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Amiel-Cowell Thinking About Inequality Yoram Amiel and Frank Cowell Ruppin Institute, Israel and London School of Economics December 1998 Contents Preface xiii 1 Introduction 1 1.1ALookatInequalityAnalysis............. 1 1.2ASecondLook..................... 4 1.3AGuidetotheBook.................. 9 2 What is Inequality? The Economists’ View 11 2.1TheAxiomaticApproach................ 11 2.2InequalityRankingsandOrderings.......... 12 2.3TheTransferPrinciple................. 16 2.4IncomeandPopulation................. 17 2.5Decomposability.................... 22 2.6Summary........................ 24 3 An Investigative Strategy 27 3.1Whatareweinvestigating?............... 27 3.2Experiments....................... 31 3.3Questions........................ 33 3.4ANewApproach.................... 35 3.5ImplementingtheApproach.............. 39 3.6ABriefSummary.................... 43 4 What is Inequality? The Students’ View 45 4.1DrawinganInequalityMap.............. 45 4.2AnIntroductiontotheQuestionnaires........ 49 4.3 Inequality and Changes in Income and Population . 53 4.4 Transfers and the Structure of Inequality Comparisons 57 i ii CONTENTS 4.5DotheAnswersMakeSense?............. 63 4.6MoreonTheTransferPrinciple............ 68 5 Income and Welfare 75 5.1WhatisWelfare?.................... 75 5.2SocialWelfare...................... 77 5.2.1 Anonymity................... 78 5.2.2 Monotonicity, Dominance and the Pareto Prin- ciple........................ 81 5.2.3 RelationshiptoInequality........... 84 5.3EmpiricalResults.................... 86 5.3.1 Anonymity................... 90 5.3.2 Thetransferprinciple.............. 92 5.3.3 Principleofpopulation............. 94 5.3.4 Decomposability ................ 95 5.3.5 Additionstoincomes............... 97 5.4 Summary: Welfare Judgments and Inequality Com- parisons......................... 99 6 Income Change 105 6.1Introduction:ComparingCakes............ 105 6.2UniformEnrichment.................. 108 6.3TheDependenceHypothesis.............. 116 6.4UnbalancedEnrichment................ 122 6.5Policyappraisal..................... 133 6.5.1 Fairness, comparability and income levels. 134 6.5.2 InequalityandGrowth.............. 135 7 Poverty 137 7.1Introduction....................... 137 7.2WhatDoes“Poverty”Mean?............. 138 7.2.1 Partitioningthepopulation.......... 140 7.2.2 IdentifyingthePoor.............. 140 7.2.3 Using information about the distribution . 141 7.3ThePovertyQuestionnaires.............. 145 7.4IncomeDistributionsandPoverty........... 147 7.4.1 Monotonicity.................. 149 CONTENTS iii 7.4.2 Thetransferprinciple..............151 7.4.3 Anonymity................... 154 7.4.4 Populationreplication............. 155 7.4.5 Populationdecomposability.......... 156 7.4.6 Thefocusaxiom................ 157 7.4.7 Newpopulationmembers........... 157 7.4.8 Poverty perceptions and the poverty line . 161 7.4.9 Whathappensifbasicneedsincrease?.... 162 7.4.10 What happens if basic needs and incomes in- crease?...................... 164 7.4.11 The definitionofthepovertyline....... 167 7.5 Conclusions: the Approach to Poverty Comparisons . 168 8 A Cross-cultural Perspective 171 8.1Introduction....................... 171 8.2AStatisticalApproach................. 173 8.3PrinciplesofDistributionalJudgments........ 175 8.3.1 Transferprinciple................ 176 8.3.2 Monotonicity.................. 179 8.3.3 ThePovertyLine................ 181 8.4DirectandIndirectApproachestoInequality..... 183 8.5DoesEconomicsMatter?................ 186 8.6AnAppraisal...................... 188 9 Thinking Again about Inequality 189 9.1SecondThoughtsaboutSecondThoughts.......189 9.2ApplyingInequalityJudgments............ 190 9.3WhereNext?...................... 193 9.4AFinalWord...................... 198 A Inequality Analysis: A summary of results 201 A.1TheAxiomaticApproach................ 201 A.2InequalityandWelfareRankings........... 206 A.3PovertyComparisons.................. 208 B The Questionnaires 211 iv CONTENTS List of Figures 1.1Asimpledistributionalexperiment........... 6 1.2 A simple distributional experiment, second view. 7 1.3Asimpledistributionalexperiment,thirdview.... 8 2.1Aninequalityranking.................. 13 2.2Inequalityinatwo-personworld............ 14 2.3Inequalitycomparisonsinathree-personworld.... 15 2.4Scaleindependence................... 18 2.5Translationindependence................ 19 2.6Thepopulationprinciple................ 20 2.7Populationreplication-hasinequalityfallen?.... 21 2.8Decomposability.................... 23 3.1 Two extreme approaches to identity in the distribu- tionproblem....................... 36 3.2Theidentityproblemwithsomeinformation...... 37 4.1Theframeworkfora3-personinequalitymap.... 46 4.2 The framework for the 2-person projection of the in- equalitymap...................... 47 4.3 Verbal questions on scale and translation independence 48 4.4 Numerical problems on changes in income and popu- lation........................... 50 4.5TheQuestiononthePopulationPrinciple....... 52 4.6 Scale-independent and translation-independent iso-inequality lines........................... 54 4.7Deviationsfromscaletransformation......... 55 v vi LIST OF FIGURES 4.8 Numerical problems on the transfer principle and de- composability...................... 58 4.9 Verbal questions on the transfer principle and decom- posability . ...................... 59 4.10 The transfer principle in the two-person projection . 60 4.11 Two-dimensional projection of the inequality map . 62 4.12TheLorenzcurve.................... 64 4.13Lorenzranking..................... 65 5.1Theanonymityprinciple................ 79 5.2Anonymityandhistory................. 80 5.3Monotonicity...................... 82 5.4Dominanceandmonotonicity............. 83 5.5 The setting for the social-welfare numerical problems 87 5.6Contoursofastandardsocialwelfarefunction.... 88 5.7 This SWF satisfies the transfer principle, but not de- composability...................... 96 5.8Contoursofanon-monotonicwelfarefunction.... 100 6.1 Two pie distributions, before and after income growth 106 6.2AdditionstoIreneandJanet’sincomes........ 110 6.3EnlargementofFigure6.2............... 111 6.4Scaleindependence................... 113 6.5Translationindependence................114 6.6Intermediate-typeindependence............ 115 6.7 Transformation direction dependent upon income (1) 117 6.8 Transformation direction dependent upon income (2) 118 6.9IntroductiontoquestionnaireA3........... 120 6.10Unbalancedenrichment................. 123 6.11 Inequality and growth - firstview........... 124 6.12Inequalityandgrowth-secondview..........126 6.13ExtractfromquestionnaireA1............. 127 6.14ExtractfromquestionnaireA2............. 129 6.15 Mean income and inequality as incomes grow from $5 to$10.......................... 132 6.16 Mean income and inequality as incomes grow from $1 to$10.......................... 132 LIST OF FIGURES vii 7.1Afundamentalpartitionofthepopulation...... 139 7.2Thepovertyline,incomesandpovertygaps..... 140 7.3Countingthepoor(1)-allthepoorareequal.... 142 7.4 Counting the poor (2) - poverty is proportional to povertygap....................... 143 7.5 Counting the poor (3) - sensitivity to inequality amongst thepoor......................... 144 7.6Themonotonicityquestion............... 144 7.7Apovertyinterval.................... 146 7.8 Numerical problems in the first poverty questionnaire 148 7.9 Weak monotonicity (note - B Cisirrelevant).... 149 7.10Incometransfersandthepovertycount........→ 152 7.11Transferprinciple(weakversion)........... 153 7.12Positionsforanadditionalpopulationmember.... 158 7.13 Numerical problems in the second poverty questionnaire161 7.14 Extract from the verbal questions in the second poverty questionnaire...................... 165 viii LIST OF FIGURES List of Tables 4.1 Inequality and proportionate and absolute income dif- ferences......................... 56 4.2 The effect on inequality of cloning the distributions . 57 4.3Thetransferprinciple ................. 61 4.4Decomposability..................... 63 4.5 What happens to inequality if you add or subtract a fixedsum? ....................... 66 4.6 Agreement with the Transfer Principle for Different TypesofTransfer.NumericalResponses. ...... 69 4.7VerbalAgreementwiththetransferprinciple..... 71 4.8 Agreement with basic axioms. Summary percentages. 72 5.1 Agreement with Transitivity of Inequality and Social- welfareOrderings.................... 90 5.2TheAnonymityPrinciple................ 91 5.3 The Transfer Principle Again: Numerical Responses . 92 5.4 Transfer Principle. Verbal Responses on Social Wel- fareQuestionnaire................... 93 5.5 The effect on social welfare of cloning the distribution 95 5.6Decomposabilityofsocialwelfare? .......... 95 5.7 Agreement with Monotonicity - Numerical questions. 98 5.8 Agreement with Monotonicity - Verbal Questions . 98 5.9 Agreement with basic axioms on social welfare. Sum- marypercentages.................... 101 5.10 Support for basic principles of inequality comparisons. 102 6.1 What income change will leave inequality unchanged?. 119 ix x LIST OF TABLES 6.2 What happens to inequality when you increase peo- ple’sincomes?...................... 121 6.3 Perceived Inequality Change in the Growth Process of Figure6.13........................ 128 6.4Comparingextremesinthegrowthprocess.......131 7.1 What happens if a poor person gets $1 more income? 150 7.2 What if $1 is taken from fairly poor Irene and given toverypoorJanet? .................. 153 7.3 If we permute the incomes does poverty stay the same? 154 7.4 What
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