text: Poverty, Indices of
  Poverty, Indices of
  A poverty index measures the level of poverty in a society. In measuring the level of poverty, a poverty line or poverty threshold, usually stated in terms of income, is defined to divide the society into two separate groups. An individual is poor if that individual lives below the poverty line.
  The traditional poverty index is the headcount ratio, which is the proportion of people in a society who are living in poverty. If n is the total number of people in the society and m is the number of the poor, the headcount ratio is
  Another poverty index that sometimes is used is the income-gap ratio, which is the percentage of the average income shortfall of the poor to the poverty line. If μp is the average income of the poor and z is the poverty line, the income-gap ratio is
  Both poverty indices, however, have serious problems in measuring the poverty level of a society adequately. For example, the headcount ratio does not consider how far a poor individual is below the poverty line; an individual just below the line and an individual far below it are treated the same in the calculation. Although the income-gap ratio does not have this problem, it does not reflect how incomes are distributed among the poor: Whether they are equally poor or some of them are desperately poor, the same level of poverty is determined. Additionally, the income-gap ratio does not deal with the number of the poor; it considers only the average income of the poor. Intuitively, for the same average income of the poor, the society should have more poverty if more people fall into poverty.
  Since the publication of Amartya Sen’s 1976 work on poverty measurement, the construction and evaluation of poverty indices have followed an axiomatic approach. In this approach, ideal properties for poverty measurement are formulated as axioms and a poverty index is generated to satisfy those axioms. The following are the key axioms for poverty measurement: The focus axiom states that the poverty index is not affected by a change in a nonpoor individual’s income as long as that individual remains nonpoor. The restricted continuity axiom states that the poverty index is continuous in poor incomes in that a small change in a poor individual’s income should not lead to a large change in the poverty level. The monotonicity axiom states that all else the same, a decrease in a poor individual’s income should increase the overall level of poverty in the society. The weak transfer axiom states that all else the same, a transfer of income from a poor individual to a poorer individual should decrease the overall level of poverty. The subgroup consistency axiom states that if the society is divided into different subsocieties (such as the different states in the United States), all else the same, an increase in the poverty level of a subsociety also should increase the overall level of poverty of the society. The increasing poverty line axiom states that the poverty index should increase as the poverty line increases. Additionally, a poverty index should satisfy the unit consistency axiom: If one society (say, the United States) has more poverty than another society (say, the United Kingdom) when all incomes and the poverty line are measured in one currency unit (e.g., U.S. dollars), the conclusion should hold when another currency (e.g., British pounds) is used.
  The headcount ratio does not satisfy the monotonicity axiom, the weak transfer axiom, or the increasing poverty line axiom. The income-gap ratio violates the weak transfer axiom and the increasing poverty line axiom.
  Sen (1976) proposed the first axiomatic-based poverty index. Suppose all individuals’ incomes are arranged in increasing order and x i is the i th individual’s income, then . The Sen poverty index is
  which also can be written as
  where H is the headcount ratio, I is the income-gap ratio, and G p is the Gini coefficient of the poor incomes. The Gini coefficient is twice the area between the Lorenz curve of the poor incomes and the 45-degree diagonal line. A slightly modified version of the Sen index is the Thon poverty index:
  Other poverty indices include the Chakravarty poverty index:
  the Foster-Greer-Thorbecke poverty index:
  the Watts poverty index:
  and the Kolm poverty index:
  Both the Sen and Thon indices satisfy all the axioms listed above except the subgroup consistency axiom; the Chakravarty index, the Foster-Greer-Thorbecke index, and the Watts index satisfy all the axioms; and the Kolm index satisfies all the axioms except the unit consistency axiom. All these new poverty indices are referred to as distribution-sensitive indices because they satisfy the weak transfer axiom. The differences among these poverty indices lie in the ways in which the shortfall of each poor income to the poverty line is characterized and the ways in which those shortfalls across all individuals are aggregated into an overall index of poverty. For example, for the headcount ratio, each individual’s shortfall is assigned a value of one if the individual is poor and zero if the individual is nonpoor, and the overall poverty level is the simple average of those shortfalls (zeros and ones). For the Sen poverty index, each individual’s shortfall is recorded by if the individual is poor and zero if the individual is nonpoor, and the overall poverty is the weighted average of those shortfalls, with the weight being which is the relative rank of that individual among the poor. For the Watts poverty index, each individual’s shortfall is recorded by ln z - In xi if the individual is poor and zero if the individual is nonpoor, and the overall poverty is the simple average of those shortfalls.
  To calculate the poverty level of a society, one needs to define the poverty line and choose a poverty index. The definition of the poverty line varies from society to society. In the United States the official poverty line initially was developed by Mollie Orshansky of the Social Security Administration in 1963–1964 and adjusted each year thereafter for inflation. The U.S. poverty line is regarded as an absolute poverty line because it was calculated as the minimum amount of resources needed for living at a point in time and is not affected by changes in the entire income distribution. Because the minimum amount of resources needed for living depends on the specific society, the calculated absolute poverty line tends to vary from society to society. For example, an individual with an income at the U.S. poverty line would be regarded as rich in some developing countries. In poverty studies, especially in international comparisons of poverty, another type of poverty line—the relative poverty line—has been used. A relative poverty line is specified as a point in the distribution of income (e.g., one-half of the mean or median income), and the line is updated automatically over time for changes in the distribution.
  Because the use of any specific poverty line, whether absolute or relative, is somewhat arbitrary, a range of poverty lines often are used to check the robustness of poverty comparisons. The resulting conditions for poverty comparisons are closely related to the conditions of stochastic dominance (Foster and Shorrocks 1988) and Lorenz dominance, which is based on comparisons of Lorenz curves. Similarly, the choice of a specific poverty index in poverty comparisons is somewhat arbitrary in light of the fact that there are multiple poverty indices that satisfy the same set of poverty axioms. If, for a specific poverty line, all poverty indices that satisfy certain axioms are used in poverty comparisons, the resulting conditions are also related to the conditions of stochastic dominance (Atkinson 1987) and Lorenz dominance.
  SEE ALSO Economics, Stratification; Gini Coefficient; Income Distribution; Inequality, Income; Poor, The; Poverty; Sen, Amartya Kumar; Social Welfare Functions; Welfare Analysis
  Atkinson, Anthony B. 1987. On the Measurement of Poverty. Econometrica 55 (4): 749–764.
  Foster, James E., and Anthony Shorrocks. 1988. Poverty Orderings. Econometrica 56 (1): 173–177.
  Sen, Amartya. 1976. Poverty: An Ordinal Approach to Measurement. Econometrica 44 (2): 219–231.
  Sen, Amartya. 1997. On Economic Inequality. Expanded ed. Oxford: Clarendon.