Definitions. Modification for minima rather than maxima The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima. 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. In a perfectly equal world, no resources would need to be redistributed to achieve equal distribution: a Hoover index of 0. 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). The Atkinson index is for no value of highly sensitive to top incomes because of the common restriction that is nonnegative. Statistical interpretation of inequality is based on fiscal data and the parameter from the model of Pareto (1895) as an income inequality measure. Clarify whether real or nominal income distributions should be used. Some references give the shape parameter as =. In probability theory, the inverse Gaussian distribution (also known as the Wald distribution) is a two-parameter family of continuous probability distributions with support on (0,).. Its probability density function is given by (;,) = (())for x > 0, where > is the mean and > is the shape parameter.. In the Keynesian tradition, Richard Goodwin accounts for cycles in output by the distribution of income between business profits and workers' wages. 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 Atkinson index becomes more sensitive to changes at the lower end of the income distribution as increases. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. Some sources[which?] The more "distinguishable" they are, the lower is the "actual entropy" of a system consisting of income and income earners. Since the log-transformed variable = has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are = + = (),where () is the quantile of the standard normal distribution. The Gini index is the most frequently used inequality index. In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate.It is a particular case of the gamma distribution.It is the continuous analogue of the geometric distribution, and it has the key The Gini index is a summary statistic that measures how equitably a resource is distributed in a population; income is a primary example. Services like public health and education are also counted in. It is named for the fictional character John Galt in Ayn Rand's novel Atlas Shrugged (1957). The Theil index indicates the distributional redundancy of a system, within which incomes are assigned to income earners in a stochastic process. The ratio is calculated both within and between countries using an elastic relationship between individual emissions and income. In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: . The Bayesian interpretation of probability can be seen as an extension of propositional logic that The Galt score is a simple ratio of a company's CEO pay to the pay of that company's median worker. Since the log-transformed variable = has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are = + = (),where () is the quantile of the standard normal distribution. For the Theil index also the term "Theil entropy" had been used. The mode is the point of global maximum of the probability density function. The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the EulerMascheroni constant), and the standard deviation is / In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.. There are three variants of the Theil index. 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.[15]. 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). Cumulative distribution function. 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. The coefficient of variation is the square root of the variance of the incomes divided by the mean income. This information reduction of course is the goal of computing inequality measures, as it reduces complexity. In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.. 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. If a subsistence farmer in Uganda grows his own grain, it will count as income. The expected value of a random variable with a finite A Theil index of 0.5 characterizes systems which are close to a 74:26 distribution. A and E have to be sorted so that the values in the column "Income per individual" are lined up in ascending order. 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. Understanding Income Inequality in the United States, A website dedicated to the topic of income inequality, WIID World Income Inequality Database, World Institute for Development Economics Research, https://en.wikipedia.org/w/index.php?title=Income_inequality_metrics&oldid=1117691450, Articles with specifically marked weasel-worded phrases from October 2021, All articles with specifically marked weasel-worded phrases, Creative Commons Attribution-ShareAlike License 3.0. It is often related to wealth distribution, although separate factors influence wealth inequality. pp. The higher the number of segments (such as deciles instead of quintiles), the closer the measured inequality of distribution gets to the real inequality. While different theories may try to explain how income inequality comes about, income inequality metrics simply While different theories may try to explain how income inequality comes about, income inequality metrics simply In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions.It is often used to model the tails of another distribution. When drawing conclusion from inequality measurements, consider how we should allocate the benefits of government spending? Income distribution has always been a central concern of economic theory and economic policy. In probability and statistics, the Tweedie distributions are a family of probability distributions which include the purely continuous normal, gamma and inverse Gaussian distributions, the purely discrete scaled Poisson distribution, and the class of compound Poissongamma distributions which have positive mass at zero, but are otherwise continuous. 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. A binomial distribution with parameters n = 1 and p is a Bernoulli distribution with parameter p.; A negative binomial distribution with parameters n = 1 and p is a geometric distribution with parameter p.; A gamma distribution with shape parameter = 1 and rate parameter is an exponential distribution with rate parameter . into quintiles should be similar. Examples of such univariate distributions are: normal distributions, Poisson distributions, binomial distributions (with common success probability), negative binomial distributions (with common success probability), Applied to the analysis of income inequality within countries, "income" often stands for the taxed income per individual or per household. The probability density function (PDF) of the beta distribution, for 0 x 1, and shape parameters , > 0, is a power function of the variable x and of its reflection (1 x) as follows: (;,) = = () = (+) () = (,) ()where (z) is the gamma function.The beta function, , is a normalization constant to ensure that the total probability is 1. (See Generalized entropy index for the transformation.). 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. 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. 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. The subsections below remark on properties of these distributions. As there are various forms of "income", the investigated kind of income has to be clearly described. The parameter pairs A and E need to be sorted for the computation of the Gini coefficient. 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. 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. A company which pays its CEO many times more than its median employee will have a high Galt score. Inequality metrics measure inequality. 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. The distribution is called "folded" because probability mass to the left of x = 0 is folded over by taking the absolute value. Since the log-transformed variable = has a normal distribution, and quantiles are preserved under monotonic transformations, the quantiles of are = + = (),where () is the quantile of the standard normal distribution. [13], 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. 35, chapter 2.11 in Amartya Sen and James E. Foster: The rich get richer and the poor get poorer, "Basics and Completely Decomposable Groups", "Figure 5.8. In probability theory and statistics, the logistic distribution is a continuous probability distribution.Its cumulative distribution function is the logistic function, which appears in logistic regression and feedforward neural networks.It resembles the normal distribution in shape but has heavier tails (higher kurtosis).The logistic distribution is a special case of the Tukey lambda It is specified by three parameters: location , scale , and shape . 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. As in case of the Hoover index, the symmetrized Theil index does not change when swapping the incomes with the income earners. 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. In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: . Here the focus is on income as a resource. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives The truncated normal distribution, half-normal distribution, and square-root of the Gamma distribution are special cases of the MHN distribution. The "meaning" of the Gini index only can be understood empirically. High inequality yields high Theil redundancies. The formula of the third Theil index has some similarity with the Hoover index (as explained in the related articles). The Theil index is an entropy measure. In such societies, which are separated into two so called "a-fractiles", the Hoover index and the Gini coefficient always are similar. In probability theory, a distribution is said to be stable if a linear combination of two independent random variables with this distribution has the same distribution, up to location and scale parameters. Therefore, the Hoover index is the "non-stochastic" counterpart to the "stochastic" Theil index. The Bayesian interpretation of probability can be seen as an extension of propositional logic that wealth or income) x={x1,x2,,xn} with xi being the economic value associated with "economic agent" i. A function with the form of the density function of the Cauchy distribution was studied geometrically by Fermat in 1659, and later was known as the witch of Agnesi, after Agnesi included it as an example in her 1748 calculus textbook. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The normal distribution is the only distribution whose cumulants beyond the first two (i.e., other than the mean and variance) are zero.It is also the continuous distribution with the maximum entropy for a specified mean and variance. 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. In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.. Gini coefficients of different measures of inter provincial inequality", "Notes on Statistical Sources and Methods The Equality Trust", "Economic Inequality is Bad for Your Health", "External Research Page 2 The Equality Trust", "Putting the Gini Back in the Bottle? The truncated normal distribution, half-normal distribution, and square-root of the Gamma distribution are special cases of the MHN distribution. Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief.. A scale-free network is a network whose degree distribution follows a power law, at least asymptotically.That is, the fraction P(k) of nodes in the network having k connections to other nodes goes for large values of k as where is a parameter whose value is typically in the range < < (wherein the second moment (scale parameter) of is infinite but the first moment is finite), The measure is used for the United Nations Development Programme Human Development Indicators. The characteristic function for the (centered) Voigt profile will then be the product of the two: What effect will inflation have on absolute measures? How does the existence of a social security safety net influence the definition of absolute measures of poverty? In particular, by solving the equation () =, we get that: [] =. A third "symmetrized" Theil index (Theil-S) is the arithmetic average of the two previous indices. In case of an unequal income distribution, the shares of income available in each segment are different. The related Atkinson(1) is just 1 minus the geometric-mean of (income i)/(mean income),over the income distribution.). Additionally the Gini does not capture where in the distribution the inequality occurs. ", "Table F-1. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. 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. This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum A Theil index of 0 indicates perfect equality. [33], Another researcher has shown that in perfect markets inequality does not influence growth.[34]. Travis Hale, University of Texas Inequality Project: This page was last edited on 23 October 2022, at 02:30. In real societies people can be distinguished by their different resources, with the resources being incomes. The article economic inequality discusses the social and policy aspects of income distribution questions. The mode is the point of global maximum of the probability density function. The characteristic function for the (centered) Voigt profile will then be the product of the two: When using income metrics, it has to be made clear how income should be defined. Modification for minima rather than maxima The theory here relates to data maxima and the distribution being discussed is an extreme value distribution for maxima. Income Limits for Each Fifth and Top 5 Percent of Families (All Races): 1947 to 2010", "Income Inequality in the United States, 19132002", "Income inequality in the United States: 19131998", "Saez, E. (October, 2007). Both very high egalitarianism and very high inequality cause slow growth. In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: . 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. [Note 5] The welfare functions serve as alternatives to the median income. The probability density function (PDF) of the beta distribution, for 0 x 1, and shape parameters , > 0, is a power function of the variable x and of its reflection (1 x) as follows: (;,) = = () = (+) () = (,) ()where (z) is the gamma function.The beta function, , is a normalization constant to ensure that the total probability is 1. In the physics of heat conduction , the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. 'The Palma' as a Policy-Relevant Measure of Inequality", "Homogeneous middles vs. heterogeneous tails, and the end of the 'Inverted-U': the share of the rich is what it's all about", "Carbon Palma Ratio: A new indicator for measuring the distribution inequality of carbon emissions among individuals", "On the measurement of regional inequality: Does spatial dimension of income inequality matter? Studies differ on the effect of very low inequality. The following subsections cover them only briefly. [24], 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. 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. 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. Inequality metrics are anonymous. In probability theory and statistics, the chi distribution is a continuous probability distribution.It is the distribution of the positive square root of the sum of squares of a set of independent random variables each following a standard normal distribution, or equivalently, the distribution of the Euclidean distance of the random variables from the origin. Should it include. These consist of questionnaires with more than 200 questions. In statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions.It is often used to model the tails of another distribution. As mentioned above, taking the ratio of income share to subpopulation size corresponds to a ratio of mean subpopulation income relative to mean income. Surveys have been completed in most developing countries. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French In probability theory, the multinomial distribution is a generalization of the binomial distribution.For example, it models the probability of counts for each side of a k-sided die rolled n times. The mode is the point of global maximum of the probability density function. 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. The stable distribution family is also sometimes referred to as the Lvy alpha-stable distribution, after The score is calculated using the total compensation of the CEO, including salary, bonuses, the value of stock awards and employee stock options, as well as non-equity incentive plan compensation, and nonqualified deferred compensation. It's often reported[by whom?] Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. A random variable is said to be stable if its distribution is stable. 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. There are two equivalent parameterizations in common use: With a shape parameter k and a scale parameter . In the physics of heat conduction , the folded normal distribution is a fundamental solution of the heat equation on the half space; it corresponds to having a perfect insulator on a hyperplane through the origin. High redundancy means low entropy. Sometimes it is specified by only scale and shape and sometimes only by its shape parameter. 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. Very high inequality slows growth; moderate inequality encourages growth. In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one.. For the computation the society usually is divided into income groups. 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 table above shows a computation of inequality indices for four groups. 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 probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set {,,, };; The probability distribution of the number Y = X 1 of failures before the first success, supported on the set {,,, }. The comparison of inequality measures requires that the segmentation of compared groups (societies etc.) In probability theory and statistics, the gamma distribution is a two-parameter family of continuous probability distributions.The exponential distribution, Erlang distribution, and chi-square distribution are special cases of the gamma distribution. However, the direct impact of income inequality on the rate of productivity growth accounts for more than 55 percent of its overall total effect. 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. For both indices, such an equalization only serves as a reference, not as a goal. Often expenditure or consumption (which is the same in an economic sense) is used to measure income. As an alternative to spreadsheet computations also a, Panel Data Econometrics: Theoretical Contributions And Empirical Applications edited by Badi Hani Baltag. {\displaystyle 4.97\approx 5} R is a shift parameter, [,], called the skewness parameter, is a measure of asymmetry.Notice that in this context the usual skewness is not well defined, as for < the distribution does not admit 2nd or higher moments, and the usual skewness definition is the 3rd central moment.. [28] Underscoring the need for initiatives like the United Nations Sustainable Development Goal 10, aim at reducing inequality.