The Atkinson index (also known as the Atkinson measure or Atkinson inequality measure) is a measure of
income inequality
There are wide varieties of economic inequality, most notably income inequality measured using the distribution of income (the amount of money people are paid) and wealth inequality measured using the distribution of wealth (the amount of we ...
developed by British economist
Anthony Barnes Atkinson
Sir Anthony Barnes Atkinson (4 September 1944 – 1 January 2017) was a British economist, Centennial Professor at the London School of Economics, and senior research fellow of Nuffield College, Oxford.
A student of James Meade, Atkinson vi ...
. The measure is useful in determining which end of the distribution contributed most to the observed inequality.
Definition
The index can be turned into a
normative
Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good, desirable, or permissible, and others as bad, undesirable, or impermissible. A norm in ...
measure by imposing a
coefficient
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing
, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as
increases. Conversely, as the level of inequality aversion falls (that is, as
approaches 0) the Atkinson becomes less sensitive to changes in the lower end of the distribution. The Atkinson index is for no value of
highly sensitive to top incomes because of the common restriction that
is nonnegative.
The Atkinson
parameter is often called the "inequality aversion parameter", since it regulates the sensitivity of the implied social welfare losses from inequality to income inequality as measured by some corresponding generalised entropy index. The Atkinson index is defined in reference to a corresponding social welfare function, where mean income multiplied by one minus the Atkinson index gives the ''welfare equivalent equally distributed income''. Thus the Atkinson index gives the share of current income which could be sacrificed, without reducing social welfare, if perfect inequality were instated. For
, (no aversion to inequality), the marginal social welfare from income is invariant to income, i.e. marginal increases in income produce as much social welfare whether they go to a poor or rich individual. In this case, the welfare equivalent equally distributed income is equal to mean income, and the Atkinson index is zero.
For
(infinite aversion to inequality) the marginal social welfare of income of the poorest individual is infinitely larger than any even slightly richer individual, and the Atkinson social welfare function is equal to the smallest income in the sample. In this case, the Atkinson index is equal to mean income minus the smallest income, divided by mean income. As in large typical income distributions incomes of zero or near zero are common, the Atkinson index will tend to be one or very close to one for very large
.
The Atkinson index then varies between 0 and 1 and is a measure of the amount of social utility to be gained by complete redistribution of a given income distribution, for a given
parameter. Under the utilitarian ethical standard and some restrictive assumptions (a homogeneous population and
constant elasticity of substitution
Constant elasticity of substitution (CES), in economics, is a property of some production functions and utility functions. Several economists have featured in the topic and have contributed in the final finding of the constant. They include Tom McK ...
utility),
is equal to the income elasticity of marginal utility of income.
The Atkinson index is defined as:
:
where
is individual income (''i'' = 1, 2, ..., ''N'') and
is the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the ''arithme ...
income.
In other words, the Atkinson index is the complement to 1 of the ratio of the
Hölder generalized mean of exponent 1−ε to the arithmetic mean of the incomes (where as usual the generalized mean of exponent 0 is interpreted as the
geometric mean
In mathematics, the geometric mean is a mean or average which indicates a central tendency of a set of numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometric mean is defined as the ...
).
Atkinson index relies on the following axioms:
# The index is symmetric in its arguments:
for any permutation
.
# The index is non-negative, and is equal to zero only if all incomes are the same:
iff
for all
.
# The index satisfies the
principle of transfers: if a transfer
is made from an individual with income
to another one with income
such that
, then the inequality index cannot increase.
# The index satisfies population replication axiom: if a new population is formed by replicating the existing population an arbitrary number of times, the inequality remains the same:
# The index satisfies mean independence, or income homogeneity, axiom: if all incomes are multiplied by a positive constant, the inequality remains the same:
for any
.
# The index is subgroup decomposable.
[Shorrocks, AF (1980). The class of additively decomposable inequality indices. ''Econometrica'', 48 (3), 613–625, ] This means that overall inequality in the population can be computed as the sum of the corresponding Atkinson indices within each group, and the Atkinson index of the group mean incomes:
:::
::where
indexes groups,
, individuals within groups,
is the mean income in group
, and the weights
depend on
and
. The class of the subgroup-decomposable inequality indices is very restrictive. Many popular indices, including
Gini index, do not satisfy this property.
See also
*
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 ...
*
Generalized entropy index
The generalized entropy index has been proposed as a measure of income inequality in a population. It is derived from information theory as a measure of redundancy in data. In information theory a measure of redundancy can be interpreted as no ...
*
Gini index
Footnotes
References
* Atkinson, AB (1970) On the measurement of inequality. ''Journal of Economic Theory'', 2 (3), pp. 244–263, . The original paper proposing this inequality index.
* Allison PD (1978) Measures of Inequality, ''American Sociological Review'', 43, pp. 865–880. Presents a technical discussion of the Atkinson measure's properties. There is an error in the formula for the Atkinson index, which is corrected in Allison (1979).
* Allison, PD (1979) Reply to Jasso. ''American Sociological Review'' 44(5):870–72.
* Biewen M, Jenkins SP (2003). Estimation of Generalized Entropy and Atkinson Inequality Indices from Complex Survey Data
IZA Discussion Paper #763 Provides statistical inference for Atkinson indices.
* Lambert, P. (2002). ''Distribution and redistribution of income''. 3rd edition, Manchester Univ Press, .
* Sen A, Foster JE (1997) ''On Economic Inequality'', Oxford University Press, .
Python scriptfor a selection of formulas in the book)
World Income Inequality Database, from
World Institute for Development Economics Research
The United Nations University World Institute for Development Economics Research (UNU-WIDER) is part of the United Nations University (UNU). UNU-WIDER, the first research and training centre to be established by the UNU, is an international academ ...
Income Inequality, 1947–1998 from
United States Census Bureau
The United States Census Bureau (USCB), officially the Bureau of the Census, is a principal agency of the U.S. Federal Statistical System, responsible for producing data about the American people and economy. The Census Bureau is part of the ...
.
External links
Software:
Free Online Calculatorcomputes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset
* Free Calculator
an
downloadable scripts(
Python
Python may refer to:
Snakes
* Pythonidae, a family of nonvenomous snakes found in Africa, Asia, and Australia
** ''Python'' (genus), a genus of Pythonidae found in Africa and Asia
* Python (mythology), a mythical serpent
Computing
* Python (pro ...
and
Lua
Lua or LUA may refer to:
Science and technology
* Lua (programming language)
* Latvia University of Agriculture
* Last universal ancestor, in evolution
Ethnicity and language
* Lua people, of Laos
* Lawa people, of Thailand sometimes referred t ...
) for Atkinson, Gini, and Hoover inequalities
* Users of th
Rdata analysis software can install the "ineq" package which allows for computation of a variety of inequality indices including Gini, Atkinson, Theil.
*
MATLAB Inequality Package, including code for computing Gini, Atkinson, Theil indexes and for plotting the Lorenz Curve. Many examples are available.
*
Stata
Stata (, , alternatively , occasionally stylized as STATA) is a general-purpose statistical software package developed by StataCorp for data manipulation, visualization, statistics, and automated reporting. It is used by researchers in many fie ...
inequality packages
ineqdecoto decompose inequality by groups
to compute design-consistent variances for the generalized entropy and Atkinson indices
to obtain generalized Lorenz curve. You can type
ssc install ineqdeco etc. in Stata prompt to install these packages.
{{DEFAULTSORT:Atkinson Index
Income inequality metrics