Income inequality metrics


Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of income and economic inequality among the participants in a particular economy, such as that of a specific country or of the world in general. While different theories may try to explain how income inequality comes about, income inequality metrics simply provide a system of measurement used to determine the dispersion of incomes. The concept of inequality is distinct from poverty and fairness.
Income distribution has always been a central concern of economic theory and economic policy. Classical economists such as Adam Smith, Thomas Malthus and David Ricardo were mainly concerned with factor income distribution, that is, the distribution of income between the main factors of production, land, labour and capital. It is often related to wealth distribution, although separate factors influence wealth inequality.
Modern economists have also addressed this issue, but have been more concerned with the distribution of income across individuals and households. Important theoretical and policy concerns include the relationship between income inequality and economic growth. The article economic inequality discusses the social and policy aspects of income distribution questions.

Defining income

All of the metrics described below are applicable to evaluating the distributional inequality of various kinds of resources. Here the focus is on income as a resource. As there are various forms of "income", the investigated kind of income has to be clearly described.
One form of income is the total amount of goods and services that a person receives, and thus there is not necessarily money or cash involved. If a subsistence farmer in Uganda grows his own grain, it will count as income. Services like public health and education are also counted in. Often expenditure or consumption is used to measure income. The World Bank uses the so-called "living standard measurement surveys" to measure income. These consist of questionnaires with more than 200 questions. Surveys have been completed in most developing countries.
Applied to the analysis of income inequality within countries, "income" often stands for the taxed income per individual or per household. Here, income inequality measures also can be used to compare the income distributions before and after taxation in order to measure the effects of progressive tax rates.

Properties of inequality metrics

In the discrete case, an economic inequality index may be represented by a function I, where x is a set of n economic values x= with xi being the economic value associated with "economic agent" i.
In the economic literature on inequality four properties are generally postulated that any measure of inequality should satisfy:
  • Anonymity or symmetry
  • :This assumption states that an inequality metric does not depend on the "labeling" of individuals in an economy and all that matters is the distribution of income. For example, in an economy composed of two people, Mr. Smith and Mrs. Jones, where one of them has 60% of the income and the other 40%, the inequality metric should be the same whether it is Mr. Smith or Mrs. Jones who has the 40% share. This property distinguishes the concept of inequality from that of fairness where who owns a particular level of income and how it has been acquired is of central importance. An inequality metric is a statement simply about how income is distributed, not about who the particular people in the economy are or what kind of income they "deserve".
  • :This is generally expressed mathematically as:
  • :
  • :where P is any permutation of x;
  • Scale independence or homogeneity
  • :This property says that richer economies should not be automatically considered more unequal by construction. In other words, if every person's income in an economy is doubled then the overall metric of inequality should not change. Of course the same thing applies to poorer economies. The inequality income metric should be independent of the aggregate level of income. This may be stated as:
  • :
  • :where α is a positive real number.
  • Population independence
  • :Similarly, the income inequality metric should not depend on whether an economy has a large or small population. An economy with only a few people should not be automatically judged by the metric as being more equal than a large economy with many people. This means that the metric should be independent of the level of population. This is generally written:
  • :
  • :where is the union of x with a copy of itself.
  • Transfer principle
  • :The Pigou–Dalton, or transfer principle, is the assumption that makes an inequality metric actually a measure of inequality. In its weak form it says that if some income is transferred from a rich person to a poor person, while still preserving the order of income ranks, then the measured inequality should not increase. In its strong form, the measured level of inequality should decrease.
Other useful but not mandatory properties include:
  • Non-negativity
  • :The index I is greater than or equal to zero.
  • Egalitarian zero
  • :The index I is zero in the egalitarian case, when all values xi are equal.
  • Bounded above by maximum inequality
  • :The index I attains its maximum value for maximum inequality. This value is usually unity as the number of agents n approaches infinity.
  • Subgroup decomposability
  • :This property states that if a set of agents x is divided into two disjoint subsets then the I is expressible as:
  • :
  • :where μ and μ are the mean incomes of x and y.
  • :
  • :and the w functions are scalar weighting function of the sets y and z. In a stronger statement, wy = μy / μx and wz = μz / μx.

    Common income inequality metrics

Among the most common metrics used to measure inequality are the Gini index, the Theil index, and the Hoover index. They have all four properties described above.
An additional property of an inequality metric that may be desirable from an empirical point of view is that of 'decomposability'. This means that if a particular economy is broken down into sub-regions, and an inequality metric is computed for each sub region separately, then the measure of inequality for the economy as a whole should be a weighted average of the regional inequalities plus a term proportional to the inequality in the averages of the regions.. Of the above indexes, only the Theil index has this property.
Because these income inequality metrics are summary statistics that seek to aggregate an entire distribution of incomes into a single index, the information on the measured inequality is reduced. This information reduction of course is the goal of computing inequality measures, as it reduces complexity.
A weaker reduction of complexity is achieved if income distributions are described by shares of total income. Rather than to indicate a single measure, the society under investigation is split into segments, such as into quintiles. Usually each segment contains the same share of income earners. In case of an unequal income distribution, the shares of income available in each segment are different.
In many cases the inequality indices mentioned above are computed from such segment data without evaluating the inequalities within the segments. The higher the number of segments, the closer the measured inequality of distribution gets to the real inequality.
Quintile measures of inequality satisfy the transfer principle only in its weak form because any changes in income distribution outside the relevant quintiles are not picked up by this measures; only the distribution of income between the very rich and the very poor matters while inequality in the middle plays no role.
Details of the three inequality measures are described in the respective Wikipedia articles. The following subsections cover them only briefly.

Gini index

The Gini index is a summary statistic that measures how equitably a resource is distributed in a population; income is a primary example. In addition to a self-contained presentation of the Gini index, we give two equivalent ways to interpret this summary statistic: first in terms of the percentile level of the person who earns the average dollar, and second in terms of how the lower of two randomly chosen incomes compare, on average, to mean income.
The Gini is the sum, over all income-ordered population percentiles, of the shortfall, from equal share, of the cumulative income up to each population percentile, with that summed shortfall divided by the greatest value that it could have, with complete inequality.
The range of the Gini index is between 0 and 1, where 0 indicates perfect equality and 1 indicates maximum inequality.
The Gini index is the most frequently used inequality index. The reason for its popularity is that it is easy to understand how to compute the Gini index as a ratio of two areas in Lorenz curve diagrams. This measure tries to capture the overall dispersion of income; however, it tends to place different levels of importance on the bottom, middle and top end of the distribution. As a disadvantage, the Gini index only maps a number to the properties of a diagram, but the diagram itself is not based on any model of a distribution process. The "meaning" of the Gini index only can be understood empirically. Additionally, the Gini does not capture where in the distribution the inequality occurs. As a result, two very different distributions of income can have the same Gini index.