In other words, it is the total of employees' income divided by the national income. 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. 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. In real societies people can be distinguished by their different resources, with the resources being incomes. 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. 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. While different theories may try to explain how income inequality comes about, income inequality metrics simply A third "symmetrized" Theil index (Theil-S) is the arithmetic average of the two previous indices. The table above shows a computation of inequality indices for four groups. Cumulative distribution function. 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. The subsections below remark on properties of these distributions. What effect will inflation have on absolute measures? This information reduction of course is the goal of computing inequality measures, as it reduces complexity. 5 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. 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). 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. Normative approach has been discussed by Charles Blackorby, Walter Bossert and David Donaldson in the work "Income Inequality Measurement: The Normative Approach". Of the above indexes, only the Theil index has this property. 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. The Gini coefficient, the Hoover index and the Theil index as well as the related welfare functions[27] can be computed together in a spreadsheet. Do government programs support some income groups more than others? The mode is the point of global maximum of the probability density function. Considering the inequalities in economically well developed countries, public policy should target an 'efficient inequality range'. A random variable is said to be stable if its distribution is stable. The expected value of a random variable with a finite In such societies, which are separated into two so called "a-fractiles", the Hoover index and the Gini coefficient always are similar. "[18] It is important to understand that an increasing Theil index does not indicate an increasing entropy, instead it indicates an increasing redundancy (decreasing entropy). In probability theory and statistics, the geometric distribution is either one of two discrete probability distributions: . wealth or income) x={x1,x2,,xn} with xi being the economic value associated with "economic agent" i. In their study for the World Institute for Development Economics Research, Giovanni Andrea Cornia and Julius Court (2001) reach slightly different conclusions. For the Theil index also the term "Theil entropy" had been used. In a perfectly equal world, no resources would need to be redistributed to achieve equal distribution: a Hoover index of 0. Despite its name, the first explicit analysis of the properties of the Cauchy distribution was published by the French has If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be closed under convolution.. In particular, by solving the equation () =, we get that: [] =. 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. 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. 4.97 There are two equivalent parameterizations in common use: With a shape parameter k and a scale parameter . Travis Hale, University of Texas Inequality Project: This page was last edited on 23 October 2022, at 02:30. The ratio is calculated both within and between countries using an elastic relationship between individual emissions and income. 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 {,,, }. 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.. [31][32] The authors therefore recommend to pursue moderation also as to the distribution of wealth and particularly to avoid the extremes. (For the Theil index and the Hoover index no sorting is required.) The mode is the point of global maximum of the probability density function. 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), 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.[19]. The parameter pairs A and E need to be sorted for the computation of the Gini coefficient. 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. When drawing conclusion from inequality measurements, consider how we should allocate the benefits of government spending? For example, the attached graph shows that in the period 19672003, 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. 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. 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. 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. Applied to the analysis of income inequality within countries, "income" often stands for the taxed income per individual or per household. In the table, fields with a yellow background are used for data input. 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. has If it has a distribution from the same family of distributions as the original variables, that family of distributions is said to be closed under convolution.. As mentioned above, taking the ratio of income share to subpopulation size corresponds to a ratio of mean subpopulation income relative to mean income. 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. The Atkinson parameter is often called the "inequality aversion parameter", since it regulates the sensitivity of the implied social welfare losses from inequality to income inequality as measured by some corresponding generalised entropy index. The Gini value for households is always lower than for individuals because of income pooling and intra-family transfers. [Note 5] The welfare functions serve as alternatives to the median income. 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 statistics, the generalized Pareto distribution (GPD) is a family of continuous probability distributions.It is often used to model the tails of another distribution. report several Lorenz-curve data-points (cumulative national income-shares up to several percentiles). 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. Another common class of metrics is to take the ratio of the income of two different groups, generally "higher over lower". As a result, two very different distributions of income can have the same Gini index. For 73:27 the Theil index and the Hoover index are identical: Both are 0.46. 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 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 In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. (See Generalized entropy index for the transformation.). 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), 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 Bayesian interpretation of probability can be seen as an extension of propositional logic that [Note 2] 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. 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. The following subsections cover them only briefly. 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 For both indices, such an equalization only serves as a reference, not as a goal. 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. Additionally the Gini does not capture where in the distribution the inequality occurs. Examples of such univariate distributions are: normal distributions, Poisson distributions, binomial distributions (with common success probability), negative binomial distributions (with common success probability), The modified half-normal distribution (MHN) is a three-parameter family of continuous probability distributions supported on the positive part of the real line. However, they do not render inequality coefficients invalid. In probability theory, the expected value (also called expectation, expectancy, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average.Informally, the expected value is the arithmetic mean of a large number of independently selected outcomes of a random variable.. Some references give the shape parameter as =. Keeping these points in mind helps to understand the problems caused by the improper use of inequality measures. 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