File Name: the estimation of the lorenz curve and gini index .zip
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The data from the World Bank database and the Organization for Economic Co-operation and Development Income Distribution Database between and are used to demonstrate how the inequality index works. Furthermore, the inequality index could be applied to other scientific disciplines as a measure of statistical heterogeneity and for size distributions of any non-negative quantities. The Gini index was devised by an Italian statistician named Corrado Gini in By far, it has arguably been the most popular measure of socioeconomic inequality, especially in income and wealth distribution, given that there are well over 50 inequality indices as reported in Coulter see Eliazar, ; McGregor et al.
The use of the Gini index is not limited to the field of socioeconomics, however. According to Eliazar and Sokolov , the application of the Gini index has grown beyond socioeconomics and reached various disciplines of science.
Examples include astrophysics—the analysis of galaxy morphology Abraham et al. In effect, the Gini index is applicable to any size distributions in the context of general data sets with non-negative quantities such as count, length, area, volume, mass, energy, and duration Eliazar, However, in order to demonstrate our method for measuring inequality, we focus our analysis on the subject of income.
The Gini index can be derived from the Lorenz curve framework Lorenz, , which plots the Cartesian coordinates where the abscissa is the cumulative normalized rank of income from the lowest to the highest x and the ordinate is the cumulative normalized income from the lowest to the highest y as illustrated in Fig.
The Gini index takes values in the unit interval. The closer the index is to zero where the area A is small , the more equal the distribution of income. The closer the index is to one where the area A is large , the more unequal the distribution of income.
The advantage of the Gini index is that inequality of the entire income distribution can be summarized by using a single statistic that is relatively easy to interpret since it takes values between 0 and 1. This allows for comparison among countries with different population sizes. In addition, the data on the Gini index is easy to access, regularly updated and reported by countries and international organizations.
Despite its advantages as a statistical measure of income inequality, Atkinson and Bourguignon note that a country with lower Gini index does not always imply that income distribution in that country is more equal than that of a country with higher Gini index. This is because the Lorenz curves of the two countries may intersect, reflecting different income distributions. To obtain a complete ranking of and to quantify the difference in income inequality among countries, Atkinson devises a social welfare-based inequality index as follows:.
This index takes values between 0 and 1. The cornerstone of the Atkinson index is the concept of equally distributed equivalent level of income y EDE , which is defined as the percentage of total income that a given society would have to forego in order to have more equal shares of income among individuals in that society.
According to Atkinson , the Gini index tends to give the rankings that are similar to those reached with a relatively low degree of inequality aversion. Although the advantages of the Atkinson index are that it provides a complete ranking of income distributions and makes explicit the social welfare function underlying the income inequality measure, which could be useful for policy decisions, Cowell and McGregor et al.
To avoid the social welfare judgment, a class of generalized entropy GE indices can be used as an alternative measure for ranking income inequality when the Lorenz curves of the two countries intersect. The GE index is defined as follows:. GE 0 is referred to as the mean logarithmic deviation, which is defined as follows:. GE 1 is known as the Theil inequality index, named after the author who devised it in The Theil index is defined as follows:.
Another limitation of using the Gini index is whenever two or more countries share the same value of the Gini index but income inequality among them could be very different if taking into consideration the information on the income share held by the richest and that held by the poorest. For example, based on the data from the World Bank, in , Greece and Thailand have the same Gini index 0.
That countries share the same Gini index but differ in the income gap between the richest and the poorest indicates that the Gini index alone cannot tell the difference in income inequality among countries. Furthermore, Atkinson notes that the Gini index is more sensitive to changes in the middle of income distribution and less sensitive to changes at the top and the bottom of income distribution.
The data used to calculate these inter-decile ratios or the ratios themselves including the Palma index are regularly updated and reported along with the Gini index by international organizations, such as the World Bank, the OECD, and the Human Development Report Office as the measures of income inequality.
From a mathematical and practical point of view, these values are more difficult to interpret and compare among countries since they have no upper bound relative to other inequality indices whose values are bounded. As noted in Eliazar , indices whose values are bounded are much more tangible to human perception than those whose values are unbounded. In addition, by construction, these inter-decile ratios capture income inequality between the top and the bottom of distribution and ignore income of those in the middle of distribution.
To overcome the limitations of the Gini index and the inter-decile ratios as discussed above, we devise an alternative method for measuring inequality. Our method is quite simple.
These three indicators comprising the inequality index are selected based on availability, accessibility, and continuity of the data without the need to collect the data on income distribution at the micro level. Our inequality index I takes values in the unit interval where the closer the index is to zero, the more equal the distribution of income and the closer the index is to one, the more unequal the distribution of income.
To demonstrate our method, we use the annual data on the Gini index and the income shares in from the World Bank a , b , c containing 75 countries and from the OECD IDD a , b comprising 35 countries. The reason to use the data in is that it has more countries than those in , , and For these reasons, we would like to define our inequality index I i as follows:. Table 1 presents the results of our ranking of income inequality based on the Gini index using the World Bank database in The results indicate that the inequality index I can differentiate income inequality in case two or more countries share the same Gini index but differ in the income gap between the top and the bottom.
Thus, using our inequality index I , we can say that Greece has a higher level of income inequality than Thailand. Our results in Table 1 also show that when comparing the rankings of income inequality among countries using our inequality index I with those using the Gini index, there are 62 countries that their rankings have been changed while there are 13 countries whose rankings remain the same.
According to the World Bank database, during and , the Gini index of Mexico is around 0. In addition, our results from Table 2 show that when comparing the rankings of income inequality among countries using our inequality index I with those using the Gini index, there are 21 countries that their rankings have been changed while there are 14 countries whose rankings remain the same.
This suggests that the income inequality in Ireland is slightly higher than that in Switzerland, which could be distinguished by our inequality index I. However, our inequality index I could capture the dynamics of income inequality in Italy since the inequality index I shows a rising trend from 0. That two or more countries have the same value of the Gini index does not necessarily imply that these countries share the same level of income inequality.
Likewise, two or more countries having the same ratio of the income share held by the richest to the income share held by the poorest does not always imply that income inequality among these countries is the same either. The Gini index is known to be less sensitive to inequality at the tails of income distribution, whereas the ratios of income share of the richest to income share of the poorest do not account for inequality in the middle of income distribution.
To overcome the limitations of the Gini index and the inter-decile ratios as measures of income inequality, this study introduces a composite index for measuring inequality that does not require the micro-data of the distribution.
Our inequality index is very simple to calculate. The data on these three indicators are also available, easy to access, and regularly updated by countries and international organizations. This implies that there are other aspects of differences in income inequality among countries that our inequality index would not be able to capture.
In this way, the whole range of the Lorenz curve would be covered. This is one way to take into account the difference in income inequality in case two or more countries share the same inequality index but have dissimilar Lorenz curves.
There might be other alternative ways to account for such a difference, which await future research. Last but not least, we hope that our simple method for measuring inequality could be applied not only to socioeconomics, but also to broad scientific disciplines as a measure of statistical heterogeneity and for size distributions of any non-negative quantities.
The alternative index for any given country i can be defined as follows:. This alternative index takes the values between 0. When everyone has the same share of income, it is equal to 0. Analysis of the Sloan Digital Sky Survey early data release. Astrophys J 1 — Atkinson AB On the measurement of inequality. J Econ Theory 2 3 — Handbook of income distribution , North-Holland, Oxford, pp.
Food and Agriculture Organization of the United Nations. Bertoli-Barsotti L, Lando T How mean rank and mean size may determine the generalised Lorenz curve: with application to citation analysis. J Informetr — Cambridge University Press, New York. Google Scholar. Coulter PB Measuring inequality: a methodological handbook. Westview Press, Boulder. Cowell FA Measuring inequality, 3rd edn. Oxford University Press, New York. Cromley GA Measuring differential access to facilities between population groups using spatial Lorenz curves and related indices.
T Gis 23 6 — Das AK Quantifying photovoltaic power variability using Lorenz curve. J Renew Sustain Energy Evidence from Italy and top industrial countries. J Epidemiol Community Health — J Transp Geogr — Eliazar II A tour of inequality. Ann Phys — Phys A — Tipografia di Paolo Cuppini, Bologna. Gini C On the measurement of concentration and variability of characters.
De Santis F trans Metron — Graczyk PP Gini coefficient: a new way to express selectivity of kinase inhibitors against a family of kinases. J Med Chem 50 23 —
It was developed by the Italian statistician and sociologist Corrado Gini. The Gini coefficient measures the inequality among values of a frequency distribution for example, levels of income. A Gini coefficient of zero expresses perfect equality, where all values are the same for example, where everyone has the same income. For larger groups, values close to one are unlikely. Given the normalization of both the cumulative population and the cumulative share of income used to calculate the Gini coefficient, the measure is not overly sensitive to the specifics of the income distribution, but rather only on how incomes vary relative to the other members of a population.
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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 for representing inequality of the wealth distribution.
This paper introduces a new coordinate system for the Lorenz curve. Particular attention is paid to a special case of wide empirical validity. Four alternative methods have been used to estimate the proposed Lorenz curve from the grouped data. The well known inequality measures are obtained as the function of the estimated parameters of the Lorenz curve. In addition the frequency distribution is derived from the equation of the Lorenz curve.
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