HOME

TheInfoList



OR:

Income inequality metrics or income distribution metrics are used by social scientists to measure the distribution of
income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...
and
economic 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 w ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathema ...
simply provide a
system of measurement A system of measurement is a collection of units of measurement and rules relating them to each other. Systems of measurement have historically been important, regulated and defined for the purposes of science and commerce. Systems of measuremen ...
used to determine the dispersion of incomes. The concept of inequality is distinct from poverty and
fairness Fairness or being fair can refer to: * Justice * The character in the award-nominated musical comedy '' A Theory of Justice: The Musical.'' * Equity (law), a legal principle allowing for the use of discretion and fairness when applying justice ...
. Income distribution has always been a central concern of economic theory and
economic policy The economy of governments covers the systems for setting levels of taxation, government budgets, the money supply and interest rates as well as the labour market, national ownership, and many other areas of government interventions into the ...
. Classical economists such as
Adam Smith Adam Smith (baptized 1723 – 17 July 1790) was a Scottish economist and philosopher who was a pioneer in the thinking of political economy and key figure during the Scottish Enlightenment. Seen by some as "The Father of Economics"——� ...
,
Thomas Malthus Thomas Robert Malthus (; 13/14 February 1766 – 29 December 1834) was an English cleric, scholar and influential economist in the fields of political economy and demography. In his 1798 book ''An Essay on the Principle of Population'', Malt ...
and
David Ricardo David Ricardo (18 April 1772 – 11 September 1823) was a British political economist. He was one of the most influential of the classical economists along with Thomas Malthus, Adam Smith and James Mill. Ricardo was also a politician, and a m ...
were mainly concerned with factor income distribution, that is, the
distribution Distribution may refer to: Mathematics *Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a varia ...
of income between the main
factors of production In economics, factors of production, resources, or inputs are what is used in the production process to produce output—that is, goods and services. The utilized amounts of the various inputs determine the quantity of output according to the relat ...
, land, labour and capital. It is often related to
wealth distribution The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or economic heterogeneity. The distribution of wealth differs from the income distribution in that ...
, although separate factors influence
wealth inequality The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or heterogeneity in economics, economic heterogeneity. The distribution of wealth differs from the i ...
. 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 Economic growth can be defined as the increase or improvement in the inflation-adjusted market value of the goods and services produced by an economy in a financial year. Statisticians conventionally measure such growth as the percent rate of ...
. The article
economic 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 w ...
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 Subsistence agriculture occurs when farmers grow food crops to meet the needs of themselves and their families on smallholdings. Subsistence agriculturalists target farm output for survival and for mostly local requirements, with little or no su ...
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 The World Bank is an international financial institution that provides loans and grants to the governments of low- and middle-income countries for the purpose of pursuing capital projects. The World Bank is the collective name for the Inter ...
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 A developing country is a sovereign state with a lesser developed industrial base and a lower Human Development Index (HDI) relative to other countries. However, this definition is not universally agreed upon. There is also no clear agree ...
. 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='' 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 Fairness or being fair can refer to: * Justice * The character in the award-nominated musical comedy '' A Theory of Justice: The Musical.'' * Equity (law), a legal principle allowing for the use of discretion and fairness when applying justice ...
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) *:where ''P(x)'' is any
permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or pr ...
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(\alpha x)=I(x) *: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: *: I(x \cup x)=I(x) *:where x \cup x is the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''U ...
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: *: I(y \cup z)=w_y(y,z) I(y)+w_z(y,z)I(z)+w_\mu(y,z) I() *:where ''μ(x)'' and ''μ(y)'' are the mean incomes of ''x'' and ''y''. *: \mu_x=\frac\sum_1^n x_i *: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 (also known as Gini coefficient), the
Theil index The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. The Theil index ''T''T is the same as redundancy in information theory which i ...
, and the
Hoover index The Hoover index, also known as the Robin Hood index or the Schutz index, is a measure of income inequality. It is equal to the percentage of the total population's income that would have to be redistributed to make all the incomes equal. i.e. Th ...
. 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. (in a weaker form, it means that it should be an explicit function of sub-regional inequalities, though not necessarily linear). Of the above indexes, only the
Theil index The Theil index is a statistic primarily used to measure economic inequality and other economic phenomena, though it has also been used to measure racial segregation. The Theil index ''T''T is the same as redundancy in information theory which i ...
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 (or any other percentage of population). 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 (such as deciles instead of quintiles), the closer the measured inequality of distribution gets to the real inequality. (If the inequality within the segments is known, the total inequality can be determined by those inequality metrics which have the property of being "decomposable".) 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 Population typically refers to the number of people in a single area, whether it be a city or town, region, country, continent, or the world. Governments typically quantify the size of the resident population within their jurisdiction using a ...
;
income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...
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 compares, 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 (0% and 100%), where 0 indicates perfect equality and 1 (100%) 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 In economics, the Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution. The curve is a graph showing the proporti ...
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.


20:20 ratio

The 20:20 or 20/20 ratio compares how much richer the top 20% of populations are to the bottom 20% of a given population. This can be more revealing of the actual impact of inequality in a population, as it reduces the effect on the statistics of outliers at the top and bottom and prevents the middle 60% from statistically obscuring inequality that is otherwise obvious in the field. The measure is used for the United Nations Development Programme Human Development Indicators. The 20:20 ratio for example shows that Japan and Sweden have a low equality gap, where the richest 20% only earn 4 times the poorest 20%, whereas in the UK the ratio is 7 times and in the US 8 times. Some believe the 20:20 ratio is a more useful measure as it correlates well with measures of human development and social stability including the index of child well-being, index of health and social problems, population in prison, physical health, mental health and many others.


Palma ratio

The Palma ratio is defined as the ratio of the richest 10% of the population's share of gross national income divided by the poorest 40%'s share. It is based on the work of Chilean economist Gabriel Palma who found that middle class incomes almost always represent about half of gross national income while the other half is split between the richest 10% and poorest 40%, but the share of those two groups varies considerably across countries. The Palma ratio addresses the Gini index's over-sensitivity to changes in the middle of the distribution and insensitivity to changes at the top and bottom, and therefore more accurately reflects income inequality's economic impacts on society as a whole. Palma has suggested that distributional politics pertains mainly to the struggle between the rich and poor, and who the middle classes side with. The carbon Palma ratio, which is derived from the income Palma ratio and described as the ratio of the total emissions of the top 10% of emitters to those of the bottom 40%, is proposed as a new indicator to inform the international community and the general public about the distribution inequality of carbon emissions among individuals. The ratio is calculated both within and between countries using an elastic relationship between individual emissions and income. The results show that the carbon Palma ratios in most developing countries are generally high, implying that they should concentrate more on coordinating regional and income inequalities and primarily encouraging high emitters to reduce in order to enhance emissions and income equity at the same time. The carbon Palma ratios in developed countries are comparatively lower; however, their greater historical obligations to warming indicate that they significantly reduce emissions of all people, in order to increase national mitigation contributions systematically. On a global scale, the current carbon Palma ratio is noticeably higher than within any country, indicating an exceedingly severe inequality when individual emissions are considered outside territorial boundaries.


Hoover index

The Hoover index is the simplest of all inequality measures to calculate: It is the proportion of all income which would have to be redistributed to achieve a state of perfect equality. In a perfectly equal world, no resources would need to be redistributed to achieve equal distribution: a Hoover index of 0. In a world in which all income was received by just one family, almost 100% of that income would need to be redistributed (i.e., taken and given to other families) in order to achieve equality. The Hoover index then ranges between 0 and 1 (0% and 100%), where 0 indicates perfect equality and 1 (100%) indicates maximum inequality.


Galt score

The Galt score is a simple ratio of a company's
CEO A chief executive officer (CEO), also known as a central executive officer (CEO), chief administrator officer (CAO) or just chief executive (CE), is one of a number of corporate executives charged with the management of an organization especiall ...
pay to the pay of that company's
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
worker. A company which pays its CEO many times more than its median employee will have a high Galt score. It is named for the fictional character
John Galt John Galt () is a character in Ayn Rand's novel ''Atlas Shrugged'' (1957). Although he is not identified by name until the last third of the novel, he is the object of its often-repeated question "Who is John Galt?" and of the quest to discover ...
in
Ayn Rand Alice O'Connor (born Alisa Zinovyevna Rosenbaum;, . Most sources transliterate her given name as either ''Alisa'' or ''Alissa''. , 1905 – March 6, 1982), better known by her pen name Ayn Rand (), was a Russian-born American writer and p ...
's novel ''
Atlas Shrugged ''Atlas Shrugged'' is a 1957 novel by Ayn Rand. It was her longest novel, the fourth and final one published during her lifetime, and the one she considered her '' magnum opus'' in the realm of fiction writing. ''Atlas Shrugged'' includes elem ...
'' (1957). The score is calculated using the total compensation of the CEO, including salary, bonuses, the value of stock awards and
employee stock options Employee stock options (ESO) is a label that refers to compensation contracts between an employer and an employee that carries some characteristics of financial options. Employee stock options are commonly viewed as an internal agreement prov ...
, as well as non-equity incentive plan compensation, and
nonqualified deferred compensation In the United States, the question whether any compensation plan is qualified or non-qualified is primarily a question of taxation under the Internal Revenue Code (IRC). Any business prefers to deduct its expenses from its income, which will reduce ...
.


Coefficient of variation

The coefficient of variation is the square root of the variance of the incomes divided by the mean income. It has the advantages of being mathematically tractable and its square is subgroup decomposable, but it is not bounded from above.


Wage share

Wage share In economics, the wage share or labor share is the part of national income, or the income of a particular economic sector, allocated to wages ( labor). It is related to the capital or profit share, the part of income going to capital, which is als ...
is the ratio between
Compensation of employees {{no footnotes, date=April 2010 Compensation of employees (CE) is a statistical term used in national accounts, balance of payments statistics and sometimes in corporate accounts as well. It refers basically to the total gross (pre-tax) wages paid ...
and GDP. In other words, it is the total of employees' income divided by the national income.


Theil index

As described in a section below, Theil-L is an income-distribution's dis-entropy per person, measured with respect to maximum entropy (which is achieved with complete equality). (In an alternative interpretation of it, Theil-L is the natural-logarithm of the geometric-mean of the ratio: (mean income)/(income i), over all the incomes. The related Atkinson(1) is just 1 minus the geometric-mean of (income i)/(mean income),over the income distribution.) Because a transfer between a larger income & a smaller one will change the smaller income's ratio more than it changes the larger income's ratio, the transfer-principle is satisfied by this index. A Theil index of 0 indicates perfect equality. A Theil index of 1 indicates that the distributional entropy of the system under investigation is almost similar to a system with an 82:18 distribution. This is slightly more unequal than the inequality in a system to which the "80:20
Pareto principle The Pareto principle states that for many outcomes, roughly 80% of consequences come from 20% of causes (the "vital few"). Other names for this principle are the 80/20 rule, the law of the vital few, or the principle of factor sparsity. Manage ...
" applies. The Theil index can be transformed into an
Atkinson index The Atkinson index (also known as the Atkinson measure or Atkinson inequality measure) is a measure of income inequality developed by British economist Anthony Barnes Atkinson. The measure is useful in determining which end of the distribution cont ...
, which has a range between 0 and 1 (0% and 100%), where 0 indicates perfect equality and 1 (100%) indicates maximum inequality. (See
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 ...
for the transformation.) The Theil index is an entropy measure. As for any resource distribution and with reference to information theory, "maximum entropy" occurs once income earners cannot be distinguished by their resources, i.e. when there is perfect equality. In real societies people can be distinguished by their different resources, with the resources being incomes. The more "distinguishable" they are, the lower is the "actual entropy" of a system consisting of income and income earners. Also based on information theory, the gap between these two entropies can be called " redundancy". It behaves like a
negative entropy In information theory and statistics, negentropy is used as a measure of distance to normality. The concept and phrase "negative entropy" was introduced by Erwin Schrödinger in his 1944 popular-science book '' What is Life?'' Later, Léon Bril ...
. For the Theil index also the term "Theil entropy" had been used. This caused confusion. As an example,
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom and the United States. Sen has made contributions to welfare economics, social choice theory, economi ...
commented on the Theil index, "given the association of doom with entropy in the context of thermodynamics, it may take a little time to get used to entropy as a good thing." It is important to understand that an increasing Theil index ''does not'' indicate an increasing entropy, instead it indicates an increasing redundancy (decreasing entropy). High inequality yields high Theil redundancies. High redundancy means low entropy. But this does not necessarily imply that a very high inequality is "good", because very low entropies also can lead to explosive compensation processes. Neither does using the Theil index necessarily imply that a very low inequality (low redundancy, high entropy) is "good", because high entropy is associated with slow, weak and inefficient resource allocation processes. There are three variants of the Theil index. When applied to income distributions, the first Theil index (Theil-L) relates to how income is distributed to income earners, whereas the second Theil index (Theil-T) relates to how income-earners are distributed to incomes. A third "symmetrized" Theil index (Theil-S) is the arithmetic average of the two previous indices. The formula of the third Theil index has some similarity with the Hoover index (as explained in the related articles). As in case of the Hoover index, the symmetrized Theil index does not change when swapping the incomes with the income earners. How to generate that third Theil index by means of a spreadsheet computation directly from distribution data is shown below. An important property of the Theil index which makes its application popular is its decomposability into the between-group and within-group component. For example, the Theil index of overall income inequality can be decomposed in the between-region and within region components of inequality, while the relative share attributable to the between-region component suggests the relative importance of spatial dimension of income inequality.


Comparison of the Theil index and the Hoover index

The Theil index indicates the distributional redundancy of a system, within which incomes are assigned to income earners in a stochastic process. In comparison, the Hoover index indicates the minimum size of the income share of a society, which would have to be redistributed in order to reach maximum entropy. Not to exceed that minimum size would require a perfectly planned redistribution. Therefore, the Hoover index is the "non-stochastic" counterpart to the "stochastic" Theil index. Applying the Theil index to allocation processes in the real world does ''not'' imply that these processes are stochastic: the Theil yields the ''distance'' between an ordered resource distribution in an observed system to the final stage of stochastic resource distribution in a closed system. Similarly, applying the Hoover index does ''not'' imply that allocation processes occur in a perfectly planned economy: the Hoover index yields the ''distance'' between the resource distribution in an observed system to the final stage of a planned "equalization" of resource distribution. For both indices, such an equalization only serves as a reference, not as a goal. For a given distribution the Theil index can be larger than the Hoover index or smaller than the Hoover index: *For ''high'' inequalities the Theil index is larger than the Hoover index. This means for achieving equilibrium (maximum entropy) in a closed system, more resources would have to be reallocated than in case of a planned and optimized reallocation process, where only the necessary minimum share of resources would have to be reallocated. For an open system the export of entropy (import of redundancy) would allow to maintain the distribution dynamics driven by high inequality. *For ''low'' inequalities the Theil index is smaller than the Hoover index. Here, on the path to reaching equilibrium, a planned and optimized reallocation of resources would contribute more to the dynamics of redistribution than stochastic redistribution. This also is intuitively understandable, as low inequalities also weaken the urge to redistribute resources. People in such a system may tolerate or even foster an increase the inequality. As this is would be an increase of redundancy (an decrease of entropy), redundancy would have to be imported into (entropy would have to be exported from) the society. In that case the society needs to be an open system. In order to increase the redundancy in the distribution category of a society as a closed system, entropy needs to be exported from the subsystem operating in that economic category to other subsystems with other entropy categories in the society. For example, social entropy may increase. However, in the real world, societies are open systems, but the openness is restricted by the entropy exchange capabilities of the interfaces between the society and the environment of that society. For societies with a resource distribution which entropywise is similar to the resource distribution of a reference society with a 73:27 split (73% of the resources belong to 27% of the population and vice versa), the point where the Hoover index and the Theil index are equal, is at a value of around 46% (0.46) for the Hoover index and the Theil index.


Atkinson index

Atkinson index (also known as the Atkinson measure or Atkinson inequality measure) is a measure useful in determining which end of the distribution contributed most to the observed inequality. The Atkinson ε
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
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. 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 va ...
ε to weight incomes. Greater weight can be placed on changes in a given portion of the income distribution by choosing ε, the level of "inequality aversion", appropriately. The Atkinson index becomes more sensitive to changes at the lower end of the income distribution as ε increases. Conversely, as the level of inequality aversion falls (that is, as ε approaches 0) the Atkinson becomes less sensitive to changes in the lower end of the distribution. The Atkinson index is for no value of ε highly sensitive to top incomes because of the common restriction that ε is nonnegative.


Ratios

Another common class of metrics is to take the ratio of the income of two different groups, generally "higher over lower". This compares two ''parts'' of the income distribution, rather than the distribution as a whole; equality between these parts corresponds to 1:1, while the more unequal the parts, the greater the ratio. These statistics are easy to interpret and communicate, because they are relative (this population earns twice as much as this population), but, since they do not fall on an absolute scale, do not provide an absolute measure of inequality.


Ratio of percentiles

Particularly common to compare a given percentile to the median, as in the first chart here; compare
seven-number summary In descriptive statistics, the seven-number summary is a collection of seven summary statistics, and is an extension of the five-number summary. There are three similar, common forms. As with the five-number summary, it can be represented by a modi ...
, which summarizes a distribution by certain percentiles. While such ratios do not represent the ''overall level'' of inequality in the population as a whole, they provide measures of the ''shape'' of income distribution. For example, the attached graph shows that in the period 1967–2003, US income ratio between median and 10th and 20th percentile did not change significantly, while the ratio between the median and 80th, 90th, and 95th percentile increased. This reflects that the increase in the Gini coefficient of the US in this time period is due to gains by upper income earners (relative to the median), rather than by losses by lower income earners (relative to the median).


Share of income

A related class of ratios is "income share", the percentage of the national income received by some specified richest or poorest percentage-segment of the population. It's often reported for the highest-income 1%, etc., of the population, but the Lorenz Curve (discussed in the Gini section) graphs the cumulative income of the ''bottom''-income X% of the population, for all X from 0 to 100. An important and obvious property of the Lorenz curve is that, if country A has a higher cumulative national-income share for (say) the poorest 1%, in comparison to country B, then: # If the two countries' Lorenz curves don't cross, then country A's Lorenz curve is everywhere higher than that of country B, meaning that, for any X from 0 to 100, the poorest X% of the population have a greater national-income-share in country A than in country B. # If the two countries' Lorenz curves cross, then it's a near-certainty that, on the poor side of the crossing-point, country A has a higher Lorenz curve, meaning that, for any poorest X% of the population up to the crossing-point, country A's population has a greater national-income-share. When inequality does harm, it does most of it at the poorer end of the population. At a any given income-ordered percentile, a greater national-income-share means a greater received percentage of the equal-share (the national mean income). So the country with a greater cumulative national-income-share for the poorest 1% will have a greater cumulative received percentage of equal-share for the poorest X%, either for every percentile X, or (if the Lorenz curves cross), for the percentiles on the poorer side of the crossing-point. Some sources report several Lorenz-curve data-points (cumulative national income-shares up to several percentiles). As mentioned above, taking the ratio of income share to subpopulation size corresponds to a ratio of ''mean'' subpopulation income relative to ''mean'' income. Because income distribution is generally positively skewed, mean is higher than median, so ratios to mean are lower than ratios to median. This is particularly used to measure that fraction of income accruing to top earners – top 10%, 1%, 0.1%, 0.01%, and also "top 100" earners or the like; in the US top 400 earners is 0.0002% of earners (2 in 1,000,000) – to study concentration of income – wealth condensation, or rather income condensation. For example, in the chart at right, US income share of top earners was approximately constant from the mid-1950s to the mid-1980s, then increased from the mid-1980s through 2000s; this increased inequality was reflected in the Gini coefficient. For example, in 2007 the top decile (10%) of US earners accounted for 49.7% of total wages (4.97 \approx 5 times fraction under equality), and the top 0.01% of US earners accounted for 6% of total wages (600 times fraction under equality).


Approaches


Normative approach

Normative interpretation of inequality through inequality indexes means that there is a relationship between an inequality index and a social-evaluation ordering defined on the incomes — incomes (nominal or real) of the members of society. Incomes are typically assigned to individuals rather than households by using an adult equivalence scale. Normative approach has been discussed by Charles Blackorby, Walter Bossert and David Donaldson in the work "Income Inequality Measurement: The Normative Approach".


Statistical approach

Statistical interpretation of inequality is based on fiscal data and the
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
α from the model of Pareto (1895) as an
income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...
inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...
measure. This approach is further discussed in work "Income Inequality Measurement: The Statistical Approach" by Giovanni M. Giorgi.


Spreadsheet computations

The Gini coefficient, the Hoover index and the Theil index as well as the related welfare functions can be computed together in a spreadsheet. The welfare functions serve as alternatives to the
median In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic f ...
income. In the table, fields with a yellow background are used for data input. From these data inequality measures as well as the related welfare functions are computed and displayed in fields with green background. In the example given here, "Theil index" stands for the arithmetic mean of a Theil index computed for the distribution of income within a society to the individuals (or households) in that society and a Theil index computed for the distribution of the individuals (or households) in the society to the income of that society. The difference between the Theil index and the Hoover index is the weighting of the relative deviation D. For the Hoover index the relative deviation D per group is weighted with its own sign. For the Theil index the relative deviation D per group is weighted with the information size provided by the income per individual in that group. For the computation the society usually is divided into income groups. Often there are four or five groups consisting of a similar number of individuals in each group. In other cases the groups are created based on income ranges which leads to having different numbers of individuals in the different groups. The table above shows a computation of inequality indices for four groups. For each group the number of individuals (or households) per group A and the total income in that group E is specified. The parameter pairs A and E need to be sorted for the computation of the Gini coefficient. (For the Theil index and the Hoover index no sorting is required.) A and E have to be sorted so that the values in the column "Income per individual" are lined up in ascending order.


Proper use

# When using income metrics, it has to be made clear how income should be defined. Should it include
capital gains Capital gain is an economic concept defined as the profit earned on the sale of an asset which has increased in value over the holding period. An asset may include tangible property, a car, a business, or intangible property such as shares. ...
, imputed house rents from home ownership, and gifts? If these income sources or alleged income sources (in the case of "
imputed rent Imputed rent is the rental price an individual would pay for an asset they own. The concept applies to any capital good, but it is most commonly used in housing markets to measure the rent homeowners would pay for a housing unit equivalent to the ...
") are ignored (as they often are), how might this bias the analysis? How should non-paid work (such as parental childcare or doing one's own cooking instead of hiring a chef for every meal) be handled? Wealth or consumption may be more appropriate measures in some situations. Broader
quality of life Quality of life (QOL) is defined by the World Health Organization as "an individual's perception of their position in life in the context of the culture and value systems in which they live and in relation to their goals, expectations, standards ...
metrics might be useful. # The comparison of inequality measures requires that the segmentation of compared groups (societies etc.) into quintiles should be similar. # Distinguish properly, whether the basic unit of measurement is households or individuals. The Gini value for households is always lower than for individuals because of income pooling and intra-family transfers. And households have a varying number of members. The metrics will be influenced either upward or downward depending on which unit of measurement is used. #Consider life cycle effects. In most Western societies, an individual tends to start life with little or no income, gradually increase income till about age 50, after which incomes will decline, eventually becoming negative. This affects the conclusions which can be drawn from a measured inequality. It has been estimated (by A.S. Blinder in ''The Decomposition of Inequality'', MIT press) that 30% of measured income inequality is due to the inequality an individual experiences as they go through the various stages of life. #Clarify whether real or nominal income distributions should be used. What effect will inflation have on absolute measures? Do some groups (e.g., pensioners) feel the effect of inflation more than others? #When drawing conclusion from inequality measurements, consider how we should allocate the benefits of government spending? How does the existence of a social security safety net influence the definition of absolute measures of poverty? Do government programs support some income groups more than others? #Inequality metrics measure inequality. They do not measure possible causes of income inequality. Some alleged causes include: life cycle effects (age), inherited characteristics (IQ, talent), willingness to take chances (risk aversion), the leisure/industriousness choice, inherited wealth, economic circumstances, education and training, discrimination, and market imperfections. #Inequality metrics are anonymous. They ignore certain effects of income mobility, in which the identity of "who is rich" and "who is poor" is considered. For example, at a particular time, Alice may have $10 and Bob may have $2. At some time later, Bob may have $10 and Alice may have $2. The inequality index will be the same in both cases and rather high. However, the inequality of the average will be zero, since Alice's and Bob's average holdings are equal ($6). The $8 which has changed hands is a measure of wealth mobility and the average inequality is generally higher than the inequality of the average. Keeping these points in mind helps to understand the problems caused by the improper use of inequality measures. However, they do not render inequality coefficients invalid. If inequality measures are computed in a well explained and ''consistent'' way, they can provide a good tool for quantitative comparisons of inequalities.


Inequality, growth, and progress

Evidence from a broad panel of recent academic studies shows that there is a nonlinear relation between income inequality and the rate of growth and investment. Very high inequality slows growth; moderate inequality encourages growth. Studies differ on the effect of very low inequality. Robert J. Barro,
Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of highe ...
found in his study "Inequality and Growth in a Panel of Countries" that higher inequality tends to retard growth in poor countries and encourage growth in well-developed regions. Underscoring the need for initiatives like the United Nations
Sustainable Development Goal 10 Sustainable Development Goal 10 (Goal 10 or SDG 10) is about reduced inequality and is one of the 17 Sustainable Development Goals established by the United Nations in 2015. The full title is: "Reduce inequality within and among countries".Unite ...
, aim at reducing inequality. According to Pak Hung Mo,
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 ...
has significant negative effect on the rate of
GDP Gross domestic product (GDP) is a monetary measure of the market value of all the final goods and services produced and sold (not resold) in a specific time period by countries. Due to its complex and subjective nature this measure is ofte ...
growth. In their work "Income Inequality and Economic Growth", they found out that the most important is the transfer channel while the least important is the human capital channel. However, the direct impact of income inequality on the rate of productivity growth accounts for more than 55 percent of its overall total effect. This indicates that the effects of income inequality on economic growth are much more complicated than what we have perceived or modeled. In their study for the World Institute for Development Economics Research, Giovanni Andrea Cornia and Julius Court (2001) reach slightly different conclusions. The authors therefore recommend to pursue moderation also as to the distribution of wealth and particularly to avoid the extremes. Both very high egalitarianism and very high inequality cause slow growth. Considering the inequalities in economically well developed countries, public policy should target an 'efficient inequality range'. The authors claim that such efficiency range roughly lies between the values of the Gini coefficients of 0.25 (inequality close to the least unequal European counties) and 0.40 (close to the level of inequality of US. According to the work "Inequality of Opportunity, Income Inequality, and Economic Growth," the relationship between income inequality and economic growth is mediated by the level of equality of opportunity, as identified by intergenerational mobility. After a number of recently developed internationally comparable measures of intergenerational mobility they confirmed that the negative impact of income inequality on growth increases as intergenerational mobility decreases. The findings imply that excluding intergenerational mobility leads to misspecification, which explains why the empirical literature on income inequality and growth has been so inconclusive. Another researcher has shown that in
perfect market In economics, specifically general equilibrium theory, a perfect market, also known as an atomistic market, is defined by several idealizing conditions, collectively called perfect competition, or atomistic competition. In theoretical models wh ...
s inequality does not influence growth.Wolfgang Kitterer:
Mehr Wachstum durch Umverteilung?
(More Growth through Redistribution?)'', 2006
The precise shape of the inequality-growth curve obviously varies across countries depending upon their resource endowment, history, remaining levels of absolute poverty and available stock of social programs, as well as on the distribution of physical and human capital.


See also

*
List of countries by income equality A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby unio ...
* List of countries by wealth equality *
Income inequality in the United States Income inequality in the United States is the extent to which income is distributed in differing amounts among the American population. It has fluctuated considerably since measurements began around 1915, moving in an arc between peaks in t ...
*
Income deficit Income deficit is the difference between a single person or family's income and its poverty threshold or poverty line, when the former is exceeded by the latter. Data on the income deficits of various members of a population allow for the construct ...
*
International inequality International inequality refers to inequality between countries, as compared to global inequality, which is inequality between people across countries. International inequality research has primarily been concentrated on the rise of internati ...
*
Kuznets curve The Kuznets curve () expresses a hypothesis advanced by economist Simon Kuznets in the 1950s and 1960s. According to this hypothesis, as an economy develops, market forces first increase and then decrease economic inequality. The Kuznets curve ...
*
Poverty threshold The poverty threshold, poverty limit, poverty line or breadline is the minimum level of income deemed adequate in a particular country. The poverty line is usually calculated by estimating the total cost of one year's worth of necessities for t ...
*
Socioeconomics Socioeconomics (also known as social economics) is the social science that studies how economic activity affects and is shaped by social processes. In general it analyzes how modern societies progress, stagnate, or regress because of their loca ...
*
Sustainable Development Goals The Sustainable Development Goals (SDGs) or Global Goals are a collection of 17 interlinked objectives designed to serve as a "shared blueprint for peace and prosperity for people and the planet, now and into the future".United Nations (2017) R ...
* "
The rich get richer and the poor get poorer "The rich get richer and the poor get poorer" is an aphorism due to Percy Bysshe Shelley. In ''A Defence of Poetry'' (1821, not published until 1840) Shelley remarked that the promoters of utility had exemplified the saying, "To him that hath, ...
" *
Distributive justice Distributive justice concerns the socially just allocation of resources. Often contrasted with just process, which is concerned with the administration of law, distributive justice concentrates on outcomes. This subject has been given considera ...
– political ideal dedicated to greater income equality *
Diversity index A diversity index is a quantitative measure that reflects how many different types (such as species) there are in a dataset (a community), and that can simultaneously take into account the phylogenetic relations among the individuals distributed a ...
* Great Gatsby curve *
Human Development Index The Human Development Index (HDI) is a statistic composite index of life expectancy, education (mean years of schooling completed and expected years of schooling upon entering the education system), and per capita income indicators, w ...
*
Suits index The Suits index of a public policy is a measure of tax progressiveness, named for economist Daniel B. Suits. Similar to the Gini coefficient, the Suits index is calculated by comparing the area under the Lorenz curve to the area under a proportio ...
* Gini index *
Lorenz curve In economics, the Lorenz curve is a graphical representation of the distribution of income or of wealth. It was developed by Max O. Lorenz in 1905 for representing inequality of the wealth distribution. The curve is a graph showing the proporti ...
* The Elephant Curve


Notes


References


Literature

* A.B. Atkinson and F. Bourguignon, ed. (2000). ''Handbook of Income Distribution'', v. 1. Elsevie
table of contents
* _____,"
International Encyclopedia of the Social & Behavioral Sciences The ''International Encyclopedia of the Social & Behavioral Sciences'', originally edited by Neil J. Smelser and Paul B. Baltes, is a 26-volume work published by Elsevier. It has some 4,000 signed articles (commissioned by around 50 subject editor ...
(2001), pp. 7265–7271
Abstract.
* Yoram Amiel (Author), Frank A. Cowell: ''Thinking about Inequality: Personal Judgment and Income Distributions'', 2000 * Philip B. Coulter: ''Measuring Inequality'', 1989


External links

* Travis Hale, University of Texas Inequality Project
''The Theoretical Basics of Popular Inequality Measures''
online computation of examples
1A1B
* Samuel Murray Matheson
''Distributive Fairness Measures for Sustainable Project Selection''
1997
Luxembourg Income Study
conducts comparative income inequality research
Two Americas: One Rich, One Poor? Understanding Income Inequality in the United States

Has US Income Inequality ''Really'' Increased?

A website dedicated to the topic of income inequality

WIID – World Income Inequality Database
United Nations University
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 ...
* Software: *
Online calculator
computes the Gini Coefficient, plots the Lorenz curve, and computes many other measures of concentration for any dataset ** Online calculator
Online
(example for processing data fro

U.S. Census Bureau, 2007: Income Distribution to $250,000 or More for Households) 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 (pr ...
and Lua) for Atkinson, Gini, and Hoover inequalities *
Python script
using formulas in Amartya Sen's ''On Economic Inequality'' ** Users of th
R
data analysis software can install th

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. {{DEFAULTSORT:Income inequality metrics Welfare economics