Satya R. Chakravarty is a Professor of Economics at the Indian Statistical Institute, Kolkata, India. He is an Editor of Social Choice and Welfare and a member of the Editorial Board of Journal of Economic Inequality.

• Preface xiEndorsements xv1 Well-Being as a Multidimensional Phenomenon 11.1 Introduction 11.2 Income as a Dimension ofWell-Being and Some Related Aggregations 41.3 Scales of Measurement: A Brief Exposition 101.4 Preliminaries for MultidimensionalWelfare Analysis 111.5 The Dashboard Approach andWeights on DimensionalMetrics in a Composite Index 141.6 MultidimensionalWelfare Function Axioms 171.6.1 Invariance Axioms 171.6.2 Distributional Axioms 191.7 MultidimensionalWelfare Functions 271.8 Concluding Remarks 37References 382 An Overview of Multidimensional Economic Inequality 492.1 Introduction 492.2 A Review of One-DimensionalMeasurement 512.2.1 Normative One-Dimensional Inequality Indices 512.2.2 Subgroup-Decomposable Indices of Inequality 532.3 Multidimensional Inequality Indices 562.3.1 The Direct Approach 562.3.1.1 Axioms for a Multidimensional Inequality Index 562.3.1.2 Examples of Indices 602.3.2 The Inclusive Measure ofWell-being Approach 692.4 Concluding Remarks 77References 783 A Synthesis of Multidimensional Poverty 853.1 Introduction 853.2 A Brief Review of One-Dimensional Analysis 893.3 Preliminaries for Multidimensional Poverty Analysis 933.4 Identification of the Poor and Deprivation Counting 953.5 Axioms for a Multidimensional Poverty Metric 993.5.1 Invariance Axioms 1003.5.2 Distributional Axioms 1033.5.3 Decomposability Axioms 1083.5.4 Threshold Limit Sensitivity Axiom 1113.5.5 Technical Axioms 1123.6 Multidimensional Poverty Measurement 1133.6.1 The Dashboard Approach 1133.6.2 The Direct Approach 1173.6.3 The Inclusive Measure ofWell-being Approach 1233.7 Multidimensional Poverty Orderings 1253.7.1 A Brief Outline of One-Dimensional Orderings 1263.7.2 Multidimensional Orderings 1293.8 Dimensions ofWell-Being with Ordinal Significance and Multidimensional Poverty 1313.9 Orderings Based on Deprivations Counts 1343.10 Multidimensional Material Deprivation 1383.11 Concluding Remarks 142References 1434 Fuzzy Set Approaches to the Measurement of Multidimensional Poverty 1634.1 Introduction 1634.2 Fuzzy Membership Function 1654.3 Axioms for a Fuzzy Multidimensional Poverty Index 1734.3.1 Invariance Axioms 1754.3.1.1 Fuzzy Strong Ratio Scale Invariance 1754.3.1.2 Fuzzy Strong Translation Scale Invariance 1774.3.1.3 FuzzyWeak Focus 1774.3.1.4 Fuzzy Strong Focus 1784.3.1.5 Fuzzy Symmetry 1784.3.1.6 Fuzzy Population Replication Invariance 1794.3.2 Distributional Axioms 1794.3.2.1 Fuzzy Monotonicity 1794.3.2.2 Fuzzy Monotonicity Sensitivity 1804.3.2.3 Fuzzy DimensionalMonotonicity 1814.3.2.4 Fuzzy Transfer 1824.3.2.5 Increasing Fuzzy Poverty under Correlation-Increasing Switch 1844.3.3 Decomposability Axioms 1854.3.3.1 Fuzzy Subgroup Decomposability 1854.3.3.2 Fuzzy Factor Decomposability 1854.3.4 Fuzzy Sensitivity Axiom 1864.3.4.1 Increasing Fuzzy Poverty for Increased Membership Function 1864.3.5 Technical Axioms 1874.3.5.1 Fuzzy Boundedness 1874.3.5.2 Fuzzy Continuity 1874.4 Fuzzy Multidimensional Poverty Indices 1874.5 Fuzzy Poverty Orderings 1914.6 Concluding Remarks 195References 1955 Poverty and Time: A Multidimensional Appraisal 2015.1 Introduction 2015.2 Preliminaries 2075.3 The Block Approach 2085.3.1 IndividualMultidimensional Intertemporal Poverty Index 2085.3.2 Aggregate Multidimensional Intertemporal Poverty Index 2185.3.3 A Review of Some Related One-Dimensional Proposals 2205.4 An Exploration of the Counting Approaches to MultidimensionalIntertemporal Deprivations 2265.5 The Multidimensional Duration Approach 2295.5.1 A Review of One-Dimensional Duration-Reliant Offers 2295.5.2 Axioms for a Chronic Multidimensional Poverty Quantifier 2315.5.3 The Bourguignon–Chakravarty Approach to Chronic Multidimensional Poverty Measurement 2405.6 Intertemporal Poverty Orderings 2435.7 Concluding Remarks 245References 2466 Vulnerability to Poverty: A Multidimensional Evaluation 2516.1 Introduction 2516.2 A Review of One-DimensionalMeasurement 2546.3 Multidimensional Representation of Vulnerability to Poverty: An Axiomatic Investigation 2626.4 Concluding Remarks 269References 2717 An Exploration of Some Composite and Individualistic Indices 2777.1 Introduction 2777.2 Human Development Index 2787.3 Human Poverty Index 2847.4 Gender Inequality Index 2867.5 Better Life Index 2887.6 Active Citizenship Composite Index 2917.7 Measuring Human Opportunity: A Counting Approach 2937.8 Assessment of Progress toward Achievements in Millennium Development Goals 2957.9 Air Quality Index 3017.10 Concluding Remarks 303References 304Index 311

Satya R. Chakravarty is a Professor of Economics at the Indian Statistical Institute, Kolkata, India. He is an Editor of Social Choice and Welfare and a member of the Editorial Board of Journal of Economic Inequality.

• Preface xiEndorsements xv1 Well-Being as a Multidimensional Phenomenon 11.1 Introduction 11.2 Income as a Dimension ofWell-Being and Some Related Aggregations 41.3 Scales of Measurement: A Brief Exposition 101.4 Preliminaries for MultidimensionalWelfare Analysis 111.5 The Dashboard Approach andWeights on DimensionalMetrics in a Composite Index 141.6 MultidimensionalWelfare Function Axioms 171.6.1 Invariance Axioms 171.6.2 Distributional Axioms 191.7 MultidimensionalWelfare Functions 271.8 Concluding Remarks 37References 382 An Overview of Multidimensional Economic Inequality 492.1 Introduction 492.2 A Review of One-DimensionalMeasurement 512.2.1 Normative One-Dimensional Inequality Indices 512.2.2 Subgroup-Decomposable Indices of Inequality 532.3 Multidimensional Inequality Indices 562.3.1 The Direct Approach 562.3.1.1 Axioms for a Multidimensional Inequality Index 562.3.1.2 Examples of Indices 602.3.2 The Inclusive Measure ofWell-being Approach 692.4 Concluding Remarks 77References 783 A Synthesis of Multidimensional Poverty 853.1 Introduction 853.2 A Brief Review of One-Dimensional Analysis 893.3 Preliminaries for Multidimensional Poverty Analysis 933.4 Identification of the Poor and Deprivation Counting 953.5 Axioms for a Multidimensional Poverty Metric 993.5.1 Invariance Axioms 1003.5.2 Distributional Axioms 1033.5.3 Decomposability Axioms 1083.5.4 Threshold Limit Sensitivity Axiom 1113.5.5 Technical Axioms 1123.6 Multidimensional Poverty Measurement 1133.6.1 The Dashboard Approach 1133.6.2 The Direct Approach 1173.6.3 The Inclusive Measure ofWell-being Approach 1233.7 Multidimensional Poverty Orderings 1253.7.1 A Brief Outline of One-Dimensional Orderings 1263.7.2 Multidimensional Orderings 1293.8 Dimensions ofWell-Being with Ordinal Significance and Multidimensional Poverty 1313.9 Orderings Based on Deprivations Counts 1343.10 Multidimensional Material Deprivation 1383.11 Concluding Remarks 142References 1434 Fuzzy Set Approaches to the Measurement of Multidimensional Poverty 1634.1 Introduction 1634.2 Fuzzy Membership Function 1654.3 Axioms for a Fuzzy Multidimensional Poverty Index 1734.3.1 Invariance Axioms 1754.3.1.1 Fuzzy Strong Ratio Scale Invariance 1754.3.1.2 Fuzzy Strong Translation Scale Invariance 1774.3.1.3 FuzzyWeak Focus 1774.3.1.4 Fuzzy Strong Focus 1784.3.1.5 Fuzzy Symmetry 1784.3.1.6 Fuzzy Population Replication Invariance 1794.3.2 Distributional Axioms 1794.3.2.1 Fuzzy Monotonicity 1794.3.2.2 Fuzzy Monotonicity Sensitivity 1804.3.2.3 Fuzzy DimensionalMonotonicity 1814.3.2.4 Fuzzy Transfer 1824.3.2.5 Increasing Fuzzy Poverty under Correlation-Increasing Switch 1844.3.3 Decomposability Axioms 1854.3.3.1 Fuzzy Subgroup Decomposability 1854.3.3.2 Fuzzy Factor Decomposability 1854.3.4 Fuzzy Sensitivity Axiom 1864.3.4.1 Increasing Fuzzy Poverty for Increased Membership Function 1864.3.5 Technical Axioms 1874.3.5.1 Fuzzy Boundedness 1874.3.5.2 Fuzzy Continuity 1874.4 Fuzzy Multidimensional Poverty Indices 1874.5 Fuzzy Poverty Orderings 1914.6 Concluding Remarks 195References 1955 Poverty and Time: A Multidimensional Appraisal 2015.1 Introduction 2015.2 Preliminaries 2075.3 The Block Approach 2085.3.1 IndividualMultidimensional Intertemporal Poverty Index 2085.3.2 Aggregate Multidimensional Intertemporal Poverty Index 2185.3.3 A Review of Some Related One-Dimensional Proposals 2205.4 An Exploration of the Counting Approaches to MultidimensionalIntertemporal Deprivations 2265.5 The Multidimensional Duration Approach 2295.5.1 A Review of One-Dimensional Duration-Reliant Offers 2295.5.2 Axioms for a Chronic Multidimensional Poverty Quantifier 2315.5.3 The Bourguignon–Chakravarty Approach to Chronic Multidimensional Poverty Measurement 2405.6 Intertemporal Poverty Orderings 2435.7 Concluding Remarks 245References 2466 Vulnerability to Poverty: A Multidimensional Evaluation 2516.1 Introduction 2516.2 A Review of One-DimensionalMeasurement 2546.3 Multidimensional Representation of Vulnerability to Poverty: An Axiomatic Investigation 2626.4 Concluding Remarks 269References 2717 An Exploration of Some Composite and Individualistic Indices 2777.1 Introduction 2777.2 Human Development Index 2787.3 Human Poverty Index 2847.4 Gender Inequality Index 2867.5 Better Life Index 2887.6 Active Citizenship Composite Index 2917.7 Measuring Human Opportunity: A Counting Approach 2937.8 Assessment of Progress toward Achievements in Millennium Development Goals 2957.9 Air Quality Index 3017.10 Concluding Remarks 303References 304Index 311

Satya R. Chakravarty is a Professor of Economics at the Indian Statistical Institute, Kolkata, India. He is an Editor of Social Choice and Welfare and a member of the Editorial Board of Journal of Economic Inequality.

• Preface xiEndorsements xv1 Well-Being as a Multidimensional Phenomenon 11.1 Introduction 11.2 Income as a Dimension ofWell-Being and Some Related Aggregations 41.3 Scales of Measurement: A Brief Exposition 101.4 Preliminaries for MultidimensionalWelfare Analysis 111.5 The Dashboard Approach andWeights on DimensionalMetrics in a Composite Index 141.6 MultidimensionalWelfare Function Axioms 171.6.1 Invariance Axioms 171.6.2 Distributional Axioms 191.7 MultidimensionalWelfare Functions 271.8 Concluding Remarks 37References 382 An Overview of Multidimensional Economic Inequality 492.1 Introduction 492.2 A Review of One-DimensionalMeasurement 512.2.1 Normative One-Dimensional Inequality Indices 512.2.2 Subgroup-Decomposable Indices of Inequality 532.3 Multidimensional Inequality Indices 562.3.1 The Direct Approach 562.3.1.1 Axioms for a Multidimensional Inequality Index 562.3.1.2 Examples of Indices 602.3.2 The Inclusive Measure ofWell-being Approach 692.4 Concluding Remarks 77References 783 A Synthesis of Multidimensional Poverty 853.1 Introduction 853.2 A Brief Review of One-Dimensional Analysis 893.3 Preliminaries for Multidimensional Poverty Analysis 933.4 Identification of the Poor and Deprivation Counting 953.5 Axioms for a Multidimensional Poverty Metric 993.5.1 Invariance Axioms 1003.5.2 Distributional Axioms 1033.5.3 Decomposability Axioms 1083.5.4 Threshold Limit Sensitivity Axiom 1113.5.5 Technical Axioms 1123.6 Multidimensional Poverty Measurement 1133.6.1 The Dashboard Approach 1133.6.2 The Direct Approach 1173.6.3 The Inclusive Measure ofWell-being Approach 1233.7 Multidimensional Poverty Orderings 1253.7.1 A Brief Outline of One-Dimensional Orderings 1263.7.2 Multidimensional Orderings 1293.8 Dimensions ofWell-Being with Ordinal Significance and Multidimensional Poverty 1313.9 Orderings Based on Deprivations Counts 1343.10 Multidimensional Material Deprivation 1383.11 Concluding Remarks 142References 1434 Fuzzy Set Approaches to the Measurement of Multidimensional Poverty 1634.1 Introduction 1634.2 Fuzzy Membership Function 1654.3 Axioms for a Fuzzy Multidimensional Poverty Index 1734.3.1 Invariance Axioms 1754.3.1.1 Fuzzy Strong Ratio Scale Invariance 1754.3.1.2 Fuzzy Strong Translation Scale Invariance 1774.3.1.3 FuzzyWeak Focus 1774.3.1.4 Fuzzy Strong Focus 1784.3.1.5 Fuzzy Symmetry 1784.3.1.6 Fuzzy Population Replication Invariance 1794.3.2 Distributional Axioms 1794.3.2.1 Fuzzy Monotonicity 1794.3.2.2 Fuzzy Monotonicity Sensitivity 1804.3.2.3 Fuzzy DimensionalMonotonicity 1814.3.2.4 Fuzzy Transfer 1824.3.2.5 Increasing Fuzzy Poverty under Correlation-Increasing Switch 1844.3.3 Decomposability Axioms 1854.3.3.1 Fuzzy Subgroup Decomposability 1854.3.3.2 Fuzzy Factor Decomposability 1854.3.4 Fuzzy Sensitivity Axiom 1864.3.4.1 Increasing Fuzzy Poverty for Increased Membership Function 1864.3.5 Technical Axioms 1874.3.5.1 Fuzzy Boundedness 1874.3.5.2 Fuzzy Continuity 1874.4 Fuzzy Multidimensional Poverty Indices 1874.5 Fuzzy Poverty Orderings 1914.6 Concluding Remarks 195References 1955 Poverty and Time: A Multidimensional Appraisal 2015.1 Introduction 2015.2 Preliminaries 2075.3 The Block Approach 2085.3.1 IndividualMultidimensional Intertemporal Poverty Index 2085.3.2 Aggregate Multidimensional Intertemporal Poverty Index 2185.3.3 A Review of Some Related One-Dimensional Proposals 2205.4 An Exploration of the Counting Approaches to MultidimensionalIntertemporal Deprivations 2265.5 The Multidimensional Duration Approach 2295.5.1 A Review of One-Dimensional Duration-Reliant Offers 2295.5.2 Axioms for a Chronic Multidimensional Poverty Quantifier 2315.5.3 The Bourguignon–Chakravarty Approach to Chronic Multidimensional Poverty Measurement 2405.6 Intertemporal Poverty Orderings 2435.7 Concluding Remarks 245References 2466 Vulnerability to Poverty: A Multidimensional Evaluation 2516.1 Introduction 2516.2 A Review of One-DimensionalMeasurement 2546.3 Multidimensional Representation of Vulnerability to Poverty: An Axiomatic Investigation 2626.4 Concluding Remarks 269References 2717 An Exploration of Some Composite and Individualistic Indices 2777.1 Introduction 2777.2 Human Development Index 2787.3 Human Poverty Index 2847.4 Gender Inequality Index 2867.5 Better Life Index 2887.6 Active Citizenship Composite Index 2917.7 Measuring Human Opportunity: A Counting Approach 2937.8 Assessment of Progress toward Achievements in Millennium Development Goals 2957.9 Air Quality Index 3017.10 Concluding Remarks 303References 304Index 311