Inequality measures: Gini, Atkinson and Generalized Entropy As opposed to the Lorenz criterion, inequality measures provide a complete

Inequality measures: Gini, Atkinson and Generalized Entropy As opposed to the Lorenz criterion, inequality measures provide a complete

ranking. All pairs of distributions can be compared. Several inequality measures have been proposed in the literature. However, not all of them are desirable from either a technical or a policy perspective. There is now a broad consensus regarding four properties that an inequality measure should satisfy in order to be used in practice and give an accurate representation of inequality, which are briefly introduced in Box 6.3.

Box 6.3 Key properties of inequality measures There are four basic properties that an inequality measure should satisfy. These are:

symmetry, replication invariance, scale invariance and transfer. Suppose that we represent an income distribution among n individuals with a vector y = (y 1 , y 2 ,..., y n ), where each element y i is the income of individual i = 1,...,n. Let I(y) denote the level of inequality of distribution y according to a given inequality measure I.

• By symmetry (also called anonymity), an inequality measure should be invariant to

who receives each income (for example, I(2,4) = I(4,2)). • By replication invariance (also called population principle), an inequality measure should be invariant to replications of the population such that the level of inequality does not depend on the population size (for example I(2,4) = I(2,2,4,4)).

• By scale invariance (also called zero-degree homogeneity or relative income

principle) an inequality measure should be invariant to proportional changes of all incomes in a distribution such that the level of inequality does not depend on the total income in a distribution (for example I(2,4) = I(4,8)).

• By transfer (also called the Pigou–Dalton principle following Pigou 1912 and Dalton 1920), an inequality measure should increase whenever there is a regressive transfer (from a poorer to a richer person, preserving total income) between two individuals (for example I(1,5) > I(2,4)) (and inequality should decrease whenever there is a progressive transfer).

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A measure satisfying these four basic properties is called a relative inequality measure. Foster (1985) proved that only the inequality measures that satisfy the four men- tioned properties will rank two distributions in the same way as the Lorenz criterion; they are said to be Lorenz-consistent inequality measures. That means that whenever the Lorenz criterion judges a distribution A more equal than another B, all relative inequality measures will agree with this judgement. However, whenever the Lorenz criterion cannot decide (because the Lorenz curves cross), the relative inequality measures can still provide

a ranking, but the ranking of different measures may differ, as each one satisfies different

additional properties. 6

The Gini coefficient Among the ‘good’ measures of inequality (that is, those satisfying the four basic properties), is the Gini coefficient (Gini, 1912), which is the most

common measure of inequality, and is reported by most countries. 7 In this measure, all possible pairs of incomes in the distribution are compared. The higher the Gini coefficient, the more unequal the distribution.

There are different equivalent formulas for the Gini. One particularly easy method makes use of the Lorenz curve (see Box 6.4). The Gini is the ratio of the area between the diagonal (equal distribution) and the Lorenz curve, to the entire area in the triangular region below the diagonal. So when distributions are perfectly equal, the area between the diagonal and the Lorenz curve is zero (since the Lorenz curve is the diagonal), therefore the Gini coefficient is zero. When distributions are perfectly unequal the area between the Lorenz curve and the diagonal is the entire triangular region below the diagonal, hence the Gini coefficient is one.

While the Gini is clearly the best known measure, it does have some weaknesses – for example it cannot be decomposed into an inequality between population groups component and an inequality within population groups

component. 8 Two other prominent inequality measures are the Atkinson Index, and the Theil Indices. The last ones are part of a class of measures known as Generalized Entropy measures, and which can be decomposed.

There are other common ways of representing inequality such as the ratio of the share of income owned by the richest 90 per cent of the population to the poorest 10 per cent (the 90:10 ratio; it can be any other two richest and poorest shares), and the percentage of income that goes to the bottom 20 per cent of the population, which are easy to interpret. However, they are less rigorous and informative as they ignore the parts of the distribution not specifically considered in the measures as well as changes that may occur within the considered shares (they violate one of the four basic properties: transfer).

POVERTY AND INEQUALITY MEASUREMENT

Formula for the Gini coefficient Following the notation introduced in Box 6.3, if vector y represents an income distribution among n individuals, the Gini coefficient can be expressed with the following formula:

i=l y i /n). Note that it adds the absolute value of the difference between all the possible pairs of incomes y i and y j , and divides it by 2 because each income

difference is counted twice, and by total population and income (so that the Gini is replication and scale invariant).