On the Curvature Properties of “Long” Social Welfare Functions
Round 1
Reviewer 1 Report
I have attached my report in the pdf below.
Comments for author File: 
Comments.pdf
Author Response
(1) The section has been halved. We agree that the shortening should improve the focus.
(2) Before Lemma 3, at the end of section 4, we added the following explanation “In other words, the household that was relatively richer before the price change maintains the relative ordering after the change. Therefore, any measure that redistributes income through price (or tax) policies does not have an effect on the ordering of the individuals (Casajus 2016). A similar concept is expressed in Casajus (2016), a reference we added to the bibliography.
(3) We carefully proofread the text and followed your suggestions. In particular
(a) Omitted when shortening Section 2.
(b) Footnote 7 has been added
(c)-(f) Done. Thank you much!
(4) It is a correct observation. This paper builds on results developed in Menon, Pagani and Perali 2016. The SWF W(V,p|ρ) is quasiconvex if the properties ES of the equivalence scales functions, PA and MD are respected. To ensure the empirical regularity of the SWF these properties should be tested when estimating a SWF as also remarked in the conclusions.
Reviewer 2 Report
Dear Authors,
I enjoyed reading your paper and I support your call to bring the long social welfare functions back into the economics profession. The paper is well written and your argumentation seems solid. I have two minor observations that may improve your paper:
1. It would be better to explain in some general terms in the introduction of your paper the potential importance of these welfare functions for empirical research and then discuss in your concluding section in more detail what the implications and opportunities are. You do this now in the early part of your paper but this tends to get lost as the rest of your paper consists of the technical part. It would also be good if you could give a bit more detail on what it would entail to apply these functions empirically.
2. Please go through your paper again to take out some minor English and style errors.
Author Response
Point 1. Introduction, Section 2 and Conclusions have been substantially revised following the suggestions in the comment.
Point 2. We revised the text correcting English and style errors.
Reviewer 3 Report
Authors characterize curvature properties of social welfare functions that use indirect utilities to represent welfare. It is a good short expose on the topic and is very well written.
Comments for author File: Comments.pdf
Author Response
Point 1. We added both in the abstract and introduction that we use the definition of generalized concavity to show the conditions necessary for the long social welfare function of interest to be decreasing and quasi-convex with respect to prices.
Point 2. In the revised version we tried to be more direct and explicit in bringing to the fore the neglect of a fundamental property of SWFs such as quansiconvexity.
Point 3. A natural application of SWF is its maximization either in the primal (quantity) or the dual (price) space. For example, in the Jorgenson and Slesnick paper “Measuring Individual Economic Well-Being and Social Welfare within the Framework of the System of National Accounts”https://scholar.harvard.edu/files/jorgenson/files/jorgenson_schreyer_measuring_individual_economic_well-being_3_24_05_16riwresubmission.pdf the authors at page 26 list the properties of SWFs but do not include curvature properties. The application they propose is descriptive and does not require maximization. Our objective was to contribute this missing element about the curvature characteristics in the set of listed properties to lend full theoretical legitimacy to long SWFs.
Point 4. We corrected typos.
Round 2
Reviewer 1 Report
I have gone through the revised version. Authors have edited the manuscript in response to my report and other reviewers' reports in the first round appropriately.
The current version of the paper should be accepted for publication.
