Abstract
What would be the analogue of the Lorenz quasi-ordering when the variable of interest is continuous and of a purely ordinal nature? We argue that it is possible to derive such a criterion by substituting for the Pigou–Dalton transfer used in the standard inequality literature what we refer to as a Hammond progressive transfer. According to this criterion, one distribution of utilities is considered to be less unequal than another if it is judged better by both the lexicographic extensions of the maximin and the minimax, henceforth referred to as the leximin and the antileximax, respectively. If one imposes in addition that an increase in someone’s utility makes the society better off, then one is left with the leximin, while the requirement that society welfare increases as the result of a decrease of one person’s utility gives the antileximax criterion. Incidentally, the paper provides an alternative and simple characterisation of the leximin principle widely used in the social choice and welfare literature.
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Notes
While this operation certainly reduces inequality in a two-person society, things are less obvious when the population consists of more than two individuals. There are indeed good reasons to consider that the fact of bringing two individuals closer in terms of their incomes has the effect of moving them farther apart from the other individuals in the society: for more on this, see, e.g., Magdalou and Moyes (2009).
In this respect, our analysis framework is similar to that used by Tungodden (2000) but our approach differs significantly from his.
Actually, the same holds true for the Lorenz criterion that is the intersection of weak supermajorisation—or, equivalently, generalised Lorenz dominance—and weak submajorisation.
The adjunction of strong Pareto to the anonymity and Hammond equity conditions suffices for rendering the social preference relation complete. This has to be contrasted to what happens in the standard framework: imposing that the social welfare function is monotone increasing in addition to being Schur-concave does not make the corresponding dominance criterion complete.
This is readily inferred from the fact that, in a cardinal framework, a progressive transfer is but a particular case of a Hammond progressive transfer.
Questionnaire studies have highlighted the fact that a large proportion of respondents do not subscribe to the view according to which a progressive transfer reduces inequality (see for instance Amiel and Cowell 1999). In this respect, a notion like that of a uniform on the right progressive transfer proposed by Magdalou and Moyes (2009), which is closely related to the reduction of deprivation, is more likely to fit the views expressed by the interviewed public.
It must be noted however that, for all the other pairs of individuals \(\{i,j\}\) such that \(i < j\), we have, either\(u_{i}^{(5)}< u_{i}^{(6)}< u_{j}^{(6)} < u_{j}^{(5)}\), or\(u_{i}^{(5)} = u_{i}^{(6)} < u_{j}^{(6)} = u_{j}^{(5)}\).
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Acknowledgements
This paper forms parts of the research projects The Measurement of Ordinal and Multidimensional Inequalities (Contract No. ANR-16-CE41-0005) and The Preferences for Redistribution: Foundations, Representation and Implications for Social Decisions (Contract No. ANR-15-CE26-0004) of the French National Agency for Research whose financial support is gratefully acknowledged. It has also been supported by LabEx Entrepreneurship (Contract No. ANR-10-LABX-11-01). We are indebted to Alain Chateauneuf, an associate editor and two anonymous referees for very useful comments and suggestions when preparing this version. Needless to say, none of the persons mentioned above should be held responsible for remaining deficiencies.
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Gravel, N., Magdalou, B. & Moyes, P. Inequality measurement with an ordinal and continuous variable. Soc Choice Welf 52, 453–475 (2019). https://doi.org/10.1007/s00355-018-1159-8
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DOI: https://doi.org/10.1007/s00355-018-1159-8