Income Distribution

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Income Distribution

Income distribution captures the proportion of total income accruing to different individuals that accounts for the total income distributed among them during a certain time period. If the unit of analysis is the country, the income distribution for a certain year would show how much of the country’s yearly income went to each person. It may be of interest to capture the country’s distribution of income within and across groups of individuals as well. Groups can be formed according to many criteria, depending on the issue under analysis. Criteria can include geographic (cities or states), gender, social, and ethnic considerations, to name a few. When groups of individuals are considered, it is typically of interest to distinguish between the income distribution within and across groups. For example, how much of a country’s income goes to the urban and to the rural population corresponds to the across-group distribution of income. How much went to each person living in rural areas and how much went to each person living in urban areas is a measure of the distribution of income within groups.

Income distribution can be represented in different ways. It can be represented by an (income) density function, f (y ), where y stands for income; f (y ) dy gives the number of individuals earning income in the [y, y + dy ] interval. Integrating the income-density function to infinity yields the total population, N, while integrating yf (y ) gives total income, Y. Mean income, µ, is given by Y/N.

A plot of the income-density function would be the most comprehensive way of representing income distribution, but this is rarely possible in practice if the unit of analysis is a country. Histograms may provide an approximation, but these representations are not usually used when analyzing the dispersion in the distribution of income. The most widely used graphic representation of income distribution—while by no means the only one possible—is the Lorenz curve. To construct a Lorenz curve, individuals are first ranked by income in ascendant order: y_i≤; y_{i +1}, i = 1, …, N. Then, the cumulative income up to rank j is computed: s_j =y ₁ + y ₂ + y_j, j = 1, …, N. The Lorenz curve corresponds to the plot of (j/N, s_j/Y ), j = 1, …, N. That is, it shows on the horizontal axis the cumulative population share (from 0 to 1) and on the vertical axis the corresponding income share (also from 0 to 1). It enables the determination of the share of income going to each quartile, quintile, decile, or centile—or any other breakdown of the population.

The Lorenz curve gives a visually compelling idea of the extent of inequality in the distribution of income. The line of perfect equality corresponds to the 45-degree line that goes from the origin to (1,1). That is, the poorest 10 percent of the population have 10 percent of income, then the poorest 20 percent have 20 percent of income, and so forth. The Lorenz curve lies always on or below the line of perfect equality, with “perfect inequality” (all income going to the richest individual) corresponding to the line that makes the lower and right-hand side of the square that encompasses the line of perfect inequality. That is, the shares of income going to all the shares of the population are zero, and 100 percent of income goes to the single individual at the top. Thus, the closer the Lorenz curve is to the line of perfect equality, the more equal the distribution of income; the farther it is, the higher the level of inequality.

Thus, intuitively, the area that lies between the line of perfect equality and the Lorenz curve provides an indication of the extent of inequality. If this area is very small, the Lorenz curve is very close to the area of perfect equality, and inequality is low. If this area is large, then the Lorenz curve is further away from the line of perfect equality, and inequality is high. The ratio of this area to the area of the triangle limited by the lines of perfect equality and perfect inequality is a number that falls always between zero (when the Lorenz curve coincides with the line of perfect equality) and one (when the Lorenz curve coincides with the line of perfect inequality and the area under the Lorenz curve is exactly the same as the area of the triangle). This number is the Gini coefficient, one of the most widely used measures of inequality—and one of the many possible summary measures of income distribution, whereby distribution is summarized by a single number.

While intuitively appealing, the Gini coefficient has a major drawback. As noted, in studies of inequality there is often an interest in grouping individuals and determining the shares of inequality that correspond to asymmetries in the distribution of income across and within groups. It is possible to decompose the Gini coefficient into withingroup and between-group components, but this decomposition is not “perfect,” in the sense that the components do not add up to total inequality—a third term, a residual, is typically required to be added to the other two terms to get to total inequality. The only summary measures of inequality that are perfectly decomposable correspond to a class known as entropy-based measures of inequality, which were pioneered by the Dutch econometrician Henri Theil (1924–2000). He drew inspiration from the entropy-based analysis of information, and thus the name of this class of measures.

The most widely used entropy-based measure of inequality is the Theil index. It is bound between zero and the natural logarithm of the size of the population (and not by one, like the Gini coefficient). The Theil index is the weighted sum of the natural logarithm of the ratio of the income shares to the population shares, where the weights are income shares. When applied to measure inequality across individuals, the “population share” is merely 1/N, so the Theil index is the weighted sum of the natural logarithm of the ratio of each individual’s income, y_j/Y, to 1/N, where the weights are y_j/Y. When individuals are grouped, the between-group component of the Theil index is the weighted sum of the natural logarithm of the ratio of each group’s income, Y_j/Y, to that group’s population share, n_j/N, where the weights are Y_j/Y (n_j and Y_j are, respectively, the group’s j population and total income). The within-group component of total inequality is given by the weighted sum of the within-group Theil index for each group, where the weights are the income shares, Y_j/Y. Total inequality measured using the Theil index is the sum of the between-group and within-group components.

SEE ALSO Economic Growth; Gini Coefficient; Inequality, Income; Inequality, Political; Justice, Distributive; Kuznets Hypothesis; Theil Index; Wealth