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Measuring Inequality Measuring Inequality Frank A. Cowell December 2009 http://darp.lse.ac.uk/MI3 ii Abstract Part of the series LSE Perspectives in Economic Analysis, pub- lished by Oxford University Press This book is dedicated to the memory of my parents. Contents Preface xi 1 First Principles 1 1.1 A preview of the book . 3 1.2 Inequality of what? . 4 1.3 Inequality measurement, justice and poverty . 7 1.4 Inequality and the social structure . 12 1.5 Questions . 13 2 Charting Inequality 17 2.1 Diagrams . 17 2.2 Inequality measures . 23 2.3 Rankings . 30 2.4 From charts to analysis . 36 2.5 Questions . 36 3 Analysing Inequality 39 3.1 Social-welfare functions . 40 3.2 SWF-based inequality measures . 48 3.3 Inequality and information theory . 52 3.4 Building an inequality measure . 60 3.5 Choosing an inequality measure . 65 3.6 Summary . 70 3.7 Questions . 71 4 Modelling Inequality 75 4.1 The idea of a model . 76 4.2 The lognormal distribution . 77 4.3 The Pareto distribution . 84 4.4 How good are the functional forms? . 91 4.5 Questions . 95 iii iv CONTENTS 5 From Theory to Practice 99 5.1 The data . 100 5.2 Computation of the inequality measures . 108 5.3 Appraising the calculations . 124 5.4 Shortcuts: …tting functional forms1 . 131 5.5 Interpreting the answers . 138 5.6 A sort of conclusion . 143 5.7 Questions . 144 A Technical Appendix 149 A.1 Overview . 149 A.2 Measures and their properties . 149 A.3 Functional forms of distribution . 152 A.4 Interrelationships between inequality measures . 160 A.5 Decomposition of inequality measures . 161 A.6 Negative incomes . 166 A.7 Estimation problems . 168 A.8 Using the website . 174 B Notes on Sources and Literature 177 B.1 Chapter 1 . 177 B.2 Chapter 2 . 179 B.3 Chapter 3 . 182 B.4 Chapter 4 . 186 B.5 Chapter 5 . 190 B.6 Technical Appendix . 195 1 This section contains material of a more technical nature which can be omitted without loss of continuity. List of Tables 1.1 Four inequality scales . 9 3.1 How much should R give up to …nance a £ 1 bonus for P? . 43 3.2 Is P further from Q than Q is from R? . 59 3.3 The break-down of inequality: poor East, rich West . 63 3.4 The break-down of inequality: the East catches up . 64 3.5 Which measure does what? . 72 4.1 Pareto’s and “average/base”inequality . 88 4.2 Pareto’s for income distribution in the UK and the USA . 94 5.1 Distribution of Income Before Tax. USA 2006. Source: Internal Revenue Service . 112 5.2 Values of Inequality indices under a variety of assumptions about the data. US 2006 . 118 5.3 Approximation Formulas for Standard Errors of Inequality Mea- sures..................................126 5.4 Atkinson index and coe¢ cient of variation: IRS 1987 to 2006 . 128 5.5 Individual distribution of household net per capita annual in- come. Czechoslovakia 1988. Source: see Appendix B . 129 5.6 Average income, taxes and bene…ts by decile groups of all house- holds. UK 1998-9. Source: O¢ ce for National Statistics . 145 5.7 Observed and expected frequencies of household income per head, Jiangsu, China . 147 A.1 Inequality measures for discrete distributions . 151 A.2 Inequality measures for continuous distributions . 153 A.3 Decomposition of inequality in Chinese provinces, Rural and Ur- ban subpopulations . 165 A.4 Source …les for tables and …gures . 175 v vi LIST OF TABLES List of Figures 1.1 Two Types of Inequality . 8 1.2 An Inequality Ranking . 8 1.3 Alternative policies for Fantasia . 15 2.1 The Parade of Dwarfs. UK Income Before Tax, 1984/5. Source: Economic Trends, November 1987 ................... 19 2.2 Frequency Distribution of Income Source: as for Figure 2.1 . 20 2.3 Cumulative Frequency Distribution. Source: as for Figure 2.1 . 21 2.4 Lorenz Curve of Income. Source: as for Figure 2.1 . 22 2.5 Frequency Distribution of Income (Logarithmic Scale).Source: as for Figure 2.1 .............................. 23 2.6 The Parade with Partial Equalisation . 25 2.7 The High-Low Approach . 28 2.8 The Parade and the Quantile Ranking . 31 2.9 Quantile ratios of earnings of adult men, UK 1968-2007. Source: Annual Survey of Hours and Earnings . 32 2.10 Ranking by Shares. UK 1984/5 Incomes before and after tax. Source: as for Figure 2.1 ........................ 33 2.11 Lorenz Curves Crossing . 34 2.12 Change at the bottom of the income distribution . 35 2.13 Change at the top of the income distribution . 35 3.1 Social utility and relative income . 44 3.2 The relationship between welfare weight and income. 45 3.3 The Generalised Lorenz Curve Comparison: UK income before tax . 47 3.4 Distribution of Income and Distribution of Social Utility . 49 3.5 The Atkinson and Dalton Indices . 50 3.6 The Theil Curve . 54 3.7 Theil’sEntropy Index . 56 3.8 A variety of distance concepts . 58 3.9 Lorenz Curves for Equivalised Disposable Income per Person. Switzerland and USA. 69 vii viii LIST OF FIGURES 3.10 Inequality Aversion and Inequality Rankings, Switzerland and USA. Source: as for Figure 3.9 .................... 70 4.1 The Normal Distribution . 78 4.2 The Lognormal Distribution . 79 4.3 The Lorenz curve for the Lognormal distribution . 81 4.4 Inequality and the Lognormal parameter 2 . 82 4.5 The Pareto Diagram.Source: as for Figure 2.1 . 85 4.6 The Pareto Distribution in the Pareto Diagram . 86 4.7 Paretian frequency distribution . 87 4.8 The Lorenz curve for the Pareto distribution . 89 4.9 Inequality and Pareto’s ...................... 90 4.10 The Distribution of Earnings. UK Male Manual Workers on Full- Time Adult Rates. Source: New Earnings Survey, 2002 . 92 4.11 Pareto Diagram. UK Wealth Distribution 2003. Source: Inland Revenue Statistics ........................... 93 4.12 Pareto’s : USA and UK. Source: see text . 95 5.1 Frequency Distribution of Income, UK 2005/6, Before and After Tax. Source: Inland Revenue Statistics . 101 5.2 Disposable Income (Before Housing Costs). UK 2006/7. Source: Households Below Average Income, 2008 . 103 5.3 Disposable Income (After Housing Costs). UK 2006/7. Source: Households Below Average Income, 2008 . 104 5.4 Income Observations Arranged on a Line . 110 5.5 Frequency Distribution of Disposable Income, UK 2006/7 (After Housing Costs), Unsmoothed. Source: as for Figure 5.3 . 111 5.6 Estimates of Distribution Function. Disposable Income, UK 2006/7. (After Housing Costs), Moderate Smoothing. Source: as for Fig- ure 5.3 . 113 5.7 Estimates of Distribution Function. Disposable Income, UK 2006/7. (After Housing Costs), High Smoothing. Source: as for Figure 5.3 114 5.8 Frequency distribution of income before tax. US 2006. Source: Internal Revenue Service . 115 5.9 Lower Bound Inequality, Distribution of Income Before Tax. US 2006. Source: Internal Revenue Service . 116 5.10 Upper Bound Inequality, Distribution of Income Before Tax. US 2006. Source: Internal Revenue Service . 117 5.11 The coe¢ cient of variation and the upper bound of the top interval.120 5.12 Lorenz Co-ordinates for Table 5.1 . 121 5.13 Upper and Lower Bound Lorenz Curves . 122 5.14 The “split histogram”compromise. 124 5.15 Lorenz Curves –Income Before Tax. USA 1987 and 2006. Source: Internal Revenue Service . 127 5.16 The Atkinson index for grouped data, US 2006. Source: as for Table 5.1. 128 LIST OF FIGURES ix 5.17 The Atkinson Index for Grouped Data: First interval deleted. Czechoslovakia 1988 . 130 5.18 The Atkinson Index for Grouped Data: All data included. Czechoslo- vakia 1988 . 131 5.19 Fitting the Pareto diagram for the data in Table 5.1 . 136 5.20 Fitting the Pareto diagram for IRS data in 1987 (values in 2006 dollars) . 137 5.21 The minimum income growth to o¤set a 1% growth in inequality 140 A.1 Relationships Between Functional Forms . 159 A.2 Density Estimation with a Normal Kernel . 171 x LIST OF FIGURES Preface “It is not the business of the botanist to eradicate the weeds. Enough for him if he can tell us just how fast they grow.” – C. Northcote Parkinson (1958), Parkinson’s Law The maligned botanist has a good deal to be said for him in the company of rival gardeners, each propagating his own idea about the extent and the growth of thorns and thistles in the herbaceous border, and each with a patent weed- killer. I hope that this book will perform a similar role in the social scientist’s toolshed. It does not deal with theories of the development of income distribu- tion, of the generation of inequality, or of other social weeds, nor does it supply any social herbicides. However, it does give a guide to some of the theoretical and practical problems involved in an analysis of the extent of inequality thus permitting an evaluation of the diverse approaches hitherto adopted. In avoiding patent remedies for particular unwanted growths, one …nds 6useful analogies in various related …elds –for example, some techniques for measuring economic in- equality have important counterparts in sociological and political studies. Thus, although I have written this as an economist, I would like to think that students in these related disciplines will be interested.
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