# On the Measurement of Inequality

TL;DR: In this paper, the problem of comparing two frequency distributions f(u) of an attribute y which for convenience I shall refer to as income is defined as a risk in the theory of decision-making under uncertainty.

Abstract: Measures of inequality are used by economists to answer a wide range of questions. Is the distribution of income more equal than it was in the past? Are underdeveloped countries characterised by greater inequality than advanced countries ? Do taxes lead to greater equality in the distribution of income or wealth? However, despite the wide use of these measures, relatively little attention has been given to the conceptual problems involved in the measurement of inequality and there have been few contributions to the theoretical foundations of the subject. In this paper, I try to clarify some of the basic issues, to examine the properties of the measures that are commonly employed, and to discuss a possible new approach. In the course of this, I draw on the parallel with the formally similar problem of measuring risk in the theory of decisionmaking under uncertainty and make use of recent results in this fie1d.l The problem with which we are concerned is basically that of comparing two frequency distributions f(u) of an attribute y which for convenience I shall refer to as income. The conventional approach in nearly all empirical work is to adopt some summary statistic of inequality such as the variance, the coefficient of variation or the Gini coefficientwith no very explicit reason being given for preferring one measure rather than another. As, however, was pointed out by Dalton 50 years ago in his pioneering article [3], underlying any such measure is some concept of social welfare and it is with this concept that we should be concerned. He argued that we should approach the question by considering directly the form of the social welfare function to be employed. If we follow him in assuming that this would be an additively separable and symmetric 1 My interest in the question of measuring inequality was originally stimulated by reading an early version of the paper by Rothschild and Stiglitz [13], to which I owe a great deal.

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### Cites background from "On the Measurement of Inequality"

...The Atkinson (1970) index resembles the Gini coefficient in that it satisfies the transfers principle and is compositionally invariant....

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### "On the Measurement of Inequality" refers background in this paper

...3 See Rothschild and Stiglitz [13], Hadar and Russell [ 5 ], and Hanoch and Levy [6]....

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### "On the Measurement of Inequality" refers methods in this paper

...Then by applying the results of Pratt [l 11, Arrow [ 2 ], and others, we can see that this requirement (which may be referred to as constant (relative) inequality-aversion) implies that U(y) has the form...

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