## 4.1 Which inequality measure should be used?

The variety of inequality measures begs a question: which inequality measure should be used? In fact, this is a very important question. However, the nature of it is not statistical or mathematical, but ethical. This section aims to clarify and, while not proposing a “perfect measure”, to provide the reader with some initial guidance about which measure to use.

The most general way to analyze if one distribution is more equally distributed than another is by the Lorenz curve. When $$L_A(p) \geqslant L_B(p), \forall p \in [0,1]$$, it is said that $$A$$ is more equally distributed than $$B$$. Technically, we say that $$A$$ (Lorenz) dominates $$B$$.7 Krämer (1998Krämer, Walter. 1998. “Measurement of Inequality.” In Handbook of Applied Economic Statistics, edited by Amman Ullah and David E. A. Giles, 1st ed., 39–62. Statistics: A Series of Textbooks and Monographs 155. New York: Marcel Dekker.) and Mosler (1994Mosler, Karl. 1994. “Majorization in Economic Disparity Measures.” Linear Algebra and Its Applications 199: 91–114. https://doi.org/https://doi.org/10.1016/0024-3795(94)90343-3.) provide helpful insights to how majorization, Lorenz dominance, and inequality measurement are connected. On the topic of majorization, Hardy, Littlewood, and Pólya (1934Hardy, G. H., J. E. Littlewood, and G. Pólya. 1934. Inequalities. 2nd ed. Cambridge University Press.) is still the main reference, while Marshall, Olkin, and Arnold (2011Marshall, Albert W., Ingram Olkin, and Barry C. Arnold. 2011. Inequalities: Theory of Majorization and Its Applications. 2nd ed. Springer Series in Statistics. Springer.) provide a more modern approach. In this case, all inequality measures that satisfy basic properties8 Namely, Schur-convexity, population invariance, and scale invariance. will agree that $$A$$ is more equally distributed than $$B$$.

When this dominance fails — i.e., when Lorenz curves do cross — Lorenz ordering is impossible. Then, under such circumstances, the choice of which inequality measure to use becomes relevant.

Each inequality measure is a result of a subjective understanding of how to rank the fairness of a distribution. As Dalton (1920, 348Dalton, Hugh. 1920. “The Measurement of the Inequality of Incomes.” The Economic Journal 30 (September). https://doi.org/10.2307/2223525.) puts it, “the economist is primarily interested, not in the distribution of income as such, but in the effects of the distribution of income upon the distribution and total amount of economic welfare, which may be derived from income.” The importance of how economic welfare is defined is once again expressed by Atkinson (1970Atkinson, Anthony B. 1970. “On the Measurement of Inequality.” Journal of Economic Theory 2 (3): 244–63. https://ideas.repec.org/a/eee/jetheo/v2y1970i3p244-263.html.), where an inequality measure is directly derived from a class of welfare functions. Even when a welfare function is not explicit, such as in the Gini index, there is an implicit, subjective judgement of the impact of inequality on social welfare.

The idea of what is a fair distribution is a matter of Ethics, a discipline within the realm of Philosophy. Yet, as Fleurbaey (1996, Ch.1Fleurbaey, Marc. 1996. Théories Économiques de La Justice. Économie Et Statistiques Avancées. Paris: Economica.) proposes, the analyst should match socially supported moral values and theories of justice to the set of technical tools for policy evaluation.

Although this can be a useful principle, a more objective answer is needed. By knowing the nature and properties of inequality measures, the analyst can further reduce the set of applicable inequality measures. For instance, choosing from the properties listed in Frank Alan Cowell (2011, 74Cowell, Frank Alan. 2011. Measuring Inequality. 3rd ed. London School of Economics Perspectives in Economic Analysis. New York: Oxford University Press.), if we require group-decomposability, scale invariance, population invariance, and that the estimate falls within $$[0,1]$$, we might resort to the Atkinson index.

Even though the discussion can go deep in technical and philosophical aspects, this choice also depends on the public. For example, it would not be surprising if a public official has not encountered the Atkinson index; however, they might be familiar with the Gini index. The same goes for publications: journalists have been introduced to the Gini index and can find it easier to compare and, therefore, write about. Also, we also admit that the Gini index is relatively straightforward, if compared to many other measures.

In the end, the choice is mostly subjective and there is no consensus of which method offers the “ideal” inequality measure.9 In fact, there is much discussion about the properties of an ideal inequality measure. We must remember that this choice is only problematic for relative inequality measures if Lorenz curves cross; otherwise, the choice among relative inequality measures is a less substantial issue.