Ethically Robust Comparisons of Bidimensional Distributions with an Ordinal Attribute

We provide foundations for robust normative evaluation of distributions of two attributes,\r\none of which is cardinally measurable and transferable between individuals and the other is\r\nordinal and non-transferable. The result that we establish takes the form of an analogue to the standard Hardy, Littlewood, and Pólya (1934) theorem for distributions of one cardinal\r\nattribute. More specifically, we identify the transformations of the distributions which\r\nguarantee that social welfare increases according to utilitarian unanimity provided that the utility function is concave in the cardinal attribute and that its marginal utility with respect to the same attribute is non-increasing in the ordinal attribute. We establish that this\r\nunanimity ranking of the distributions is equivalent to the Bourguignon (1989) ordered\r\npoverty gap quasi-ordering. Finally, we show that, if one distribution dominates another\r\naccording to the ordered poverty gap criterion, then the former can be derived from the latter by means of an appropriate and finite sequence of such transformations