in the rankings, none is correct. For example, when we look at the positions of Sweden (in 2005 it is supposed to be 8th), it is in fact 19th.

9. Also, the Authors write that the average Gini IOP value in 2005 is 20%. I think that it is much lower – 15-17%.

10.  The Authors should provide the interpretation of the presented indexes. I suppose that the lower their value is, the better (lower inequality), but I would like to know, what for example the value of the Gini index 0.099 for Germany in 2005 and the value of the MLD index of about 0.04 for Germany in 2005 mean.

11.   The Authors should be consequent when they present the figures – charts for the Gini index contains values for each country, while charts for the MLD index do not.

12. The article lacks the discussion of obtained results and the Conclusions section. The last paragraph on page 12 can be considered as a very weak form of concluding remarks. However, the manuscript should contain separate section for conclusions, some policy recommendations, limitations of the research and directions for future research.

Author Response

Please see the attachment.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

no further comments

Reviewer 2 Report

Comments and Suggestions for Authors

The Authors addressed all my comments. They made revisions and explained reasons if the changes have not been made. I am very satisfied with their revisions.