Top

Social Choice and Welfare

Published in:

20-03-2018 | Original Paper

Income inequality measurement: a fresh look at two old issues

Author: Brice Magdalou

Published in: Social Choice and Welfare | Issue 3/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

The literature on income inequality measurement is split into (1) the ethical approach, from which the Atkinson–Kolm–Sen and Kolm–Pollak classes of indices are derived, and (2) the axiomatic approach, which mainly leads to the generalised entropies. This paper shows how to rationalise, under utilitarianism, the generalised entropies as ethical indices. In this framework a generalised entropy index is consistent with the principle of transfers if and only if the underlying utility function is increasing. This unconventional interpretation explains the strange behaviour of the generalised entropies for some values of the inequality aversion parameter, as identified by Shorrocks (Econometrica 48:613–625, 1980). In that case, the underlying utility function is convex. Then, it provides a solution to escape the so-called Hansson–Sen paradox (Hansson in Foundational problems in the special sciences. Reidel Publishing Company, Dordrecht, 1977; Sen in Personal income distribution. North-Holland, Amsterdam, pp 81–94, 1978) that affects the standard ethical indices and which corresponds to a counterintuitive increase in inequality as a result of a concave transformation of the utility function. A normalised version of the generalised entropies behaves appropriately after such a transformation.

previous article The role of aggregate information in a binary threshold game

next article Working time regulation, unequal lifetimes and fairness

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Available only for authorised users

Continuous differentiability in Definitions 1 and 2 is actually not necessary, even if satisfied by all the indices in the literature. It is just a simplification for ease of reading thereafter.

We thus operate in a simplified framework by considering that the situation of an individual is fully described by their income, and that personal welfare is cardinally measurable and interpersonally comparable (see Sen 1976, for a discussion).

Again, continuous differentiability is usually not required. The symbol \(|\!|\) is used to indicate that a divergence measure is different from a distance measure, as it needs to neither be symmetric nor satisfy the triangle inequality.

Actually this property has been introduced by Kolm (1976) and called the principle of diminishing transfers.

From an empirical perspective, one can argue that it is not meaningful to investigate the impact of a utility transformation on the index as most of the applications assume a fixed utility function (and thus a fixed level of inequality aversion). Nevertheless, some papers adopt a dual approach by considering that each country, for instance, is characterised by its own degree of inequality aversion (see Lambert et al. 2003). In that case, inequality comparisons require taking account of these different views on inequality aversion, which makes empirically relevant the Hansson–Sen paradox.

This definition immediately extends to the general case of unequal-mean distributions. We illustrate here the normalisation for the relative generalised entropies \(G_r\) (see Appendix A), where admissible incomes are the strictly positive real numbers, recalling that \(G_r(\varvec{x}) {{\mathrm{=}}}G_{\phi _r}(\hat{\varvec{x}})\) with \(\hat{\varvec{x}} {{\mathrm{=}}}\varvec{x}/ \mu (\varvec{x})\) and \(\phi _r\) as defined in (A.3). Consider for instance that the distributions under comparison are \(\varvec{x}=(1,3)\) and \(\varvec{y}=(2,18)\), such that \(\hat{\varvec{x}} = (0.5,1.5)\) and \(\hat{\varvec{y}} = (0.2,1.8)\). The normalisation is here concerned by the lowest observable reduced income, namely 0.2. Hence one can choose the normalised index \(G_{\phi _r} / U'(0.2)\). For the absolute generalised indices, the same reasoning applies to the centred incomes.

Atkinson A (1970) On the measurement of inequality. J Econ Theory 2:244–263CrossRef

Atkinson A, Brandolini A (2015) Unveiling the ethics behind inequality measurement: Dalton’s contribution to economics. Econ J 125:209–234CrossRef

Boissonnat J, Nielsen F, Nock R (2010) Bregman Voronoi diagrams. Discrete Comput Geom 44:281–307CrossRef

Bosmans K, Cowell F (2010) The class of absolute decomposable inequality measures. Econ Lett 109:154–156CrossRef

Bourguignon F (1979) Decomposable income inequality measures. Econometrica 47:901–920CrossRef

Cowell F (1980) On the structure of additive inequality measures. Rev Econ Stud 47:521–531CrossRef

Dalton H (1920) The measurement of the inequality of incomes. Econ J 30:348–361CrossRef

Dasgupta P, Sen A, Starrett D (1973) Notes on the measurement of inequality. J Econ Theory 6:180–187CrossRef

Hansson B (1977) The measurement of social inequality. In: Butts R, Hintikka J (eds) Foundational problems in the special sciences. Reidel Publishing Company, Dordrecht

Hardy G, Littlewood J, Polya G (1934) Inequalities. Cambridge University Press, London

Kolm S (1976) Unequal inequalities II. J Econ Theory 13:82–111CrossRef

Kolm S-C (1969) The optimal production of social justice. In: Margolis J, Guitton H (eds) Public economics. Macmillan, London, pp 145–200CrossRef

Lambert P, Millimet D, Slottje D (2003) Inequality aversion and the natural rate of subjective inequality. J Public Econ 87:1061–1090CrossRef

Magdalou B, Nock R (2011) Income distributions and decomposable divergence measures. J Econ Theory 146:2440–2454CrossRef

Pigou A (1912) Wealth and welfare. Maxmillan, London

Pollak R (1971) Additive utility functions and linear Engel curves. Rev Econ Stud 38:401–414CrossRef

Pratt J (1964) The class of additively decomposable inequality measures. Econometrica 32:122–136CrossRef

Sen A (1973) On economic inequality. Clarendon Press, OxfordCrossRef

Sen A (1976) Welfare inequalities and Rawlsian axiomatics. Theor Decis 7:243–262CrossRef

Sen A (1978) Ethical measurement of inequality: some difficulties. In: Krelle W, Shorrocks A (eds) Personal income distribution. North-Holland, Amsterdam, pp 81–94

Shorrocks A (1980) The class of additively decomposable inequality measures. Econometrica 48:613–625CrossRef

Shorrocks A (1984) Inequality decomposition by population subgroups. Econometrica 52:1369–1385CrossRef

Shorrocks A, Foster J (1987) Transfer sensitive inequality measures. Rev Econ Stud 54:485–497CrossRef

Title: Income inequality measurement: a fresh look at two old issues
Author: Brice Magdalou
Publication date: 20-03-2018
Publisher: Springer Berlin Heidelberg
Published in: Social Choice and Welfare / Issue 3/2018
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-018-1121-9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Other articles of this Issue 3/2018

Working time regulation, unequal lifetimes and fairness

A concept of sincerity for combinatorial voting

Multiwinner analogues of the plurality rule: axiomatic and algorithmic perspectives

Cost asymmetry and incomplete information in a volunteer’s dilemma experiment

The role of aggregate information in a binary threshold game

Increasing discriminatory power in well-being analysis using convex stochastic dominance