Report

Income Inequality Here we see the idea of income distribution and the Lorenz Curve 1 Quintile % of income cumulative % Bottom 20% 4 4 Second 20% 10 14 Middle 20% 16 30 Fourth 20% 23 53 Fifth 20% 47 100 Here we have a table of income as distributed across the economy for a given year. Grouping is done on quintiles – population is divided into fifths in a special way – here by income. Put people in order, from low to high income. Then starting from the lowest income person, take the first 20% of people and see what income % they have. Then go to the next 20% and so on. 2 Cumulative % of income 1 .8 Lorenz curve .6 A .4 .2 B .2 .4 .6 .8 1 Cumulative % of households 3 The Lorenz curve is a graph of the cumulative percent of income by quintiles. Actually, in the graph each percent is expressed in decimal form. So, for example, 20% is .2 The degree to which the curve is bowed out is an indication of the income inequality in the economy, with a larger bow meaning more inequality. The Gini coefficient is the ratio of area A to area A + B. The Gini has a coefficient value between 0 and 1, with 0 meaning perfect equality of income. Note that area A + B = .5(1)(1) = .5 because A + B is a triangle and the area of a triangle is ½ of base times height. Area A is found by working with areas of rectangles and triangles. 4 Table 7-2 in the book has US income distribution for 2006. I reproduce that table here (note it is in decimal form – what the author calls “shares of income”). Quintile, by the way, meansbreak up into groups of 20%. Quintile First 20% Second Third Fourth Fifth Share of Income .034 .086 .145 .229 .505 Cumulative Share .034 .120 .265 .494 1.00 The 1.00 in the cumulative share looks like it should be .999, but that is due to rounding the values in the real data to 3 decimals in the table here. 5 1 This is the Lorenz curve. .8 .6 .4 .2 .2 .4 .6 .8 1 1 .8 .6 .4 .2 .2 .4 a quintile at a time. .6 .8 We need this area for the calcul ation of the Gini coeffi cient. Let’s 1 do it 1 .8 .6 .4 .2 .2 .4 .6 .8 1 In the first quintile we want this part of the triangle made up of the boundary of the quintile. The area we want is then = .5(.2)(.2) - .5(.2)(.034) = .02 .0034 = .0166 This area is made up of the whole triangle and then subtracting out the smaller triangle Remember the area of a triangle is ½ of the base times the height. 9 1 .8 .6 .4 .2 D E F In the second quintile, from .2 to .4 we need to add areas D, E, and F. (in class I made a mistake here) .2 .4 .6 .8 1 D = .5(.2)(.2), E = .2(.2 - .12), F = .5(.12 - .034)(.2), so D+E+F=.0446 1 .8 .6 .4 .2 G H I In the third quintile, from .4 to .6 we need to add areas G, H, and I. .2 .4 .6 .8 1 G = .5(.2)(.2), H = .2(.4 - .265), I = .5(.265 - .12)(.2), so G+H+I=.0615 1 .8 .6 .4 J K L In the fourth quintile, from .6 to .8 we need to add areas J, K, and L. .2 .2 .4 .6 .8 1 J = .5(.2)(.2), K = .2(.6 - .494), I = .5(.494 - .265)(.2), so J+K+L=.0641 1 .8 .6 .4 .2 .2 .4 .6 .8 1 M In the 5th quintile I really only want area N but it may be easier to calculate M + N + O and then subtract out M and O. So N .2(1 - .494) - .5(.2)(.2) - .5(1 - .494)(.2) = .0306 O The Gini coefficient here is (.0166 + .0446 + .0615 + .0641 + .0306)/.5 = .2174/.5 = .4348 Note the Gini coefficient ranges from 0 to 1 and the closer to 1 the more unequal is the income across the community. 14