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 (which is the same in an economic sense) 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(x), where x is a set of n economic values (e.g., wealth or income) x={x1,x2,...,xn} 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:
I ( P ( x ) ) = I ( x ) {\displaystyle I(P(x))=I(x)}
where P(x) 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 (or multiplied by any positive constant) 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:
I ( α x ) = I ( x ) {\displaystyle I(\alpha x)=I(x)}
where α is any 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:
I ( x ⊔ x ) = I ( x ) {\displaystyle I(x\sqcup x)=I(x)}
where x ⊔ x {\displaystyle x\sqcup x} 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(x) is greater than or equal to zero. Egalitarian zero The index I(x) is zero in the egalitarian case, when all values xi are equal. Bounded above by maximum inequality The index I(x) attains its maximum value for maximum inequality. (all xi are zero except one) 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 (y and z) then the I(x) is expressible as:
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