_Table 1: Comparison of various inequality measures_

|Inequality measure|Commonly<br>used|Easy to<br>explain|Lorenz<br>consistent|Subgroup<br>decomposable|
|---|---|---|---|---|
|Gini coefficient|Yes|No|Yes|No|
|Mean log deviation|No|No|Yes|Yes|
|Theil index|No|No|Yes|Yes|
|Kuznets ratios|Yes|Yes|No|No|
|Quantile ratios|Yes|Yes|No|No|

Commonly used is difficult to track formally given that the Kuznets ratios and quantile ratios come in various forms. Due to their frequent use in inequality research, such as in Piketty (2014), we classify them as commonly used. Among the other three measures, Google returns about 5,000,000 hits for the Gini coefficient (or Gini index) while the Theil Index and mean log deviation receive around 100,000 and 15,000 hits, respectively.<sup>3</sup> This leads us to classify the Gini as commonly used and the other two measures as not.

Kuznets ratios and quantile ratios benefit from being expressed in units that are easy to explain. Most people can easily understand what it means if the top 10% has twice the income of the bottom 40%. By contrast, the other three measures do not have very intuitive scales or interpretations. One of the simplest (though still not straightforward) explanations of the Gini coefficient is that it represents the average transfer needed to make two people in a country equal, expressed relative to the country’s mean income (see Mahler & Baur (2023) for a visualization of this). For example, a Gini coefficient of 50 means that to equalize the incomes of two randomly chosen individuals, on average 50% of the country’s mean income should be transferred from the richer to the poorer individual. Many other interpretations of the Gini coefficient exist (Pyatt 1969, Sen 1973, Yitzhaki 1979, Lerman & Yitzhaki 1984).

Lorenz consistency indicates whether a measure satisfies certain theoretically desirable properties (Foster 1985), such as the Pigou-Dalton transfer principle, which (in its strong form) states that inequality must fall if income is transferred from a wealthier individual to a poorer individual. This does not hold for Kuznets ratios as transfers between individuals not considered in the groups chosen would not influence the inequality measure. For example, the Palma ratio ignores the middle class and is unaffected by what happens in the distribution between the 40<sup>th</sup> and 90<sup>th</sup> percentiles. Among the inequality indices that are Lorenz sensitive, they differ by which parts of the distribution they place larger weights on. The Gini coefficient is rank-informed, which means that the index changes in response to transfers depending on the rank of the involved. As a result of being rank-informed, the Gini coefficient is generally more sensitive to transfers in the middle of distributions (where there is more population mass, and hence more possibilities of re-rankings) than in the ends (Atkinson 1970, Sen 1973). In addition, compared to the mean log deviation, the Gini coefficient tends to be relatively more responsive to the top than the bottom.

Subgroup decomposability means that it is possible to decompose total inequality into inequality between groups of a country (say, geographical regions) and within groups of a country (Bourguignon 1979). The Theil index and mean log deviation were developed with this attribute in mind. The Gini coefficient, Kuznets ratios, and quantile ratios, by contrast, are not subgroup decomposable.

As can be seen, no measure satisfies all criteria, and in general there is a trade-off between selecting a measure with desirable theoretical properties and a measure that is easy to understand and familiar to many. The Gini index somewhat falls in the middle. It is likely familiar to the widest audience with a long history of use, while also satisfying basic theoretical properties, yet it is difficult to interpret.

> 3 The mean log deviation is also called a Theil index, which complicates the analysis, but should reinforce the results. The mean log deviation and the Theil index correspond to GE0 and GE1 in the General Entropy (GE) class.

4