Lorenz Curve

11 Key excerpts on "Lorenz Curve"

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  Societal Problems as Public Bads

  • Nan Dirk de Graaf, Dingeman Wiertz, Nan de Graaf(Authors)
  • 2019(Publication Date)
  • Routledge

    (Publisher)

  Figure 7.1 . This curve essentially plots how much of the total national income is earned by the poorest X percent of the country’s population. If income is distributed completely equally across all citizens, the curve will be a straight diagonal line: e.g., the poorest ten percent of the population earn ten percent of the national income, the poorest 50 percent earn 50 percent of the national income, the poorest 90 percent earn 90 percent of the national income. The larger the degree of curvature of the Lorenze curve, the more unequal the income distribution: e.g., the poorest ten percent of the population earn 0.0001 percent of the national income, the poorest 50 percent earn one percent of the national income, the poorest 90 percent earn three percent of the national income (leaving 97 percent of the national income for the richest ten percent). As such, the Lorenz Curve provides a convenient way of eyeballing how much inequality there is within a group, a society, or across a selection of countries.

  Figure 7.1     The Lorenz Curve and the Gini coefficient

  The Gini coefficient or Gini index , in turn, provides a quantitative measure of inequality. It ranges from zero to one, where zero corresponds to a situation of complete equality (sticking to the example of the income distribution within a country, this would mean that every citizen earns the same income) and one corresponds to a situation of complete inequality (one citizens earns all of the income and everyone else earns nothing). In between zero and one, higher values correspond to a greater degree of inequality. Therefore, the Gini coefficient can broadly be interpreted as the percentage of national income that needs to be redistributed in order to achieve a perfectly equal distribution.

  The Gini coefficient bears a direct relationship to the Lorenz Curve. That is, it can be calculated by dividing the size of area A (i.e., the area between the diagonal line of equality and the Lorenz Curve) by the sum of areas A plus B; see Figure 7.1 . In equation form: Gini = A / (A + B)

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  Principles of Microeconomics for AP® Courses

  • Steven A. Greenlaw, Timothy Taylor(Authors)
  • 2015(Publication Date)
  • Openstax

    (Publisher)

  Figure 14.8 The Lorenz Curve A Lorenz Curve graphs the cumulative shares of income received by everyone up to a certain quintile. The income distribution in 1980 was closer to the perfect equality line than the income distribution in 2011—that is, the U.S. income distribution became more unequal over time. Every Lorenz Curve diagram begins with a line sloping up at a 45-degree angle, shown as a dashed line in Figure 14.8. The points along this line show what perfect equality of the income distribution looks like. It would mean, for example, that the bottom 20% of the income distribution receives 20% of the total income, the bottom 40% gets 40% of total income, and so on. The other lines reflect actual U.S. data on inequality for 1980 and 2011. 318 Chapter 14 | Poverty and Economic Inequality This OpenStax book is available for free at http://cnx.org/content/col11858/1.4 The trick in graphing a Lorenz Curve is that you must change the shares of income for each specific quintile, which are shown in the first column of numbers in Table 14.6, into cumulative income, shown in the second column of numbers. For example, the bottom 40% of the cumulative income distribution will be the sum of the first and second quintiles; the bottom 60% of the cumulative income distribution will be the sum of the first, second, and third quintiles, and so on. The final entry in the cumulative income column needs to be 100%, because by definition, 100% of the population receives 100% of the income.

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  Principles of Economics 3e

  • Steven A. Greenlaw, David Shapiro, Daniel MacDonald(Authors)
  • 2022(Publication Date)
  • Openstax

    (Publisher)

  We show it as a dashed line in Figure 15.8. The points along this line show what perfect equality of the income distribution looks like. It would mean, for example, that the bottom 20% of the income distribution receives 20% of the total income, the bottom 40% gets 40% of total income, and so on. The other lines reflect actual U.S. data on inequality for 1980 and 2020. The trick in graphing a Lorenz Curve is that you must change the shares of income for each specific quintile, which we show in the first and third columns of numbers in Table 15.6, into cumulative income, which we 372 15 • Poverty and Economic Inequality Access for free at openstax.org show in the second and fourth columns of numbers. For example, the bottom 40% of the cumulative income distribution will be the sum of the first and second quintiles; the bottom 60% of the cumulative income distribution will be the sum of the first, second, and third quintiles, and so on. The final entry in the cumulative income column needs to be 100%, because by definition, 100% of the population receives 100% of the income. Income Category Share of Income in 1980 (%) Cumulative Share of Income in 1980 (%) Share of Income in 2020 (%) Cumulative Share of Income in 2020 (%) First quintile 4.2 4.2 3.0 3.0 Second quintile 10.2 14.4 8.1 11.1 Third quintile 16.8 31.2 14.0 25.1 Fourth quintile 24.7 55.9 22.6 47.7 Fifth quintile 44.1 100.0 52.2 100.0 TABLE 15.6 Calculating the Lorenz Curve In a Lorenz Curve diagram, a more unequal distribution of income will loop farther down and away from the 45-degree line, while a more equal distribution of income will move the line closer to the 45-degree line. Figure 15.8 illustrates the greater inequality of the U.S. income distribution between 1980 and 2020 because the Lorenz Curve for 2020 is farther from the 45-degree line than the Lorenz Curve for 1980. The Lorenz Curve is a useful way of presenting the quintile data that provides an image of all the quintile data at once.

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  Imagine There's No Country

  Poverty, Inequality, and Growth in the Era of Globalization

  • Surjit S. Bhalla(Author)
  • 2002(Publication Date)
  • Peterson Institute for International Economics

    (Publisher)

  By definition, therefore, the Lorenz Curve starts at zero and ends at unity (figure 3.1 plots such a curve for three different years—1960, 1980, and 2000). If incomes are equally distributed, then the Lorenz Curve is described by the diagonal (i.e., the share of income at each point is equal to the share of population). If all the income in society accrued to just one person, then the Lorenz Curve would be the zero line (A to B) and then at the end the share would ‘‘spike’’ up to 1 (B to C). The Gini coefficient 2 is equal to 1 in the latter case (perfect inequality) and 0 in the former (perfect equality). For the curve shown in the figure (world distribution in 2000), the Gini coefficient is 0.651 (or 65.1). 2. The Gini coefficient is the ‘‘area between the Lorenz Curve and the diagonal, relative to the whole triangle below the diagonal, or half the mean difference relative to the mean’’ (Atkinson and Brandolini 1999, 6). 32 IMAGINE THERE’S NO COUNTRY Figure 3.1 World Lorenz Curve, 1960, 1980, and 2000 1960 1980 100 80 60 40 20 0 100 60 percent income percent population 40 20 A B C 80 2000 Note: World income inequality is computed by the simple accounting procedure (SAP) method of aggregating individual country data on distributions of incomes into a world distribution of individuals; see chapter 11 for the details. Sources: Deininger and Squire (1996); World Income Inequality Database, available at http://www.wider.unu.edu/wiid; Asian Development Bank (2002); Milanovic and Yitzhaki (2001). Measuring Intertemporal Inequality Along with the Gini, researchers have used other definitions of inequal-ity—the share in incomes of a particular quintile (e.g., the bottom 20 percent) or the ratio in incomes of different groups (e.g., the ratio in incomes of the top vs. the bottom decile). This logic has even been extended to the study of incomes of the richest country relative to the poorest country, as we saw in the previous chapter.

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  Tackling Inequality

  • R. Layard(Author)
  • 1999(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  For people with incomes low enough to experience positive net benefits, the Lorenz Curve rises, since they all gain and total income falls; so their relative income share rises. But the share of the richest man may rise if his proportional loss is less than the proportional loss in the economy as a whole. However if there is some income V * such that for all incomes greater than V * the proportional loss exceeds that for the economy as a whole, then the Lorenz Curve will rise for all incomes above V * . And, if the proportional loss in the whole economy is not great, then V * will lie close to the income for which net benefits are zero, and the Lorenz Curve is likely to lie above its old level throughout its range. 16. Again, if condition (3.3) does not hold, this does not strictly follow. 17. It is perhaps a more convenient one than the traditional two-person classroom illustration that is sometimes used. In this, person 1 is richer than person 2 and therefore bears more costs. We then show that redistribution occurs if the project is efficient and B\IB 2 < C\IC 2 . The proof is B\ - C\ = Ci — Bi (the efficiency condition)

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  Economic Inequality and Poverty

  International Perspectives

  • Lars Osberg(Author)
  • 2017(Publication Date)
  • Routledge

    (Publisher)

  x Lorenz-dominates y ). And this has been formally proven:

  (T1) For any two arbitrary distributions x, y, with the same number of members and μx = μy , whose members are ranked by income to give ordered distributions x' and y' respectively, then x' is obtained from y' by a finite sequence of progressive transfers if and only if x Lorenz-dominates y.

  The result is particularly useful because the condition on Lorenz Curves is so easily checked. The problem is, of course, that inequality comparisons will be indecisive wherever Lorenz Curves cross (see the dotted line in Figure 1.5 ), and the result cannot be applied where the distributions have different means. Before considering some of the strategies adopted as a consequence of this, it will be fruitful to redevelop the results within a different framework.

  Social Welfare, Inequality Rankings, and Lorenz Curves

  Those familiar with welfare economics will know that it is standard practice to evaluate the desirability of policy outcomes in terms of their effects on social welfare. This in turn is measured by using some form of social welfare function (SWF) to aggregate outcomes across all persons in the population. It has been seen in the preceding sections that different inequality measures incorporate different systems for weighting individual incomes, and so in this sense are just like some form of SWF. This suggests that an alternative approach could be to start with a SWF, specified to incorporate certain social values, and then to derive an inequality measure from this, knowing that it must reflect these same properties.

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  Pareto Distributions

  • Barry C. Arnold(Author)
  • 2015(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  He identifies two boundary curves L 1 ( x 1 ) = L ( x 1 , ∞ ) and L 2 ( x 2 ) = L ( ∞ , x 2 ) and shows that suitable projections of these curves represent the marginal Lorenz Curves of X 1 and X 2 (with the usual definitions). Other projections produce the correlation curves of Blitz and Brittain. In addition, Taguchi considers related surfaces obtained by changing the sense of one or both of the in-equalities X 1 ≤ x 1 and X 2 ≤ x 2 in the k = 2 case of (4.2.84). In the latter sections of Taguchi (1987), a variety of Gini indices associated with two-dimensional random variables are defined and their values are computed for several well-known bivariate distributions. Such Gini indices were earlier discussed in Taguchi (1981). An alternative multivariate extension of the Lorenz Curve was proposed by Ko-shevoy (1995). For full details of this zonoid approach, see Mosler (2002). As men-tioned earlier, to extend the Lorenz Curve concept to higher dimensions, it is desirable to have definition of the Lorenz Curve which is free of allusions to order statistics and/or quantile functions since such ideas do not extend readily to higher dimen-sions. In the one-dimensional setting, the Koshevoy approach can be described as follows. COMMON MEASURES OF INEQUALITY OF DISTRIBUTIONS 143 Consider a set of non-negative numbers (individuals’s incomes perhaps) x 1 , x 2 , ..., x n with a corresponding ordered set x 1: n ≤ x 2: n ≤ ... ≤ x n : n . The Lorenz Curve for this set of numbers is a linear interpolation of the ( n + 1 ) points in the unit square consisting of the points ( 0 , 0 ) , ( 1 , 1 ) and the points i n , ∑ i j = 1 x j : n ∑ n j = 1 x j : n ! , i = 1 , 2 , ..., n -1 . (4.2.97) If we think of the x i ’s as the incomes of n individuals in a population, then the second coordinate of (4.2.97) corresponds to the proportion of the total income ac-counted for by the poorest i individuals in the population.

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  Unemployment, Inequality and Poverty in Urban China

  • Hiroshi Sato, Shi Li(Authors)
  • 2006(Publication Date)
  • Taylor & Francis

    (Publisher)

  Figure 4.1 indicates that the 1988 Lorenz Curve dominates that of 1995, which in turn dominates that of 1999. This again confirms that income inequality increased over the period.

  To understand whether the increase in income inequality affected social welfare over the period studied, Figure 4.2 presents the generalized Lorenz Curve for 1988, 1995 and 1999 (left-hand side) and the difference between the curves of 1988 and 1995 and the curves of 1995 and 1999 (right-hand side). The vertical axis represents the cumulative percentage of per capita household income multiplied by the mean. It shows the total resources being accessed by each percentile of the population. If one generalized Lorenz Curve lies everywhere above another, it is said that the higher curve is preferable to the lower curve with regard to social welfare as every percentile of the population distribution has access to more resources.

  Figure 4.2 (A) illustrates the comparison between the 1988 and 1995 distributions. It is clear that although income inequality had increased by 1995, the income growth over the period more than compensated for the inequality increase, as social welfare in 1995 is everywhere greater than in 1988.

  Figure 4.1 Lorenz Curves for real per capita household disposable income, 1988, 1995 and 1999.

  Figure 4.2 Generalised Lorenz Curve. (A) 1988 and 1995. (B) 1995 and 1999.

  Figure 4.3 Distribution of households with unemployed members across income deciles.

  Figure 4.2

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  Income Distribution in Less Developed Countries

  • R. M. Sundrum(Author)
  • 2003(Publication Date)
  • Routledge

    (Publisher)

  c – b the corresponding measure for higher incomes. We can then measure the pattern of inequalities in the distribution by the ratio

  Table 3.4

  Correlation of inequality measures with Gini index

  Figure 3.8

  In terms of the diagram in Figure 3.4 , where P represents a given distribution and E the corresponding egalitarian distribution, the quantity w is indicated by the slope of the line joining P to E. Therefore this measure represents a different aspect of income distribution from overall inequality, which is represented by the distance PE for the Hirschman index and the horizontal distance between P and E for the Gini index.

  A more sophisticated measure of this aspect of income distributions has been suggested by Kakwani (1980: Chap. 17). Starting from the Lorenz Curve of Figure 3.3 , he makes a transformation of co-ordinates such that the diagonal becomes the horizontal axis; then the Lorenz Curve may have different shapes, as illustrated in Figure 3.9 .

  In this diagram, the curve in Figure 3.9a is skewed towards the point (0, 0) and may be said to be negatively skewed, while the curve in Figure 3.9b is skewed towards the point (1, 1) and may be said to be positively skewed. To measure the extent of skewness, Kakwani uses the ratio of the parameters α and β of the beta-distribution fitted to the Lorenz Curve after transforming it as described above. This measure of the skewness of the Lorenz Curve is closely related to the measure w defined above. The average values of the two measures for various groups of countries are shown in Table 4.2

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  Economic Justice in American Society

  • Robert E. Kuenne(Author)
  • 2014(Publication Date)
  • Princeton University Press

    (Publisher)

  The second characteristic revealed by a study of table 9.1 is the essen-tial stability of the pattern over this forty-one-year period. Figure 9.4 makes the point graphically by superimposing the 1947 and 1987 Lorenz Curves for family incomes and revealing their near-coincidence, despite the forty years that separate them. As will be shown, this impression is somewhat misleading, since it masks the existence of a slight trend to-ward more inequality, especially in the post-Vietnam War period, al-though the Lorenz Curves in the figure give some evidence of it. Never-theless, the 1987 share data differ from those of 1947 a percentage point or so at the most, despite the rather extensive changes in transfer poli-cies and otherwise over what is a rather lengthy period of time. The Gini coefficient also moves by only .016 points, revealing no dramatic shift one way or another. 1-0 9 -0.8-a °· 7 > 6 -I > •a G 0.4-5 5 0.3-0.2-01-C ,*' y' 02 y' ,'' .... ···'·''''' ,,γ / -*''' J .·...;:#· . . -Λ * * : : : : θ'.4 θ'.6 o!8 Cumulative Family Quintiles Lorenz Curve, 1947 -+•-Equality Curve ••*•· Lorenz Curve, 1987 FIGURE 9.4. Lorenz Curves, All-Race Family Money Income Shares, by Deciles, 1947 and 1987 POSTWAR U.S. INCOME DISTRIBUTION 239 TABLE 9.2 Descriptive Trend Lines Fitted to Quintile Family Money Income Shares and to Gini Coefficients, 1947-1987 1Q. 2Q_ 3Q. 4Q. 5Q. Gini ( y , ) ( y j (J 3 ) ( y j ( y j Top 5% C .37501 4.95334 11.99411 17.01334 23.00870 42.98139 17.61205 (121.25) (34.78) (158.60) (190.41) (159.51) (162.39) (79.38) t -.00124 -.05648 .05499 .19910 .20086 -.33935 -.30785 (-1.82) (-1.81) (6.29) (6.27) (3.92) (-3.61) (-3.90) t 2 -.00002 -.00663 -.00217 -.01610 -.01651 .02149 .02134 (-.63) (3.63) (-10.29) (-4.90) (-3.11) (2.21) (2.61) i 3 .000002 -.00014 Percentage Income Shares Percentage Income Shares FIGURE 9.7. Second Quintile Shares, All-Race Family Money Income, 1947-1987 FIGURE 9.6. First Quintile Shares, All-Race Family Money Income, 1947-1987

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  Handbook of Applied Economic Statistics

  • Aman Ullah(Author)
  • 1998(Publication Date)
  • CRC Press

    (Publisher)

  The implied Gini coefficient is G(x) = 1-'!:_B (_l_, _l_-1) (){ (){ fJ (35) where B( ·, ·) is the familiar Beta function, so this approach amounts to first fitting (34) to the observed points of a true Lorenz Curve via some nonlinear least-squares technique and approximating the true Gini coefficient (35). However, as shown by Schader and Schmid (1994), the approximations thus obtained often fall outside the Gastwirth bounds (for curves of the form (34) and for other families as well), so the consensus is that such do not work also Slottje 1990). VI. MULTIDIMENSIONAL AND POVERTY There are various additional research areas which impinge on inequality, such as the welfare ranking of income distributions (Shorrocks 1983, Thistle 1989), the mea-surement of interdistributional inequality (Butler and McDonald 1987, Dagum 1987), or the parametric modeling of income distributions and the inequality orderings that are implied by the parameters (Chipman 1985, Wilfling and Kramer 1993), which I do not cover here. Rather, I pick the issues of multidimensional inequality and poverty to show how the concepts introduced above can be extended. Take poverty. Although among certain sociologists, this is taken to be almost synonymous to inequality (relative deprivation), the majority consensus is that *RP-member that the Lorenz Curve has the slope X(i)/X in the interval ((i-1)/n, ijn), so its (left-hand) slope in p; is a;(X. 56 KRAMER poverty is something different, and needs different measures for quantification. Fol-lowing Sen (1976), this is usually done by first identifying a poverty line z below which an income unit is considered poor, and then combining the poverty character-istics of different units into an overall measure of poverty. Two obvious candidates are the head-count ratio n* H(x) = ~ n where n* is the number of units below the poverty line, and the income gap ratio (:~7) i.e., the normalized per unit percentage shortfall of the poor.

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