Social justice
Tools often used around the world to measure inequality, such as the Gini index, can act in unexpected ways in certain situations. This is highlighted by economists F. A. Cowell and E. Flachaire.
Elon Musk, the boss of Tesla and SpaceX, earns 500 dollars per second, which is what half the world earns in a month. Since the 1980s, economists have been paying close attention to the sharp rise in global inequality. How can it be measured? Should we take into consider income, assets, the average, the median, or the inter-decile range? A. Atkinson and T. Piketty have looked at the highest incomes, S. Jenkins at the dynamics of household incomes in the UK, and A. Krueger on the rise of inequality in the United States.
However, there is one measure of inequality that has been the object of consensus for many years: the Gini index. Developed by the Italian statistician Corrado Gini in the early 20th century, this index takes the form of a value between 0 and 1, with zero representing perfect equality and 1 the greatest inequality. This measure is now widely used around the world to analyse inequalities between countries, and over time. According to the World Bank, in 2021 Sweden has a coefficient of 0.298, France 0.315 and Brazil 0.529. In China, it rose from 0.437 in 2010 to 0.371 in 2020, signalling a decline in inequality.
Economists Frank A. Cowell and Emmanuel Flachaire show that the Gini index can behave in unexpected ways. By increasing the income of a person who is wealthier than the average for the group studied, a measurement with the Gini index reduces inequality, rather than increasing it. The researchers investigate this anomaly and study the performance of other inequality measures in their article Inequality Measurement and the Rich: Why Inequality Increased More Than We Thought, published in 2024 in the journal The Review of Income and Wealth.
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Properties of the indices
Using an axiomatic approach, the authors study the properties of the index and those of two other measures: the Theil index and the mean logarithmic deviation (MLD), both developed by Henri Theil, a Dutch professor of econometrics in the 1960s. They range from 0, which represents perfect equality, to infinity, which represents the greatest inequality.
The Gini, Theil and MLD indices measure the distance between the observed distribution of incomes and a situation of perfect equality where all individuals receive the same income. The distance is defined differently depending on the measure used. The Gini index is based on absolute differences, while the Theil and MLD indices use logarithmic differences. Furthermore, the Theil index is more sensitive to variations between high incomes than the other two indices.
All three indices are unaffected by scale, meaning that if the income of each person is multiplied by two, as happens in the case of inflation, for example, inequality remains unchanged. This principle of invariance to scale stipulates that comparisons of inequality should not be affected by proportional changes in income or wealth. It is respected by dividing incomes by their mean, which allows inequality to be compared independently of the absolute scale of the data.
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All three comply with the Pigou-Dalton principle of transfer: any transfer of income from a richer person to a poorer person should not result in an increase in inequality. For example, if one of the wealthiest people gives €1,000 to one of the poorest people, with all other incomes unchanged, inequality is not likely to increase.
The MLD index is the only one to respect a third principle, called 'distance monotonicity’: by which, the enrichment of an individual who is wealthier than the level of perfect equality should not reduce inequalities, just as the impoverishment of a poorer individual may. Surprisingly, this is not the case with the indices of Gini and Theil. With these indices, total inequality can decline when the rich are made wealthier. The example below is a perfect illustration: in situation B, the 5th individual sees their income increase by €20. This situation is judged to be less unequal by the Gini and Theil indices, whereas the MLD index judges it to be more unequal, since it corresponds to a situation where an individual with an above-average income becomes richer.
Mean or median?
Generally speaking, Frank A. Cowell and Emmanuel Flâchaire sought to find out whether other indices exist that can simultaneously respect the principle of invariance to scale and the principle of distance monotonicity. They demonstrate that the traditional indices defined with incomes divided by their median, rather than with incomes divided by their mean, allow these two principles to be respected. This has an important statistical impact, because the mean is sensitive to extreme values, unlike the median.
According to the French Observatoire des Inégalités, the average monthly income in the world in 2021 will be $1,392. This includes a number of immense fortunes, which results in a rising average. Currently, Elon Musk, the boss of Tesla and SpaceX, embodies an extreme value: he has asked his shareholders for remuneration totalling 56 billion dollars for the period from 2018 to 2022, or 1.16 billion dollars per month - more than 800,000 times the world average. At the other end of the spectrum, the 2.5 billion adults making up the poorest half of the world's population earn less than €558 a month. On average, the poorest 50% earn just 233 euros a month - six times less than the global average. If the world were perfectly egalitarian, each adult would earn €1,400 a month.
Calculations based on the median are more stable. But inequality indices based on income divided by the median do not always respect the Pigou-Dalton principle of transfer. Ultimately, the researchers show that there is a single measure that respects all the desired principles of scale invariance, the Pigou-Dalton principle as well as distance monotonicity - and the coup de grâce is that it does not involve division by the mean or the median: the MLD index.
Inequality has increased more than previously thought
In an analysis of the evolution of inequality in the United States and the United Kingdom from the 1960s to 2015, the Gini and Theil indices show a fairly similar curve in the two countries. Conversely, the MLD index shows a much more marked increase in inequality in the United States than in the United Kingdom, especially from the 1990s onwards. These results show that an analysis based solely on the Gini index does not capture the rise in inequality in the United States as clearly as the MLD index.
Deciding which measures of inequality to employ is crucial when devising policies to reduce inequality. In the studies of Thomas Piketty, it is the proportion of total income held by a fraction of the highest earners, such as the top 1% or top 10%, that is measured. This measure is intuitive and readily understood by the general public. According to the World Inequality Database, in 2022, the richest 1% will receive 19.6% of the world's income. More than half of this income will go to the richest 10%, and only 8% to the poorest half. However, these metrics do not consider the distribution of income within either the 1% or the 99%. In other words, the same value can correspond to very different situations, where there are many or few poor people among the population.
The Gini, Theil or MLD indices measure the distribution of income across an entire population, they allow to analyse the dynamics of inequality and make comparisons between countries. However, their values are not easily interpreted, and their application can lead to significantly different results. So when it comes to measuring inequality, while a perfect index does not exist, it is nevertheless important to be familiar with the properties of the indices used.
