The concept of inequality is distinct from that of poverty^{[1]} and fairness. 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.
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.
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.
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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 poor subsistence farmer in Uganda grows her 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 socalled "living standard measurement surveys"^{[2]} 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.
In the economic literature on inequality four properties are generally postulated that any measure of inequality should satisfy:
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 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.
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 lots of people. This means that the metric should be independent of the level of population.
DaltonPigou, or the transfer principle  this 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.
Among the most common metrics used to measure inequality are the Gini index (also known as Gini coefficient), 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 subregions, 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 (in a weaker form, it means that it should be an explicit function of subregional inequalities, though not necessarily linear). 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, e.g. 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 amount of segments (e.g. deciles instead of quintiles), tho 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.
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 diagrams. 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.
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.
A Theil index of 0 indicates perfect equality. (As for presenting the Theil index, usually no percentage notation is used.) 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^{[3]}. This is slightly more inequal than the inequality in a system to which the "80:20 Pareto principle" applies.^{[4]} The Theil index can be transformed into an Atkinson index, which has a range between 0 and 1 (0% and 100%), where 0 indicates perfect equality and 1 (100%) indicates maximum inequality.
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"^{[5]}. It behaves like a negative entropy. Thus, the "meaning" of the Theil index is, that the Theil index is such a redundancy.
For the Theil index also the term "Theil entropy" had been used. This caused confusion. As an example, Amartya Sen commented on the Theil index, that it is "an interesting measure of inequality", however, "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."^{[6]} With regard to such amazement 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 relates to systems within which incomes are stochastically distributed to income earners, whereas vice versa the second Theil index relates to systems within which income earners are stochastically distributed to incomes.
A third "symmetrized" Theil index is the arithmetic average of the two previous indices. Interestingly, 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.
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 "nonstochastic" 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. In contrary, 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. In contrary, the Hoover index yields the distance between the resource distribution in an observed system to the final stage of a planned "equalizatin" 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:
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 the 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)^{[8]}, 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.
The Gini coefficient, the Hoover index and the Theil index as well as the related welfare functions^{[9]} can be computed together in a spreadsheet^{[10]}. The welfare functions serve as alternatives to the median income.
Group  Members per Group 
Income per Group 
Income per Individual 
Relative Deviation 
Accumulated Income 
Gini  Hoover  Theil 

1  A_{1}  E_{1}  Ē_{1} = E_{1}/A_{1}  D_{1} = E_{1}/ΣE  A_{1}/ΣA  K_{1} = E_{1}  G_{1} = (2 * K_{1}  E_{ 1}) * A_{1}  H_{1} = abs(D_{1})  T_{1} = ln(Ē_{1}) * D_{ 1} 
2  A_{2}  E_{2}  Ē_{2} = E_{2}/A_{2}  D_{2} = E_{2}/ΣE  A_{2}/ΣA  K_{2} = E_{2} + K_{1}  G_{2} = (2 * K_{2}  E_{ 2}) * A_{2}  H_{2} = abs(D_{2})  T_{2} = ln(Ē_{2}) * D_{ 2} 
3  A_{3}  E_{3}  Ē_{3} = E_{3}/A_{3}  D_{3} = E_{3}/ΣE  A_{3}/ΣA  K_{3} = E_{3} + K_{2}  G_{3} = (2 * K_{3}  E_{ 3}) * A_{3}  H_{3} = abs(D_{3})  T_{3} = ln(Ē_{3}) * D_{ 3} 
4  A_{4}  E_{4}  Ē_{4} = E_{4}/A_{4}  D_{4} = E_{4}/ΣE  A_{4}/ΣA  K_{4} = E_{4} + K_{3}  G_{4} = (2 * K_{4}  E_{ 4}) * A_{4}  H_{4} = abs(D_{4})  T_{4} = ln(Ē_{4}) * D_{ 4} 
Totals  ΣA  ΣE  Ē = ΣE/ΣA  ΣG  ΣH  ΣT  
Inequality Measures 
Gini = 1  ΣG/ΣA/ΣE  Hoover = ΣH / 2  Theil = ΣT / 2  
Welfare Function 
W_{G} = Ē * (1  Gini)  W_{H} = Ē * (1  Hoover)  W_{T} = Ē * (1  Theil) 
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 amount of individuals in each group. In other cases the groups are created based on income ranges which leads to having different amounts of individuals in the different groups. The table above shows a computation of inequality indices for four groups. For each group the amount 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 has the be sorted so that the values in the column „Income per individual“ are lined up in ascending order.
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 at least within a research project.
The question whether inequality or equality is beneficial for economic growth and progress has occupied the minds of the greatest scientific thinkers as well as policy makers. There is evidence from a broad panel of recent academic studies shows that there is a non linear relation between income inequality and the rate of growth and investment.
Robert J. Barro, Harvard University 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.^{[11]}. In their study for the World Institute for Development Economics Research, Giovanni Andrea Cornia and Julius Court (2001) reach analogous conclusions.^{[12]} 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.
Income inequality diminishes growth potential through the erosion of social cohesion, increasing social unrest and social conflict causing uncertainty of property rights. Extreme inequality can effectively reduce access to productivity enhancement measures, or cause such measures to be allocated inefficiently toward those who already have, or can no longer absorb such measures.
On the other hand, The World Bank World Development Report 2000/2001^{[13]} shows, that inequality and growth are not related. Inequality neither drives growth nor does it impair growth. Other research (W.Kitterer^{[14]}) also shows, that in perfect markets inequality does not influence growth. In real markets redistribution contributes to 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 25 (the inequality value of a typical Northern European country) and 40 (that of countries such as China^{[15]} and the USA^{[16]}).
The precise shape of the inequalitygrowth 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.
