Social Values and Social Indicators: Essays in Normative Economics and Measurement (Themes in Economics) 9811604274, 9789811604270

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Themes in Economics Theory, Empirics, and Policy

S. Subramanian

Social Values and Social Indicators Essays in Normative Economics and Measurement

Themes in Economics Theory, Empirics, and Policy

Series Editors Satish Kumar Jain , Jawaharlal Nehru University, New Delhi, India Karl Ove Moene, Max Planck Institute, Munich, Germany Anjan Mukherji, Jawaharlal Nehru University, New Delhi, India

The series aims to publish monographs and edited volumes, both theoretical and empirical, with important policy implications, on topics of contemporary interest. Volumes in this series are envisaged to be one of the following kinds: (i) Research dealing with important economic theory topics; (ii) Rigourous empirical work on issues of contemporary importance; and(iii) Edited volumes of selected papers, either dealing with a single topic of contemporary interest, or of papers presented in economic theory conferences and workshops. Collections of previously published papers, unless integrated into a connected text, are not encouraged. Some of the topics of contemporary interest that are envisaged to be explored in this series are: • • • • •

poverty, income inequality, eminent domain (land acquisition in particular), some theoretical aspects of the functioning of the market mechanism, economic and social implications of affirmative action.

The list is by no means exhaustive.

More information about this series at http://www.springer.com/series/15590

S. Subramanian

Social Values and Social Indicators Essays in Normative Economics and Measurement

S. Subramanian Independent Scholar Chennai, Tamil Nadu, India

ISSN 2730-5597 ISSN 2730-5600 (electronic) Themes in Economics ISBN 978-981-16-0427-0 ISBN 978-981-16-0428-7 (eBook) https://doi.org/10.1007/978-981-16-0428-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

To the women in my life, Prabha and Daya, with love and gratitude for their tireless (and vocal) belief in the improvability of their husband and father

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

3

4

1 6

Social Values and Social Choice

‘Instrumentalism’ and Friedman’s Methodology: A Short Objection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Sort of Paretian Liberalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Liberalism in a Conventional Social Choice Framework . . . . . . . . 3.2.1 Sen’s Dilemma: Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Some Limitations of Conventional ‘Social Choice’ Formulations of Individual Liberty . . . . . . . . . . . 3.3 Towards an Alternative Framework of Social Choice . . . . . . . . . . 3.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Preference Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Personal Feature Choice Profiles . . . . . . . . . . . . . . . . . . . . 3.3.4 Rights-Waiving Profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.5 Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Paretianism and Libertarianism Revisited . . . . . . . . . . . . . . . . . . . . 3.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Liberty, Equality, and Impossibility: Some General Results in the Space of ‘Soft’ Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Equity and Libertarian Principles . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 The Crisp Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 The Soft Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 12 13 13 16 16 17 19 19 19 20 21 21 21 24 25 27 27 28 32 32 34

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5

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4.4 On the Possibility of Egalitarian Liberalism . . . . . . . . . . . . . . . . . . 4.5 Choice Functions and Exact Choice . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Liberty, Equity, and the Possibility of Consistent Choice . . . . . . . 4.7 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

35 38 41 43 44

The Arrow Paradox with Fuzzy Preferences . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The Asymmetric and Symmetric Components of the Fuzzy Weak Preference Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Fuzzy Aggregation Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Two Possibility Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47

Equality, Priority, and Distributional Judgements . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 On the Robustness of Egalitarianism to Parfit’s Objections . . . . . 6.3 On the Practical Relevance of the Equality–Priority Distinction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Egalitarian Judgements and Additive Separability . . . . . . . . . . . . . 6.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 57 59

Two Logical and Normative Issues Relating to Measurement in the Social Sciences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Social Indicators: Outcomes and Processes . . . . . . . . . . . . . . . . . . . 7.3 Description, Prescription, and Measurement . . . . . . . . . . . . . . . . . . 7.4 Concluding Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II 8

48 51 52 55 56

63 65 68 69 71 71 72 74 76 76

Inequality and Poverty Measurement: Questioning Some Axiomatic Foundations

Social Groups and Economic Poverty: A Problem in Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Measuring Poverty in a Stratified Society . . . . . . . . . . . . . . . . . . . . 8.3 Formalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Two General Possibility Results for Poverty Indices . . . . . . . . . . . 8.5 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6 ‘Group-Sensitive’ Poverty Indices: An Example . . . . . . . . . . . . . . 8.7 Two Implications of ‘Group Sensitivity’ in a Poverty Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 82 84 86 88 89 93 96 97

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9

Reckoning Inter-group Poverty Differentials in the Measurement of Aggregate Poverty . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Poverty Aggregation When Inter-group Inequality is Intrinsically Dis-Valued . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Properties of Poverty Measures When the Distribution of Subgroup Poverty Matters . . . . . . . . . 9.4.2 Two Poverty Measures Which Are Sensitive to the Inter-group Distribution of Poverty . . . . . . . . . . . . . 9.4.3 The Group Poverty Profile and the Group Poverty Lorenz Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.4 Aggregate Poverty Adjusted for Its Unequal Inter-group Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 Poverty Measurement in the Presence of a ‘Group-Affiliation’ Externality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 An ‘Externality-Adjusted’ Poverty Measure . . . . . . . . . . . . . . . . . . 10.3.1 Accommodating Group-Affiliation Externalities: One Specific Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2 Partitioning the Population on the Basis of Multiple Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 An Application to a Non-income Dimension: Measuring Literacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Some Implications of ‘Grouping’ for Poverty Rankings, Anti-poverty Policy, and Poverty Axiomatics . . . . . . . . . . . . . . . . . 10.4.1 Poverty Rankings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 Anti-poverty Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.3 Poverty Axiomatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Revisiting the Normalization Axiom in Poverty Measurement . . . . . 11.1 Introduction: Classifying the Population by Poverty Status . . . . . 11.2 The Sen Normalization Under an Inclusive Definition of the Poor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Axiom N* and Its Rationalization . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 Variants of the Sen and the Foster–Greer–Thorbecke Poverty Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

99 99 100 101 102 102 103 104 107 109 109 111 111 114 115 115 117 119 120 120 120 121 122 123 125 125 129 131 134

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11.5 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 12 The Focus Axiom and Poverty: On the Coexistence of Precise Language and Ambiguous Meaning in Economic Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 Focus and the Possibility of Coherent Aggregation . . . . . . . . . . . . 12.3 Where Focus is Comprehensively Respected: Examples of Measures Which Assess the ‘Quantity of Poverty’ . . . . . . . . . . 12.4 Where Focus is Comprehensively Violated: Examples of Measures Which Assess the ‘Poorness’ of a Society . . . . . . . . . 12.5 Where Focus is Irrelevant: Further Examples of Measures Which Assess the ‘Poorness’ of a Society . . . . . . . . . . . . . . . . . . . . 12.6 Concluding Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Focus: Pro or Con? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

139 140 142 145 145 147 150 152 152 153

13 Assessing Inequality in the Presence of Growth: An Expository Essay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2 Concepts and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.3 The Lorenz Curve and the Gini Coefficient: Beyond Pure Relativism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.4 A ‘Linear Invariance’ Approach to Intermediate Inequality Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5 The Krtscha Approach to Intermediate Inequality Measures . . . . 13.6 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On the Parabolic Shape of the Krtscha Iso-Inequality Curve . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166 169 174 175 175 176

14 Revisiting an Old Theme in the Measurement of Inequality and Poverty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 On Intermediate Measures of Inequality and Poverty . . . . . . . . . . 14.2.1 Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.2 Poverty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 179 180 180 182 187 187

155 155 159 163

15 Inequality Measurement with Subgroup Decomposability and Level-Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 15.2 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

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15.3 Some Observations on Subgroup Decomposability and Level-Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 15.4 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Chapter 1

Introduction

This volume is a product of a post-retirement two-year National Fellowship—which the author is happy to acknowledge—awarded to him by the Indian Council of Social Science Research. The book is a collection of essays written (mainly) over the last decade or so, and dealing, in one way or another, with the place of values in economic analysis. The centrality of values in the collection is not surprising, given that the thematic concerns informing the essays in the book relate principally to methodological issues in economic enquiry, to the normatively constrained aggregation of personal preferences into collective choice, and to problems of logical coherence and ethical appeal in the axiom systems underlying the measurement of economic and social phenomena such as poverty, inequality, and literacy. While many of the essays are more or less ‘technical’ in nature, it is the author’s hope that they are all explicitly motivated by considerations that go beyond the formalisms of presentation to an involvement with the role of moral reasoning in economic analysis. In particular, it would be a cause for satisfaction to the author if the work done in the last decade or so of his official career, as reflected in this selection of essays, could be seen to emphasize the importance of ‘ought propositions’ in a science which is all too often regarded as being wholly and exclusively ‘positive’ in its orientation. In what follows, the chapters in the book are briefly introduced. If the discussion is somewhat cryptic, that is because the objective here is only to provide a very quick account of the principal thematic focus of each chapter, leaving it to the chapter itself for a detailed elaboration of its concerns. Chapter 2 (“‘Instrumentalism’ and Friedman’s Methodology: A Short Objection”) is a rather simple and straightforward critique of the perceived role of assumptions in Friedman’s (1953) ‘methodology of positive economics’. ‘Instrumentalist’ interpretations of economic theories have been held to provide justification for the rightness of being indifferent to the ‘realism’ or otherwise of the assumptions underlying a theory. In this view, what is important for an economist engaged in formulating policy is to have a theory which is capable of generating accurate predictions. This note asserts, with the help of a very simple example, that theories based on unrealistic assumptions © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_1

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1 Introduction

are—precisely contrary to its claim—inimical to the interests of an instrumentalist view of theories. Tests of predictive accuracy can be narrow or broad. When narrow, the likelihood is high of there being a large number of theories passing the test—but this could also raise the chances of survival of (unrealistic) theories which may have unintended and undesirable policy outcomes. On the other hand, when the scope of the test of predictive accuracy is widened, the set of contending theories is likely to be small, with the survivors restricted to those based on ‘realistic’ assumptions. Chapters 3–5 deal with the social choice problem of aggregating individual rankings of social states into a collective ranking. Chapter 3 (“A Sort of Paretian Liberalism”) reviews Sen’s (1970) well-known result on ‘the impossibility of a Paretian liberal’. In so doing, it takes stock of certain difficulties associated with conventional social choice theoretic formulations of the notion of individual liberty. An alternative formulation, in the spirit of an ‘outcome-oriented’ version of Nozick’s (1974) approach to individual liberty as the freedom to fix certain personal features of the world, is advanced. An ‘extended’ version of a conventional social choice function, called a ‘social selection function’ (SSF), is introduced: the domain of the SSF is enlarged to include, apart from preference profiles, what are called ‘personal choice profiles’ and ‘rights-waiving profiles’. Within this framework, it is noted that a version of Sen’s original impossibility result can be recovered. It is also pointed out that there is an alternative, ethically plausible construction which can be placed on the notion of ‘Paretian liberalism’, and it is demonstrated that this is a coherent construction. Chapters 4 and 5 deal with the aggregation of ‘fuzzy’ rather than ‘exact’ preferences—the former deals with Sen’s (1970) ‘liberal paradox’, and the latter with the ‘Arrow (1963) paradox’. Chapter 4 (“Liberty, Equality, and Impossibility: Some General Results in the Space of ‘Soft’ Preferences”) is concerned with examining the mutual compatibility of the ethical principles of equity and liberty in a social choice framework of ordinally formulated vague preferences. With sufficiently weakened versions of the liberty and equity principles, one can secure an existence result in a ‘relation-functional’ setting. However, difficulties tend to reappear in a ‘choicefunctional’ setting, when one subscribes to the notion that while preference may be vague, choice must perforce be exact. Chapter 5 (“The Arrow Paradox with Fuzzy Preferences”) considers a new factorization of a fuzzy weak binary preference relation into its asymmetric and symmetric parts. Arrow’s general possibility theorem is then examined within the resulting framework of vague individual and social preferences. The outcome of this exercise is compared with some earlier results available in the literature on the Arrow paradox with fuzzy preferences. Chapter 6 (“Equality, Priority, and Distributional Judgements”) undertakes an assessment of the substantive significance of Derek Parfit’s distinction between Prioritarianism and Egalitarianism. In providing a brief critique of Parfit’s arguments, the essay draws on the author’s own earlier work and that of Thomas Christiano and Will Braynen, John Broome, and Marc Fleurbaey. It considers issues relating to the ‘levelling down objection’, the ‘Divided World example’, and the distinction between ‘absolute’ and ‘relative’ valuations of individual benefit. It is

1 Introduction

3

contended that ‘levelling down’ presents a difficulty only for ‘pure telic egalitarianism’, not for ‘pluralist telic egalitarianism’; that one can have an egalitarian rationalization for favouring equality in the distribution of a smaller sum of well-being over inequality in the distribution of a larger sum even in a ‘Divided World’; and that, while a particular ‘absolute’/‘relative’ dichotomy is relevant for a particular ‘distribution-invariance’/‘distribution-sensitivity’ dichotomy, the resulting distinction is useful for differentiating two types of Egalitarian rather than for differentiating a non-Egalitarian principle such as Prioritarianism from Egalitarianism. Chapter 7 (“Two Logical and Normative Issues Relating to Measurement in the Social Sciences”) is an extract from a longer essay and is based on the premise that measurement is very important for the social sciences. However, it also enjoins care on the practitioner’s part in his or her engagement with the project of measurement. It deals, in particular, with two often overlooked issues with which quantification in the social sciences should be concerned: (1) social indicators in relation to the contrast between outcomes and processes; and (2) measurement which tends to depend on the derivation of ‘ought’ propositions from ‘is’ propositions. ‘Decomposition analysis’ in measurement is found to play a role in both issues. The rest of the book is devoted to essays on social indicators, principally indicators of poverty and inequality. The focus here is on some of the elementary axiomatic foundations of measurement, on the view that axioms which are routinely accepted as plausible and innocuous have a way of turning out to be less than wholly persuasive when we consider contexts of measurement that are a little different from the ‘canonical’ settings in which measurement is customarily carried out. This is true, in inequality and poverty measurement, for the properties of ‘transfer’, ‘symmetry’, and ‘focus’ when we move away from the conventional emphasis on interpersonal disparities to inter-group disparities in achievement. Chapters 8–10 deal with poverty, and Chap. 15 deals with inequality, from an inter-group perspective. The first of these chapters (“Social Groups and Economic Poverty: A Problem in Measurement”) points to some elementary conflicts between the claims of interpersonal and intergroup justice as they manifest themselves in the process of seeking a real-valued index of poverty which is required to satisfy certain seemingly desirable properties. It indicates how ‘group-sensitive’ poverty measures, similar to the Anand and Sen (1995) ‘Gender-Adjusted Human Development Index’ and the Subramanian and Majumdar (2002) ‘Group-Disparity Adjusted Deprivation Index’, may be constructed. Some properties of a specific ‘group-sensitive’ poverty index are appraised, and the advantage of having a ‘flexible’ measure which is capable of effecting a trade-off between the claims of interpersonal and inter-group equality is spelt out. The implications of directly incorporating group disparities into the measurement of poverty for poverty comparisons and anti-poverty policy are also discussed. This theme is pursued further in Chap. 9 (“Reckoning Inter-group Poverty Differentials in the Measurement of Aggregate Poverty”). In a heterogeneous population which can be partitioned into well-defined subgroups, it is plausible that the extent of measured aggregate poverty should depend upon the distribution of poverty across the subgroups. In particular, a judgement against an unequal inter-group distribution of poverty can be upheld as an intrinsic social virtue. The aggregate measure

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of poverty, in line with this view, would then lend itself to ‘penal adjustment’ in order to reflect the extent of inter-group disparity in the distribution of poverty that obtains. In the present paper, this approach to poverty measurement is examined with specific reference to the advancement of a diagrammatic aid to analysis called the group poverty profile. The latter is a virtual transplantation, to the present context, of the notion of a deprivation profile that has been explored and analysed by A. F. Shorrocks in a different context (Shorrocks 1995, 1996). Chapter 10 (“Poverty Measurement in the Presence of a ‘Group Affiliation Externality’”) considers the implications for poverty measurement of the observed fact that any individual’s level of deprivation is a function not only of her own income, but of the general level of prosperity of the group to which she is affiliated. Individual deprivation functions are specialized to a form which reflects this ‘group-affiliation’ externality, and the resulting poverty measure is studied with respect to its properties and its implications for poverty rankings. Mainstream approaches to measuring deprivation tend to neglect group-related externalities in favour of a certain thoroughgoing ‘individualism’. This paper is a preliminary attempt at filling this gap. The apparent straightforwardness of the ‘normalization’ axiom in poverty measurement is subjected to scrutiny in Chap. 11 (“Revisiting the Normalization Axiom in Poverty Measurement”). It is contended in this essay that the ‘normalization’ axiom associated with Sen’s (1976) poverty index—and this, indeed, holds for most extant measures of poverty—entails an uncomfortable implication when we adopt a strong, or inclusive, definition of the poor. The paper suggests that we may not always be at liberty to adopt a weak definition. The available alternative then is to change the form of the poverty measure. Accordingly, a modification of the Sen normalization axiom which leads to a variant of Sen’s index, together with a variant also of the Foster et al. (1984) poverty measures, is advanced and discussed. The derivation of the new normalization axiom benefits from Basu’s (1985) decomposition of the Sen axiom. Despite the formal rigour that attends social and economic measurement, the substantive meaning of particular measures could be compromised in the absence of a clear and coherent conceptualization of the phenomenon being measured (Chap. 12: “The Focus Axiom and Poverty: On the Co-existence of Precise Language and Ambiguous Meaning in Economic Measurement”). A case in point is afforded by the status of a ‘focus axiom’ in the measurement of poverty. ‘Focus’ requires that a measure of poverty ought to be sensitive only to changes in the income distribution of the poor population of any society. In practice, most poverty indices advanced in the literature satisfy an ‘income-focus’ but not a ‘population-focus’ axiom. This, it is argued in the paper, makes for an incoherent underlying conception of poverty. The paper provides examples of poverty measures which either satisfy both income focus and population focus or violate both, or which effectively do not recognize a clear dichotomization of a population into its poor and non-poor components, and suggests that such measures possess a virtue of consistency, and coherent meaning, lacking in most extant measures of poverty available in the literature. Two other axioms in the measurement of inequality and poverty which tend to be routinely regarded as unproblematic are the ‘invariance’ axioms of scale and

1 Introduction

5

population. In a great deal of empirical work on distributional analysis, the only sorts of inequality measures employed are ‘relative’ ones, namely those that satisfy the scale- and replication-invariance properties. Scale invariance requires that an equiproportionate change in incomes should leave the extent of measured poverty and inequality unchanged, while replication invariance demands the same outcome for population replications at each level of income. Closer examination suggests that these two axioms are actually in the nature of elephants in the drawing room of distributional analysis. Scale invariance in inequality measurement is investigated in Chap. 13 (“Assessing Inequality in the Presence of Growth: An Expository Essay”), which argues the case for a more plural approach to assessments of the conditions under which distributions with different means can be interpreted as reflecting equal inequality. It advocates the importance of considering both ‘absolute’ and ‘intermediate’, as opposed to exclusively ‘relative’, conceptualizations of inequality. In particular, it is suggested that there is a strong case for the employment of intermediate measures of inequality in assessing overtime changes in inequality and, through that route, the inclusiveness or otherwise of the process of growth in per capita income. The purpose of the paper is twofold: exposition and persuasion. Chapter 14 (“Revisiting an Old Theme in the Measurement of Inequality and Poverty”) explores this theme in the context of both inequality and poverty measurement and considers specific centrist measures that are variants of the well-know Gini coefficient of relative inequality and the Foster– Greer–Thorbecke family of relative poverty indices. The essay advocates the routine use of ‘centrist’ measures, and in the process, it revisits some old debates on the logical adequacy and normative appeal of measures of inequality and poverty that are either wholly relative or wholly absolute. The implication of these issues for the diagnosis of magnitudes and trends in inequality and poverty is illustrated by means of a couple of simple empirical examples drawn from Indian data. Chapter 15 (“Inequality Measurement with Subgroup Decomposability and LevelSensitivity”) deals with two properties which it may be desirable for an inequality measure to possess. Subgroup decomposability is a very useful property in an inequality measure, and level-sensitivity, which requires a given level of inequality to acquire a greater significance the poorer a population is, is a distributionally appealing axiom for an inequality index to satisfy. In this essay, which is largely in the nature of a recollection of important results on the characterization of subgroup decomposable inequality measures, the mutual compatibility of subgroup decomposability and level-sensitivity is examined, with specific reference to a classification of inequality measures into relative, absolute, centrist, and unit-consistent types. Arguably, the most appealing combination of properties for a symmetric, continuous, normalized, transfer-preferring, and replication-invariant (S-C-N-T-R) inequality measure to satisfy is that of subgroup decomposability, centrism, unit consistency, and level-sensitivity. The existence of such an inequality index is (as far as this author is aware) yet to be established. However, it can be shown, as is done in this paper, that there does exist an S-C-N-T-R measure satisfying the (plausibly) next-best combination of properties—those of decomposability, centrism, unit consistency, and level-neutrality.

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Taken together, the essays in this volume could be said to address issues in economic analysis with a philosophical slant. It is unlikely that the reader will agree with all, or even most, of the positions advanced and defended by the author. But the elicitation of ‘consensus’ from its readers is too large an ambition for any book to entertain. A more modest outcome than ‘consensus’, but nevertheless one that is at least as satisfying, would be an acknowledgement from the reader that the issues dealt with in the book are informed by normative considerations that deserve to be addressed.

References Anand S, Sen A (1995) Gender inequality in human development: theories and measurement, Occasional Paper, 19, Human Development Report Office. New York: UNDP. Cited in UNDP, 1995 Arrow, KJ (1963) Social choice and individual values, 2nd edn. John Wiley and Sons, New York Basu K (1985) Poverty measurement: a decomposition of the normalization axiom. Econometrica 53(6):1439–1443 Foster J, Greer J, Thorbecke E (1984) A class of decomposable poverty indices. Econometrica 52(3):761–65 Friedman, M (1953) The methodology of positive economics. Essays in Positive Economics. University of Chicago Press (1963), Chicago, pp. 3–41 Nozick R (1974) Anarchy, State and Utopia. Oxford, Blackwell Sen AK (1970) The impossibility of a Paretian liberal. J Polit Econ 78(1):152–159 Sen AK (1976) Real National Income. Review of Economic Studies 43(1):19–39 Shorrocks, AF (1995) Revisiting the Sen poverty index. Econometrica 63(5):1225–1230 Shorrocks AF (1996) Deprivation profiles and deprivation indices. In: Jenkins S, Kapteyn A, Vaan Praag B (eds) The distribution of welfare and household production: international perspectives. Cambridge University Press, London Subramanian S, Majumdar M (2002) On measuring deprivation adjusted for group disparities. Social Choice Welfare 19(2):265–280

Part I

Social Values and Social Choice

Chapter 2

‘Instrumentalism’ and Friedman’s Methodology: A Short Objection

Abstract ‘Instrumentalist’ interpretations of economic theories have been held to provide justification for the rightness of being indifferent to the ‘realism’ or otherwise of the assumptions underlying a theory. In this view, what is important for an economist engaged in formulating policy is to have a theory which is capable of generating accurate predictions. This note asserts, with the help of a very simple example, that theories based on unrealistic assumptions are—precisely contrary to its claim—inimical to the interests of an instrumentalist view of theories. Tests of predictive accuracy can be narrow or broad. When narrow, the likelihood is high of there being a large number of theories passing the test—but this could also raise the chances of survival of (unrealistic) theories which may have unintended and undesirable policy outcomes. On the other hand, when the scope of the test of predictive accuracy is widened, the set of contending theories is likely to be small, with the survivors restricted to those based on ‘realistic’ assumptions. Keywords Friedman’s methodology · ‘Instrumentalism’ · Theories · Realism of assumptions …Ten false philosophies will fit the universe; ten false theories will fit Glengyle Castle. But we want the real explanation of the castle and the universe... —Father Brown in G. K. Chesterton’s ‘The Honour of Israel Gow’ (The Innocence of Father Brown)

Lawrence Boland (1979) urges the view that Milton Friedman’s proposed ‘methodology of positive economics’ (Friedman 1963) ought to be seen primarily as advocating an ‘instrumentalist’ interpretation of theories. In particular, if a theory generates conclusions or predictions that are compatible with known facts, then such a theory is a useful instrument for a policy-oriented economist. As he puts it (Boland 1979; pp. 508–509): This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2013): “Instrumentalism’ and Friedman’s Methodology: A Short Objection’, Quality and Quantity: International Journal of Methodology, 47(1): 577–580. https://doi.org/10.1007/s11135-011-9480-7. An earlier version originally appeared in the house journal, the Bulletin, of the Madras Institute of Development Studies (Subramanian 1989). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_2

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2 ‘Instrumentalism’ and Friedman’s Methodology: A Short Objection So long as a theory does its intended job, there is no apparent need to argue in its favour (or in favour of any of its constituent parts). For some policy-oriented economists, the intended job is the generation of true or successful predictions. In this case, a theory’s predictive success is always a sufficient argument in its favour. This view of the role of theories is called ‘instrumentalism’. It says that theories are convenient and useful ways of (logically) generating what have turned out to be true (or successful) predictions or conclusions. Instrumentalism is the primary methodological point of view expressed in Friedman’s essay. …[I]f the object of building or choosing theories (or models of theories) is only to have a theory or model that provides true predictions or conclusions, a priori truth of the assumptions is not required if it is already known that the conclusions are true or acceptable by some conventionalist criterion. Thus, theories do not have to be considered true statements about the nature of the world, but only convenient ways of systematically generating the already known ‘true’ conclusions.

In terms of this ‘instrumentalist’ defence of what Paul Samuelson (1963) has called the ‘F-twist’, one presumes that for a ‘policy-oriented economist’ the truth of the conclusions generated by a theory is a sufficient argument in favour of the theory, precisely because such a theory then serves the purpose of initiating (rationalizable) policy interventions. But then an inescapable corollary to this proposition seems to be that one cannot any longer be indifferent to the realism or otherwise of the assumptions underlying a theory. This point is argued with the help of a simple example. A widely observed phenomenon in the context of Indian agriculture is that the rental share in crop-share tenancy arrangements displays a certain constancy at the value of one-half. Two alternative ‘explanations’—which I shall call E 1 and E 2 , respectively—for this observed phenomenon are provided below. E 1 : The landlord and the tenant (designated L and T, respectively) are both self-interested and altruistic utility-maximizers: their utility functions, UL and UT , respectively, are taken to be identically the same, and given by: UL = UT = U (YL , YT ) = (YL )σ + (YT )σ , where YL (respectively, YT ) is the amount of the total output Y that goes to the landlord (respectively, the tenant), and σ is a positive real number less than unity. In this view of the matter, each agent’s utility is an increasing and strictly concave function of her own, and of the other agent’s, receipt of output, and each agent values her own income exactly as she does the income of the other agent. If r is the rental rate, then it is clear that YL = r Y and YT = (1 − r )Y , so that UL = UT = U (YL , YT ) = [r σ + (1 −r )σ ]Y σ . If the landlord is assumed to choose the rental rate in such a way as to maximize his own utility, it follows that the solution to this elementary optimization problem will be reflected in a value of one-half for r . (As it happens, this also is the value of r which the tenant, as a utility-maximizer equipped with the utility function that has been credited to her, would have chosen had she been given the option.) E 2 : Output Y is assumed to be an increasing, strictly concave, and indefinitely differentiable function of the amount of labour (L) inputted, and will, in fact, be taken to be represented by the specialized form Y (L) = L 1/2 . The tenant’s opportunity cost of labour is w per unit, and if r is the rental share, the tenant’s surplus is given by s = (1 − r )Y (L) − wL = (1 − r )L 1/2 − wL . It is easily verified that the tenant’s

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surplus-maximizing choice of L is given by L ∗ = ((1 − r )/2w)2 , so that the optimal level of output will be given by Y (L ∗ ) = (1 − r )/2w. The landlord’s surplus is then, clearly, r (1 − r )/2w; if the landlord is assumed to have the power of determining the rental share, and if he is credited with profit-maximizing behaviour, then clearly his surplus r (1 − r )/2w will be maximized by a choice of r equal to one-half. Let us suppose that E 1 is the ‘true’ explanation of the observed phenomenon of a rental share of one-half. Note, however, that E 2 ‘predicts’ as well as E 1 . If now a policy-maker proceeds on the presumed ‘truth’ of E 2 , and decides that profitmaximizing landlords can afford a squeeze in their unearned surpluses, he may be guided to legislate a rental rate of, say, one-fourth, with a view to offering tenants a ‘better deal’. In terms of what is actually happening ‘on the ground’, however, the only effect of this policy will be to force tenants to a lower level of utility (it can be verified that the tenant’s—as well as the landlord’s—utility will decline by an amount of [(3 − 3σ )/4σ )]Y σ when the rental rate declines from one-half to one-fourth). Briefly, a theory of selfish and profit-maximizing landlords and tenants (E 2 ) ‘predicts’ as well as a theory of self-interested-cum-altruistic and utility-maximizing landlords and tenants (E 1 ). Rationalization, on the grounds of E 2 , of a policy-intervention intended to make certain agents better off, only succeeds in rendering all agents worse off (when E 1 is the ‘true’ explanation). In the face of this, how does one maintain—particularly if one is adopting an instrumentalist position—indifference to the realism or otherwise of the assumptions of a theory? It can be objected that the example above achieves its ends simply by placing too restrictive an interpretation on the notion of ‘prediction’. A theory must be construed to fail the ‘predictive test’—it may be argued—if policies based on it generate outcomes that were unintended, or are incompatible with other conclusions that may come to light after the policy has been implemented. In this sense, E 2 in the example just discussed must be deemed a failure. (But note also that the failure is a product of indifference to the realism or otherwise of the assumptions underlying the theory; and it is a moot point whether an economy can be expected to bear the costs of possibly repeated failures within such a heuristic approach.) However, by assigning a much broader scope to the notion of ‘predictive success’, one must strain the possibilities of coincidence in order to accommodate a multiplicity of theories (based on ‘false’ assumptions) which precisely replicate each other’s true predictions. In the context of the example discussed earlier, if the set of ‘known true conclusions’ is restricted to the single observation that the rental share is one-half, then both E 1 and E 2 are eligible for being regarded as satisfactory theories; but if the set of ‘known true conclusions’ is expanded to admit the observation that tenants do not favour a rental share of less than one-half, then E 2 drops out as an eligible theory. In other words, considerations of probability must dictate that when theories are whittled away, one by one, owing to failure of the ‘predictive test’ (when the scope of the latter is not confined to a few ‘immediate’ implications), then the tendency must be towards a unique theory which is based on realistic assumptions. (This point of view, incidentally, is in no way incompatible with David Miller’s denial of the possibility of attaining ‘warranted/justified knowledge’, on which—for a particularly compelling and lucid piece of reasoning—see Miller 1999).

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To summarize: if the predictive test of a theory is a ‘narrow’ one, then a theory based on unrealistic assumptions might pass it—but at the cost of initiating policies that have unintended (and presumably undesirable) outcomes; if the predictive test is a ‘broad’ one, then the possibility that a theory based on unrealistic assumptions will survive it is weak. The share-cropping example discussed in this note highlights both aspects of the problem.

References Boland LA (1979) A critique of Friedman’s critics. J Econ Literature 17(2):503–-522 (1979) Friedman M (1963) The methodology of positive economics. In: Friedman M (ed) Essays in positive economics. University of Chicago Press, Chicago, pp 3–41 Miller D (1999) Being an absolute sceptic. Science 284(5420):1625–1626 Samuelson PA (1963) Problems of methodology—discussion, american economic review. Papers Proc 53(2):231–236 Subramanian S (1989) “Instrumentalism” and Friedman’s methodology: a brief comment. Bull Madras Inst Dev Stud 19(11):571–573

Chapter 3

A Sort of Paretian Liberalism

Abstract This paper reviews Amartya Sen’s well-known result on ‘the impossibility of a Paretian liberal’. In so doing, it takes stock of certain difficulties associated with conventional social choice theoretic formulations of the notion of individual liberty. An alternative formulation, in the spirit of an ‘outcome-oriented’ version of Nozick’s approach to individual liberty as the freedom to fix certain personal features of the world, is advanced. An ‘extended’ version of a conventional social choice function, called a ‘social selection function’ (SSF), is introduced: the domain of the SSF is enlarged to include, apart from preference profiles, what are called ‘personal choice profiles’ and ‘rights-waiving profiles’. Within this framework, it is noted that a version of Sen’s original impossibility result can be recovered. It is also pointed out that there is an alternative, ethically plausible construction which can be placed on the notion of ‘Paretian liberalism’, and it is demonstrated that this is a coherent construction. Keywords · Minimal liberalism · Gibbard liberalism · Preference profiles · Personal choice profiles · Rights-waiving profiles · Social choice function · Social selection function · Paretian liberalism

3.1 Introduction The theme of this conference is governance. While the objective of those presiding over the governance of a society is frequently hard to reconcile with any suggestion of benignity, it is at least conceivable that governors (even of state legislatures) are sometimes inspired by honourable motives. At any rate, we shall assume that the goal of governance is good, rather than bad, governance. If such is the case, then governance will be concerned with fostering such values and institutions as are conducive to the realization of the right and the good. The ethical principle of efficiency, as captured in the notion of Pareto optimality, is one such good; and the This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2006) A Sort of Paretian liberalism, The Journal of International Trade and Economic Development, 15:3, 311–324. https://doi.org/10.1080/09638190600871628. Thanks are owed to Taposik Banerjee for helpful comments on an earlier version of the paper. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_3

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ethical principle of negative freedom, as captured in the notion of individual liberty, is one such right. It would appear to be unexceptionable to demand of good governance that it should promote mechanisms for the attainment of the most extensive possible collective efficiency and personal liberty. One goal of good governance, therefore, could be that of securing the possibility of what Amartya Sen (1970a, b) has called Paretian liberalism. Before one can address questions of governance-related mechanisms, institutions, and stratagems for the realization of a goal, one must first address the prior question of whether the goal under review is a coherent one. The concern, in this paper, will be precisely—and only—with this question: is the notion of Paretian liberalism a conceptually meaningful, i.e. contradiction-free, one? The importance of the question resides in the fact that it has, in the work of Sen (1970a, b), elicited an answer in the negative: he has demonstrated that there is a well-defined sense in which an untrammelled pursuit of the principles of Paretianism and liberalism is a lost cause—one which is most realistically consigned to the realm of logical impossibility. A simple example, which has some implications for governance and development, may clarify the issue. For any developing economy, the aggregate level of saving is an important determinant of the economy’s growth prospects and also has implications for deriving the appropriate ‘accounting’ rate of interest relevant for cost–benefit analyses of public sector projects (see Sen 1961, 1967, and Marglin 1963). It is reasonable to expect that saving decisions will be presided over by a system of governance which is informed by respect for both individual liberty and collective efficiency. But is this always possible? To see why it may not be (see Subramanian 1987), imagine a two-person society comprising individuals 1 and 2 respectively, and let the social states available be given by a = (N, N), b = (S, N), and c = (S, S), where the first element of each social state describes 1’s activity and the second element 2’s activity, with N standing for ‘does not save’, and S standing for ‘saves’. It is conceivable that each person, left to himself, would prefer not saving to saving, in line with the dictates of a certain sort of ‘private rationality’, and yet each might also prefer a state in which both persons save to one in which neither does, resulting in the well-known problem of atomistic rationality leading to collective suboptimality in the presence of mutual inter-dependencies. Specifically, we shall assume that person 1 prefers state a to state b, that 2 prefers b to c, and that both individuals prefer c to a. Note that the states a and b differ only with respect to a feature which is personal to individual 1, while the states b and c differ only with respect to a feature which is personal to individual 2, and Sen’s notion of liberty would require that these personal preferences should be socially respected. In view of 1’s preference over the pair (a, b) and 2’s preference over the pair (b, c), libertarian considerations will dictate that the states b and c are not socially chosen, while the Pareto principle, which would seek reflection of unanimous preferences in social choice, will dictate—given that both individuals prefer c to a—that a is not socially chosen. The dictates of liberalism and Paretianism have, between them, ensured the impossibility of coherent collective choice. The objective of this paper will be to examine if some plausible notion of Paretian liberalism can nevertheless be rescued from the nihilism of Sen’s result.

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Given that an ‘ideal’ Paretian liberal is a contradiction in terms, it is meaningful to ask if one can identify ‘compromise candidates’ for the epithet of ‘Paretian liberal’— candidates that emerge from a recognition of the fact that the pursuit of efficiency may be fruitfully guided by considerations of liberty, just as the pursuit of liberty may be fruitfully guided by considerations of efficiency. Efforts at such identification have, of course, been made in the past. For example, Vallentyne (1988, 1989) presents a case for what he calls a ‘rights-based axiological theory’, in which respect for individual rights serves as a mechanism for narrowing down social choice from within the set of possible Pareto optima: this leads to the formulation of a certain kind of ‘libertyconstrained Paretianism’. As for the alternative possibility, perhaps the best-known case for according priority to libertarian rights, and seeking efficiency within the restrictions imposed by this scheme of prioritization, is due to Nozick (1974): this could lead to the formulation of a certain kind of ‘Pareto-constrained liberalism.’ The present paper explores yet another way of conceptualizing the notion of Paretian liberalism. The object is not to seek a ‘resolution’ of Sen’s dilemma; rather it is, while deferring to his impossibility result, to seek an alternative, constrained version of the notion. In the process of formulating this version, the conventional social choice formulation of liberty is critiqued; an alternative approach to liberty as ‘freedom of choice’ is developed; and the particular form of ‘Paretian liberalism’ which is eventually advanced is informed also by a provision for ‘rights-waiving’, after the manner suggested by Gibbard (1974). A Paretian liberal of the type described in this paper would endorse the following: that social choice should be directed towards selection of a liberty-respecting state which is also Pareto optimal, if such a state exists. Failing this, individual members of the society should be invited to declare whether they would or would not be willing to waive their rights in favour of a Pareto-efficient outcome which Pareto dominates the liberty-respecting social state. The understanding should be (a) that such an outcome will be socially chosen provided only that every person votes to waive her or his rights in its favour, and (b) that, in the absence of unanimous voting in favour of rights-waiving, it is the rights-respecting state that will be socially chosen. Under this scheme, the demands of both Paretianism and liberalism will be satisfied whenever it is feasible to do so; where this is not possible, Pareto-efficient social choice will be prioritized—but subject only to a unanimous subscription to what Basu (1984) calls the ‘meta-right to give up rights’, in the absence of which, it is liberty-respecting social choice that will be upheld. Both principles, it can be claimed, will have been given their due in this version of Paretian liberalism. It should be emphasized that this paper draws on a number of earlier contributions to the literature, in a bid to present a synthetic view of the appealing features of these various contributions: in this sense, the paper cannot make any large claims to originality. It is organized as follows. Section 3.2 offers a critique of conventional social choice formulations of the liberty principle. Section 3.3 deals with a set of formal definitions of the notions of personal selection profiles, preference profiles, and rights-waiving profiles, which enter into the construction of what is called a social selection function. In Sect. 3.4, an Unrestricted Domain principle, an individual liberty principle, and a Pareto principle are defined as restrictions on the

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social selection function; Sen’s ‘impossibility of a Paretian liberal’ result is recovered within this framework; additionally, a new ‘Pareto-Sensitive Liberty’ principle is defined, and an alternative notion of ‘Paretian liberalism’ is advanced, and defended, as a plausible and constructive way of interpreting the notion. Section 3.5 offers a concluding discussion in which the ‘sort of Paretian liberalism’ advanced in Sect. 3.4 is appraised from both ‘pragmatic’ and ‘ethical’ perspectives.

3.2 Liberalism in a Conventional Social Choice Framework 3.2.1 Sen’s Dilemma: Recall N = {1, . . . , i, . . . , n}, n ≥ 2 is the finite set of individuals constituting society. X (#X ≥ 4) = {x, y, z, …} is the finite set of all conceivable social states. Following Gibbard (1974), for every i ∈ N , there is assumed to exist a personal issue X i = q(i) {xi1 , xi2 , . . . , xi } which is a non-empty finite set, with at least two elements, of features of the possible social states that may be regarded as being ‘private’ or ‘personal’ to individual i. The cardinality of X i is denoted by q(i). Wholly for reasons of subsequent notational convenience, we shall make the assumption that the personal issues of all individuals are of the same cardinality, viz. q(i) = q for all i ∈ N . The set of all conceivable social states is given by the Cartesian product X = X 1 × X 2 ×. . .× X n . A typical social state x ∈ X can be written as a list x = (x1 , . . . , xi , . . . , xn ), where, for all i, x i is a description of i’s private feature in the state x. The null setexcluded power set of X is written X. For each individual i ∈ N , there is assumed to exist a ‘weak’ ordering Ri on X, and Pi is the asymmetric component of Ri . R is the set of all possible weak orders on X, and Rn is the n-fold Cartesian product of R. Every element σ ∈ Rn is a preference profile, or n-tuple (Ri ) of individual preference orders on X, one preference order for each individual. A social choice function (SCF) is a mapping C :  → X such that, for every σ ∈ (⊆ Rn ) : φ = C(σ ) ⊆ X . For every σ ∈ , C(σ ) is the set of socially chosen alternatives from X, given the preference profile σ , or the choice set of X. Three restrictions one could impose on an SCF are Unrestricted Domain (Condition U), the Weak Pareto Principle (Condition P), and Sen’s (1970a, b) principle of Minimal Liberalism (Condition ML). These are defined below. An SCF C :  → X satisfies Condition U if and only if  = Rn , viz. the domain of the choice function admits every logically possible preference profile, and it satisfies Condition P if and only if for every σ ∈ : C(σ ) ⊆ E(σ ),

3.2 Liberalism in a Conventional Social Choice Framework

17

where E(σ ) ≡ {x ∈ X |¬[∃y ∈ X : (∀i ∈ N : y Pi x)]} . That is, the weak Pareto principle confines social choice to the set E(·) of efficient, or Pareto-undominated, alternatives in X. To define Sen’s principle of Minimal Liberalism, it is useful, following Gibbard (1974), to first define, for every i ∈ N , the set of all pairs of i-variants that can be constituted from X, as the set Di ≡ {(x, y) ∈  X 2 ∀ j ∈ N \{i} : x j = y j and xi = yi } . Thus, x and y are i-variants if they differ only with respect to a feature which is personal to individual i. Minimal Liberalism can now be defined as follows. An SCF C :  → X satisfies Condition ML if and only if for every σ ∈ : ∃ j, k ∈ N and ∃[(x, y) ∈ D j and (w, z) ∈ Dk ] such that     x P j y resp., y P j x → y ∈ / C(σ ) resp., x ∈ / C(σ ) and     w Pk z resp., z Pk w → z ∈ / C(σ ) resp., w ∈ / C(σ ) . That is, Minimal Liberalism requires that for each of at least two individuals there should be at least one pair of alternatives in her ‘protected sphere’ such that the concerned individual is decisive in the social choice over the pair. Sen’s ‘impossibility of a Paretian liberal’ result can be stated as follows: there exists no SCF C :  → X satisfying Conditions U, P and ML (see Sen 1970a). The proof essentially revolves around invoking permissible preference configurations in which ‘nosy’ externalities play a part: the result exemplifies the possibility that the choice set can be systematically emptied out by the conflict between ‘private’ preferences (seeking reflection in pair-wise choices through the operation of Condition ML) and ‘all-things-considered’ preferences (seeking reflection in pair-wise choices through the operation of Condition P).

3.2.2 Some Limitations of Conventional ‘Social Choice’ Formulations of Individual Liberty Sen envisaged his principle of Minimal Liberalism as a necessary condition for guaranteeing at least a minimum area of autonomy for an individual in deciding the outcome in relation to some purely personal affair of his, such as whether the person should sleep on his back or on his belly. Unfortunately, the scope of Condition ML seems to be so minimal that it might actually fail to guarantee even the extremely restricted individual autonomy it was designed to secure (on which see also Gardenfors 1981). To see this, consider the following simple example revolving around a two-person society in which for each person i (i = 1, 2), i’s personal issue X i is given by X i = {xi1 , xi2 }, with xi1 (respectively, xi2 ) standing for i’s sleeping on his back (respectively, stomach). The set X of social states is {x, y, w, z}, where x ≡ (x11 , x21 ), y ≡ (x12 , x21 ), w ≡ (x11 , x22 ), and z ≡ (x12 , x22 ). Suppose that other

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things remaining the same each person prefers sleeping on his back to sleeping on his belly: x P1 y, w P1 z, x P2 w, and y P2 z; and let σ ∗ be a preference profile with which these individual preferences are compatible. Suppose further that person 1 is assigned decisiveness over the pair of alternatives (x, y) and person 2 over the pair / C(σ ∗ ). All that (x, w). Then, by Condition ML, one must have: y ∈ / C(σ ∗ ) and w ∈ Condition ML does is to expel the states y and w from the set of socially chosen outcomes. It is, however, perfectly compatible with the requirement of Condition ML to have C(σ ∗ ) = {z}, that is, for society to choose an alternative in which each person, contrary to his wishes in this wholly private matter, ends up sleeping on his stomach. From a purely formal point of view, it is desirable to place as weak restrictions as possible on the social choice rule in deriving an impossibility result: Sen’s result on the impossibility of a Paretian liberal is strong precisely because it employs such a weak condition of individual liberty as Minimal Liberalism. However, from a substantive point of view, it must be a source of concern that the principle of liberty employed does not restrict the collective choice rule sufficiently to prevent the emergence of social outcomes that one would, typically, associate with illiberalism. From this perspective, Condition ML is a less than wholly satisfactory principle of liberty. This view can be countered by pointing out that that a simple way out of the problem is to accord each individual i libertarian privileges over not just one pair but over every pair of i-variants in her protected sphere. This leads to a strengthened version of Sen’s Minimal Liberalism, as proposed by Gibbard (1974). This strengthened version, which may be christened Gibbard Liberalism (GL), is defined below. An SCF C :  → X satisfies Condition GL if and only if for every σ ∈ : ∃ j, k ∈ N such that ∀(x, y) ∈ D j and ∀(w, z) ∈ Dk :     x P j y resp., y P j x → y ∈ / C(σ ) resp., x ∈ / C(σ ) and     w Pk z resp., z Pk w → z ∈ / C(σ ) resp., w ∈ / C(σ ) . In the context of the example discussed earlier, Condition GL assures us that / C(σ ∗ ), x P2 w → w ∈ / C(σ ∗ ), and each of w P1 z and y P2 z implies x P1 y → y ∈ ∗ ∗ z∈ / C(σ ), leaving us with C(σ ) = {x}—a state in which each individual sleeps on his back, in conformity with his wishes in the matter, and in conformity with what one would expect a libertarian rule to bring about. Unfortunately, all libertarian stories presided over by Condition GL do not share this happy ending. In particular, with a slightly different specification of individual preferences, as captured in a preference configuration σ ∗∗ , for which x P1 y, z P1 w, w P2 x, and y P2 z, Condition GL would dictate C(σ ∗∗ ) = φ. This is the substance of Gibbard’s (1974) result on what one might call ‘the impossibility of a liberal liberal’, namely, that there exists no SCF C :  → X satisfying Conditions U and GL. Of course, it can be objected that the case for recognizing a right to individual liberty is not very compelling when individual preferences in protected spheres fail to be ‘privately unconditional’ (Gibbard 1974), or are ‘other oriented’ (Sen 1976). However, the

3.2 Liberalism in a Conventional Social Choice Framework

19

objection is arguably not very convincing. For one thing, personal motivations underlying preferences are presumably personal, and not a libertarian basis for curtailing libertarian rights (Sugden 1985). For another, Gibbard’s paradox can arise when individual preferences in protected spheres fail to be unconditional for reasons—such as commitment to a defendable moral principle—which are not necessarily inspired by dubious motivations (Subramanian 1995). Briefly, formulating liberty in terms of the requirement of a certain correspondence between individual preferences in protected spheres and social choice is not unproblematic. A weak condition of liberty such as Condition ML appears too weak to be compatible with a substantive guarantee of even a minimal degree of individual autonomy in deciding on purely private matters. A move towards a certain ‘natural’ strengthening of Condition ML, in the direction of a libertarian principle such as Condition GL, precipitates problems of internal consistency. In view of these difficulties, there would appear to be some advantages to pursuing an alternative route to formulating the demands of individual liberty. One such approach, which leads to the formulation of an ‘outcome-oriented’ version of the Nozickian conception of rights, with emphasis on individual liberty as being constituted in the ‘freedom to choose’, is explored in the following section. Motivationally similar approaches have been explored by, among others, Sugden (1978, 1985), Gardenfors (1981), Mezetti (1987), and Gaertner et al. (1992).

3.3 Towards an Alternative Framework of Social Choice 3.3.1 Motivation In this section, an alternative mechanism of social choice, captured in the notion of a social selection function (SSF), is advanced. The domain of an SSF is more expansive than that of an SCF: it goes beyond preference profiles, to include what are called personal feature choice profiles and rights-waiving profiles. Each of the ingredients of an SSF is dealt with in what follows, and this leads up to the definition of an SSF. It will be subsequently argued that considerations of efficiency and respect for individual liberty can be more adequately accommodated in an SSF scheme of aggregation than in an SCF scheme.

3.3.2 Preference Profiles The notion of a preference profile has already been dealt with in Sect. 3.1, but is here repeated for purposes of consolidation and continuity. Each individual i ∈ N is assumed to have a ‘weak’ ordering Ri on X, and Pi is the asymmetric component of Ri . R is the set of all possible weak orders on X, and Rn is the n-fold Cartesian

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product of R. Every element σ ∈ Rn is a preference profile—an n-tuple (Ri ) of individual preference orders on X, one preference order for each individual.

3.3.3 Personal Feature Choice Profiles For each individual i ∈ N , there is assumed to exist a personal feature choice function q defined on her personal issue X i (≡ {xi1 , xi2 , . . . , xi }). This function is defined as a mapping f i : X i → {0, 1} such that p

∃t ∈ {1, . . . , q} : f i (xit ) = 1, and ∀ p = t : f i (xi ) = 0. That is, in every individual’s personal issue there is exactly one feature which is assigned a value of unity by the individual’s personal choice function—signifying that this is the feature she would like to declare as her personal choice in her private domain—while each of the remaining features, which the individual would like to declare as being rejected, is assigned a value of zero.1 It is as well, here, to define, for each i ∈ N , the inverse personal feature choice function f i−1 : {0, 1} → X i , which is a mapping such that ∀r ∈ {0, 1} : f i−1 (r ) = xit for any t ∈ {1, . . . , q} if and only if f i (xit ) = r. Each individual i ∈ N can have any one of q personal feature choice lists, a typical choice list being a q-vector eit in which the t-th component (1 ≤ t ≤ q) is unity and all other components are zero: such a choice list conveys the information that the t-th feature in individual i’s personal issue is assigned a value of 1 by her personal choice function f i , while each of the remaining features is assigned a value of zero. For any i ∈ N , the set of all possible personal choice lists i can have is given by q

Fi = {ei1 , ei2 , . . . , ei }.

1

An anonymous referee points out that the personal feature choice function has been designed in such a way that every individual is allowed to choose just one feature from his personal issue, which s/he finds ‘too strict’, and also compatible with being ‘forced’ into a unique choice when the individual is indifferent among features. The referee, of course, is correct in pointing out that under the restriction of a singular choice of personal feature by each individual, there will be one, and only one, social state which is rights-respecting: the set of rights-respecting states will cease being a singleton set if individuals are permitted to declare more than one personally favoured feature, as might happen in the presence of indifference among features. While agreeing with this, I would suggest that nothing of substance is lost, and something of simplicity is gained, by restricting each individual’s choice from her personal issue to a single feature: after all, there is nothing particularly coercive in the invitation to an individual to make up her mind (Perhaps indifference can be resolved by the simple expedient of tossing a coin or some variant thereof.).

3.3 Towards an Alternative Framework of Social Choice

21

Finally, define the set F = F1 × . . . × Fi × . . . × Fn . Elements of F, written as f , f , etc., are personal feature choice profiles, that is, collections of personal choice lists—one list for each individual.

3.3.4 Rights-Waiving Profiles While the notions of individual rights and Pareto efficiency have not yet been defined, it is still possible to conceive that individuals are capable of deciding whether or not they are willing to waive their liberty in personal matters in favour of outcomes that are unanimously preferred to the ‘liberty-respecting’ outcome. In line with this, each person i ∈ N is assumed to have a rights-waiving set M = {Y, N}: the element Y (standing for ‘Yes’) signifies that the individual is willing to waive his libertarian rights, and the element N (standing for ‘No’) signifies that the individual is not willing to waive his rights. Let M stands for the n-fold Cartesian product of M. Then, a typical element m of M is a rights-waiving profile, namely a collection of Y ’s and N’s, indicating which individuals are willing to waive their rights and which are not. A distinguished member of M is the profile m*, for which mi = Y for all i. That is, m* is the profile in which every individual is willing to waive his rights.

3.3.5 Aggregation A social selection function (SSF) is a mapping B : F∗ ×  × M∗ → X, such that, for every f ∈ F∗ (⊆ F), σ ∈ (⊆ Rn ) and m ∈ M∗ (⊆ M) : φ = B( f, σ, m) ⊆ X . For every ( f, σ, m) ∈ F∗ ×  × M∗ , B( f, σ, m) is the set of socially selected alternatives from X, or the selection set of X.

3.4 Paretianism and Libertarianism Revisited Versions of the Unrestricted Domain, Pareto, and Liberty principles invoked in the context of a social choice function can be appropriately reformulated within the context of a social selection function. The corresponding restrictions on an SSF are labelled Condition U*, Condition P*, and Condition L* respectively, and are defined below. An SSF B : F∗ ×  × M∗ → X satisfies Condition U* if and only if F∗ = F,  = Rn , and M∗ = M, viz. the domain of the choice rule admits every logically possible combination of personal choice profile, preference profile, and rights-waiving profile;

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Condition P* if and only if, for every f ∈ F∗ , σ ∈  and m ∈ M*: B( f, σ, m) ⊆ E( f, σ, m), where X |¬[∃y ∈ X : (∀i ∈ N : y Pi x)]} ; and

E( f, σ, m)

≡

{x

∈

Condition L* if and only if, for every f ∈ F∗ , σ ∈  and m ∈ M*: B( f, σ, m) = {x ∗ ( f, σ, m)}, where x* is a social state with the property that, for all i ∈ N :xi∗ = f i−1 (1). Condition L* is the only one which may require some explanation. L* demands that each person can signal that feature of the social state in her personal issue which she favours through her personal feature choice function; and when every person has done this, the socially selected state should be one which is a combination of personal features that precisely reflects each person’s choice. Thus, L* is one specific (‘outcome-oriented’) formalization of the general approach to modelling rights which Nozick (1974; pp. 165–166) advocates, when he says: The trouble stems from treating an individual’s right to choose among alternatives as the right to determine the relative ordering of these alternatives within a social ordering…A more appropriate view of individual rights is as follows. Individual rights are co-possible; each person may exercise his rights as he chooses. The exercise of these rights fixes some features of the world.

Or, as Gartner, Pattanaik and Suzumura (1992, p. 167) put it: …the individual enjoys the power to determine a particular aspect or feature…of the social alternative, and when he makes his choice with respect to this particular aspect, his choice imposes restrictions on the final social outcome in so far as, in the final social outcome, that particular aspect must be exactly as he chose it.

This way of formulating a libertarian principle avoids the difficulties associated with conventional social choice formulations such as Condition ML and Condition GL, which were reviewed in Sect. 3.2. However, there is no reason to believe that Sen’s original dilemma of an impossible Paretian liberal stands resolved by adopting an approach to modelling rights such as is captured in Condition L*. In particular, it is not hard to see that there exists no SSF B : F∗ ×  × M∗ → X satisfying Conditions U*, P*, and L*. To see this, one only has to note that there exist logically possible combinations of f , σ , and m, such that x*(f , σ, m) is Pareto-dominated by some other social state. If, by a ‘Paretian liberal’, one means somebody who can successfully defend the notion that there exists an SSF which satisfies Conditions U*, P*, and L*, then such a ‘Paretian liberal’ cannot be found. Is there an alternative, plausible way of describing a ‘Paretian liberal’? To this end, consider the following Pareto-Sensitive Liberty principle, or Condition P*L*. An SSF B : F∗ ×  × M∗ → X satisfies Condition P*L* if and only if, for every f ∈ F∗ , σ ∈  and m ∈ M*: B( f, σ, m) = {x ∗ ( f, σ, m)} if (i) x ∗ ( f, σ, m) ∈ E( f, σ, m) or (ii) x ∗ ( f, σ, m) ∈ / E( f, σ, m) and m = m ∗ ;

3.4 Paretianism and Libertarianism Revisited

23

⊆ E ( f, σ, m) if x ∗ ( f, σ, m) ∈ / E( f, σ, m) and m = m ∗ , where  E ( f, σ, m) ≡ {x ∈ E( f, σ, m)∀i ∈ N : (x Pi x ∗ ( f, σ, m))} . That is, the Pareto-Sensitive Liberty principle demands that the socially selected alternative should be the ‘rights-respecting’ state x* if either x* is Pareto optimal, or, when x* is not Pareto optimal, if there is no unanimous vote in favour of rightswaiving; and it demands that final social selection be confined to a subset of those Pareto-efficient alternatives which Pareto-dominate x* if every person in society is willing to waive his rights in favour of one of these alternatives. This setting— aside from the particular manner in which liberalism has been formalized, and also the provision for rights-waiving—is very much in the spirit of Vallentyne’s (1989) advocacy of ‘how to combine Pareto optimality with liberty considerations.’ If, by a ‘Paretian liberal’, one means somebody who can successfully defend the notion that there exists an SSF which satisfies Conditions U* and P*L*, then does such a ‘Paretian liberal’ exist? The following theorem answers this question. Theorem There exists an SSF B : F∗ ×  × M∗ → X which satisfies Conditions U*and P*L*.2 Proof We know that, by definition, for all f ∈ F∗ , σ ∈  and m ∈ M*, x*(f , σ, m) exists. Therefore, if either (a) x* ∈ E(f , σ, m) or (b) x* ∈ / E(f , σ, m) and m = m*, / E(f , B( f, σ, m) ≡ {x ∗ } is non-empty. It remains to consider the case in which x* ∈ σ, m) and m = m*: we need to show then that E (f , σ, m) is non-empty. Suppose, to the contrary, that E (f , σ, m) is empty. Then, by definition of E (f , σ, m), there does not exist any alternative in E(f , σ, m) which is Pareto-preferred to x*. But then x* must itself be Pareto optimal, which contradicts x* ∈ / E(f , σ, m). This completes the proof of the theorem. (Q.E.D.) 2

The referee points out that a social selection function constrained by Conditions U* and P*L* can yield an ‘…outcome [that] seems unrealistic and we are yet to provide an explanation for such a choice.’ The problem, s/he suggests, arises from treating the three arguments of the SSF—the personal feature choice profile, the preference profile, and the rights-waiving profile—as independent. The referee illustrates with the help of the following example: ‘[Suppose] N = {1, 2}, and (∀i ∈ N )(X i = {b, s}). We have four possible social states: x = (s, s), y = (s, b), z = (b, s) and w = (b, b). Here we have X = {x, y, z, w}. Let both individuals hold the following preference ordering, xPyPzPw. Let both individuals assign value 1 to the feature b and 0 to s. Also, let m = m*. Our assumptions are consistent with U*. We have x* = w and the SSF satisfying U* and P*L* gives us, B( f, σ, m) = {w}. Here w will be the selected social state which is the least preferred alternative in the preference orderings of both individuals. The outcome seems unrealistic and we are yet to provide an explanation for such a choice.’ The explanation—if one is required—is that the persons in the example seem to be a bit unreal! Interpreting ‘s’ to mean ‘sleeping on stomach’, and ‘b’ to mean ‘sleeping on back’, the preference profile suggests that, left to himself, each person prefers sleeping on his stomach to sleeping on his back, and yet, each also declares his uniquely favoured personal feature to be that of sleeping on his back! This is at least a little eccentric, and it is this eccentricity which ‘explains’ a social choice in which both individuals wind up copping their least preferred alternative w. If we wish to prevent such ‘unrealistic’ outcomes, we may have to impose some sort of domain restriction; but the whole point of Condition U* is, precisely, to avoid proscribing ‘eccentricity’, or even lunacy!.

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3.5 Concluding Observations The theorem above upholds the coherence of ‘a sort of Paretian liberal’. In assessing any Pareto-consistent libertarian claim (to use Gibbard’s (1974) phrase), it is important, as Sen (1976) has pointed out, to make an appraisal from both ‘ethical’ and ‘pragmatic’ perspectives. Pragmatic considerations would come into play in assessing rights-waiving strategies. For a person with a libertarian outlook, one’s libertarian rights are not ‘traded away’ for Pareto-superior outcomes: for such a person, the rights-waiving decision would always be to choose the element N from her rightswaiving set {Y,N}. For a person with a ‘Paretian’ outlook, here is a naïve line of reasoning: if he expected that at least one person in society would refuse to waive her rights, then it wouldn’t matter whether he himself chose Y or N from his rightswaiving set, for in either case the socially chosen outcome would be x*; on the other hand, if he expected everybody else also to be a ‘Paretian’ like himself and to vote Y, his own best strategy would be to vote Y: Y would thus be his dominant-strategy choice. Briefly, the rights-waiving profile would always be m* only in a ‘Paretian’ society, by which is meant a society all of whose members are ‘Paretians’. And it is only in a ‘Paretian’ society that the pragmatic problem would arise. This is the problem of how to ensure that people will not unilaterally defect from the agreement to confine social choice to the set of alternatives in E . This problem of ‘incentive compatibility’ presents serious difficulties for Gibbard’s (1974) ‘rights-waiving’ resolution of Sen’s dilemma, as has been highlighted by Kelly (1976) and, definitively, by Basu (1984).3 The problem persists for the ‘sort of Paretian liberalism’ advanced in this note, but, arguably, in diluted form: it arises, potentially, only in a wholly ‘Paretian’ society, bereft of a single individual with any libertarian impulse (in fact, the requirement that the society be ‘Paretian’ is only necessary, not sufficient, for the ‘pragmatic problem’ to arise: while I desist from pursuing this discussion further here, some additional notes on the problem are available, on request, with the author). Of perhaps greater interest from a normative perspective is the ‘ethical’ content of alternative conceptualizations of ‘Paretian liberalism’. A difficulty with a Sen-type construction is that it tends to make individuals prisoners of their private preferences, without giving them the option of being guided by their all-things-considered preferences. The point is underlined by Barry (1984; p. 19) when he advocates caution against confusing ‘…two quite different ideas: that people should never fail to act on their personal preferences in what “directly concerns them”, and that people should not be required to violate their personal preferences in what “directly concerns them”. The second is, indeed, an authentically liberal idea. But the first is not, as Sen suggests, an essential part of any reasonable conception of liberalism.’ Condition P*L* does 3

‘Such a [possibility of strategic] manipulation’, as the referee points out, ‘is also possible when an individual chooses her feature alternative in order to construct her personal feature choice function.’ Indeed, the possibility extends also to strategic misrepresentation of preferences. But this paper confines itself to a very brief discussion of the possibility of strategic behaviour only in the context of the rights-waiving decision. Sufficient unto the day….

3.5 Concluding Observations

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not insist that ‘people should never fail to act on their personal preferences in what “directly concerns them”’: rather, it accords them what Basu (1984) calls the ‘right to give up rights.’ Respect for such a meta-right would be of particular salience in a community of ‘honest’, or ‘non-strategic’ Paretians: in such a situation, people would not be deprived of the freedom of renouncing their libertarian rights in favour of a commonly preferred outcome, the realization of which is not threatened by the ‘pragmatic problem’ alluded to earlier. At the same time, Condition P*L* does not swing to the other extreme: it does not suggest as Gibbard’s ‘Pareto-consistent libertarian claim’ seems to—that ‘rights-waiving’ across the board must be an automatic response to the Pareto suboptimality of a ‘libertarian’ outcome. Here, there is considerable appeal in Sen’s (1976) objection to the apparent requirement that Paretianism should be prioritized over libertarianism in every instance of conflict, and that people should forsake their libertarian ethics by waiving their rights in favour of utility gain. If Condition P*L* upholds the right to give up rights, it also upholds the right not to give up rights: it should be noted that rights-waiving, in this scheme, is accorded legitimacy only when it is unanimous. Thus, the ‘sort of Paretian liberalism’ advanced in this note tries to find a way out of Sen’s dilemma by according some weight to both Paretianism and libertarianism: it is not a resolution of the dilemma, only an alternative, constructive interpretation of the notion of ‘Paretian liberalism’, which, it is hoped, has the merit of a measure of ethical appeal.

References Barry B (1984) Lady Chatterly’s lover and doctor fisher’s bomb party: liberalism, pareto optimality and the problem of objectionable preferences. In: Elster J, Hylland A (eds) Foundations of social choice theory. Cambridge University Press Basu K (1984) The right to give up rights. Economica 51 Gardenfors P (1981) Rights, games and social choice. Nous 15(3):341–356 Gaertner W, Pattanaik PK, Suzumura K (1992) Individual rights revisited. Economica 51 Gibbard A (1974) A pareto-consistent libertarian claim. J Econ Theor 7 Kelly JS (1976) Rights-exercising and a pareto-consistent libertarian claim. J Econ Theor 13(1):138– 153 Mezetti C (1987) Paretian efficiency, rawlsian justice and the Nozick theory of rights. Soc Choice Welfare 4(1):25–37 Nozick R (1974) Anarchy, state and utopia. Blackwell, Oxford Sen A (1970a) Collective choice and social welfare. Holden Day, San Francisco Sen A (1970b) The impossibility of a Paretian liberal. J Polit Econ 72 Sen A (1976) Liberty, unanimity and rights. Economica 43 Subramanian S (1995) Preference motivations and libertarian dilemmas. Manchester School Econ Soc Stud 63(2):167–174 Sugden R (1978) Social choice and individual liberty. In: Artis MJ, Nobay AR (eds) Contemporary economic analysis: Papers presented at the conference of the association of university teachers of economics, April 1977, Croom Helm, London Sugden R (1985) Liberty, preference and choice. Econ Philos 1 Vallentyne P (1988) Rights based paretianism. Can J Philos 18:527–544 Vallentyne P (1989) How to combine pareto optimality with liberty considerations. Theor Decis 27(3):217–240

Chapter 4

Liberty, Equality, and Impossibility: Some General Results in the Space of ‘Soft’ Preferences

Abstract This paper is concerned with examining the mutual compatibility of the ethical principles of equity and liberty in a social choice framework of ordinally formulated vague preferences. With sufficiently weakened versions of the liberty and equity principles, one can secure an existence result in a ‘relation-functional’ setting. However, difficulties tend to re-appear in a ‘choice-functional’ setting, when one subscribes to the notion that while preference may be vague, choice must perforce be exact. Keywords Soft binary preference relation · Soft extended binary preference relation · Modified equity · Minimal liberalism · Generalized soft aggregation rule · H-rationalizability of a choice function

4.1 Introduction This conference is on the themes of growth, inequality, and institutions. The present paper is concerned with these themes, but at a certain level of generality and abstraction that is not immediately—but nevertheless foundationally—related to aspects of reform and delivery in the space of policy formulation. Specifically, in contemplating economic and political phenomena such as growth, inequality, and institutions, it is important to investigate the possibility of pursuing certain desired social virtues from the perspectives of constitutional provision, organizational design, and policy initiative. Not the least important aspect of this investigation is the one relating to issues of logical coherence mediating the quest for valued social desiderata. Two such desiderata are embodied in the social goals of personal liberty and interpersonal equality. How far, and in what sense, these principles are mutually compatible is a problem in social choice. It is precisely this collective choice-theoretic problem that is examined in this paper, in a framework of some considerable generality This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder. S. Subramanian (2010) Liberty, equality, and impossibility: some general results in the space of ‘soft’ preferences, Journal of Economic Policy Reform, 13:4, 325–341. https://doi.org/10.1080/17487870.2010.523970. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_4

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4 Liberty, Equality, and Impossibility: Some General Results …

entailing an analysis of preference aggregation when preferences—both individual and social—are seen as being vague. In conventional social choice theory, both individual and collective preferences are taken to be ‘exact’ or ‘crisp’. In several works over the last couple of decades, many familiar impossibility results in collective choice theory have been generalized in a setting wherein the traditional choice framework has been relaxed to allow for vagueness in preference relations. ‘Arrow-type’ problems of aggregation, and related issues, have been investigated by, among others, Barrett et al. (1986), Dutta (1987), Barrett and Pattanaik (1990), Banerjee (1994), and Richardson (1998). A fuzzy version of a ‘Sen-type’ liberal paradox has been considered by, among others, Subramanian (1987) and Dimitrov (2004). In a preponderance of cases, the concern has been with modelling vague preferences within a ‘cardinal’ framework of fuzzy sets. One early exception to this rule is the work of Barrett and Pattanaik (hereafter B-P 1990): these authors undertake an ‘ordinal’ reformulation of ‘cardinal’ fuzzy sets, drawing on a version of Goguen’s (1967) L-fuzzy set theory. The ‘ordinalized’ version of fuzzy sets is referred to by Basu et al. (1992) as ‘soft’ sets. In the present paper, I examine, within a framework of ‘soft’ preferences, the compatibility between Sen’s (1970) principle of liberty and (a modified version of) Hammond’s (1976) principle of equity. The modified equity principle is formulated in ‘non-welfarist’ terms, and—like the original version— exploits information on interpersonal utility comparisons. Despite this, and even in a relaxed framework of ‘soft’ preferences, the results on the possibility of ‘liberal egalitarianism’ are found to be essentially discouraging. This paper is organized as follows. Section 2 discusses basic concepts and definitions and introduces some preliminaries relating to soft binary preference relations. Section 3 is concerned with the formulation of the equity and libertarian principles in a framework of soft preferences. The basic (relation-functional) results of the paper are set out in Sect. 4. In Sect. 5, the question of exact choice based on soft preferences is addressed; and in Sect. 6, the liberty–equality result is discussed in a choice-functional setting. Concluding observations are offered in Sect. 7.

4.2 Basic Concepts X (# X ≥ 4) = {x, y, z, . . .} is the finite set of all conceivable social states or alternatives, and N (#N ≥ 2) = {1, . . . , i, . . . , n} is the finite set of individuals constituting society. For all i ∈ N , X i (# X i ≥ 2) is individual i’s personal issue, i.e. a non-empty set of features of the possible social states that may be regarded as being personal to i (see Gibbard 1974). (One could also include a set X 0 of ‘public’ features of the possible social states: this is an avoidable complexity which I shall, accordingly, avoid.) The set  X of all conceivable states is given by the Cartesian product of the X i , viz. X = i∈N X i . A typical social state x ∈ X can be written as a list x = (x1 , . . . , xi , . . . , xn ) where for all i ∈ N , xi is a description of person i’s condition in the alternative x. A couple of definitions, which are of relevance to

4.2 Basic Concepts

29

the formulation of the equity and liberty principles (to be discussed in the following section), are now presented. Definition 4.1 (j-variants) A pair of states x, y ∈ X will be said to be j-variants if and only if x i = yi ∀i ∈ N \{ j}&x j = y j . For all j ∈ N , we shall let D j stand for the set of all possible pairs of j-variants that can be constituted from X. Definition 4.2 (j,k - variants) A pair of states x, y ∈ X will be said to be j, k-variants if and only if x i = yi ∀i ∈ N \{ j, k}&xi = yi ∀i ∈ { j, k}. For all j, k ∈ N , we shall let D j,k stand for the set of all possible pairs of j, k-variants that can be constituted from X. Also, for all x ∈ X and all j, k ∈ N , we shall let x−( j,k) stand for the list (x1 , ..., xn ) with the jth and kth components, x j and xk respectively, suppressed. I now introduce the notion of vague binary preference relations. In a ‘cardinal’ settheoretic framework, it is conventional to define a fuzzy binary preference relation as a function P : X × X → [0, 1] such that for all x, y ∈ X , P(x, y) is interpreted as ‘the degree of confidence’ with which x is preferred to y, with P(x, y) taking some value in the closed interval [0, 1]. The specification of a precise degree of confidence with which any x is preferred to any y, on a cardinal scale going from zero to one, may be thought to do some injustice to precisely the notion of indeterminacy supposed to be captured by a fuzzy preference relation. An ‘ordinal’ reformulation of fuzziness would permit comparisons of levels of vagueness without any commitment to specification of precise degrees of vagueness on a cardinal scale. This serves as the motivation for the ‘ordinally’ reformulated ‘soft’ set-theoretic framework developed by Basu et al. (1992). In this paper, I shall rely on the ‘soft’ sets approach adopted by B-P (1990), which draws on a version of Goguen’s (1967) L-fuzzy set theory. To this end, consider the following. Let L(#L ≥ 2) be a finite ordered set {d1 , ..., d M }, the elements of which are ranked by the exact antisymmetric ordering Q, whose asymmetric part is denoted ¯ M−1 Q... ¯ Qd ¯ 1 . A typical element d ∈ L (see B-P, 1990) is ¯ so that d M Qd by Q, ¯  to be construed as representing a ‘degree of belonging’: thus, ∀d, d  ∈ L, d Qd implies that the ‘degree of belonging’ as represented by d is greater than the ‘degree of belonging’ as represented by d  . d M will be taken to represent ‘belonging with complete confidence’ and d1 to represent ‘not belonging with complete confidence’. Definition 4.3 (Soft Binary Preference Relation) A soft binary preference relation (SBPR) P on X is a function P : X × X → L. Definition 4.4 (Exact or Crisp Binary Preference Relation) In Definition 4.3, if L = {d M , d1 }, then the function P is an exact or crisp binary preference relation on X. I now introduce the notion of a soft extended preference relation, drawing on Arrow’s (1977) notion of ‘extended sympathy’, which allows for ordinal interpersonal comparisons of utility whereby objects of the type ‘being person j in state x’ and ‘being person k in state y’ can be submitted to preference ranking.

30

4 Liberty, Equality, and Impossibility: Some General Results …

Definition 4.5 (Soft Extended Binary Preference Relation) A soft extended binary  : (X × N ) × (X × preference relation (SEBPR) P˜ on the set X × N is a function P N ) → L. i is individual i’s SEBPR on For all i ∈ N , Pi is individual i’s SBPR on X and P X × N . In this paper, I shall employ the notation ‘Pi (x, y)’ interchangeably with i ((x, i), (y, i))’; in general, the latter notation will be preferred. Also, the notation ‘ P  for all j, k ∈ N and all x, y ∈ X , the statement ‘ P((x, j), (y, k)) = d M ’ will  k)’, while, similarly, for all x, y ∈ X , the statement also be written as ‘(x, j) P(y, ‘P(x, y) = d M ’ will be written equivalently as ‘x P y’. Let  be the set of all SBPRs on X. Let 0 be the set of all P ∈  such that: for all x ∈ X : P(x, x) = d1 (irreflexivity);

(4.2.1)

for all distinct x, y ∈ X : P(x, y) = d M → P(y, x) = d1 (asymmetry); and (4.2.2) for all distinct x, y, z ∈ X : P(x, z)Qm(P(x, y), P(y, z)) where m(P(x, y), P(y, z)) = P(x, y) if P(y, z)Q P(x, y) and m(P(x, y), P(y, z)) = P(y, z) if P(x, y)Q P(y, z) (maxmin transitivity of strict preference).

(4.2.3)

We shall let C stand for the set of all P ∈  such that P is crisp; and we define the set 0C ≡ 0 ∩ C . Remark 4.6 In terms of the relationship of the set 0C to traditional crisp preferences, it should be noted that elements of 0C are what we would conventionally refer to as strict partial orderings (irreflexive, asymmetric, and transitive binary relations). Remark 4.7 There is no necessarily unique notion of transitivity of strict preferences in a fuzzy framework. In (4.2.3), I have presented the ‘maxmin’ transitivity rule widely employed in the literature (see B-P 1990). Next, I discuss a certain distinguished subset of the set of all soft extended binary preference relations, analogously with the preceding discussion of strict soft binary  be the set of all SEBPRs on X × N , and let  0 be the set preference relations. Let  ∈   satisfying (4.2.4)–(4.2.6) below: of all P  for all x ∈ X and all i ∈ N : P((x, i), (x, i)) = d1 (irreflexivity);  for all distinct(x, j), (y, k) ∈ X × N : P((x, j), (y, k))  = d M → P((y, k), (x, j)) = d1 (asymmetry); and

(4.2.4)

(4.2.5)

4.2 Basic Concepts

31

for all distinct (x, j), (y, k)(z, ) ∈ X × N :    P((x, j), (z, ))Qm( P((x, j), (y, k)), P((y, k), (z, ))), where   m( P((x, j), (y, k)), P((y, k), (z, )))   P((y, k), (z, ))Q P((x, j), (y, k)), and

 P((x, j), (y, k))

=

if

  m( P((x, j), (y, k)), P((y, k), (z, )))    = P((y, k), (z, )) if P((x, j), (y, k))Q P((y, k), (z, )) (maxmin transitivity of strict preference).

(4.2.6)

∈   is crisp; and we C stand for the set of all P  such that P We shall let  define the set of all crisp extended preference relations on X which are transitive by 0C ≡  0 ∩  C .  Finally, I consider the question of aggregating individual soft preferences into a collective soft preference. It will be assumed that society’s problem is to arrive at a collective soft ranking of alternative social states on the basis of information provided by the set of individual soft extended preference relations on the set X × N . Definition 4.8 (Generalized Soft Aggregation Rule) A generalized soft aggre ⊆   and   )#N →  , where φ =  gation rule (GSAR) is a function f : ( φ =  ⊆ , such that, for every #N -tuple of individual soft extended preference i ) in its domain, the aggregation rule f specifies a unique soft preference relations ( P ranking of X. 



Remark 4.9 There are at least two reasons why it may be of interest to consider rules that take values in soft (social) preferences. The first is that there may be some intrinsic merit to allowing for the possibility that collective judgments are, from the point of view of descriptive realism and normative reasonableness, fuzzy and imprecise in a way which traditional aggregation theory, with its emphasis on preferences that are unvaryingly undivided over pairs of alternatives, denies. The second reason is an essentially technical one, relevant for the special case in which the domain of the aggregation rule consists only of n-tuples of crisp preferences: given an aggregation rule with domain D (of crisp preference n-tuples) and range R, a theorem to the effect that there exists no rule f : D → R satisfying a set of properties c1 , . . . , c K is a weaker result than a theorem that asserts that there exists no rule f : D → R  satisfying c1 , . . . , c K when R  ⊇ R; since the set of soft social preferences is a superset of the set of exact social preferences, there is some mileage—in terms of greater generality—to be gained from proving an impossibility theorem in a framework wherein the range of the aggregation rule is expanded from exact social preferences to soft social preferences. We may wish to restrict the GSAR by requiring it to satisfy what we may regard as desirable properties in an aggregation mechanism. In particular, society may be interested in constraining the collective choice mechanism to satisfy the ethical principles

32

4 Liberty, Equality, and Impossibility: Some General Results …

of equity and liberty. Whether this is a reasonable expectation will be investigated in the rest of this paper. But first, a formulation of the equity and the liberty principles.

4.3 The Equity and Libertarian Principles 4.3.1 The Crisp Framework Hammond (1976) provides a social choice-theoretic approximation of Sen’s (1973) weak equity axiom which is essentially a rule for an optimal distribution of income between two individuals, given the incomes of all other individuals. The weak equity axiom demands that in distributing a given income between two individuals, a larger share should go to the more disadvantaged individual. Hammond translates this requirement, in the social choice framework, in terms of the following principles of ‘equity’ and ‘weak equity’. (In the definitions provided below, for all i, Ii is the symmetric component—representing ‘indifference’—of the ‘weak’ preference  is the strict extended preference ordering of an hypothetical ‘ethical ordering Ri ; P observer’; R is the ‘weak’ social preference ordering; and P is the asymmetric factor of R). Definition 4.1.1 (Equity) Equity demands that for all x, y ∈ X and all j, k ∈ N ,   if x P j y, y Pk x, [∀i ∈ N \{ j, k} : x Ii y], and (x, k) P(x, j) and (y, k) P(y, j), then x P y. Definition 4.1.2 (Weak Equity) The weak equity principle is derived from the Equity principle by replacing ‘x P y’ in Definition 4.1.1 by ‘x Ry’. Hammond (1976; p. 795) explains his formulation of the equity and weak equity principles, based on Sen’s Weak Equity Axiom, as follows (in the ensuing quotation, I have resorted to some very minor notational changes from the original): Suppose that y denotes an equal distribution of income, and x an alternative distribution in which j’s income has risen and k’s income has fallen, with all other incomes remaining the same [emphasis added]. Then Sen’s axiom effectively requires that, if x is close enough to y, then x is socially preferred to y. This is assuming that j has a lower level of welfare than k in state y, i.e. that j enjoys fewer advantages than k. One might now try to extend this principle to social choices that are more general than choices of income distribution. Then, the obvious and essential features of the social choice examined in the previous paragraph are as follows: ˜ (i) x P j y, y Pk x, and for all i ∈ / { j, k}, x Ii y; (ii) (y, k) P(y, j); and (iii) x is close to y. Only condition (iii) is imprecise. There are a number of ways one might try to make it precise. One way is to insist that x must re-distribute income so that j, who enjoyed fewer advantages  than k in state y still enjoys fewer advantages than k in state x, i.e. (iii) (x, k) P(x, j). [This is the Equity Principle of Definition 4.1.1.] But in fact, one can weaken this slightly to require that if conditions (i), (ii) and (iii) are satisfied, then x should be weakly preferred to y by society. [This is the Weak Equity Principle of Definition 4.1.2.]

4.3 The Equity and Libertarian Principles

33

The difficulty with Hammond’s translation of Sen resides in the italicized part of the sentence in the first paragraph of the above quotation. In applying the equity principle in order to decide whether to respect j’s or k’s preference over the pair of states x, y, the requirement that the personal features of all individuals other than j and k be the same in both states x and y seems to have been imperfectly captured by Hammond’s requirement that all individuals other than j and k be indifferent as between x and y. While individuals may (from a normative point of view) be expected to be indifferent between states which are invariant with respect to their personal features, it is not clear that they will, in practice, be so indifferent. In translating Sen’s weak equity Axiom into its social choice counterpart, there does seem to be a case for imposing more structure on the pair of social states under comparison—by requiring, typically, that x and y be a pair of j, k-variants (for a definition of which see Sect. 4.2). There also appears to be a case for dropping Hammond’s ‘indifference’ requirement: the preferences of individuals other than j and k should simply not be relevant for determining the social preference over the pair of states x, y (note that this stricture does not apply to their extended preferences). In line with this reasoning, I advance the following modified equity and weak equity principles: Definition 4.1.3 (Modified Equity) Modified Equity demands that for all x, y ∈ X and all j, k ∈ N , if  (x, y) ∈ D j,k , x P j y, y Pk x and i (y, j) , then x P y. i (x, j) and (y, k) P ∀i ∈ N : (x, k) P Definition 4.1.4 (Modified Weak Equity) Modified weak equity is derived from modified equity by replacing ‘x P y’ in Definition 4.1.3 by ‘x Ry’. Remark 4.1.5 Note that in the modified versions of the equity principles, the fact of j being the more disadvantaged individual is represented by the requirement that every individual prefers being person k to being person j in both the states x and y: this is one way of basing social preference on individual extended preference orderings, rather than on a single extended preference ordering purporting to be that of an impartial ‘ethical observer’. Remark 4.1.6 Notice also that Hammond’s formulation of the equity principle is such as to make it satisfy Arrow’s (1963) condition of ‘neutrality’, which essentially requires that if information on individual utilities with respect to any pair of states {x, y} is identical to information on individual utilities with respect to any other pair of states {w, z}, then the social ranking of x vis-à-vis y should be identical to the social ranking of w vis-à-vis z. (This property Sen 1979 also calls ‘welfarism’.) However, because of the structure imposed on pairs of alternatives to qualify them for comparison in terms of the modified equity axiom the latter violates the neutrality axiom: modified equity is a non-welfarist principle (i.e. a principle which is sensitive to considerations other than solely the utility content of alternative states of the world). Next is the libertarian principle. Sen’s (1970) principle of minimal liberty requires that each of at least two individuals in society should be decisive over at least one

34

4 Liberty, Equality, and Impossibility: Some General Results …

pair of alternatives each, with—in line with Gibbard’s (1974) interpretation—the alternatives in each pair differing only in a feature of personal relevance to the concerned individual. Formally, we have: Definition Minimal liberalism requires that ∃ j, k ∈   4.1.7 (Minimal Liberalism) N and ∃ {x, y} ∈ D j , {w, z} ∈ Dk such that if x P j y (respectively, y P j x), then x P y (respectively, y P x), and if w Pk z (respectively, z Pk w), then w Pz (respectively, z Pw). Precisely in the spirit of the weakened version of the equity principle, one can have a weakened version of the minimal liberty principle which allows only for a weak veto on assigned pairs of alternatives: Definition 4.1.8 (Weak minimal liberalism requires  Minimal Liberalism) Weak  that ∃ j, k ∈ N and ∃ {x, y} ∈ D j , {w, z} ∈ Dk such that if x P j y (respectively, y P j x), then x Ry (respectively, y Rx), and if w Pk z (respectively, z Pk w), then w Rz (respectively, z Rw).

4.3.2 The Soft Framework The following definitions are reasonably straightforward translations, in a framework of soft preferences, of the corresponding definitions in the crisp framework: Definition 4.2.1 (Modified Equity) A GSAR satisfies modified equity (ME) if and i )i∈N in the domain of the GSAR: only if for all x, y ∈ X , all j, k ∈ N , and all ( P j ((x, j), (y, j)) = d M & P j ((y, j), (x, j)) = d1 , if {x, y} ∈ D j,k , P k ((x, k), (y, k)) = d1 , and ∀i ∈ N : P i ((x, k), (x, j)) = k ((y, k), (x, k)) = d M & P P i ((x, j), (x, k)) = P i ((y, j), (y, k)) = d1 , then i ((y, k), (y, j)) = d M & P P P(x, y) Q¯ P(y, x) = d1 . Definition 4.2.2 (Modified Weak Equity) Modified weak equity (MWE) is derived from modified equity by replacing ‘P(x, y)Q P(y, x) = d1 ’ in Definition 4.2.1 by ‘P(x, y)Q P(y, x) = d1 ’. Definition 4.2.3 (Minimal Liberalism) A GSAR satisfies minimal liberalism (ML) if and only if there exist at least two distinct individuals j and k and two distinct i )i∈N in doubletons of alternatives {x, y} ∈ D j and {w, z} ∈ Dk such that for all ( P the domain of the GSAR:   j ((y, j), (x, j)) = d1 j ((x, j), (y, j)) = d M and P (P    j ((y, j), (x, j)) = d M and P j ((x, j), (y, j)) = d1 resp., P     → P(x, y)Q P(y, x) = d1 resp., P(y, x)Q P(x, y) = d1 ; and

(4.2.1)

4.3 The Equity and Libertarian Principles

35

  k ((w, k), (z, k)) = d M and P k ((z, k), (w, k)) = d1 (P (resp., [ P˜k ((z, k), (w, k)) = d M & P˜k ((w, k), (z, k)) = d1 ]) → [P(w, z) Q¯ P(z, w) = d1 ] (resp., [P(z, w) Q¯ P(w, z) = d1 ]).

(4.2.2)

Definition 4.2.4 (Weak Minimal Liberalism) Weak minimal liberalism (WML) is obtained from ML by replacing ‘[P(x, y) Q¯ P(y, x) = d1 ] (resp., [P(y, x) Q¯ P(x, y) = d1 ])’ in (3.2.1) by ‘[P(x, y)Q P(y, x) = d1 ] (resp., [P(y, x)Q P(x, y) = d1 ])’ and ‘[P(w, z) Q¯ P(z, w) = d1 ](resp., [P(z, w) Q¯ P(w, z) = d1 ])’ in (3.2.2) by ‘[P(w, z)Q P(z, w) = d1 ](resp., [P(z, w)Q P(w, z) = d1 ])’. Remark 4.2.5 All references to the equity and liberty principles will be with respect to the generalized ‘soft’ Definitions 4.2.1–4.2.4. Given the preceding inventory of concepts and definitions, the main (relationfunctional) results of the paper can now be stated.

4.4 On the Possibility of Egalitarian Liberalism The following proposition is true. ˜ 0C )#N → 0C satisfying conditions Theorem 4.1 There exists no GSAR f : ( ML and ME if X contains a subset A = {x, y, w, z}such that {x, y} ∈ D j and {w, z} ∈ Dk for some j, k ∈ N , and either (i) (ii)

#({x, y} ∩ {w, z}) = 1; or = {x, y} ∩ {w, z} = {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j {{x, w}, {x, z}, {y, w}, {y, z}} ∩ Dk = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) .

Remark 4.2 A proof of the theorem is omitted since Theorem 4.1, which is in the crisp framework, is implied by and subsumed under its soft generalization, Theorem 4.4, which will be presently stated. Remark 4.3 Condition (ii) in the statement of Theorem 4.1 is essentially a purely technical condition, intended to guarantee that {x, z} and {y, w} are pairs of j,kvariants. To see that this is indeed the case, note that if (WLOG) j = 1 and k = 2, and we replace each of x−( j,k) , y−( j,k) , w−( j,k) , and z −( j,k) by, say, , then we can write x, y, z and w as, respectively, x = (x1 , x2 , ), y = (x1 , x2 , ), z = (z 1 , z 2 , ), and w = / D1 implies not [x1 = z 1 , x2 = z 2 ] which, (z 1 , z 2 , ); so that, for instance, {x, z} ∈ in turn, must imply one of the following: (i) x1 = z 1 , x2 = z 2 ; (ii) x1 = z 1 , x2 = z 2 ; and (iii) x1 = z 1 , x2 = z 2 . (ii) would imply that x = z, which cannot be the case since {x, y} ∩ {w, z} = φ. (iii) would imply that x and z are a pair of 2-variants, which also

36

4 Liberty, Equality, and Impossibility: Some General Results …

cannot be the case since it is given that {{x, w}, {x, z}, {y, w}, {y, z}}∩D2 = φ. We are therefore left with (i), namely the inference that x and z are a pair of 1,2-variants. It can be similarly inferred that y and w are a pair of 1,2-variants. ˜ 0C )#N → 0 satisfying conditions ML Theorem 4.4 There exists no GSAR f : ( and ME if X contains a subset A = {x, y, w, z}such that {x, y} ∈ D j and {w, z} ∈ Dk for some j, k ∈ N , and either (i) (ii)

#({x, y} ∩ {w, z}) = 1; or = {x, y} ∩ {w, z} = {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j {{x, w}, {x, z}, {y, w}, {y, z}} ∩ Dk = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) .

Proof Two cases must be distinguished: (i) {x, y} and {w, z} have exactly one alternative in common, say y = w; and (ii) each of the alternatives x, y, w, and z is distinct, {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j = {{x, w}, {x, z}, {y, w}, {y, z}} ∩ Dk = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) . In what follows, I shall invoke individual preferences which, it can be verified, are compatible with the domain restrictions specified in the statement of the theorem. Case (i) Consider the following pattern of individual preferences: P˜ j ((x, j), (y, j)) = d M , P˜ j ((y, j), (x, j)) d M , P˜k ((z, k), (y, k)) = d1 ;

=

d1 ; P˜k ((y, k), (z, k))

=

P˜ j ((x, j), (z, j)) = d1 , P˜ j ((z, j), (x, j)) = d M ; P˜ j ((x, j), (z, j)) = d1 , P˜ j ((z, j), (x, j)) = d M ; P˜k ((x, k), (z, k)) = d M , P˜k ((z, k), (x, k)) = d1 ; ∀i ∈ N : P˜i ((x, k), (x, j)) = P˜i ((z, k), (z, j)) = d M & P˜i ((x, j), (x, k)) = P˜i ((z, j), (z, k)) = d1 . By condition ML for individual j with respect to the pair {x, y}: P(x, y) Q¯ P(y, x) = d1 .

(4.4.4.1)

By condition ML for individual k with respect to the pair {y, z}: P(y, z) Q¯ P(z, y) = d1 .

(4.4.4.2)

By maxmin transitivity over the triple {x, y, x}: P(x, z)Qm(P(x, y), P(y, z)) whence, in view of (4.4.4.1) and (4.4.4.2), ¯ 1. P(x, z) Qd

(4.4.4.3)

However, since {x, y} ∈ D j and {y, z} ∈ Dk , it follows that {x, z} ∈ D j,k , so that by condition ME with respect to the pair {z, x}, one has: P(z, x) Q¯ P(x, z) = d1 .

(4.4.4.4)

4.4 On the Possibility of Egalitarian Liberalism

37

(4.4.4.3) and (4.4.4.4) are mutually incompatible. Case (ii) Let individual preferences be as follows: P˜ j ((x, j), (y, j)) = d M , P˜ j ((y, j), (x, j)) = d1 ; P˜k ((w, k), (z, k)) = d M , ˜ Pk ((z, k), (w, k)) = d1 ; P˜ j ((y, j), (w, j)) = d M , P˜ j ((w, j), (y, j)) = d1 ; P˜k ((y, k), (w, k)) = d1 , ˜ Pk ((w, k), (y, k)) = d M ; ∀i ∈ N : P˜i ((y, k), (y, j)) = P˜i ((w, k), (w, j)) = d M & P˜i ((y, j), (y, k)) = ˜ Pi ((w, j), (w, k)) = d1 ; P˜ j ((x, j), (z, j)) = d M , P˜ j ((z, j), (x, j)) = d1 ; P˜k ((x, k), (z, k)) = d1 , P˜k ((z, k), (x, k)) = d M ; ∀i ∈ N : P˜i ((x, j), (x, k)) = P˜i ((z, j), (z, k)) = d M & P˜i ((x, k), (x, j)) = P˜i ((z, k), (z, j)) = d1 . As was noted in Remark 4.3, when x, y, w, and z are distinct, {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j = {{x, w}, {x, z}, {y, w}, {y, z}} ∩ Dk = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) , then {x, z} ∈ D j,k and {y, w} ∈ D j,k . By condition ML for person j with respect to the pair {x, y}: P(x, y) Q¯ P(y, x) = d1 .

(4.4.4.5)

By condition ME with respect to the pair {y, w}: P(y, w) Q¯ P(w, y) = d1 .

(4.4.4.6)

By maxmin transitivity over the triple {x, y, w}: P(x, w)Qm(P(x, y), P(y, w)) whence, in view of (4.4.4.5) and (4.4.4.6), we have: ¯ 1. P(x, w) Qd

(4.4.4.7)

By condition ML for person k with respect to the pair {w, z}: P(w, z) Q¯ P(z, w) = d1 .

(4.4.4.8)

By maxmin transitivity over the triple {x, w, z}: P(x, z)Qm(P(x, w), P(w, z)) whence, in view of (4.4.4.7) and (4.4.4.8), we have: ¯ 1. P(x, z) Qd

(4.4.4.9)

By condition ME with respect to the pair {z, x}: P(z, x) Q¯ P(x, z) = d1 .

(4.4.4.10)

(4.4.4.9) and (4.4.4.10) are mutually incompatible, and this completes the proof of the theorem. 

38

4 Liberty, Equality, and Impossibility: Some General Results …

One can now obtain a possibility result if the liberty and equity principles are diluted to their respective weakened versions: indeed, now the domain of the aggregation rule can also be expanded to accommodate #N-tuples of ‘properly’ soft extended preference relations. The following proposition is true. ˜ 0 )#N → 0 satisfying conditions WML Theorem 4.5 There exists a GSAR f : ( and WE. Proof Construct the following GSAR f ∗: ˜ 0 )#N : P ∗ (x, y) = d1 . ∀x, y ∈ X, ∀( P˜i )i∈N ∈ ( It is easy to see that f ∗ satisfies condition MWE. Let { j, k} be any pair of individuals and {x, y} any pair of alternatives such that P˜ j ((x, j), (y, j)) = d M , P˜ j ((y, j), (x, j)) = d1 , P˜k ((x, k), (y, k)) = d1 , P˜k ((y, k), (x, k)) = d M and ∀i ∈ N : P˜i ((x, k), (x, j)) = P˜i ((y, k), (y, j)) = d M & P˜i ((x, j), (x, k)) = ˜ Pi ((y, j), (y, k)) = d1 . Then, by construction of f ∗, P ∗ (x, y)(= d1 )Q P ∗ (y, x)(= d1 ), as required by condition MWE. It is also clear that f ∗ satisfies condition WML. For, again, let j be any individual and {x, y} any pair of alternatives such that P˜ j ((x, j), (y, j)) = d M and P˜ j ((y, j), (x, j) = d1 ; then, by construction of f ∗, P ∗ (x, y)(= d1 )Q P ∗ (y, x)(= d1 ), as required for condition MWL to be satisfied. It is immediate that P∗ is irreflexive and also satisfies asymmetry by default (note that by construction of f ∗, P ∗ (x, y) = d M can happen for no x, y ∈ X ). Finally, consider any triple of alternatives {x, y, z}. By construction of f ∗, P ∗ (x, z)(= d1 )Qm[P ∗ (x, y)(= d1 ), P ∗ (y, z)(= d1 )]: thus, P∗ satisfies maxmin transitivity. This completes the proof of the theorem.  Remark 4.6 While the existence result of Theorem 4.5 is certainly more encouraging than the impossibility result of Theorem 4.4, the full implication of the result— in terms of the possibility of consistent social choice—becomes transparent only when we move from a ‘relation-functional’ to a ‘choice-functional’ framework of preference aggregation. It seems reasonable to believe—see, for example, Dutta (1987)—that while preferences can be vague, choice, of necessity, must be exact (after all, one chooses or one does not choose: it is hard to confer any sensible interpretation on the notion of ‘degrees’ of choice); and it is the question of choice functions and exact choice that is addressed in the following section.

4.5 Choice Functions and Exact Choice There is an extensive literature on the theory of exact choice in a framework of exact preferences; standard references would include Arrow (1959), Richter (1966),

4.5 Choice Functions and Exact Choice

39

and Sen (1971). Considerable recent work is also available on exact choice in a framework of fuzzy preferences; of particular interest are the papers by Basu (1984), Dutta, Panda and Pattanaik (1986), Barrett, Pattanaik and Salles (1990), Dutta (1987), and Basu, Deb and Pattanaik (1992, especially Sect. 6). I begin with a definition of an exact choice function. In what follows, X will stand for the power set of X, that is the set of all subsets of the set of conceivable alternatives. Definition 4.1 (Crisp Choice Function) A crisp choice function (CCF) is a mapping C : X → X such that for all A(= φ) ∈ X , φ = C(A) ⊆ A. Remark 4.2 Definition 4.1 tells us that a (crisp) choice function is a function C which, for every non-empty set A in the power set of X assigns a non-empty subset of A, C(A), which is the set of chosen elements from A, or the choice set of A. The issue of rationalizability of a choice function revolves around the investigation of a systematic link between the choice set of A and a binary preference relation, in terms of which the choices made from A can be ‘explained’. One particular notion of rationalizability of a CCF by a soft binary preference relation—that of H-rationalizability (see Dutta, Panda and Pattanaik 1986; Dutta 1987)—is discussed below. Let A ∈ X , and let P be any SBPR. For every a ∈ A, let a ∗A ∈ A\{a} be defined such that P(a, b)Q P(a, a ∗A )∀b ∈ A\{a}. In some straightforward sense, a ∗A can be interpreted as a’s ‘closest rival’ in A, in terms of the preference relation P. Now define the set B(A, P) ≡ {x ∈ A P(x, x ∗A )Q P(y, y ∗A )∀y ∈ A} . Stated informally, an alternative x will belong to the set B(A, P)—which it is convenient to interpret as the set of ‘P-best elements of A’—if and only if the extent to which x is preferred to its ‘nearest rival’ is at least as much as the extent to which any other alternative y in A is preferred to its (y’s) ‘nearest rival’. The notion of H-rationalizability can now be precisely defined. Definition 4.3 (H-Rationalizability) A CCF C is H-rationalizable by a SBPR P if and only if for all A ∈ X , C(A) = B(A, P). (That is, C is H-rationalizable by P if the choice set of A consists only of those elements which are P-best in A.) One of a number of ‘consistency properties’ of choice functions, intensively investigated in the literature by, among others, Bordes (1976) and Sen (1977), is a strong ‘expansion’ consistency property called β + , which is now defined. Definition 4.4 (Expansion Consistency: Property $$\beta ˆ{ + }$$) Property β + requires of a CCF that for all x, y ∈ X and all A, B ∈ X such that A ⊆ B: [x ∈ C(A)&y ∈ A] → [y ∈ C(B) → x ∈ C(B)]. (That is, if x belongs to the choice set of a set A which contains y, then y cannot belong to the choice set of B which is an expansion of A without x also belonging to the choice set of B).

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4 Liberty, Equality, and Impossibility: Some General Results …

A result concerning Property β + which is of importance for future use is presented below. This result is a close parallel of one stated and proved, in a ‘cardinal’ framework of fuzzy preferences, in Dutta, Panda, and Pattanaik (1987: see their Proposition 5.4(a)). Lemma 4.5 Let Cbe a CCF. If Cis H-rationalizable by a SBPR P ∈ 0 , then Csatisfies Property β + . Proof I shall suppose the lemma to be false and derive a contradiction. Let C be a CCF which is H-rationalizable by a SBPR P ∈ 0 . Let x, y ∈ X and A, B ∈ X with A ⊆ B such that x ∈ C(A), y ∈ A, y ∈ C(B) and x ∈ / C(B): this will be shown to lead to contradiction. Since x ∈ C(A) and y ∈ A, and C is H-rationalizable by P, P(x, x ∗A )Q P(y, y ∗A ).

(4.5.5.1)

Similarly, since y ∈ C(B), x ∈ B, x ∈ / C(B), and C is H-rationalizable by P, P(y, y B∗ ) Q¯ P(x, x B∗ ).

(4.5.5.2)

P(y, y ∗A )Q P(y, y B∗ ).

(4.5.5.3)

Note next that since A ⊆ B,

From (4.5.5.1), (4.5.5.2), and (4.5.5.3), we obtain P(x, x ∗A ) Q¯ P(x, x B∗ ).

(4.5.5.4)

Since P ∈ 0 , P satisfies maxmin transitivity; consequently, maxmin transitivity over the triple {x, y, x B∗ } requires that P(x, x B∗ )Qm(P(x, y), P(y, x B∗ )).

(4.5.5.5)

P(y, x B∗ )Q P(x, y).

(4.5.5.6)

Suppose now that

Then, by (4.5.5.5), we must have P(x, x B∗ )Q P(x, y).

(4.5.5.7)

But note that, by definition of x ∗A , P(x, y)Q P(x, x ∗A ).

(4.5.5.8)

From (4.5.5.7) and (4.5.5.8), we obtain: P(x, x B∗ )Q P(x, x ∗A ) which, however, contradicts (4.5.5.4). Therefore, (4.5.5.6) must be false, and one must have:

4.5 Choice Functions and Exact Choice

P(x, y) Q¯ P(y, x B∗ )

41

(4.5.5.9)

which, in view of (4.5.5), leads to: P(x, x B∗ )Q P(y, x B∗ ).

(4.5.5.10)

But note that, by definition of y B∗ , P(y, x B∗ )Q P(y, y B∗ ).

(4.5.5.11)

From (4.5.5.10) and (4.5.5.11), we obtain: P(x, x B∗ )Q P(y, y B∗ ) which, however, contradicts (4.5.5.2).



Lemma (5.1) Has an important role to play in the choice-functional approach to aggregating individual preferences, to which I now turn. Definition 4.6 (Crisp Collective Choice Function) A crisp collective choice func˜  )#N → X (where φ =  ˜  ⊆ ) ˜ such that, tion (CCCF) is a mapping C : X × (  #N ˜ ) , φ = C(A, ( P˜i )i∈N ) ⊆ A. for all A(= φ) ∈ X and for all ( P˜i )i∈N ∈ ( The notion of H-generation of a crisp collective choice function by a generalized soft aggregation rule (see Dutta 1987) is now considered. ˆ (φ = ˜  )#N →  Definition 4.7 (H-generation of a CCF by a GSAR) Let f : (  ˜ ⊆ ) ˜ and φ =  ˆ ⊆ ) be a GSAR, and let C : X × ( ˜  )#N → X be a  CCF. Then, C will be said to be H-generated by f if and only if for all A ∈ X ˜  )#N , C is H-rationalizable by f , that is, C(A, ( P˜i )i∈N ) = and all ( P˜i )i∈N ∈ ( ˜ B(A, f (( Pi )i∈N )), where B(A, f (( P˜i )i∈N )) is the set of ‘ f -best’ elements in A. The choice-functional approach to aggregating preferences when the aggregating mechanism is required to satisfy the principles of equity and liberty, is considered in the next section.

4.6 Liberty, Equity, and the Possibility of Consistent Choice The following proposition is true. ˜ 0 )#N → 0 be a GSAR which satisfies conditions WML Theorem 4.1 Let f : ( and MWE (we know, from Theorem 4.5, that such an f exists), and let Cbe a CCF which is H-generated by f . Suppose X contains a subset A = {x, y, w, z}such that {x, y} ∈ D j and {w, z} ∈ Dk for some j, k ∈ N , and either (i) (ii)

#({x, y} ∩ {w, z}) = 1; or = {x, y} ∩ {w, z} = {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j {{x, w}, {x, z}, {y, w}, {y, z}} ∩ Dk = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) .

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4 Liberty, Equality, and Impossibility: Some General Results …

˜ 0 )#N such that the choice set of A would Then, there exists a profile ( P˜i )i∈N ∈ ( have to admit every equity-dominated and every liberty-dominated social state in A. Proof Two cases must be distinguished: (i) the pairs {x, y} and {w, z} have exactly one alternative in common, say y = w; and (ii) x, y, w, and z are all distinct, with {{x, w}, {x, z}, {y, w}, {y, z}} ∩ D j = {{x, w}, {x, z}, {y, w}, {y, z}} = φ and x−( j,k) = y−( j,k) = w−( j,k) = z −( j,k) . Case (i) Consider exactly the same pattern of individual preferences as has been employed in the proof of case (i) of Theorem 4.4. It is easy to see, given the pattern of individual preferences, that x is an equity-dominated state (dominated by z), while y and z are liberty-dominated states (dominated by x and y, respectively). It must now be proved that: if x ∈ / C(A), then C(A) = φ; if y ∈ / C(A), then C(A) = φ; and if z∈ / C(A), then C(A) = φ. Suppose, to the contrary, that—first—x ∈ / C(A) and C(A) = φ. Then at least one of the following must be true: (a) y ∈ C(A) or (a ) z ∈ C(A). By WML for person j with respect to the pair {x, y}, P(x, y)Q P(y, x) whence, since C is H-generated by f , x ∈ C({x, y}). Recalling that the range of f is 0 , we know from Lemma 4.5 that C satisfies Property β + . Therefore, [x ∈ C({x, y}) and y ∈ C(A)] implies x ∈ C(A), contrary to supposition. Hence, (a) cannot be true. Suppose, next, (a ) to be true. By WML for person k with respect to the pair {y, z}, P(y, z)Q P(z, y), whence y ∈ C({y, z}). By Property β + , [y ∈ C({y, z}) and z ∈ C(A)] implies y ∈ C(A) which, as we have already seen, leads to contradiction. Hence, (a ) is also false, and C(A) = φ, as desired. Next suppose y ∈ / C(A) and C(A) = φ. Then, at least one of the following is true: (b) x ∈ C(A) or (b ) z ∈ C(A). Suppose, first, (b) to be true. By condition MWE with respect to the pair of alternatives {z, x}, P(z, x)Q P(x, z), whence z ∈ C({x, z}). Since, again by Lemma 4.5, C satisfies Property β + , [z ∈ C({x, z}) and x ∈ C(A)] implies z ∈ C(A). Note, further, that by WML for k over the pair {y, z}, P(y, z)Q P(z, y), so y ∈ C({y, z}). Again by Property β + , [y ∈ C({y, z}) and z ∈ C(A)] implies y ∈ C(A), contrary to supposition. Therefore, (b) is false. Suppose (b ) to be true. But we have already seen that z ∈ C(A) must imply, contrary to supposition, that y ∈ C(A). Therefore, (b ) is also false, that is, C(A) = φ, as desired. Finally, suppose z ∈ / C(A) and C(A) = φ. Then, at least one of the following must be true: (c) x ∈ C(A) or (c ) y ∈ C(A). Suppose (c) to be true. By MWE with respect to {z, x}, P(z, x)Q P(x, z); hence z ∈ C({x, z}). By Property β + , [z ∈ C({x, z}) and x ∈ C(A)] implies z ∈ C(A), which is a contradiction. Thus, (c) must be false, leaving us with (c ). By WML for j with respect to {x, y}, P(x, y)Q P(y, x), leading to: x ∈ C({x, y}) and, by Property β + , to x ∈ C(A); but we have just seen that x ∈ C(A) cannot be true. Therefore, (c ) is also false, leaving us with C(A) = φ, as desired. Case (ii) By invoking exactly the same configuration of individual preferences as has been done in the proof of case (ii) of Theorem 4.4, case (ii) of the present theorem

4.6 Liberty, Equity, and the Possibility of Consistent Choice

43

can be easily proved along the lines of the proof of case (i). The proof is here omitted in order to avoid tedious repetitiveness.  Remark 4.2 The impossibility result of Theorem 4.4 can be avoided by weakening the liberty and equity axioms (Theorem 4.5). While preferences can be vague, it seems reasonable to require that choice be exact; and Theorem 4.1 shows that the price to be paid for the possibility result embodied in Theorem 4.5 is that the equity and liberty principles could altogether lose their cutting edge: if social choice is not to be vacuous, then for certain preference profiles the choice set would have to admit all alternatives that are equity- and liberty-dominated. (In this connection, it is instructive to consider Sen’s 1976 remarks on the consequences of weakening the liberty principle along the lines suggested by Karni 1978 in the context of the Pareto-liberty paradox.)

4.7 Concluding Observations In this paper, I have examined the problem of aggregating individual preferences when the preference aggregation mechanism is required to satisfy the ethical principles of individual liberty and equity. The problem has been investigated in both ‘relation-functional’ and ‘choice-functional’ terms. The results of these exercises have been essentially negative. This may or may not possess ‘surprise-value’, but it is worth emphasizing that the results are discouraging despite the facts that: (a)

(b)

(c)

the conventional tight framework of exact or crisp social preferences has been relaxed by allowing for ordinal soft social preferences: this easing-up does not prevent a generalization of the impossibility result from the crisp framework to its soft counterpart; the equity principle explicitly allows for interpersonal comparability of utility: the Arrow–Sen programme of ‘extended sympathy’, designed to enrich the informational basis of preference aggregation, is not of great assistance in preventing the libertarian dilemma discussed in the paper (see also, in this connection, Kelly 1976); and Hammond’s equity principle has been reformulated in such a way that the modified axiom employed in this paper is no longer a ‘welfarist’ principle: from an interpretational point of view, the blame for the ‘impossibility of an egalitarian liberal’ cannot—unlike in the case of Sen’s (1970) ‘impossibility of a Paretian liberal’—be laid at the door of ‘welfarism’.

In pursuance of the last issue, it may be noted that Pattanaik (1988) makes a similar point—to the effect that libertarian dilemmas are not necessarily always a reflection of the inadequacies of welfarism. This he does by pointing to the possibility of a problem of internal consistency of libertarian values, interpreted as a conflict between ‘group’ and ‘individual’ liberties (in which connection, see also Riley 1989,

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4 Liberty, Equality, and Impossibility: Some General Results …

1990). Indeed, the ‘impossibility of egalitarian liberalism’ considered in this paper could itself be construed as an instance of internal inconsistency of libertarian values of the type exemplified by incompatibility between ‘group’ and ‘individual’ rights— if the ‘rules of association’ of the group constituted by any two individuals required that the group’s collective preference will be determined by considerations of equity (rather than, as is more commonly presumed, by considerations of unanimity). Finally, it is important to underline the fact that the number of individuals in society has, throughout this paper, been assumed to be at least two. In a substantive sense, the significance of the ‘impossibility of an egalitarian liberal’ would greatly diminish if we were to restrict the number of individuals in society to be strictly greater than two: for, with #N ≥ 3, it can be verified that the equity principle (in its strong form) would run into problems of internal consistency—which would tend to reduce the force of an inconsistency between the equity and the liberty principles. But in a framework which does allow for a two-person society, there would appear to be a genuine ‘equity-liberty’ dilemma.

References Arrow KJ (1959) Rational choice functions and orderings. Economica NS 26(101):121–127 Arrow KJ (1963) Social choice and individual values, 2nd edn. Wiley, New York Arrow KJ (1977) Extended sympathy and the possibility of social choice. Am Econ Rev 67(1):219– 225 Banerjee A (1994) Fuzzy preferences and arrow-type problems in social choice. Soc Choice Welfare 11(2):121–130 Basu K (1984) Fuzzy revealed preference theory. J Econ Theor 32(2):212-227 Basu K, Deb R, Pattanaik PK (1992) Soft sets: an ordinal formulation of vagueness with some applications to the theory of choice. Fuzzy Sets Syst 45(1):45–58 Bordes G (1976) Consistency, rationality and collective choice. Rev Econ Stud 43(3):451–457 Dimitrov D (2004) The paretian liberal with intuitionistic fuzzy preferences: a result. Soc Choice Welfare 23(1):149–156 Dutta B (1987) Fuzzy preferences and social choice. Math Soc Sci 13(3):215–229 Dutta B, Panda SC, Pattanaik PK (1986) Exact choice and fuzzy preferences. Math Soc Sci 11(1):53– 68 Gibbard A (1974) A pareto consistent libertarian claim. J Econ Theor 7(4):388–410 Goguen JA (1967) L-fuzzy sets. J Math Anal Appl 18(1):145–174 [Cited in Barrett and Pattanaik,1990.] Hammond PJ (1976) Equity, arrow’s conditions, and Rawls. Difference principle. Econometrica 44(4):793–804 Karni E (1978) Collective rationality, unanimity and liberal ethics. Rev Econ Stud 45(2):571–574 Kelly JS (1976) Rights-exercising and a Pareto-Consistent libertarian claim. J Econ Theor 13(1):138–153 Pattanaik PK (1988) On the consistency of libertarian values. Economica NS 55(220):517–524 Richardson G (1998) The structure of fuzzy preferences: social choice implications. Soc Choice Welfare 15(3):359–369 Richter M (1966) Revealed preference theory. Econometrica 34(3):635–645 Riley J (1989) Rights to liberty in purely private matters, Part 1. Econ Philos 5:121–166 Riley J (1990) Rights to liberty in purely private matters, Part 2. Econ Philos 6:27–64 Sen AK (1970) The impossibility of a paretian liberal. J Polit Econ 78(1):152–159

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Sen A (1970) Collective choice and social welfare. Holden Day, San Francisco Sen AK (1971) Choice functions and revealed preference. Rev Econ Stud 38(3):307–317 Sen AK (1973) On economic inequality. Oxford University Press, Clarendon Sen AK (1976) Liberty, unanimity and rights. Economica NS 43(171):217–245 Sen AK (1979) Personal utilities and public judgments or what’s wrong with welfare economics? Econ J 89(355):537–558 Sen AK (1977) Social choice theory: a re-examination. Econometrica 45(1):53-89 Subramanian S (1987) The liberal paradox with fuzzy preferences. Soc Choice Welfare 4(3):213– 218

Chapter 5

The Arrow Paradox with Fuzzy Preferences

Abstract This note considers a new factorization of a fuzzy weak binary preference relation into its asymmetric and symmetric parts. Arrow’s General Possibility Theorem is then examined within the resulting framework of vague individual and social preferences. The outcome of this exercise is compared with some earlier results available in the literature on the Arrow paradox with fuzzy preferences. Keywords Fuzzy weak binary preference relation · Fuzzy strict binary preference relation · Fuzzy indifference relation · Fuzzy aggregation rule · Dictatorship · oligarchy

5.1 Introduction Barrett et al. (1986) have considered the classical Arrow (1963) problem of aggregation of individual preferences, in a context wherein both personal and public preferences are taken to be vague. The results of these exercises have been, on the whole, discouraging, with the impossibilities in the exact framework more or less carrying over to their fuzzy counterparts. In the work just cited, an important component of the methodology employed has been the treatment of the fuzzy binary relation of strict preference as a primitive. Dutta (1987), on the other hand, has started out with the fuzzy relation of ‘weak’ preference and then derived, through axiomatic rationalization, the asymmetric and symmetric parts of this relation. He then proceeds, within this framework of fuzzy individual and social preferences, to review Arrowtype theorems of aggregation. Alternative avenues of decomposition of the weak preference relation into its strong preference and indifference parts have also been explored by, among others, Banerjee (1994), Richardson (1998), and Dasgupta and

This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2009) The Arrow paradox with fuzzy preferences, Mathematical Social Sciences, 58:2, 265-271, DOI: https://doi.org/10.1016/j.mathsocsci.2009.05.001. I am grateful to Prasanta Pattanaik and to two anonymous referees of Mathematical Social Sciences for very helpful comments on earlier versions. The usual disclaimer applies. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_5

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5 The Arrow Paradox with Fuzzy Preferences

Deb (1999); the ‘aggregational’ implications of these exercises have been summarized and reviewed in a very comprehensive survey by Barrett and Salles (2006). A further variation on this theme is offered in the present paper. Inasmuch as it is a variation on a theme, the paper will not dwell elaborately on an explication of concepts and definitions. Also, the focus of this paper is on the problem, somewhat specifically, of social choice; consequently, potentially interesting issues pertaining to the properties of different fuzzy preference relations (strong, weak, and indifference), and their interrelationship, will receive, at best, only perfunctory attention. In what follows, I too start out with a fuzzy weak preference relation and, with a slightly differently specified axiom system to that employed by Dutta, derive the corresponding strict preference and indifferent relations. Within this framework of fuzzy preferences, the ‘relation-functional’ versions of some Arrow-type aggregation problems are re-examined. The results, on the whole, are encouraging, in terms of offering a way out of both Arrow’s ‘dictatorship’ result and Gibbard’s (1969) ‘oligarchy’ result.

5.2 The Asymmetric and Symmetric Components of the Fuzzy Weak Preference Relation X = {x, y, z, . . .} is the finite set of all conceivable alternatives, with X containing at least three elements. N = {1, ..., i, ..., n} is the finite set of individuals constituting society, with N containing at least two members. A fuzzy weak binary preference relation (FWBPR) is a function R : X × X → [0, 1], while an exact weak binary preference relation (EWBPR) is a function R : X × X → {0, 1}. An FWBPR R on X is (a) reflexive iff for all distinct x ∈ X : R(x, x) = 1; (b) connected iff for all distinct x, y ∈ X : R(x, y)+ R(y, x) ≥ 1; and (c) maxmin (or M-) transitive iff for all distinct x, y, z ∈ X : R(x, z) ≥ min[R(x, y), R(y, z)]. A fuzzy weak binary preference ordering (FWBPO) is an FWBPR which is reflexive, connected, and M-transitive. Dutta (1987) has shown that extracting the strict preference (P) and the indifference (I ) relations from an FWBPR R on X is an unfruitful exercise when R, P, and I are governed by the rules which hold for them in the exact framework. More precisely, he demonstrates that if R is a connected FWBPR satisfying (i) R = P ∪ I , viz.∀x, y ∈ X : R(x, y) = max[P(x, y), I (x, y)]; (ii) I is symmetric, viz. ∀x, y ∈ X : I (x, y) = I (y, x); (iii) P is (strongly) asymmetric, viz. ∀x, y ∈ X : P(x, y) > 0 → P(y, x) = 0; and (iv) P ∩ I = φ, viz. ∀x, y ∈ X : min[P(x, y), I (x, y)] = 0; then, either R is an EWBPR or ∀x, y ∈ X : R(x, y) = R(y, x) = I (x, y) = I (y, x). In order to find some meaningful way out of this result, one or more of axioms (i)–(iv) listed in the statement of the result must be relaxed. Dutta himself regards axiom (iv) as a natural candidate for relaxation and demonstrates the truth of the following proposition: if R

5.2 The Asymmetric and Symmetric Components …

49

is a connected FWBPR satisfying axioms (i)–(iii) above and also axiom (iv ), namely ∀x, y ∈ X : R(x, y) = R(y, x) → P(x, y) = P(y, x), then, (∗) ∀x, y ∈ X : P(x, y) = R(x, y) if R(x, y) > R(y, x); = 0, otherwise; and I (x, y) = min[R(x, y), R(y, x)] In what follows, I pursue an alternative route to weakening the axiom system (i)– (iv). In particular, it is easy to see that axiom (iii) is just about the strongest version of asymmetry one can invoke—requiring, as it does, that no two alternatives can be strictly preferred to each other with any positive degree of confidence. Barrett et al. (1986) employ a weaker condition of asymmetry in terms of which if any alternative x is preferred to any other alternative y with complete confidence, then y may not be preferred to x with any positive degree of confidence: ∀x, y ∈ X : P(x, y) = 1 → P(y, x) = 0. This formulation is compatible with a situation in which, for a pair of alternatives x,y, P(x, y) = P(y, x) = 0.99; if now the extent to which x is strictly preferred to y were to rise by a marginal degree from 0.99 to 1.0, then asymmetry would demand that P(y, x) must decline abruptly from 0.99 to zero. This sort of discontinuously precipitous decline in P(y, x) for a marginal increase in P(x, y)— a point observed by Richardson (1998)—is not a very appealing property of the asymmetric relation. A possible mitigating factor is that the discontinuity property just noted is a somewhat ‘localized’ phenomenon, confined to values of P(x, y) and P(y, x) that are in the ‘neighbourhood’ of 1. Further, some sort of ‘discontinuity problem’ is probably an inescapable price one has to pay for the ‘fuzzification’ of preference relations. In this connection, consider the formulation of the strict preference and indifference relations in (*). Notice that, under this formulation, one can have a situation in which for any pair of alternatives x, y, R(x, y) = 0.7 and R(y, x) = 0.69, so that P(x, y)[= R(x, y)] = 0.7 and P(y, x) = 0; if now there is a small increase in R(y, x) from 0.69 to 0.7, then P(x, y) would have to drop suddenly from 0.7 to zero. Indeed, this problem is ‘endemic’ under the (*) formulation, in the sense that for any pair of alternatives x, y, and for all α ∈ [0, 1], one can have a situation in which P(x, y) = R(x, y) = α and R(y, x) = α −ε, where ε is some arbitrarily small positive number, but P(x, y) must decline to zero if R(y, x) should rise marginally from α − ε to α. Indeed, a similar difficulty presides over the fuzzy indifference relation also: notice that the symmetry of the indifference relation, together with axioms (i) and (iv), ensures that for any x,y, if R(x, y) = R(y, x), then it must always be the case that I (x, y) = I (y, x) = 0, whereas one can have I (x, y) = R(x, y) when R(x, y) = R(y, x). This can lead to a discontinuity of the following sort. Suppose R(x, y) = R(y, x) = 0.7, then it is possible that I (x, y) = 0.7; however, if R(x, y) were to rise marginally to 0.71, I (x, y) would have to decline discontinuously from 0.7 to zero. Again, the problem is ‘endemic’, in the sense that for any pair of alternatives x, y, and for all α ∈ [0, 1], one can have a situation in which I (x, y) = R(x, y) = R(y, x) = α, but if, other things remaining constant, R(y, x) were to decline marginally from α to α −ε, where

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ε is some arbitrarily small positive number, then I (x, y) must decline from α to zero. One possible way of avoiding this problem would be to deny any genuine fuzziness to the indifference relation, by requiring it to be an exact relation. With a modified axiom system which weakens asymmetry of the strict preference relation and imposes exactness on the indifference relation along the lines just discussed, the following proposition can be stated. Proposition 5 Let R be a connected FWBPR satisfying (i) (ii) (iii ) (iv) (v)

R = P ∪ I , viz. ∀x, y ∈ X : R(x, y) = max[P(x, y), I (x, y)]; I is symmetric, viz. ∀x, y ∈ X : I (x, y) = I (y, x); P is (weakly) asymmetric, viz. ∀x, y ∈ X : P(x, y) = 1 → P(y, x) = 0; P ∩ I = φ, viz. ∀x, y ∈ X : min[P(x, y), I (x, y)] = 0; and I is exact, viz. ∀x, y ∈ X : I (x, y) ∈ {0, 1}.

Then, (**) ∀x, y ∈ X : P(x, y) = 0 & I (x, y) = R(x, y) if R(x, y) = R(y, x) = 1; P(x, y) = R(x, y) & I (x, y) = 0 otherwise. Proof Suppose, first, that R(x, y) = R(y, x) = 1, but P(x, y) = 0. By (iv) and (ii), I (x, y) = I (y, x) = 0, whence by (i), P(x, y) = R(x, y) & P(y, x) = R(y, x), viz. P(x, y) = P(y, x) = 1, which, however, violates (iii ). Therefore, if R(x, y) = R(y, x) = 1, then P(x, y) = 0, whence, in view of (i), I (x, y) = R(x, y). Suppose, next, that R(x, y) = R(y, x) = α < 1, but P(x, y) = R(x, y). Then, by (i), I (x, y) = R(x, y) and by (ii), I (y, x) = I (x, y) = α ∈ / {0, 1}, and this violates (v). Therefore, if R(x, y) = R(y, x) < 1, then P(x, y) = R(x, y) and so, by (i) and (iv), I (x, y) = 0. Finally, suppose R(x, y) = R(y, x). WLOG, let R(x, y) > R(y, x). If now P(x, y) = R(x, y), then by (i), I (x, y) = R(x, y); by (ii) I (y, x) = I (x, y) > 0; by (iv), I (y, x) = max[P(y, x), I (y, x)]; and by (i), I (y, x) = R(y, x). Taken together, one must then have: R(x, y) = R(y, x), in contradiction of the fact that R(x, y) > R(y, x). Therefore, if R(x, y) = R(y, x), then P(x, y) =  R(x, y) & I (x, y) = 0. This completes the proof of the proposition. As was noted at the outset, Barrett et al. (1986; BPS, for short) employ the fuzzy strict preference relation P as a primitive; they require P to satisfy the following properties: for all distinct x, y, z ∈ X : P(x, x) = 0 (irreflexivity); P(x, y) = 1 → P(y, x) = 0 (BPS asymmetry); and P(x, y) > 0 & P(y, z) > 0→ P(x, z) > 0 (a transitivity condition which is weaker than max–min transitivity). We shall let H1 stand for the set of all fuzzy strict preference relations P which satisfy these properties. We shall also let H2 [respectively, H3 ] stand for the set of all FWBPOs R on X such that the asymmetric and symmetric parts of R are as defined in (*) [respectively, (**)], and R is M-transitive.

5.3 Fuzzy Aggregation Rules

51

5.3 Fuzzy Aggregation Rules Let T be a set of all fuzzy binary preference relations (FBPRs) on X. In what follows, we shall be specifically concerned with three distinguished subsets of T, which we have called H1 , H2 , and H3 respectively. n  (T , T ( = φ) ⊆ T ) A fuzzy aggregation rule (FAR) is a function f : T → T such that, for every n-tuple of individual fuzzy binary preference relations (h i ) in its domain, f specifies a unique social fuzzy binary preference relation h in its range. n Elements of T , which are preference profiles, will be designated (Ri )i∈N , (Ri )i∈N , etc., when T = H2 or H3 ; and (Pi )i∈N , (Pi )i∈N , etc., when T = H1 . We shall also write R for f ((Ri )i∈N ), R  for f ((Ri )i∈N ), and P for f ((Pi )i∈N ) or for the asymmetric part of R, P  for f ((Pi )i∈N ) or for the asymmetric part of R  , etc. Some restrictions one may wish to impose on an FAR are defined below (these are standard conditions in the social choice literature and will therefore be dealt with summarily). n  satisfies An FAR f : T → T 











n



Neutrality (Condition N) iff ∀(h i )i∈N , (h i )i∈N ∈ T and for all x, y, w, z ∈ X : [h i (x, y) = h i (w, z)∀i ∈ N & h i (y, x) = h i (z, w)∀i ∈ N ] implies [h(x, y) = h  (w, z) & h(y, x) = h  (z, w)]; n



Independence of Irrelevant Alternatives (Condition I) iff ∀(h i )i∈N , (h i )i∈N ∈ T and for all distinct x, y ∈ X : [h i (x, y) = h i (x, y)∀i ∈ N & h i (y, x) = h i (y, x)∀i ∈ N ] implies [h(x, y) = h  (x, y)&h(y, x) = h  (y, x)]; n



Anonymity (Condition A) iff ∀(h i )i∈N , (h i )i∈N ∈ T and for all distinct x, y ∈ X : [h i (x, y) = h σ (i) (x, y)∀i ∈ N where σ is any permutation of the elements of N] implies [h(x, y) = h  (x, y)&h(y, x) = h  (y, x)]; Non-Dictatorship 1 (Condition D1) iff there does not exist j ∈ N such that ∀(h i )i∈N and for all distinct x, y ∈ X : P j (x, y) > 0 & P j (y, x) = 0 implies P(x, y) > 0 & P(y, x) = 0; Non-Dictatorship 2 (Condition D2) iff there does not exist j ∈ N such that ∀(h i )i∈N and for all distinct x, y ∈ X : P j (x, y) > 0 implies P(x, y) > 0; Non-Dictatorship 3 (Condition D3) iff there does not exist j ∈ N such that ∀(h i )i∈N and for all distinct x, y ∈ X : P j (x, y) = 1 implies P(x, y) = 1; Non-Oligarchy 1 (Condition O1) iff there does not exist a coalition C such that ∀(h i )i∈N and for all distinct x, y ∈ X : (i) PC (x, y) > 0 & PC (y, x) = 0 implies P(x, y) > 0, and (ii) for any j ∈ C : P j (x, y) > 0 & P j (y, x) = 0 implies P(y, x) = 0, where a coalition is any non-empty subset of N, and, for all distinct x, y ∈ X , [PC (x, y) > 0 & PC (y, x) = 0] is employed as a shorthand for [Pi (x, y) > 0 & Pi (y, x) = 0∀i ∈ C];

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Non-Oligarchy 2 (Condition O2) iff there does not exist a coalition C such that ∀(h i )i∈N and for all distinct x, y ∈ X : (i) PC (x, y) > 0 implies P(x, y) > 0, and (ii) for any j ∈ C : P j (x, y) > 0 implies P(y, x) = 0, where a coalition is any non-empty subset of N, and, for all distinct x, y ∈ X , [PC (x, y) > 0] is employed as a shorthand for [Pi (x, y) > 0∀i ∈ C]; Non-Oligarchy 3 (Condition O3) iff there does not exist a coalition C such that ∀(h i )i∈N and for all distinct x, y ∈ X : (i) PC (x, y) = 1 implies P(x, y) = 1, and (ii) for any j ∈ C : P j (x, y) = 1 implies P(y, x) < P(x, y), where a coalition is any non-empty subset of N, and, for all distinct x, y ∈ X , [PC (x, y) = 1] is employed as a shorthand for [Pi (x, y) = 1∀i ∈ C]; and Pareto Criterion (Condition P) iff ∀(h i )i∈N and for all distinct x, y ∈ X : P(x, y) ≥ mini∈N Pi (x, y). It is useful to note the following. First, each of the three non-dictatorship conditions, as also each of the three non-oligarchy conditions, is an inherently plausible fuzzy version of the corresponding principle of social choice in the exact framework. Second, the three non-dictatorship Conditions D1, D2, and D3 are mutually independent, just as the three non-oligarchy Conditions O1, O2, and O3 are mutually independent. Third, a member of a coalition C such as has been described in the non-oligarchy Conditions 1, 2, and 3 is a vetoer. Under the definition contained in Condition O3, for instance, a person j who can ensure, for any pair of alternatives x, y, that the extent to which y is socially preferred to x can never equal or exceed the extent to which x is socially preferred to y just by virtue of his preferring x to y with complete confidence, and no matter what the preferences of others are, is a vetoer. Fourth, Condition N implies (without being implied by) Condition I, while Condition A implies (without being implied by) each of Conditions D1, D2, and D3. From BPS’ (1986) work, we know that there exists no FAR f : H1n → H1 satisfying Conditions I, D1, and P, and, indeed, that there exists no FAR f : H1n → H1 satisfying Conditions I, O1, and P: both Arrow’s ‘dictatorship’ result and Gibbard’s ‘oligarchy’ result are recovered in BPS’ fuzzy framework. Dutta’s (1987) formulation avoids the Arrovian impossibility but not the Gibbardian problem: he demonstrates that there exists an FAR f : H2n → H2 which satisfies Conditions I, D2, and P, but no f : H2n → H2 which satisfies Conditions I, O2, and P. The fuzzy framework we employ delivers us from both the Arrow and Gibbard impasses, as discussed next.

5.4 Two Possibility Theorems The following is a strong possibility result. Proposition 5 There exists an FAR f : H3n → H3 which satisfies Conditions N, A, and P.

5.4 Two Possibility Theorems

53 

Proof Construct the following FAR f : 

∀x, y ∈ X, ∀(Ri )i∈N ∈ H3n : R (x, y) = 1 if x = y; = [1 + mini∈N Ri (x, y)]/2, otherwise. Note first that by construction of f , R is reflexive. Moreover, by construction of f , for all distinct x, y ∈ X,R (x, y) ≥ 1/2, which ensures that R is connected. To see that R is M-transitive, consider the following. Let {x, y, z} ⊆ X be any triple of distinct alternatives, and let mini∈N Ri (x, z) = Rk (x, z) for some k ∈ N . Then, since the Ri are M-transitive, one must have: 











Rk (x, z) ≥ min[Rk (x, y), Rk (y, z)]

(5.1)

Suppose (a)Rk (x, y) ≥ Rk (y, z). Then, by virtue of (4.1), Rk (x, z) ≥ Rk (y, z)[≥ mini∈N Ri (y, z)]. Further, Rk (x, y) ≥ Rk (y, z) implies that Rk (x, z)≥ min[mini∈N Ri (x, y), mini∈N Ri (y, z)]: this is true irrespective of whether mini∈N Ri (x, y) ≥ Rk (y, z) or mini∈N Ri (x, y) < Rk (y, z). If (a) is not true, it must be the case that (b) Rk (y, z) > Rk (x, y) so that, in view of (4.1), Rk (x, z) ≥ Rk (x, y) [≥ mini∈N Ri (x, y)]. Further, Rk (x, z) ≥ Rk (x, y) implies that Rk (x, z) ≥min[mini∈N Ri (x, y), mini∈N Ri (y, z)]: this is true irrespective of whether mini∈N Ri (y, z) ≥ Rk (x, y) or mini∈N Ri (y, z) < Rk (x, z). We have thus proved that for every triple of distinct alternatives {x, y, z} : mini∈N Ri (x, z) ≥ min[mini∈N Ri (x, y), mini∈N Ri (y, z)], whence [1 + mini∈N Ri (x, z)]/2 ≥ min[{1 + mini∈N Ri (x, y)}/2, {1 + mini∈N Ri (y, z)}/2]] or, equivalently, by construction of f , R (x, z) ≥ min[R (x, y), R (y, z)], as required by M-transitivity. To see that f satisfies Condition P, note that for any pair of distinct alternatives x, y ∈ X, if mini∈N Pi (x, y) > 0, then, by construction of f , P (x, y) = R (x, y) =[1 + mini∈N Ri (x, y)}/2]≥mini∈N Ri (x, y) = mini∈N Pi (x, y). The proof of the proposition is completed by noting that f obviously satisfies both neutrality and anonymity.  

















Theorem 2 is a strong possibility result and implies that the Arrow paradox can be circumvented in the fuzzy framework under review: this follows from recalling that Condition N implies Condition I and Condition A implies Condition D3. Having said this, it must be added that the aggregation rule f is a far from satisfactory one, in (at least) one sense—the following one, as pointed out by an anonymous referee. Consider a pair of alternatives x, y, and two situations 1 and 2. In Situation 1, Ri (x, y) = ε∀i ∈ N , where ε is some arbitrarily small positive number. In situation 2, R1 (x, y) = ε & Ri (x, y) = 1∀i ∈ N \{1}. Our normal instincts would suggest that R(x, y) should be larger in Situation 2 than in Situation 1. However, f will not discriminate between Situations 1 and 2: it will dictate that, in both situations, R (x, y) = (1 + ε)/2. The discomfort one experiences with such a judgement is reminiscent of the discomfort one experiences when confronted by a poverty index π which measures the extent of poverty in a society by identifying it with the 





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income-gap ratio of the poorest of the poor individuals in that society. Imagine two communities of a million persons each, in each of which 500,000 individuals earn an income which is less than the poverty line of rupees 1000 per month; in society 1, let us suppose that the income distribution of the poor is given by the 500,000-vector (0,0,…,0), and in society 2, by the 500,000-vector (0,999,…,999). The corresponding vectors of income-gap ratios (or proportionate shortfalls of incomes from the poverty line) are, respectively, (1,1,…,1) and (1,0.001,…,0.001). Our normal instinct would be to certify that there is a good deal more poverty in the first society than in the second; however, the poverty measure π , which estimates the extent of poverty in terms of the income-gap ratio of the worst-off person, will certify that the extent of poverty in both societies is the same, at 1. The difficulty with π is that it endorses a sort of Rawlsian ‘dictatorship of the weakest’. What π does in the context of poverty measures, f does in the context of FARs! The next result shows that an aggregation rule from H3n to H3 can get around the Gibbardian ‘oligarchy’ problem as well. 

Proposition 5 There exists an FAR f : H3n → H3 which satisfies Conditions I, O3, and P. 

Proof Consider again the FAR f constructed in the proof of Theorem 2. We already know that f satisfies I and P, and that R is reflexive, connected, and M-transitive, so it only remains to prove that f is non-oligarchic. Suppose, to the contrary, that there exists an oligarchic coalition C. Now consider (Ri )i∈N ∈ H3n and x, y ∈ X , such that: Ri (x, y) = 1 & Ri (y, x) = 0∀i ∈ S1 , Ri (x, y) = 0 & Ri (y, x) = 1∀i ∈ S2 , and S1 and S2 are non-empty subsets of N satisfying S1 ∪ S2 = N and S1 ∩ S2 = φ. By construction of f , R (x, y) = [1 + mini∈N Ri (x, y)]/2 = 1/2 and R (y, x) = [1 + mini∈N Ri (y, x)]/2 = 1/2, whence P (x, y) = P (y, x) = 1/2. Let j ∈ C. Either j ∈ S1 or j ∈ S2 . Suppose (a) j ∈ S1 . Since j ∈ C, j must be a vetoer; since P j (x, y) = 1, one must have P (y, x) < P (x, y), which, however, is not the case. So (a) cannot be true, and it must be the case that (b) j ∈ S2 . Again, since j ∈ C, j must be a vetoer; since P j (y, x) = 1, one must have P (x, y) < P (y, x), which, however, is also not the case. Thus, assuming f to be oligarchic results in contradiction, and the proposition is proved.  

























An interesting ‘technical’ question which arises in the context of the preceding theorem is the one of why Dutta’s (1987) fuzzy formulation falls foul of the Gibbardian ‘oligarchy’ outcome, in contrast to the existence result of Proposition 3. (This query has, in fact, been raised by an anonymous referee of the paper.) I doubt I shall be able to provide a properly and completely satisfactory answer to the question, but it is possible that the following explanation may have some intuitive suggestiveness to commend it. The explanation revolves around the ‘naturalness’ with which the sorts of FARs that secure the possibility of a non-dictatorial aggregating mechanism lend themselves, in a Dutta-like framework, to the emergence of a specific oligarchic coalition, namely the coalition constituted by the set N of all individuals in society.

5.4 Two Possibility Theorems

55

This, by contrast, is not a feature of the framework employed in the present paper. To see this more clearly, it helps to note that Dutta’s proof of the existence of an FAR f : H2n → H2 satisfying Conditions I, D2, and P revolves around the construction of an FAR f¯ which is defined as follows: ¯ x) = 1; ∀x ∈ X, ∀(Ri )i∈N ∈ H2n : R(x, ¯ ∀x, y(x = y) ∈ X, ∀(Ri )i∈N ∈ H2n : R(x, y) = 1 if ∀i ∈ N : Ri (x, y) > Ri (y, x); = β ∈ (0, 1), otherwise. 

Notice that strict preference under the FAR f¯ (as, indeed, under the FAR f in the proof of Proposition 3) essentially mirrors the Pareto dominance relation (see Dutta 1987). It is easy to see that this sort of FAR is compatible with the existence of an oligarchy which is constituted simply by the grand coalition of individuals N in the Dutta framework (though not—as demonstrated in the proof of Proposition 3—within the framework employed in this paper). Under f¯, the grand coalition can be seen to be oligarchic from the fact, first, that since f¯ satisfies the Pareto condition, ¯ y)[= 1] > 0. Second, suppose we have a configuration PN (x, y) > 0 implies P(x, n of preferences in H2 such that, for some pair of alternatives x, y, neither alternative Pareto dominates the other. That is, suppose there exist individuals j and k such that R j (x, y) > R j (y, x) and Rk (x, y) < Rk (y, x). Then, the FAR f¯ will dictate that ¯ ¯ R(x, y) = R(y, x) = β, whence—following from Dutta’s derivation of the strict ¯ ¯ preference relation from the weak one under (*)— P(x, y) = P(y, x) = 0. This suggests that every member of the grand coalition N is a vetoer. Briefly, N is a readily identifiable oligarchic coalition in the Dutta framework, whereas, given Condition O3 and the particular manner in which the strict preference relation is derived from the weak one under (**), such a result does not emerge from the framework employed in this paper (as the proof of Proposition 3 makes clear). Whatever may be the specific ‘mechanics’ which explain the specific results which emerge from specific frameworks, what compels attention is the fact that there are certain combinations of domain, co-domain, and principles of fuzzy social choice which yield impossibility outcomes, and other combinations which yield existence results; it is hard to rank some particular combination above another in terms of an unambiguously appealing set of selection criteria. Such plurality and diversity of outcomes could, in principle, lead to a measure of arbitrariness in the presentation and interpretation of results in fuzzy social choice theory.

5.5 Concluding Observations The impossibility results associated with Arrow-type preference aggregation mechanisms are sometimes replicated, and sometimes circumvented, when individual and social preferences are sought to be ‘fuzzified’, depending upon what precise

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approach to ‘fuzzification’ one takes. The transformation of impossibility into possibility results is attended by varying levels of success in alternative frameworks of vague preferences. Of more interest than this fact is the one of what ought to count for ‘success’: simply the fact of securing an existence result where earlier one had non-existence, or the overall greater intrinsic plausibility of the particular framework of vagueness that is compatible with a possibility outcome? One would imagine it is the latter criterion of success that is persuasive. Even if the case for fuzzy over exact preferences is accepted as reflecting some aspect of realistic indeterminacy or dividedness that characterizes individual and social preferences, there is a plurality of vague formulations that can be conceived of. While a small sample of this plurality has been reviewed in the present paper, and while it is submitted that this is an exercise which is not without its uses, the paper also reflects a difficulty that is, in some measure, a feature of the literature at large: the difficulty of finding compelling, rather than ad hoc, fuzzy escape routes from the nihilism of social choice results in the exact framework. The issue, which is merely flagged here, deserves independent and elaborate investigation: such a more detailed treatment is, however, beyond the rather limited scope of the present paper.

References Arrow KJ (1963) Social choice and individual values, 2nd ed. Wiley, New York Banerjee A (1994) Fuzzy preferences and arrow-type problems in social choice. Soc Choice Welfare 11(2):121–130 Barrett CR, Pattanaik PK, Salles M (1986) On the structure of fuzzy social welfare functions. Fuzzy Sets Syst 19(1):1–10 Barrett CR, Salles M (2006) Social choice with fuzzy preferences. Economics Working Paper Archive (University of Rennes 1 & University of Caen) 200615, Centre for Research in Economics and Management (CREM), University of Rennes 1, University of Caen and CNRS; forthcoming. In Arrow KJ, Sen AK, Suzumura K (eds) Handbook of social choice and welfare. Amsterdam: North-Holland. Available at http://www.lse.ac.uk/collections/VPP/VPPpdf/VPPPub lications/VPP04_02.pdf Dasgupta M, Deb R (1999) An impossibility theorem with fuzzy preferences. In: de Swart H (ed) Logic, game theory and social choice. Tilburg University Press, Tilburg [Cited in Barrett and Salles, 2006.] Dutta B (1987) Fuzzy preferences and social choice. Math Soc Sci 13(3):215–229 Dasgupta, M. and R. Deb (1999): ‘An Impossibility Theorem with Fuzzy Preferences’, in: H. de Swart (ed.): Logic, Game Theory and Social Choice. Tilburg: TilburgUniversity Press. [Cited in Barrett and Salles, 2006.] Gibbard A (1969) Intransitive social indifference and the arrow dilemma. unpublished [Cited in Barrett and Salles, 2006.] Richardson G (1998) The structure of fuzzy preferences: social choice implications. Soc Choice Welfare 15(3):359–369

Chapter 6

Equality, Priority, and Distributional Judgements

Abstract The present essay undertakes an assessment of the substantive significance of Derek Parfit’s distinction between Prioritarianism and Egalitarianism. In providing a brief critique of Parfit’s arguments, the essay draws on the author’s own earlier work and that of Thomas Christiano and Will Braynen, John Broome, and Marc Fleurbaey. It considers issues relating to the ‘Levelling Down Objection’, the ‘Divided World example’, and the distinction between ‘absolute’ and ‘relative’ valuations of individual benefit. It is contended that ‘levelling down’ presents a difficulty only for ‘Pure Telic Egalitarianism’, not for ‘Pluralist Telic Egalitarianism’; that one can have an Egalitarian rationalization for favouring equality in the distribution of a smaller sum of well-being over inequality in the distribution of a larger sum even in a ‘Divided World’; and that, while a particular ‘absolute’/‘relative’ dichotomy is relevant for a particular ‘distribution-invariance’/‘distribution-sensitivity’ dichotomy, the resulting distinction is useful for differentiating two types of Egalitarian rather than for differentiating a non-Egalitarian principle such as Prioritarianism from Egalitarianism. Keywords Prioritarianism · Egalitarianism · Levelling Down Objection · Divided world example · Additively separable welfare function

6.1 Introduction The thematic focus of this conference is, broadly, on aspects of individual and social values. The present paper is specifically concerned with the value that goes by the name of Egalitarianism. The notes which follow are, more than anything else, an account of certain confusions experienced by the present author from a reading of Parfit’s (1997) essay on priority and equality. As such, they may reflect nothing This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2015): ‘Equality, Priority, and Distributional Judgements’, in S. K. Jain and A. Mukherji (eds.): Perspectives on Growth and Inequality. Routledge: New Delhi. ISBN 9780815373308. I would like to thank, without implicating, John Broome, Thomas Christiano, Sanjay Reddy and Henry Richardson for comments on earlier versions. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_6

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more than a faulty reading of the text. On the other hand, and in the event that the reservations expressed here are not, after all, a product of misinterpretation, there may be some value in the questions raised. A preliminary part of this paper is based heavily on earlier work of mine (Subramanian 2011) and is also related to the perspective advanced in Christiano and Breynan (2008). Additionally, my appreciation of some of the issues involved has been considerably aided by the work of Broome (2003) and Fleurbaey (2001), and it will be seen that I draw on both writers—sometimes in agreement and sometimes not—in presenting my views. In terms of final outcome, my sympathies are largely with Fleurbaey’s conclusions in the matter. I should add that the entire problem of distributional judgements in an environment of uncertainty, which has been dealt with by both Broome and Fleurbaey, is neglected in the present discussion. The following is a very brief summary of Parfit’s (1997) position on equality and priority, as I understand it. Parfit’s difficulty with Egalitarianism seems to reside in his perception that there is, ultimately, no purely Egalitarian justification available, in certain situations, for the reasonable judgement we may make that a particular equal distribution of a smaller sum of well-being is preferable to a particular unequal distribution of a larger sum. In Parfit’s view, the case in favour of Egalitarianism must rest in one of two views: (a) that an equal distribution of well-being is intrinsically good; and (b) that it is right to pursue equality in the cause of justice or fairness (or some other value). The first view is a Telic Egalitarian view, while the second is a Deontic Egalitarian view. Telic Egalitarianism, in Parfit’s judgement, falls foul of what he calls the Levelling Down Objection: this is the objection that one cannot hold that it is in any respect good to move towards equality by just pulling down the better-off person’s well-being level to that of the worse-off person. What, then, of the alternative avenue of justification, Deontic Egalitarianism? Parfit, in this context, conceives of a scenario called the Divided World situation, in which the inequality of a distribution neither has any unfavourable effects nor is attributable to injustice or unfairness or some similar casualty of a desired value: in the Divided World, one cannot have a Deontic Egalitarian argument for favouring an equal distribution of a smaller sum of well-being over an unequal distribution of a larger sum. If Egalitarianism—of either the Telic or Deontic variety—is of no help in rationalizing such a preference, one must resort to some other view for achieving a satisfactory rationalization; and the view which Parfit advances is the Priority View, which requires that ‘benefiting people matters more the worse off these people are’ (Parfit 1997; p. 213). The Priority View is distinguished from the Egalitarian view on grounds that the former is concerned only with people’s absolute levels of well-being, while the latter is concerned with the relativities arising from interpersonal comparisons of well-being. Briefly, the grounds for equality are more pertinently located in priority than in equality itself. In what follows, we have, first, a description of what is claimed to be a plausible view of Pluralist Telic Egalitarianism which is not vulnerable to the Levelling Down Objection and which can, in fact, satisfactorily address the Divided World situation. Second, the validity of linking the status of belief in equality with the

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‘relative versus absolute’ basis of the distinction between Egalitarianism and Prioritarianism is reviewed and found wanting. Third, it is suggested that there is a specific ‘distribution-invariance’/‘distribution-sensitivity’ dichotomy for which a specific ‘absolute’/‘relative’ dichotomy is relevant: the resulting distinction, it is claimed, is useful for differentiating two types of Egalitarian rather than for differentiating Prioritarians from Egalitarians. It is concluded that the distinction between Prioritarianism and Egalitarianism claimed by Parfit is not one of profound conceptual significance, as far as a critique of Egalitarianism is concerned.

6.2 On the Robustness of Egalitarianism to Parfit’s Objections Parfit (1997) draws a distinction between two kinds of Telic Egalitarianism, which he calls, respectively, Pure Telic Egalitarianism and Pluralist Telic Egalitarianism, both of which are united in endorsing the principle of equality, which upholds the view that ‘it is in itself bad if some people are worse off than others’ (Parfit 1997; p. 204). The distinction between the two kinds of Telic Egalitarianism is identified in the following terms by Parfit (1997; p. 205): if we cared only about equality, we would be Pure Egalitarians. If we cared only about utility, we would be utilitarians. Most of us accept a pluralist view: one that appeals to more than one principle or value. According to Pluralist Egalitarians, it would be better both if there was more equality and if there was more utility. In deciding which of two outcomes would be better, we give weight to both these values.

I have claimed elsewhere (Subramanian 2011; p. 10) that the following descriptions of the categories of Pure and Pluralist Telic Egalitarianism are both inherently plausible and compatible with Parfit’s differentiation (as set out in the quote above) of the two notions: Pure Telic Egalitarianism requires that, given any two equi-dimensional distributions of well-being, the more equal distribution be judged to be the better one. Pluralist Telic Egalitarianism requires that, given any two equi-dimensional distributions of well-being with the same sum-total of well-being, the more equal distribution be judged to be the better one, and given any two equal equi-dimensional distributions of well-being, the distribution with the larger sum-total of well-being be judged to be the better one. It is noted in Subramanian (2011) that while Pure Telic Egalitarianism, as defined above, is certainly vulnerable to the Levelling Down Objection, this is not true of Pluralist Telic Egalitarianism, which—as defined above—is simply not committed to comparing distributions of different sums of well-being. By holding the sum-total of well-being constant, one avoids commitment to a needlessly strong version of Pluralist Telic Egalitarianism, a version which requires that if any one distribution is more equal than another, then the former must be judged to be better than the

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latter in at least the one respect of equality. Such a formulation of Pluralist Telic Egalitarianism makes the notion easy game for the Levelling Down Objection. Parfit achieves his ends precisely by saddling Pluralist Telic Egalitarianism with this strong baggage: why must all Egalitarians accept this interpretation of their ethic? The point is not to assert that Pluralist Telic Egalitarianism will survive the Levelling Down Objection under Parfit’s ‘unrestricted’ characterization (which of course it will not, and which Pure Telic Egalitarianism does not), but rather to point out, precisely, that such a characterization is neither the only nor the most reasonable one available. Putting it differently, one’s moral intuition, it seems, can be very clear about favouring equal over unequal distributions of a given sum-total of well-being: why must one, as a person that cares for equality at all, necessarily be seized of a similar certitude regarding the superiority, even if only in one respect, of (for example) an equal distribution of a small sum of well-being over a (Pareto-dominating) unequal distribution of a larger sum of well-being? In particular, it does not appear to be fair (or logical) to criticize an Egalitarian because she confines her Egalitarian impulses to a restricted domain of application, and, at the same time, trip her up with the Levelling Down Objection when she expands the domain of application! More significantly, the ‘value’ of equality can be focused upon precisely under a ceteris paribus clause (one in which, specifically, the aggregate level of well-being is kept constant). The matter is amenable to elaboration with the help of an example. Suppose one were inclined to judge sculptures favourably by, inter alia, assigning some intrinsic value to the size of the sculptures. This should surely not require one to say that in the respect of size, at least, a misshapen but large sculpture has something to commend it vis-à-vis a delicate but small sculpture! One would imagine that an eminently acceptable way of attaching intrinsic value to size is to say that sculpture A is a better one than sculpture B if A is a blown-up version of B (i.e. A is identical to B except for a scale factor greater than one). Why is the ‘intrinsic’ nature of this value in any way compromised by restricting A to be a scaled-up version of B? (Indeed, and symmetrically, one can imagine many persons holding exactly the contrary intrinsic value of ‘small is beautiful’ by requiring that A is better than B whenever A is a scaled-down version of B.) Analogously, people of inegalitarian persuasion will typically have no systematically intrinsic preference for equal over unequal distributions of a given sum of well-being. It is worth noting here that Parfit’s Levelling Down Objection has been dealt with, in the foregoing, by defining Pluralist Telic Egalitarianism in such a way that it avoids any entailment of Levelling Down as an acceptable principle. The two-person distribution of well-being (50,50) does not, in a concession to Levelling Down, have to be declared superior to the distribution (50,100) in at least the one respect of equality, because the Pluralist Telic Egalitarian principle, as it has been here defined, is not obliged to pronounce judgement on comparisons of distributions which have different aggregate levels of well-being. This defence of Egalitarianism is in contrast to the argument resorted to by Temkin (2002), who questions the view that the Levelling Down Objection is indeed objectionable: in his view, the Levelling Down Objection has acquired a certain quality of unexceptionableness entirely because of a near-universal (and, presumably, inadequately examined) subscription to what he

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calls The Slogan, more commonly known to economists as the Pareto principle, and which Broome has called ‘the Principle of Personal Good’. Here is how Temkin states The Slogan (Temkin 2002; p. 126): The Slogan. One situation cannot be worse (or better) than another if there is no one for whom it is worse (or better).

Temkin contests the prevailing view of the unassailable moral appeal of The Slogan, and so calls into question the position that there is something necessarily repugnant about Levelling Down. The argument in the present paper simply sidesteps the question of whether The Slogan deserves the status of a Sacred Cow: it avoids what is arguably an avoidable controversy. While the definition of Pluralist Telic Egalitarianism employed in this essay does not entail the Levelling Down Objection, it also does not entail The Slogan—though it is not inimical to The Slogan. A simple restatement of the Pluralist Telic Egalitarian principle for a two-person society, as enunciated in this article, should clarify the issue. First, suppose  to be a binary relation of strict betterness. The version of Pluralist Telic Egalitarianism (PTE) which has been advanced in this paper can be shown to subscribe, without contradiction, to the following three principles, which I call α, β, and γ , respectively. Principles α and β, between them, characterize PTE, while Principle γ is, essentially, The Slogan. Principle α. For all well-being levels x, a, b > 0, if a = b and a + b = 2x, then (x, x)  (a, b). (That is, for any two-person society, given any aggregate level of well-being, an equal distribution of the aggregate is better than an unequal distribution.) Principle β. For all well-being levels x, y > 0, if x > y, then (x, x)  (y, y). (That is, for any two-person society, an equal distribution of well-being levels is better than another equal distribution of well-being levels, if aggregate well-being is greater in the first case than in the second.) Principle γ (The Slogan). For all well-being levels x, c, d > 0, if c > d, then (x, c)  (x, d). (That is, for any two-person society, if one person’s well-being level is raised while keeping the other person’s level constant, then this is an improvement). Given three two-person distributions of well-being (10,100), (10,10), and (55,55), by Principle α, (55, 55)  (10, 100), and by Principle γ , (10, 100)  (10, 10). Principles α and γ in conjunction with transitivity of  imply also that (55, 55)  (10, 10), which, precisely, is what Principle β demands. Briefly, the Egalitarian argument in this paper gets around the Levelling Down Objection by showing that there is an eminently reasonable way of describing Pluralist Telic Egalitarianism so that it does not either entail the Objection nor call into question the appeal of The Slogan. (The reader is invited to see also, in this context, the work of Christiano and Braynen 2008.) If Parfit finds Telic Egalitarianism unsatisfactory as a basis for upholding the intrinsic value of equality, he also finds Deontic Egalitarianism unsatisfactory as a basis for upholding the instrumental value of equality. His Divided World Example (Parfit 1997) is designed to demonstrate that there are certain situations in which Deontic Egalitarianism is of no avail in upholding equality as a virtue promoting other valued principles such as justice or fairness. Parfit invites us to consider two

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worlds, World 1 and World 2, each containing the same number of persons, and completely cut off from one another, so that neither World is aware of the other’s existence. Consider two alternative distributions of well-being, in the first of which each person in World 1 experiences a well-being level of 100 and each person in World 2 a well-being level of 200, while in the second distribution, each person in each World experiences a well-being level of 145 (all in appropriately chosen units). The two distributions can be represented by the ordered vectors (100,200) and (145,145), respectively. Many of us would be tempted to pronounce that (145,145)  (100,200), from the consideration that, even though the aggregate level of well-being in the distribution (145,145) is a little lower than in the distribution (100,200), this fact is more than compensated for by the feature that the first distribution is an equal one and the second distribution a very unequal one. Yet, since the two Worlds are Divided, and neither World has any knowledge of the existence of the other, one cannot appeal to the requirements of justice or fairness or any allied instrumental cause for pronouncing the more equal distribution to be better than the less equal one; nor—given the complete insulation of each World from the other—can one appeal to any adverse effects which inequality might have as a rationalization of the preference for (145,145) over (100,200). In Parfit’s view, Telic Egalitarianism falls foul of the Levelling Down Objection, while Deontic Egalitarianism falls foul of the Divided World Example—whence the inability of Egalitarianism to support the moral intuition which favours an equal distribution of a smaller sum of well-being over an unequal distribution of a larger sum of well-being, as in the case of the distributions (145,145) and (100,200) just considered. Hence also Parfit’s contention that, in view of this failure of Egalitarianism, one needs some other principle to justify one’s preference for equality—and the principle he proposes is the Priority View. It can be argued, however, that the Divided World Example does not constitute any sort of conclusive case against Egalitarianism. In Subramanian (2011), we are invited to consider a third hypothetical distribution (150,150), such that each person in each of the two Worlds experiences a well-being level of 150 units. Recalling the constitutive elements of Pluralist Telic Egalitarianism as we have defined it, in terms of the Principles α and β, it may be noted that (150,150)  (100,200) by Principle α, and (150,150)  (145,145) by Principle β. Call the distributions (150,150), (100,200), and (145,145) a, b, and c, respectively. For any pair of distributions x and y, let P(x, y) stand for the ‘extent to which’ x is preferred to y. Let Q be an exact asymmetric ordering such that, for all distributions x, y, w, z, P(x, y)QP(w, z) will be taken to mean that the extent to which x is preferred to y is greater than the extent to which w is preferred to z. What we are speaking of is a form of ordinal comparison of intensities of preference. Returning to the distributions a =(150,150), b = (100,200), and c =(145,145) defined earlier, it is entirely conceivable that P(a, b)QP(a, c), from the consideration that, at the same level of aggregate well-being, there is considerably more inequality in b than in a, while at the same level of equality, there is only a relatively smaller level of aggregate well-being in c than in a. To suggest that P(a, b)QP(a, c) is also to suggest that c lies above b in one’s preference ranking. Pluralist Telic Egalitarianism, as it has been defined in this essay, is not vulnerable to the

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Levelling Down Objection; and, further, it is compatible, even in a Divided World, with the judgement that an equal distribution of a smaller sum of well-being (such as c) is preferable to an unequal distribution of a larger sum of well-being (such as b). Briefly, the Levelling Down Objection and the Divided World Example do not, after all, between them dispose of the case for Egalitarianism.

6.3 On the Practical Relevance of the Equality–Priority Distinction As we have seen in the previous section, the Levelling Down Objection and the Divided World Example, between them, serve as grounds for Parfit to advance the notion that, in some cases, our preference for equal over unequal distributions must be located in a non-Egalitarian view of the world. While the force of this claim has been questioned in the preceding discussion, this is not a sufficient reason for denying the possibility that a preference for equality could be grounded in some nonEgalitarian view. Parfit claims that the Priority View is such a non-Egalitarian view. It is the view that ‘benefiting people matters more the worse off these people are’ (Parfit 1997; p. 213). In what way is the Priority View distinct from Egalitarianism? Parfit says: ‘Egalitarians are concerned with relativities: with how each person’s level compares with the level of other people. On the Priority View, we are concerned only with people’s absolute levels’ (1997; p. 214). He also says: ‘…on the Priority View, benefits to the worse off matter more, but that is only because these people are at a lower absolute level. It is irrelevant that these people are worse off than others. Benefits to them would matter just as much even if there were no others who were better off’ (1997; p. 214). It is worth asking if these particular verbal distinctions translate into any operational distinctions of outcome when we are confronted with some specific practical problem of distribution, such as that of allocating a benefit of fixed size among a set of people constituting society. By way of example, let x = (10,20) be an ordered two-person distribution of well-being levels, or ‘benefits’. Suppose we had a budget of 10 units of benefit to distribute. Fairly straightforwardly, both an Egalitarian and a Prioritarian would transfer the entire budget to the person with a well-being level of 10 in distribution x, to arrive at the distribution x = (20, 20). Let us call a person who subscribes to the Priority View without believing in equality a ‘Pure Prioritarian’. A Pure Prioritarian might be expected to advance something like the following claim: ‘in an exercise of optimal transfers, both an Egalitarian and I would recommend a movement from x to x ; but this is not a reason for you to confuse (Pure) Prioritarians and Egalitarians: the Egalitarian and I may make the same prescription, but he does it in order to achieve equality, whereas I do it in order to give priority to the worse off’. This, as far as it goes, is a valid claim of distinction; but it is not at all clear how far the distinction actually goes. Setting aside the Pure Telic Egalitarian (who

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is altogether too much of a sitting duck), one would like to know if the difference in motivation between a Pure Prioritarian and an Egalitarian that Parfit claims ever causes any divergence in distributional judgements of what is better or what ought to be done, and if the divergence can be traced to the belief of the one and the nonbelief of the other in equality. Without pre-judging the issue, it seems fair to suggest that, from Parfit’s own (1997) account, it is difficult to deduce any such divergence: one is therefore led to find Fleurbaey’s position understandable when he says of the distinction between (Pure) Prioritarianism and Egalitarianism that ‘… [it] mostly has to do with the reasons for, rather than the content of, judgments about distributions…’ (Fleurbaey 2001; p. 1). If this is so, one could be speaking of a distinction without a difference. It may be objected here that inasmuch as utilitarianism could also support equality in a practical exercise of distribution, that fact should not serve as a basis for the pronouncement that utilitarian and Egalitarian judgements differ only with respect to their reasons and not with respect to their content. In response to this objection, it is relevant to ask if utilitarianism invariably prescribes distributional equality. The assumption that all individuals are equipped with the same utility function is the force behind what Sen (1973) has called utilitarianism’s ‘ill-deserved reputation’ for Egalitarianism. Typically, a physically challenged person (PCP) may be deemed to be a less efficient pleasure machine than an able-bodied person (ABP) in transforming income into utility. Typically, utilitarianism will prescribe both an income and a total-utility distribution which allocates a larger share to ABP than to PCP. This being the case, there is no reason whatever for failing to see a distinction between Egalitarianism (or Prioritarianism), on the one hand, and utilitarianism, on the other, ‘in practice’. This is the well-known problem of ‘heterogeneity’. But the problem is not only with heterogeneity. Consider a case in which both individuals 1 and 2 in a two-person √ society have the same utility function U defined on income x: Ui (x) = U (x) = x, i = 1, 2. (This is an increasing, strictly concave utility function, of the type favoured in canonical discussions of the nature of utility.) Suppose a total income of 50 is to be distributed between the two persons. The optimal utilitarian distribution will be x* = (25,25), and aggregate welfare, by the utilitarian calculus, will be 10 utils. Consider another distribution y = (0,100) of a total income of 100: aggregate welfare for this distribution, by the utilitarian calculus, is also 10 utils. A utilitarian is obliged to be indifferent between the distributions x* and y. There is little reason for the view that Prioritarians and Egalitarians are obliged to agree with the utilitarian. One should not have much difficulty in conceding that the distinction between Egalitarianism and utilitarianism is also an overdrawn and functionally not very useful one if it were indeed the case that Egalitarianism and utilitarianism invariably prescribed the same distributions in practice: only, they do not do so. Hence, the view that when it comes to Egalitarianism and Prioritarianism—unlike in the case of Egalitarianism (or Prioritarianism) and utilitarianism—one could be speaking of a distinction without a difference. As it happens, Parfit’s ‘absolute/relative’ distinction can indeed, under a certain interpretation of that distinction, lead to a divergence in distributional judgements, although it is questionable if the divergence should be traced to the status of belief

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in equality (as will be discussed in the following section). Specifically, it does not, in any way, appear to be a necessity of logic that a Prioritarian—understood to mean someone that subscribes to the Priority View—should not believe in equality. Hence, the earlier suggestion that if what is involved is only a matter of definition, then it would be convenient to call somebody who believes in the Priority View without believing in equality a ‘Pure Prioritarian’. It will be contended, in what follows, that any possible divergence in distributional judgements that may arise between ‘relativists’ and ‘absolutists’ does not necessarily imply a disagreement on the appeal of equality, such that ‘absolutists’ must necessarily be seen as adopting a stand in favour of a principle of priority which is distinct from, and incompatible with, a principle of equality.

6.4 Egalitarian Judgements and Additive Separability While a Prioritarian is free to renounce any belief in equality, it is, I believe, perfectly open to an Egalitarian to find the Priority View (‘benefiting people matters more the worse off these people are’) appealing. Indeed, if Prioritarianism requires that ‘benefits to the worse off should be given more weight’ (Parfit 1997; p. 213), then it is difficult to imagine that any Egalitarian can quarrel with Prioritarianism on this ground. Certainly, and as already stated, a Prioritarian need not believe in equality: a person with such a view we have called a Pure Prioritarian. It appears to be safe to suggest, minimally, that the intersection between ‘Non-Pure’ Prioritarians and Egalitarians is a non-empty set. Let us use the term ‘P-Egalitarians’ to describe Egalitarians belonging to this intersection. While all P-Egalitarians may be expected to agree that benefits to the worse off should be given more weight, there can be disagreement about precisely what sort of weighting structure should be employed in order to realize this objective. Parfit’s ‘absolute-relative’ distinction could conceivably be related to this weighting structure question, but the distinction would be relevant for differentiating between two sorts of P-Egalitarian rather than for differentiating Prioritarians and Egalitarians. Specifically, it is useful, here, to invoke the economist’s device of the social welfare function. Suppose income, for specificity, to be the space in which advantage or benefit is assessed. Let a = (a1 , . . . , ai , . . . , an ) be a non-decreasingly ordered n-vector of incomes, where ai is the income of the ith poorest person, and n is an element of the set N of positive integers. Let Xn be the set of non-decreasingly ordered n-vectors of income, and define X to be the set ∪n∈N Xn . For every a ∈ X, N (a) will stand for the set of individuals whose incomes are represented in a, and n(a) for the dimensionality of a. Social welfare is a function W : X → R (where R is the set of real numbers), such that, for every a ∈ X, W (a) specifies a real number which is supposed to represent the amount of social welfare associated with the income vector a. Typically, W is taken to be an increasing function of each person’s income. For all a ∈ X and i ∈ N (a), we shall let v(ai ; a) stand for the valuation function which assigns a real number to the ith poorest person’s income, with the number

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signifying the social valuation placed on her income. The valuation function v is said to be menu-independent if, for all a, b ∈ X, j ∈ N (a) and k ∈ N (b), a j = bk implies v(a j ; a) = v(bk ; b). The valuation function v is said to be menu-dependent if ∃[a, b ∈ X, j ∈ N (a)&k ∈ N (b)] such that a j = bk and v(a j ; a) = v(bk ; b). One way of writing a social welfare function is in terms of an additive average of the valuations of all individuals’ incomes, that is, for all a ∈ X: W (a) = [1/n(a)]Σi∈N (a) v(ai ; a).

(6.1)

For all a ∈ X, W (a), will be said to be an additively separable social welfare function if and only if W (a) = [1/n(a)]Σi∈N (a) v(ai ; a)

(6.2)

and v is menu-independent. Let ri (a) ≡ (n(a) + 1 − i) be the rank order of the ith poorest person’s income in a. Consider a particular income valuation function, given by v B (ai ; a) = ri (a)ai for all i ∈ N (a). Employing this valuation function in (6.1), we obtain what we call the Borda (rank order weighted) social welfare function W B : W B (a) = [1/n(a)][n(a)a1 + . . . + (n(a) + 1 − i)ai + . . . + an ].

(6.3)

Consider a slight variation of the valuation function v B , given by v DW (ai ; a) = for all i ∈ N (a). If we employed this valuation function in (6.1), then we would obtain a social welfare function W DW , attributable to Donaldson and Weymark (1980), and given by:

ri2 (a)ai

W DW (a) = [1/n(a)][n 2 (a)a1 + . . . + (n(a) + 1 − i)2 ai + . . . + an ].

(6.4)

Suppose we took the income valuation function—call it v A —to be an identically increasing and strictly concave transform of ai for all i, such that v A (ai ; a) = (1/λ)aiλ for all i ∈ N (a), where λ is any number greater than zero and less than one. Then, the social welfare function in (6.1) can be written (after the Atkinson 1970 fashion) as: W A (a) = (1/n(a)λ)(a1λ + . . . + aiλ + . . . + anλ ).

(6.5)

W A is an example of an additively separable function, because v A (ai ; a) is menuindependent, while W B and W DW are examples of functions which, while they are additive, are not separable, because v B (ai ; a) and v DW (ai ; a) are menu-dependent. It is a property of an additively separable social welfare function that, given any two (ordered) income vectors a and b, the valuation placed on the jth poorest person’s income in the vector a will be exactly the same as the valuation placed on the kth poorest person’s income in the vector b if the two individuals should happen to share the same income. Thus if W A , for example, is the welfare function employed,

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and if the jth poorest person in some n-dimensional distribution a receives the same income a as the kth poorest person in some other m-dimensional distribution b, then the valuation v A (a j ; a) placed on the jth poorest person’s income in a is identical to the valuation v A (bk ; b) placed on the kth poorest person’s income in b: v A (a j ; a) = v A (bk ; b) = (1/λ)a λ . The Borda social welfare function is not, however, additively separable, and if W B is the welfare function employed, then it is easy to check that v B (a j ; a) = (n(a) + 1 − j)a, v B (bk ; b) = (n(b) + 1 − k)a and (unless n(a) − j = n(b) − k), v B (a j ; a) = v B (bk ; b). The same is also obviously true of the welfare function W DW . Notice that when the social welfare function is of the additively separable type, the valuation placed on a person’s income depends only on the absolute level of that income, whereas when the social welfare function is of the Borda or Donaldson– Weymark type, the valuation placed on a person’s income is also influenced by the relative place occupied by that income level in the distribution. Valuation, in an additively separable welfare function, is thus ‘distribution-invariant’ in a manner in which valuation, in a Borda-type welfare function, is not. It is this sort of ‘absolute-relative’ distinction which Parfit conceivably has in mind when he seeks to differentiate Prioritarianism from Egalitarianism; and, indeed, Broome (2003) insists that the distinction between Prioritarianism and Egalitarianism is to be found in an identification of Prioritarians with additively separable welfare functions and of Egalitarians with welfare functions that are not additively separable. However, in my view, a distinction of this nature is applicable not in the context of Prioritarians and Egalitarians, but in the context of the class of Egalitarians earlier referred to as ‘P-Egalitarians’. Differences in distributional judgement can certainly arise between ‘absolute valuationists’ and ‘relative valuationists’, but these, in my view, are differences between two types of Egalitarian rather than between Prioritarians and Egalitarians. Thus, I am inclined to agree with Fleurbaey, when he says: ‘…insofar as [the distinction between Prioritarianism and Egalitarianism] bears on the content of distributional judgments, it merely draws a line within egalitarianism’ (Fleurbaey 2001; p. 1). In particular, it seems reasonable to suggest that P-Egalitarians can be categorized into two types: P1 -Egalitarians, who insist that the welfare function be additively separable, and P2 -Egalitarians, who insist that the welfare function should not be additively separable (of which two examples are the W B and the W DW functions): these two varieties of Egalitarian have a simple disagreement on the structural form in which benefits to the worse off should be given more weight. That the distinction between P1 - and P2 -Egalitarians could be a non-trivial one is reflected in the following considerations. While a welfare function can be seen as a kind of ‘gain function’, an inequality measure can be seen as a kind of ‘loss function’, and the latter can, through suitable manipulation, be derived as a sort of obverse of the former. Typically, inequality measures derived from additively separable welfare functions would be ‘subgroup consistent’, whereas inequality measures derived from welfare functions that are not additively separable would violate subgroup consistency, which is the requirement that, when a population is partitioned into mutually exclusive and collectively exhaustive subgroups, an increase in inequality in any one subgroup should

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increase overall inequality. Theil’s inequality measure is an example of a subgroup consistent measure, while the Gini coefficient is an example of a measure which is not subgroup consistent. The properties of subgroup consistency and a stronger version of it, called decomposability, have for long been known to economists working on the measurement of inequality (see, in particular, Shorrocks 1980, 1988). Whether or not subgroup consistency is a compulsively desirable property in an inequality index is also a fairly long-standing controversy among economists (Sen and Foster 1997 has an instructive review of the issues involved). In a particular comparison of two distributions x and y, it is possible that the Theil index pronounces x to display more inequality than y, and the Gini index reverses this ranking. This could be a cause for a potential conflict in the inequality judgements of P1 - and P2 -Egalitarians. Of course, this could also be a cause for a conflict between Pure Prioritarians and P2 -Egalitarians, but—relevantly—not for a conflict between Pure Prioritarians and P1 -Egalitarians. It therefore appears to be misplaced to suggest that Prioritarians should be distinguished from Egalitarians by identifying the latter with a belief in equality and the former with a lack of such belief. In fact, divergences in welfare (and therefore inequality) judgements of distributions can also arise among those committed exclusively to welfare functions that are not additively separable, as the following reveals. Imagine two 4-vectors of income b = (24, 26, 40, 50) and c = (20, 30, 44, 46) which share the same mean income of 35. It is easily verified that W B (b) = W B (c) = 76, while W DW (b)(= 207) > W DW (c)(= 203). This sort of rank reversal seems to be inadequate reason to suggest that, as between adherents of Borda-type welfare functions and adherents of Donaldson–Weymark-type welfare functions, one lot ought to be identified with Egalitarianism and the other with some distinct, nonEgalitarian principle. (Indeed, economists will be quick to recognize that the distinction between the two sets of evaluators resides, rather simply, in the differing appeal which Kolm’s (1976) ‘principle of diminishing transfers’ has for them.) Briefly, divergences in the inequality ranking of distributions do not necessarily deserve to be invested with any more significance than as manifestations of differing varieties and degrees of ‘equity-consciousness’. It seems to be neither warranted nor profitable to counter-pose Prioritarianism as a principle which is distinct and separate from Egalitarianism.

6.5 Concluding Observations In this paper, an attempt has been made to argue that a reasonable interpretation of Egalitarianism enables it to survive the charge of its being vulnerable to the Levelling Down Objection and the Divided World Example. This is not in any way to suggest that a Prioritarian can be coerced into believing in equality—of course this is not the case. That said, it is not clear that subscription to the Priority View without belief in equality constitutes grounds for viewing the distinction between

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Prioritarianism and Egalitarianism as being either practically relevant for distributional judgements or conceptually significant for foundational analysis. It is hard to determine if there exists any Prioritarian argument which establishes a convincing case against an Egalitarian endorsement of an Atkinson-type social welfare function. Broome invites us to consider such an interpretation—in an effort, I believe, to confer some precise meaning on Parfit’s ‘relative-absolute’ distinction. One can however hold, in response to this interpretation, that there is nothing inherently ‘Egalitarian’ about rejecting subgroup consistency, as one might need to argue in order to insist that Egalitarians may not subscribe to additively separable welfare functions. Putting it differently, while some Egalitarians may abjure additive separability, not all Egalitarians should be required to do so. In particular, it would be reasonable to expect, from those that advance the Priority View as a distinct and substantively important concept vis-à-vis the Egalitarian view, to provide evidence of this claim; the present paper questions precisely the Prioritarian success of this enterprise. Additionally, and very markedly differently from Prioritarian efforts, this essay does bring out the practical consequences of alternative weighting schemata, in terms of success or failure in satisfying subgroup consistency, decomposability, and transfer sensitivity—all of which are issues in respect of which Egalitarians can have differences of opinion among themselves, without having to see these differences as entailing the elicitation of deference to some distinct and counter-posed ethic such as ‘Prioritarianism’. On balance, Fleurbaey’s view of the matter is compelling, when he says: ‘…[the Prioritarian-Egalitarian] distinction mostly has to do with the reasons for, rather than the content of, judgments about distributions; [and]…insofar as it bears on the content of distributional judgments, it merely draws a line within egalitarianism’ (Fleurbaey 2001; p. 1).

References Atkinson AB (1970) On the measurement of inequality. J Econ Theor 2(3):244–263 Broome J (2003) Equality versus priority: a useful distinction. forthcoming In: Murray C, Winkler D (eds) ‘Goodness’ and ‘Fairness’: ethical issues in health resource allocation. World Health Organization. Available at http://users.ox.ac.uk/~sfop0060/ Christiano T, Braynen W (2008) Inequality, injustice and leveling down. Ratio 21(4):392–420 Donaldson D, Weymark JA (1980) A single-parameter generalization of the Gini indices of inequality. J Econ Theor 22(1):67–86 Fleurbaey M (2001) Equality versus priority: how relevant is the distinction? forthcoming In: Murray C, Winkler D (eds) ‘Goodness’ and ‘Fairness’: ethical issues in health resource allocation. World Health Organization. Available at http://cerses.shs.univ-paris5.fr/marc-fleurbaey_eng.htm Kolm SC (1976) Unequal inequalities I. J Econ Theor 12(3):416–454; Unequal inequalities II. J Econ Theor 13(1):82–111 Parfit D (1997) Equality and priority. Ratio (new series) 10(3):202–221 Sen AK (1973) On economic inequality. Oxford University Press, Clarendon Sen A, Foster JE (1997) On economic inequality: expanded edition with a substantial annexe. Clarendon Press, Oxford Shorrocks AF (1980) The class of additively decomposable inequality measures. Econometrica 48(3):613–625

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Shorrocks AF (1988) Aggregation issues in inequality measurement. In: Eichhorn W (ed) Measurement in economics: theory and applications in economic indices. Physica Verlag, Heidelberg, p 1988 Subramanian S (2011) Are egalitarians really vulnerable to the levelling down objection and the divided world example? J Philos Econ IV(2):5–14 Temkin L (2002) Equality, priority, and the levelling down objection. In Clayton A, Williams M (eds) The Ideal of equality. Palgrave Macmillan, Basingstoke and New York, pp 126–161.

Chapter 7

Two Logical and Normative Issues Relating to Measurement in the Social Sciences

Abstract This essay is based on the premise that measurement is very important for the social sciences. However, it also enjoins care on the practitioner’s part in his or her engagement with the project of measurement. It deals, in particular, with two often overlooked issues with which quantification in the social sciences should be concerned: (1) social indicators in relation to the contrast between outcomes and processes; and (2) measurement which tends to depend on the derivation of ‘ought’ propositions from ‘is’ propositions. Keywords Outcomes · Processes · Basic judgements · Non-basic judgements · Positive values · Normative values · Language

7.1 Introduction One need hardly mention some of the standard problems to which measurement is vulnerable. There is the problem of non-availability of data. There is the problem of data which are available but not necessarily reliable. There is the problem of reliable data which are available but must nevertheless be utilized in the service of protocols of quantification that must respect the laws of statistics. To these we might add the problem of ‘fake science’—disciplines such as phrenology and eugenics which have aided the cause of some of the most egregious practices known to us of categorizing and characterizing groups of human beings in terms of intellectual ability or physical capability or personal attractiveness. Among the greatest casualties of fake science is the damage and abuse visited upon the disciplines of statistics, This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2019) Some Logical and Normative Issues Relating to Measurement in the Social Sciences, Journal of Quantitative Economics, 17:4, 937–948, https://doi.org/10.1007/s40953-019-00187-7. The chapter is a part of the text of the C. Chandrasekaran Memorial Lecture, delivered at The International Institute for Population Sciences (IIPS) in Mumbai on 11 March 2019. In preparing the lecture, I have drawn considerably on earlier work in which I have been involved, either singly or in collaboration. For discussions over the years on the issues dealt with in the lecture, I am grateful to D Jayaraj, Kaushik Basu, Nicole Hassoun, Sanjay Reddy, and Mala Lalvani. This lecture was published as Paper No. 18 in the IIPS Working Papers Series (IIPS: Mumbai, April 2019). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_7

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zoology, and demography. The issue has been dealt with in Stephen Jay Gould’s wonderfully principled book on what he calls ‘the mismeasurement of man’ (Gould 1997). I mention these difficulties only in order to admit that I shall not be speaking on any of them. What I do intend to speak on are two specific problems relating to the formulation and interpretation of social indicators. Specifically, my concern will be with (1) social indicators in relation to the contrast between outcomes and processes; and (2) measurement which tends to depend on the derivation of ‘ought’ propositions from ‘is’ propositions. I shall endeavour to illustrate the problems I speak of with examples drawn mainly from economics.

7.2 Social Indicators: Outcomes and Processes Social indicators are, by their very nature, what one might call ‘end-state’ descriptions of states of affairs. They refer, as a general rule, to outcomes; and perhaps naturally, we tend to interpret these outcomes, and therefore the social indicators associated with them, as ‘Good Things’ or ‘Bad Things’. This is rather in the spirit of the famous lampoon called 1066 And All That, perpetrated by the writers Sellar and Yeatman in the 1930s, on the manner in which history was taught (and for all one knows, still is) in British schools of the time. A typical example is a vignette from one of their history examination papers: ‘Would it have been a Good Thing if Wolfe had succeeded in writing Gray’s Elegy instead of taking Quebec?’ We, as social scientists, have a similar tendency to interpret social indicators in terms of their being ‘Good Things’ or ‘Bad Things’. Consider, for instance, the headcount ratio (HCR) of poverty, which is simply the proportion of a society’s population with incomes less than the poverty line. In time-series analyses of poverty, we are inclined to pronounce an observed decline in the headcount ratio as a Good Thing. Similarly, consider the social indicator encompassed in the sex ratio (SR) of a population, which is simply the proportion of females in a society’s population. In time-series analyses of the SR, we would be inclined to certify a diminution in the SR as a Bad Thing. In such interpretations of what a social indicator tells us, we concentrate only on outcomes while completely neglecting the possible processes that might have led to these outcomes. Thus, Kanbur and Mukherjee (2007) bring to our attention the possibility that a particular observed decline in the HCR of poverty might have been caused not by policy or processes aimed at extricating people from poverty, but because of excess mortality among the poor occasioned precisely by ill-health arising from poverty. Surely, such a decline in the HCR is a Bad Thing, not a Good Thing (even if it connotes a decline in poverty in terms of the chosen indicator of deprivation)! Kanbur and Mukherjee, in the work cited earlier, discuss how the prior history of a society’s demographics must be taken into account in order to judge observed poverty outcomes. This is a discussion beyond the scope of this paper, but what is relevant to note is that if, in our normative judgements, we concentrate

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exclusively on outcomes without regard for processes, then we would run the risk of interpreting all poverty indicators which are increasing in the HCR as signifying a Good Thing if poverty is eliminated by simply (physically) eliminating the poor. What of the notion that a decline in the sex ratio of a population is necessarily a Bad Thing? To address this question, it is useful to note that the SR is a function of three factors: the age structure (AS) of the population; the past history of sex ratios at birth (SRB); and the relative age-specific survival ratios for females (RASF). Jayaraj and Subramanian (2000) construct an elementary arithmetical example involving two points of time with the following relevant demographic features. First, the cumulative age distribution function in the second period is assumed to first-order stochastically dominate that of the first period. We shall assume that the changed age structure has been brought about by relatively benign developmental factors at work, including in particular reduced death (especially infant mortality) rates, vastly reduced birth rates, and a general increase in longevity. Second, we shall take it that the RASF declines with age but is higher, at each age, in the latter period compared to the former period, and this again due to developmental factors that have been beneficial to women. Third, the SRB is assumed to decline, owing to the favourable developmental factor of improved maternal status in respect of health and nutrition leading to reduced pregnancy wastage and, through that route (given that the male embryo is more prone to mortality in utero), to a reduced sex ratio at birth. Notice that the postulated changes in the AS of the population and in the RASF ratios would cause the SR of the population to increase, while the reduction in the SRB would cause the SR to decline. It is conceivable that the reduction in the SRB has a greater downward impact on the SR than the upward impact of the changes in its other two determinants, leading to an overall net decline in the SRB. Nevertheless, all of the factors at work in this declining SR have been developmentally favourable to women—and there is therefore little reason to infer that the observed decline in the SR has been a Bad Thing! From a philosophical point of view, the failure to distinguish between outcomes and processes in interpreting social indicators has something to do with Sen’s (1967) classification of prescriptive judgements into basic and non-basic judgements. Basic judgements are invariant with respect to the precise factual circumstances in which they are made. The judgement that ‘killing is wrong’ would be a basic judgement if it were to hold irrespective of whether the killing has been done for reasons of gain or for reasons of self-defence. Non-basic judgements, by contrast, allow for the possibility that they could change with the circumstances in which they are made. In the context of judgements made on the strength of the information transmitted by social indicators, we often tend to make what we believe are basic judgements, when reflection would suggest that our judgements ought, more sensibly, to be seen as being non-basic in nature. The carelessness involved in this aspect of measurement is less rather than more excusable if the circumstances in which our judgement would change were more rather than less plausible, foreseeable, and commonplace.

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7.3 Description, Prescription, and Measurement It is an old dictum of philosophy, dating back to David Hume, that one must be wary of deriving ‘ought’ propositions wholly from ‘is’ propositions. It may appear to be stressing the obvious to warn against making such a mistake, but it is surprising that the mistake is a common enough one among social scientists to warrant remarking. The problem is particularly evident in quantifications involving the methodology of ‘decomposition’, of which I consider a couple of examples in what follows. One common application of the technique of decomposition relates to the analysis of poverty reduction over two points in time. Specifically, there are procedures by which a given reduction in a real-valued measure of poverty can be ‘decomposed’ into a change attributable to the increase in average real per capita income and a change attributable to the variation in the distribution of incomes over the two points in time under comparison. The first component of change is usually referred to as the ‘growth effect’ of the change in poverty, and the second component as the ‘inequality effect’. Such a decomposition of poverty change into its growth and inequality components and an assessment of the relative contributions of the two effects to the overall change are unquestionably valuable as a positive exercise—one which is descriptively informative and instructive as to causation. Some of the earliest empirical applications of the decomposition technique to an analysis of changes in India’s poverty are those of Kakwani and Subbarao (1990), Jain and Tendulkar (1990), and Datt and Ravallion (1992). Dollar and Kraay (2002) observed that ‘growth is good for the poor’ and have reiterated the sentiment (‘growth is still good for the poor’) in Dollar et al. (2013, 2014). In their 2013 paper, the authors conduct a cross-country econometric analysis on a large data set, and their calculations suggest that over 75% of the variations in country growth performances have been accounted for by variations in the growth (of per capita mean income) across these countries. A lesson drawn by Dollar et al. (2014; p. 4) from their decomposition analysis is the following: ‘The main policy message of our work is the importance of overall economic growth for improvements in social welfare. Inequality may be a “hot” current topic, but inequality changes in most countries over the past thirty years have been small, while differences in average growth performance have been large’. Similar views are expressed in Patillo et al. (2005; pp. 35, 36) when they say: ‘The key finding that emerges from poverty decompositions is that the bulk of the variations across countries in the rate of poverty reduction is due to variation in overall growth… Since growth is the most important long-run driver of poverty reduction, pro-poor growth policies overlap with growth policies’. While it may be a matter of observed fact that the growth effect dominates in one or some or several decomposition exercises, does it follow that therefore growth must be regarded as the policy instrument best suited for bringing about reductions in poverty? Notice that the first part of the preceding statement is a positive or descriptive observation, while the second part is a normative or prescriptive one. This is clearly an instance of an ‘ought’ proposition being derived from an ‘is’

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proposition, in violation of the Human warning against committing such a mistake. Although the mistake under review appears to enjoy a certain popular vogue in the economic development literature, it is also true that some economists have resisted it, as evident in the work of—among others—Dhongde (2002), Subramanian (2010), dos Santos and da Cruz Vieira (2013), Basu (2013), Subramanian and Lalvani (2018), and Basu and Subramanian (2019). To see what is involved, let us employ Basu’s (2001, 2006, 2013) ‘quintile income statistic’ Q—which is simply the average income of the poorest 20% of a population—as our indicator of poverty. Then, if s is the income share of the poorest 20% (a rudimentary indicator of inequality), and m is mean per capita average income, it is easy to see, as in Rosenblatt and MacGavock (2013) and Basu and Subramanian (2019), that Q = 5sm

(7.1)

In some particular empirical exploration or even in a large number of such explorations, it may emerge that changes in Q are largely accounted for by changes in m, while changes in s account for a small or even negative share of changes in Q. This does not immediately justify the prescriptive statement that we ought to concentrate on promoting growth, rather than on reducing inequality, in order to reduce poverty. All that our empirical exercises suggest is that the weight of redistributive effort in overall poverty reduction has been small, not that such effort is relatively ineffective. Equation (7) suggests that s and m can, in principle, be combined in indefinitely many ways in order to achieve a particular target value of Q: some of these combinations will stress the importance of growth, and others will stress the importance of redistribution. What policy tack we choose to follow will depend upon our assessment of feasibility constraints and normative notions of justice. A similar problem presides over the decomposition of measures of inequality into a ‘within-group’ component and a ‘between-group’ component. All inequality measures are not amenable to such decomposition: the Gini coefficient is an example of such a measure which is not subgroup decomposable. The set of decomposable indices essentially boils down to the class of ‘Generalised Entropy Measures’, of which the coefficient of variation and the Theil inequality measures are examples (Shorrocks 1984, 1988, 2013). As a positive exercise aimed at assessing the relative contributions of the two components—‘within-group’ and ‘between-group’—to total inequality, the decomposition procedure is unexceptionable. But the discovery that for some particular partitioning of the population into subgroups the betweengroup component is relatively small is not a conclusive case against taking group inequality seriously. Such an inference is somewhat crudely ‘contribution-oriented’ in motivation, in the sense that it is not overly informed by a sense of the intrinsic unfairness of group disparity, but rather by a sense of how much, in purely quantitative terms, group disparity contributes to aggregate inequality. Furthermore, as the number of groups into which the population is partitioned increases, between-group inequality also increases. Indeed, in principle, it is possible for the partitioning of the population to be so ‘fine’ or ‘granular’, that each person by herself or himself

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could be regarded as a separate group—in which case between-group inequality will account for all of the observed inequality, and within-group inequality for none of it! Briefly, an interpretation of decomposition results which is not informed by a sense of the political salience of the ‘grouping’ of the population we have resorted to could be an unhelpful approach to measurement. This is a matter of sociology and politics and only incidentally of arithmetic.

7.4 Concluding Thoughts As the great logician Frank Ramsey said, ‘…we can make several things clearer, but we cannot make anything clear’. Since we are doomed to failure when it comes to making anything ‘clear’, I can only hope that in this lecture I might have met with a little success in making ‘clearer’ at least a few things drawn from a difficult topic. I believe measurement in the social sciences is very important—one need only look at the life and work of a person such as Dr C. Chandrasekaran to see this. That is also why one needs to be careful, when dealing with the enterprise of measurement, to take serious note of the notions of outcomes and processes, and facts and values. It is wise to heed that old proverb: ‘He that forsakes measure, measure shall forsake him’.

References Basu K (2001) On the goals of development. In: Meier GM, Stiglitz JE (eds) Frontiers of development economics: the future in perspective. Oxford University Press, New York Basu K (2006) Globalization, poverty, and inequality: what is the relationship? What can be done? World Dev 34(8):1361–1373 Basu K (2013) Shared prosperity and the mitigation of poverty: In Practice and in precept. World Bank Policy Research Working Paper, No. 6700. http://ssrn.com/abstract=2354167 Basu K, Subramanian S (2019) Inequality, growth, poverty, and lunar eclipses: policy and arithmetic. Dev Change 51(2):352–370 Datt G, Ravallion M (1992) Growth and redistribution components of changes in poverty measures. J Dev Econ 38(2):275–295 Dhongde S (2002) Measuring the impact of growth and income distribution on poverty in India’. Mimeo, Department of Economics, University of California at Riverside. Available at: http://cit eseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.198.3867&rep=rep1&type=pdf Dollar D, Kraay A (2002) Growth is good for the poor. J Econ Growth 7(3):195–226 Dollar D, Kleineberg T, Kraay A (2013) Growth still is good for the poor. In: World bank policy research working paper 6568. http://documents.worldbank.org/curated/en/496121468149 676299/pdf/WPS6568.pdf Dollar D, Kleineberg T, Kraay A (2014) Growth, inequality, and social welfare: cross-country evidence. In: Economic policy sixtieth panel meeting, Einaudi institute for economics and finance, Rome. http://www.economic-policy.org/wp-content/uploads/2014/10/Dollar-Kraay-Kleineberg. pdf

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Dos Santos VF, da Cruz Vieira W (2013) Effects of growth and reduction of income inequality on poverty in Northeastern Brazil, 2003–2008. Economia Aplicada 17(4). http://dx.doi.org/10. 1590/S1413-80502013000400006. Available at: http://www.scielo.br/scielo.php?script=sci_art text&pid=S1413-80502013000400006 Gould SJ (1997) The mismeasure of man. Penguin Books, London Jain LR, Tendulkar SD (1990) The role of growth and distribution in the observed change in headcount ratio measure of poverty: a decomposition exercise for India. Indian Econ Rev 25(2):165– 205 Jayaraj D, Subramanian S (2009) The wellbeing implications of a change in the sex-ratio of a population. Soc Choice Welfare 33(1):129–150 Kakwani N, Subbarao K (1990) Rural poverty and its alleviation in India. Econ Polit Wkly 25:A2– A16 Kanbur SR, Mukherjee D (2007) Premature mortality and poverty measurement. Bull Econ Res 19(4):339–359 Patillo C, Gupta S, Carey K (2005) Sustaining growth accelerations and pro-poor growth in Africa. In: IMF working paper WP/05/195. International Monetary Fund, Washington, D. C. Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.511.2979&rep=rep1&type=pdf Rosenblatt D, McGavock T (2013) A note on the simple algebra of the shared prosperity indicator. Policy Research Working Paper; no. WPS 6645, World Bank, Washington, D. C. Available at: http://documents.worldbank.org/curated/en/894291468337199450/A-note-on-the-simple-alg ebra-of-the-shared-prosperity-indicator Sen A (1967) The nature and classes of prescriptive judgements. Philos Q 17(66):46–62 Shorrocks AF (1984) Inequality decomposition by population sub-groups. Econometrica 52(6):1369–1385 Shorrocks AF (1988) Aggregation issues in inequality measurement. In: Eichhorn W (ed) Measurement in economics: theory and applications in economic indices. Physica Verlag, Heidelberg Shorrocks AF (2013) Decomposition procedures for distributional analysis: a unified framework based on the Shapley value. J Econ Inequality 11(1):99–126 Subramania S (2010) Introduction. In: Jayaraj D, Subramanian S (2010) Poverty, inequality, and population: essays in development and applied measurement. Oxford University Press, Delhi Subramanian S, Lalvani M (2018) Poverty, growth, inequality: some general and India-specific considerations. Indian Growth Dev Rev 11(2):136–151

Part II

Inequality and Poverty Measurement: Questioning Some Axiomatic Foundations

Chapter 8

Social Groups and Economic Poverty: A Problem in Measurement

Abstract In the context of poverty comparisons for homogeneous populations, axioms such as the principle of ‘symmetry’ and the principle of ‘transfer’ are routinely regarded as innocuous and un-contentious. However, and as this paper shows, these same principles tend to become rather more problematic when poverty comparisons are undertaken in the context of heterogeneous populations that can be partitioned into well-defined social groups. Keywords Homogeneous populations · Heterogeneous populations · Interpersonal inequality · Intergroup inequality · Symmetry · Transfer · Impossibility result

8.1 Motivation This paper combines the themes of poverty and inequality, within a measurement setting, with a view to elucidating some of the complications that can arise, and how these might be addressed, when we allow for a certain elementary obtrusion of considerations of ‘society’ into routinely mainstream notions of the ‘economy’. Specifically, the concern is with reckoning aspects of distributive justice from a group perspective, in addition to the more standardly individualistic perspective, with an emphasis on the sorts of conflicts which these alternative perspectives could engender, and how these conflicts might be reconciled in the process of seeking a real-valued measure of income poverty. The two perspectives of distributive justice just alluded to are handily described by Stewart (2002) in the terms, respectively, of horizontal inequality and vertical inequality. Much of received theorizing has been concerned almost exclusively with vertical inequality, which has tended to confine horizontal inequality, in a relative sense, to the unhappy status of what Stewart (op. This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2006) Social Groups and Economic Poverty: A Problem in Measurement, in M. McGillivray (ed.): Inequality, Poverty, and Wellbeing, 143–161, Palgrave Macmillan: London. I am grateteful to Kaushik Basu, to the late S. Guhan, and to D. Jayaraj, Prasanta Pattanaik, and A.F. Shorrocks for helpful discussions of either earlier versions or specific concerns of the paper. I am indebted to Anne Ruohonen and Adam Swallow for their final editing of the paper. The usual caveat applies. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_8

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cit.) calls ‘a neglected aspect of development’. The question of why groups deserve a great deal more analytical and empirical attention than they would appear to have received in the discourse on poverty, inequality, and development has been dealt with fairly exhaustively in Stewart’s work and therefore represents ground that one does not need to cover here again. Reference, in this context, must also be made to earlier work, notably from the viewpoint of measurement, by Anand and Sen (1995), Jayaraj and Subramanian (1999), Majumdar (1999), Majumdar and Subramanian (2001), and Subramanian and Majumdar (2002). It is perhaps important to stress that the analytical content of this paper—whether in the matter of the existence results it advances or the specific poverty measures it discusses—is not motivated by any illusion as to either its revelation or novelty value. The arguments in this paper are, on the whole, uniformly simple and obvious. It is perhaps precisely because of this obviousness that attention needs to be drawn to the pervasive reality and centrality of groups in any assessment of social welfare; the motivational concern of this paper is, simply, to point to the obvious so that it is not overlooked. There is nothing very paradoxical in this: it is just another instance of Edgar Allan Poe’s purloined letter. If the notion of horizontal inequality has met with traditionally little engagement in exercises dealing with the assessment of overall deprivation or well-being, this fact probably has much to do with the common failure of being blind to what stares one in the face (excepting, of course, those instances of a deliberate ideological opposition to the notion of groups and their relevance in the scheme of things). The motivational objective of this paper, therefore, is to highlight an issue for reasons which arise not from its complexity, but from a combination of its importance and its relative historical neglect.

8.2 Measuring Poverty in a Stratified Society An issue of potential interest in the measurement of poverty has to do with the way in which poverty is distributed across different well-defined subgroups within the population.1 Foster and Shorrocks (1991) have advanced and studied a property of poverty indices which they call subgroup consistency and which demands that, other things equal, an increase in any subgroup’s poverty should increase overall poverty. In motivating their discussion of this property, the authors (1991: 687) state: ‘Subgroup consistency may … be regarded as a natural analogue of the monotonicity condition of Sen (1976), since monotonicity requires that aggregate poverty fall … if one person’s poverty is reduced, ceteris paribus, while subgroup consistency demands that aggregate poverty fall if one subgroup’s poverty is reduced, ceteris paribus’. In this connection, it is immediately tempting to seek also an analogy between the 1

For analyses of poverty measurement when different groups are perceived to have different needs, as reflected in variations in subgroup poverty lines, see Atkinson (1987) and Keen (1992). While the concern in these papers, as in the present one, is with reckoning subgroup poverty in the measurement of aggregate poverty, the underlying motivations are rather different.

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conventional transfer axiom and a corresponding one which could be defined for subgroups. The transfer condition requires that, ceteris paribus, a progressive rank-preserving transfer between two poor individuals should be accompanied by a reduction in poverty. In a similar spirit, one could require—speaking loosely for the moment— that aggregate poverty should decline with a move towards equalization, through income redistribution, of subgroup poverty levels, other things remaining the same. The requirement is formalized, in this note, through the postulation of a property called subgroup sensitivity. The relevance of subgroup sensitivity is captured in the following illustration. Suppose poverty to be measured by the simple headcount ratio. Imagine that the population is partitioned into two subgroups, A and B, where A stands for a historically disadvantaged social group, say, and B stands for the rest. Suppose the headcount ratio of poverty for subgroup A to be 0.7 and that for subgroup B to be 0.3. If subgroup A’s share in total population is 50 per cent, then the headcount ratio for the population as a whole would be 0.5(=0.5 * 0.7 + 0.5 *0.3). If now there is a pure redistribution of income from subgroup B to subgroup A, whereby A’s headcount ratio is reduced to 0.6 while B’s headcount ratio is raised to 0.4, then we may be disposed to judge that such a movement towards equalization of poverty across subgroups should lead to an overall reduction in measured poverty. This, precisely, is the sort of judgement that would be endorsed by the axiom of subgroup sensitivity. Such a possibility, however, is not accommodated by the headcount ratio which, in the context of the present example, continues to remain at 0.5(=0.5 * 0.6 + 0.5 * 0.4). This simple example points to a possible limitation underlying conventional approaches to the measurement of poverty. The difficulty in question resides in the fact that certain axioms for poverty measurement have been advanced on the implicit assumption that there is no more than one group—that constituted by the population as a whole—which needs to be reckoned in an overall assessment of the extent of poverty in a society. This assumption is particularly salient in the so-called symmetry and transfer axioms. The former property demands, in essence, that in making poverty comparisons across income profiles, the personal identities of individuals should be of no account. This property is also a standard feature of the literature in social choice theory, where it more commonly goes by the name of the ‘anonymity’ axiom. One can immediately see that if the identity of an individual is linked to the fact of group affiliation, then a poverty index which is sensitive to the group composition of a population could well militate against the requirement of anonymity imposed by the symmetry axiom. The axiom in question is widely regarded as being completely innocuous and self-evidently desirable from an ethical point of view: it is, indeed, so much taken for granted that its social choice version—anonymity—has come in for specifically targeted criticism in a carefully argued assessment by Loury (2000), who refers to the anonymity axiom as a stark example of ‘liberal neutrality’. What could constitute a possible objection to symmetry/anonymity which, after all, echoes a requirement that is a feature of many liberal constitutions—the requirement that no person may be discriminated against on the grounds of birth, race, class, caste, religion, or sex?

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Here is an objection: one may wish to discriminate in favour of members of an historically oppressed and consequently currently disadvantaged group; but in order to discriminate in favour of somebody, one will have to discriminate against somebody else on the ground of the latter’s group affiliation—an avenue of redress for the former individual which is denied by the symmetry axiom. Briefly, symmetry cannot be reconciled with group-based principles of distributive justice such as are embodied in provisions like ‘compensatory discrimination’ or ‘affirmative action’. The preceding discussion suggests that symmetry is an unquestionably desirable property when one is assessing inequality or poverty or welfare in the context of a homogeneous population; however—and possibly because of repeated, mechanical use—often the qualifying attribute of homogeneity seems implicitly to be forgotten when the axiom is invoked. Indeed, one of the few authors who are careful to rationalize the symmetry axiom on the grounds of its appeal in the context of homogeneous populations is Shorrocks (1988). One could, of course, suppose that symmetry is so widely specified as a desirable property only because of a formally unstated assumption as to the homogeneity of a population; but somehow, this is a less than convincing explanation when the axiom is also routinely invoked in the context of exercises which explicitly accommodate groups into the analysis and therefore are concerned with heterogeneous populations. Similarly, where the transfer axiom is concerned, one can see that considerations of inter-group equality could, in specific cases, conflict with considerations of interpersonal equality. Regressive transfers between members of a homogeneous group may naturally be taken to reduce welfare and enhance inequality and poverty; but should the same outcome necessarily hold for transfers between individuals belonging to different groups in a heterogeneous population? It is these sorts of difficulties attending the measurement of poverty that are sought to be made transparent in this paper. The rest of the chapter is organized as follows. Section 8.3 deals with certain relevant preliminary formalities of concepts and definitions. Section 8.4 presents a couple of general possibility theorems on poverty indices which highlight the complications that can arise from taking the issue of group-wise poverty distribution seriously. Section 8.5 offers a brief interpretation and assessment of the results presented in the previous section. Examples from the literature of ‘group-sensitive’ poverty measures are discussed in Sect. 8.6, while Sect. 8.7 carries a brief discussion of the implications of ‘group sensitivity’ for budgetary intervention in poverty redress exercises. Section 8.8 concludes.

8.3 Formalities2 Let N be the set of all positive integers. For every n ∈ N , let Xn be the set of nonnegative vectors {(x1 , . . . , xi , . . . , xn )} and define X to be the set ∪n∈N Xn . A 2

This section is heavily dependent on Jayaraj and Subramanian (1999: especially pp. 197–200).

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typical element of the set X is an income vector x, a typical element x i of which stands for the income of person i(i ∈ N ). For every x ∈ X, n(x) stands for the dimensionality of x. The poverty line, z, is a positive level of income such that individuals with incomes less than z are certified to be poor. For every (z, x) ∈ T × X (where T is the set of positive reals), x P (z, x) will stand for the vector of poor incomes; x R (z, x) will stand for the vector of non-poor incomes and μ P (z, x) will stand for the average income of the poor population. A vector x ∈ X will be said to be derived as a permutation of a vector y ∈ X if x = y for some permutation matrix ; xo is the ordered version of x if xo is derived from x by a permutation for 0 (i = 1, . . . , n(x) − 1).3 For all x, y ∈ X, it will be said that x vector which xi0 ≤ xi+1 dominates y—written x ∨ y—if x and y are equi-dimensional, xi0 ≥ yi0 for all i and xi0 > yi0 for some i; x will be said to be derived from y through an increment to a person’s income if x and y are equi-dimensional, xi = yi for all i = k and xk > yk for some k; x will be said to be derived from y through a permissible progressive transfer if x and y are equi-dimensional, xi = yi for all i = j, k for some j, k satisfying y j < yk , x j = y j + δ and xk = yk − δ where 0 ≤ δ ≤ (yk − y j )/2. Since a major concern of this paper is with the notion of reference groups, I turn now to this latter issue. For every n ∈ N , let Gn be the set of all possible partitions of the set {1,…,n}, and define G to be the set ∪n∈N Gn . A typical element of the set G is a partition g of the population, and a typical element of g is a subgroup, denoted by the running index j. (It is immediate, of course, that for any partition g of the population, the number of subgroups must be at least one, and cannot exceed the number of individuals in the population.) Notice that any g ∈ G is induced by some appropriate grouping of the population, by which is meant some well-defined scheme of categorization (such as by height, age, gender, caste, religion), in accordance with which the population can be classified in a mutually exclusive and completely exhaustive fashion. (It is possible, of course, that two or more groupings can induce the same partitioning of a given population: for example, if in some society the only illiterate individuals all happen to be females, then a grouping according to gender [{male, female}] will yield up the same partition as a grouping according to literacy status [{literate, illiterate}].) For all x ∈ X and g ∈ G, the pair (x, g) will be said to be compatible if and only if g partitions a population of the same size as the dimensionality of x. Given any compatible pair (x, g) belonging to X × G, subgroup j’s vector of incomes will be represented by xj , for every j belonging to g. Two polar cases of grouping are of interest. The first is what one may call the ‘atomistic grouping’, which induces the finest partition ga = {{1},…,{i},…,{n}} of {1,…,n}: this is the case of ‘complete heterogeneity’, such as might be precipitated by a classification according to ‘fingerprint type’. The second is what one may call the ‘universal grouping’, which induces the coarsest partition gu = {{1,…,n}} of {1,…,n}: this is the case of ‘complete homogeneity’, such as might be precipitated by a classification according, say, to membership to the human race. We are now in a position to define a poverty index, by which shall be meant a mapping P : T ×X ×G → R (where R is the real line), such that, for all z ∈ T , and 3

See Foster and Shorrocks (1991).

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all compatible (x, g) ∈ X × G, P(z, x, g) is a unique real number which is intended to signify the extent of poverty that obtains in the regime (z, x, g). To invest P with more structure, we need to constrain it with a set of properties that we may require the poverty index to satisfy. What follows is a restricted set of just six axioms, of which the sixth4 is a relatively recent addition to the stock of known axioms. Symmetry (Axiom S). For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if x is derived from y by a permutation, then P(z, x, g) = P(z, y, g). Monotonicity (Axiom M). For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if x is derived from y by an increment to a poor person’s income, then P(z, x, g) < P(z, y, g). Respect for Income Dominance (Axiom D). (See Amiel and Cowell 1994.) For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if x P ∨ y P , then P(z, x, g) < P(z, y, g). [Note: Amiel and Cowell (1994) point out that Axioms M and D are independent, but are rendered equivalent in the presence of the symmetry axiom]. Transfer (Axiom T). For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if x R = y R and x P is derived from y P by a permissible progressive transfer, then P(z, x, g) < P(z, y, g). Subgroup Consistency (Axiom SC). (See Foster and Shorrocks 1991.) For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if xj and yj are of the same dimensionality for all j ∈ g and [P(z, xj , gu ) = P(z, yj , gu ) for all j ∈ g\{k} and P(z, xk , gu ) < P(z, yk , gu ) for some k ∈ g], then P(z, x, g) < P(z, y, g). Subgroup Sensitivity (Axiom SS). For all z ∈ T , all x, y ∈ X, and all g ∈ G such that (x, g) and (y, g) are compatible pairs, if (i) xj and yj are of the same dimensionality for all j ∈ g; (ii) μ P (z, x) = μ P (z, y); and (iii) xj = yj for all j ∈ g\{s, t} for some s, t satisfying P(z, xt , gu ) < P(z, yt , gu ) ≤ P(z, ys , gu ) < P(z, xs , gu ), then it is the case that P(z, x, g) > P(z, y, g). [That is, other things remaining the same, if by a pure redistribution of poor incomes the relatively disadvantaged subgroup s becomes less poor and the relatively advantaged subgroup t becomes poorer, while maintaining the relative poverty rankings of the two subgroups, then overall poverty should decline.] What is the class of poverty indices which satisfy the property of subgroup sensitivity in conjunction with some combination of other desirable properties discussed earlier? This question is addressed in the next section.

8.4 Two General Possibility Results for Poverty Indices The following two propositions are true.

4

The axiom of ‘subgroup sensitivity’ has been advanced in Jayaraj and Subramanian (1999).

8.4 Two General Possibility Results for Poverty Indices

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Proposition 8.1. There exists no poverty index P : T × X × G → R satisfying Axioms S, M, and SS. Proof What follows is a proof by contradiction. Consider a situation in which z = 50. Let g be such that the population is partitioned into exactly two subgroups which are indexed 1 and 2, respectively. Consider a pair of income vectors x, y such that n(xP ) = n(yP ) = n(xR ) = n(yR ) = 2. Further, assume that x1R = x2R = y1R = y2R = (60, 60) and that x1P = (10, 20); x2P = (30, 40); y1P = (10, 30); and y2P = (20, 40). It is immediately clear that μ P (z, x) = μ P (z, y)(= 25).

(8.4.1.1)

Further, since P satisfies Axioms S and M, it must satisfy Axiom D as well; by Axiom D, given that n(x1 ) = n(x2 ) = n(y1 ) = n(y2 ) and x2P ∨ y2P ∨ y1P ∨ x1P , we have:         P z, x2 , gu < P z, y2 gu < P z, y1 gu < P z, x1 , gu .

(8.4.1.2)

In view of (8.4.1.1) and (8.4.1.2), Axiom SS will dictate that P(z, x, g) > P(z, y, g).

(8.4.1.3)

Next, notice that x = (x1P , x2P , x1R , x2R ) is just a permutation of y = if the person with income 20 in x1P swaps places with the person with income 30 in x2P , then x1P becomes y1P and x2P becomes y2P . By Axiom S, one must have: (y1P , y2P , y1R , y2R ):

P(z, x, g) = P(z, y, g).

(8.4.1.4)

(8.4.1.3) and (8.4.1.4) are mutually incompatible, and this completes the proof of the proposition (Q.E.D.). Proposition 8.2. There exists no poverty index P : T × X × G → R satisfying Axioms D, T, and SS. Proof Again we have a proof by contradiction. As in the proof of Proposition 8.1, imagine a situation in which z = 50, and g is such as to partition the population into two subgroups, 1 and 2. Let x and y be two income vectors satisfying n(xP ) = n(yP ) = n(xR ) = n(yR ) = 2, and let it be the case that x1R = x2R = y1R = y2R = (60, 60); x1P = (10, 20); x2P = (20, 30); y1P = (10, 25); and y2P = (15, 30). Notice first that μ P (z, x, g) = μ P (z, y, g)(= 20).

(8.4.2.1)

Further, since n(x1 ) = n(x2 ) = n(y1 ) = n(y2 ), and x2P ∨ y2P ∨ y1P ∨ x1P , Axiom D will require that         P z, x2 , g < P z, y2 , g < P z, y1 , g < P z, x1 , g .

(8.4.2.2)

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In view of (8.4.2.1) and (8.4.2.2), Axiom SS will dictate that P(z, x, g) > P(z, y, g).

(8.4.2.3)

Next, it is easy to see that x P = (x1P , x2P ) has been derived from y P = (y1P , y2P ) by a permissible progressive transfer (of 5 from the person with income 25 in the vector y1P to the person with income 15 in the vector y2P ). Given, additionally, that n(x) = n(y), Axiom T will demand that P(z, x, g) < P(z, y, g).

(8.4.2.4)

From (8.4.2.3) and (8.4.2.4), we obtain a contradiction. This completes the proof of the proposition (Q.E.D.).

8.5 Assessment The two non-existence results presented in the preceding section confirm the wisdom of a moral (suitably translated to the poverty measurement context) that has been upheld by Sen in his discussion of impossibility results in social choice theory. This moral (Sen 1970: 178) points to ‘the sever[ity] of the problem of postulating absolute principles … that are supposed to hold in every situation’. Confronted with an impossibility theorem, it is not always a simple matter to be able to convincingly identify the ‘villain of the piece’, namely the guilty axiom that is driving the result under review. Specifically, given the present context, and by way of illustration, neither the transfer nor the subgroup sensitivity axiom is as persuasive as either may appear in a context-free environment. For example, in particular cases the antecedents of Axiom SS can be satisfied by a very regressive transfer (from an acutely impoverished person belonging to a relatively advantaged subgroup to a much better-off individual belonging to a relatively disadvantaged subgroup), and in such cases we may find it hard to endorse Axiom SS. By the same token, any permissible progressive transfer between two poor individuals would be encouraged by the transfer axiom, and in particular cases, wherein such transfers exacerbate inter-group poverty differentials, we may find it hard to accept Axiom T. Given this general difficulty of discerning an unqualified virtue in any given axiom under all plausible circumstances, a possible way out may be to restrict the applicability of the axiom to a domain that is relatively non-controversial. In the specific instance of the symmetry axiom, for example, there may be a case for requiring only that within any subgroup, swapping incomes across members of the subgroup should leave the value of the poverty index for the subgroup unchanged. In a between-group context, one might further wish to impose the restriction that measured poverty should be invariant with respect to the precise labels that are attached to subgroups. Similarly,

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in the case of the transfer axiom, one may wish to restrict its operation, in its conventional form, only to interpersonal redistributions of income within each subgroup. Is there also a way of ensuring some requirement of equity, from a between-group perspective, in the distribution of poverty across subgroups? How this can be done is perhaps best exemplified by means of an illustration, discussed in the following section.

8.6 ‘Group-Sensitive’ Poverty Indices: An Example Let P* be the set of all symmetric, monotonic, transfer-satisfying, and decomposable poverty indices. (A poverty index is said to be decomposable—see Foster et al. 1984—if overall poverty can be expressed as a population-share weighted sum of subgroup poverty levels.) Let (x, g) be a compatible pair belonging to X × G, and let g partition the population into K distinct subgroups. The poverty line, as usual, is given by z. Consider P* ∈ P*, and let P j∗ serve as a shorthand for P*(z, xj , gu ), which is the poverty level, as measured by the index P*, of the jth subgroup (j = 1,…,K). Assume, further, that the subgroups have been indexed in non-increasing order of ∗ , j = 1, . . . , K . Let θ j be the population share of the poverty, so that P j∗ ≥ P j+1 jth subgroup and  j the proportion of the population that belongs to groups whose poverty levels are less than or equal to the poverty level of the jth subgroup. Then, two examples of aggregate poverty measures which directly incorporate considerations of ‘inter-group equity’ are the indices 1 P and 2 P below: 1 P(z, x, g) = [1/(K − 1)]

K 

[(K − 1 − j)θ j +  j ]P j∗

(8.6.1)

j=1

and ⎡ 2 P(z, x, g)



=⎣

⎤1/2 θ j (P j∗ )2 ⎦

.

(8.6.2)

j∈g

The index 1 P is of a type discussed in Jayaraj and Subramanian (1999), Majumdar and Subramanian (2001), and Subramanian and Majumdar (2002), while the index 2 P is of a type discussed in Anand and Sen (1995). In what follows, we shall confine attention to the class of indices embodied in (8.6.2); in the interests of specificity, it would be useful to particularize the index P* to a familiar poverty measure. To this end, consider the so-called Pα class of indices proposed by Foster et al. (1984) (FGT Pα ), and given, for all z ∈ T and all compatible (x, gu ) ∈ X × G, by: Pα (z, x, gu ) = (1/n(x))

 i∈Q(x)

[(z − xi )/z]α , α ≥ 0,

(8.6.3)

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where Q(x) is the set of poor individuals in x. Each member of the class of indices Pα (see Foster et al. 1984) is known to be symmetric and decomposable; the index P2 , in addition, satisfies the monotonicity and transfer axioms. Let us designate by F the index P2 . A specialization of the class of ‘between-group equity-conscious’ poverty measures encompassed in (8.6.2) is yielded by the following index, F a , which is given, for all z ∈ T , and all compatible (x, g) ∈ X × G, by ⎡



F (z, x, g) = ⎣ a

⎤1/2 θ j F j2 ⎦

.

(8.6.4)

j∈g

In what sense does F a attend to the concern for inter-group equity? One way of seeing this is, first, to note that C 2 ≡ [(1/F 2 ) j∈g θ j F j2 − 1] is the squared coefficient of variation of the group-specific poverty levels F j . in the2 distribution θ j F j = F 2 (1 + C 2 ) whence, in view of (8.6.4), and after Then, it is clear that j∈g

making the appropriate substitution, we have: F a (z, x, g) = F(1 + C 2 )1/2 .

(8.6.5)

Notice from (8.6.5) that the index F a is just the average (across subgroups) level of poverty, as measured by the index F, enhanced by a factor incorporating the squared coefficient of variation in the inter-group distribution of poverty as measured by the F j . F a is mean poverty ‘adjusted’ for inter-group inequality. (It may be noted that for the class of poverty indices subsumed in (8.6.1), the ‘adjustment’ for inequality in the inter-group distribution of poverty levels is via a ‘Gini-type’, rather than ‘coefficient of variation-type’, inequality measure.) It is not difficult to check that the index F a satisfies both the ‘within-group’ and the ‘between-group’ versions of the symmetry property discussed earlier. ‘Withingroup’, because it is a known property of the FGT Pα class of indices that each member satisfies symmetry, and each of the F j is just the FGT index realized for α = 2. ‘Between-group’, because one can see from inspection of (8.6.4) that switching around the labels of the subgroups will make no difference to the value of F a . Further, F a clearly also satisfies the ‘within-group’ version of the transfer property, since each of the F j is known to be transfer-respecting (indeed, the Pα indices all satisfy transfer for every α > 1), while in a ‘between-group’ context, as the expression for F a in (8.6.3) makes clear, it is sensitive to inter-group inequality in the distribution of poverty: when the latter (as measured by C 2 ) rises, other things equal, F a also registers an increase in value. An advantage with a poverty index such as F a resides in a specific sort of ‘flexibility’ it possesses, in that it effects a trade-off between the conventional transfer axiom and Axiom SS, by allowing the former to ‘trump’ the latter when interpersonal transfers across groups are rather more than less ‘progressive’ and allowing the latter to ‘trump’ the former when interpersonal transfers across groups are rather less than

8.6 ‘Group-Sensitive’ Poverty Indices: An Example

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more ‘progressive’. A simple example, similar to one in Jayaraj and Subramanian (1999), might help to elucidate this point. Let the poverty line z be given by 10. Let x and y be two 4-dimensional income vectors, and let go be such as to partition the population into two subgroups 1 and 2, respectively, with each subgroup having two persons in it. Similarly, let u and v be any other two 4-dimensional vectors, and let go again partition the population into the subgroups 1 and 2, with each subgroup having two members. The vectors x, y, u, and v can be written, respectively, as x = (x1 , x2 ), y = (y1 , y2 ), u = (u1 , u2 ), and v = (v1 , v2 ). Let it be the case that x1 = (1, 9), x2 = (1.5, 3.5), y1 = (0, 9), and y2 = (1.5, 4.5); and u1 = (1, 9), u2 = (1.5, 2), v1 = (0, 9), and v2 = (1.5, 3). Suppose we measure poverty by the ‘adjusted’ index F a of (5.2). Some routine computation will confirm that F(z, x1 , gu ) = 0.205; F(z, x2 , gu ) = 0.28625; F(z, y1 , gu ) = 0.2525; F(z, y2 , gu ) = 0.25625; F a (z, x, go ) = 0.2490; F a (z, y, go ) = 0.2541; F(z, u1 , gu ) = 0.205; F(z, u2 , gu ) = 0.340625; F(z, v1 , gu ) = 0.2525; F(z, v2 , gu ) = 0.303125; F a (z, u, go ) = 0.2811; and F a (z, v, go ) = 0.2790. Notice now that y is derived from x through a regressive transfer, exactly as v is derived from u through a regressive transfer; in both cases, the transfer is from a poor person belonging to a relatively advantaged group to a richer poor person belonging to a relatively disadvantaged group. The transfer axiom will dictate that F a (z, y, go ) > F a (z, x, go ), while the subgroup sensitivity axiom will dictate that F a (z, y, go ) < F a (z, x, go ); in exactly similar fashion, the transfer axiom will dictate that F a (z, v, go ) > F a (z, u, go ), while the subgroup sensitivity axiom will dictate that F a (z, v, go ) < F a (z, u, go ). What actually obtains is a situation in which F a (z, y, go ) (=0.2541) > F a (z, x, go ) (=0.2490), and F a (z, v, go ) (=0.2790) < F a (z, u, go ) (=0.2811). That is to say, in the transition from x to y, Axiom T is upheld and Axiom SS violated, while in the transition from u to v, Axiom T is violated and Axiom SS upheld. In both cases, an interpersonally regressive income transfer has been accompanied by a diminution in the inter-group poverty differential; only, in the first case the transfer has been more regressive than in the second (the income difference between those involved in the transfer is greater in the first case than in the second), and the poverty index F a has effected a trade-off in favour of the transfer axiom in the first case and a trade-off in favour of the subgroup sensitivity axiom in the second case. This does not accord ill with intuition for, as was pointed out in Sect. 8.5, in particular cases the antecedents of Axiom SS can be satisfied by a very regressive transfer (from an acutely impoverished person belonging to a relatively advantaged subgroup to a much better-off individual belonging to a relatively disadvantaged subgroup), and in such cases we may find it hard to endorse Axiom SS, [while], by the same token, any permissible progressive transfer between two poor individuals would be encouraged by the transfer axiom, and in particular cases, wherein such transfers exacerbate inter-group poverty differentials, we may find it hard to accept Axiom T.

Additionally, F a has the convenient property of precipitating the index F as a special case, which happens when the grouping employed is the universal grouping which induces the coarsest partition gu of the population. At the other extreme, when the grouping employed is the atomistic one, which induces the finest partition ga of

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the population, it can be verified that F a just becomes (P4 )1/2 , where P4 is the Pα index for α = 2. (The details are available in Jayaraj and Subramanian 1999.) Finally, a concrete empirical illustration of the information value of an index such as F a may be useful. Making use of data on the cross-country distribution of per capita gross national product (GNP) available in the United Nations Development Programme’s Human Development Report (HDR), one can construct a picture of global poverty. Such an exercise has been carried out in Subramanian (2003), and we draw on the results of that exercise. The HDR 1999 provides information on per capita GNP, in ‘purchasing power parity dollars’, for each of 174 countries for the year 1997. The global average per capita GNP works out to a little in excess of PPP$6000, and we shall take the ‘international poverty line’ to be PPP$3000 per capita per annum—which is less than one-half of the global average per capita GNP. We shall designate this poverty line by z*; let x* denote the country-wise distribution of income as presented in the HDR 1999. Since information on intracountry distribution is unavailable, we shall simply assume that each person in each country receives the country’s average per capita GNP. A grouping of countries resorted to in the HDR 1999 is a classification comprising the following seven groups: sub-Saharan Africa, Asia and the Pacific, the Arab states, Latin America and the Caribbean, Eastern Europe and the Commonwealth of Independent States, Southern Europe, and industrialized countries. Let us denote by g* the partition of the world’s population induced by this particular grouping. Poverty for each country group j will now be measured by an index which one may call the triage headcount ratio h j : for a poor country group j (i.e. a country group whose per capita GNP is less than the poverty line), h j is simply the proportion of the country group’s population that must be allocated an income of zero so that the average income of the rest of the population in the country group is enabled to rise to the poverty line level of income z*; for a non-poor country group j, we shall take h j to be zero. By this reckoning, the proportion of the world’s population—call it h—that must just cease to exist in order that every remaining person may receive an income of z* works out to 18.6%—the details are provided in Table 8.7.1. More specifically, this is the value of the triage headcount ratio corresponding to the universal grouping gu : h(z*, x*, gu ) = 0.186. What if we employed the grouping g* resorted to in the HDR 1999? 1/2

2 The adjusted triage headcount ratio h a (z ∗ , x∗ , g∗ ) ≡ θ h turns to be ∗ j j∈g j 0.247, as can be confirmed from the figures presented in Table 1. The rise in the triage headcount ratio from 18.6% to 24.7% is a substantial one and an indicator of the considerable inequality in the cross-country distribution of deprivation. If this outcome is an unattractive one, the picture, presumably, would be even worse if the grouping we employed classified the world not into seven groups, but into 174 groups—each group being represented by an individual country. Incorporating intergroup differentials into an overall assessment of deprivation certainly shows up in the deeply stratified world in which we live.

8.7 Two Implications of ‘Group Sensitivity’ in a Poverty Measure

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Table 8.1 Grouped data on global poverty (1997) Country group

Population (in millions)

Population share

Triage headcount ratio

Sub-Saharan Africa

555.20

0.0967

0.6048

Asia and the Pacific

3140.30

0.5467

0.2126

Arab states

252.30

0.0439

0.1012

Latin America and the Caribbean (including Mexico)

490.50

0.0854

0.0167

Eastern Europe and the Commonwealth of Independent States

398.90

0.0695

0.0823

Southern Europe

64.20

0.0112

0.0000

Industrialized countries

842.30

0.1466

0.0000

World

5743.70

1.0000

0.1863

Note The poverty line is taken to be PPP$3,000 per capita per annum (roughly one-half the average per capita GNP). The ‘triage headcount ratio’ is the proportion of the population that must be allowed to perish so that each member of the surviving population is enabled to just achieve a poverty line level of income Source Based on UNDP (1999)

8.7 Two Implications of ‘Group Sensitivity’ in a Poverty Measure The particular grouping of a population we resort to must be informed by an appreciation of the sociological salience of the classificatory scheme we adopt. More than one classificatory scheme may be relevant, depending on the precise context in, and purpose for which deprivation is being sought to be measured. It is therefore a matter of some importance, in making poverty comparisons, to be explicit not only about the distributions under comparison and the poverty line(s) employed, but also about the grouping invoked. A certain ranking, valid for poverty measured with a particular grouping in mind, could in principle be inverted by a ranking which is valid for poverty measured with some other grouping in mind. An example of such rank reversal has in fact already been considered in Sect. 6. Harking back to the income vectors u and v reviewed in Sect. 6, it can be verified that, when z = 10, F a (z, u, gu ) (=0.2728) < F a (z, v, gu ) (=0.2778), but F a (z, u, go ) (=0.2811) > F a (z, v, go ) (=0.2790). A first implication of working with ‘group-sensitive’ poverty measures, therefore, is that the particular grouping we employ can make a substantial difference to the evaluative outcome of poverty comparisons. Second, the grouping that is employed also has implications for ‘targeting’ in poverty redress schemes. Again, a simple numerical illustration might be helpful in explicating the idea. Let the poverty line be z  = 10, and let a be a 4-dimensional income vector (namely n(a) = 4). Let g be a partition of the population, based, let us say, on a grouping by caste, which divides it into two groups, 1 and 2, respectively.

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We shall write a = (a1 , a2 ), with a1 = (a11 , a12 ) and a2 = (a21 , a22 ), where a ji stands for the income of the ith person in the jth group (j = 1, 2 and i = 1, 2). Suppose, for specificity, that a1 = (4, 6) and a2 = (3, 9). If θ j is the population share of subgroup j(j = 1, 2), then it is clear that in the present instance θ1 = θ2 = 1/2. Let gu be an alternative partitioning of the population, induced by the universal grouping, which recognizes only one group, that constituted by the grand coalition of individuals. Suppose a budgetary allocation of T = 5 is available for poverty alleviation. Which is the best way of allocating the budget among the individuals in the population? To complicate matters, we shall imagine that there are two policymakers, A and B, of whom A—who has no time for sociological affectations—believes that gu is the only valid partitioning of the population, while B—who herself is a member of an underprivileged caste—believes that g is a meaningful partition of the population. Policymaker A is comfortable with using the poverty index F(z, x) [which, to recall, is the same as the index F a (z, x, gu )], while policymaker B is comfortable with using the index F a (z, x.g ). Let t ji , in B’s notation, be the amount of the budgetary allocation T which goes to the ith individual in the jth group (j = 1, 2 and i = 1, 2) and denote by t the vector (t11 , t12 , t21 , t22 ); further, let t1 and t2 stand for the vectors (t11 , t12 ) and (t21 , t22 ), respectively. (Of course, the individuals that B calls 11, 12, 21, and 22 will probably be called just 1, 2, 3, and 4 by A, but since the latter believes in a thoroughgoing version of the symmetry axiom, his philosophy of ‘what’s in a name?’ should be compatible with an acceptance of B’s eccentric mode of labelling the individuals.) A’s objective is to solve the following programming problem: Problem A Minimize

{t11 ,t12 ,t21 ,t22 }

F a (z  , a + t, gu ) = [1/n(a)(z  )2 ][{z  − (a11 + t11 )}2 + {z  − (a12 + t12 )}2 + {z  − (a21 + t21 )}2 + {z  − (a22 + t22 )}2 ] subject to (t11 + t12 + t21 + t22 ) ≤ T ; and 0 ≤ t11 ≤ z  − a11 , 0 ≤ t12 ≤ z  − a12 , 0 ≤ t21 ≤ z  − a21 , and 0 ≤ t22 ≤ z  − a22 . The optimal solution to this problem is the so-called lexicographic maximin solution (see Bourguignon and Fields 1990; Gangopadhyay and Subramanian 1992). The solution consists in raising the poorest person’s income to the income of the second poorest person if the budget will permit or to the highest feasible level not exceeding the second poorest person’s income; if the budgetary outlay is thereby exhausted, we stop the exercise here, and if not, the incomes of the two poorest individuals are raised to the income of the third poorest person if the budget will permit or to the highest feasible level not exceeding the third poorest person’s income; and so on, until we reach that marginal individual with whom the budget is exhausted. Given the specific numerical values we have assigned to the poverty line z  , the income vector a, and the budgetary outlay T, it can be verified that the optimal solution to Problem A is provided by t*, where t* is the transfer schedule defined by (8.7.1)

8.7 Two Implications of ‘Group Sensitivity’ in a Poverty Measure

95

below: ∗ ∗ ∗ ∗ = 2, t12 = 0, t21 = 3, and t22 = 0. t11

(8.7.1)

The resulting, post-transfer income vector is given by ∗ ∗ ∗ ∗ , a12 + t12 , a21 + t21 , a22 + t22 ) = (6, 6, 6, 9). a∗ = (a11 + t11

(8.7.2)

Next, policymaker B’s problem can be written as follows. Problem B Minimize

{t11 ,t12 ,t21 ,t22 }

F a (z  , a + t, g ) = [θ1 {F a (z  , a1 + t1 , gu )}2 + θ2 {F a (z  , a2 + t2 , gu )}2 ]1/2 subject to t11 + t12 + t21 + t22 ) ≤ T ; and 0 ≤ t11 ≤ z  − a11 , 0 ≤ t12 ≤ z  − a12 , 0 ≤ t21 ≤ z  − a21 , and 0 ≤ t22 ≤ z  − a22 . Will the transfer schedule t* presented in (8.7.1) also be an optimal solution to Problem B? It can be verified, given the numerical assumptions we have made   ∗  = 0.1281. If there is no other t , g regarding θ1 , θ2 , z  and T, that F a z  , a +  allocation t** such that F a z  , a + t∗∗ , g < 0.1281, then t* is an optimal solution to Problem B. However, consider the transfer schedule t** given by ∗∗ ∗∗ ∗∗ ∗∗ = 2.4558, t12 = 0.4558, t21 = 2.0884 and t22 = 0. t11

(8.7.3)

    It is easy to check that F a z  , a + t∗∗ , g = 0.1256 < F a z  , a + t∗ , g = 0.1281. Since poverty is lower with the transfer schedule t** than with the schedule t*, it is clear that t* is not an optimal solution to Problem B. (In fact, t** solves Problem B. This claim will not be proved here, but it may be noted that an intuitive sufficient condition for an optimum is a feasible transfer schedule which (a) respects the lexicographic maximin principle of allocation within each subgroup and (b) simultaneously ensures equalization of poverty levels between subgroups. In the present instance, note that a1 + t1 ** = (6.4558, 6.4558) and a1 + t2 ** = (5.0884, 9): the lexicographic maximin outcome obtains within each subgroup, and further, subgroup levels are equalized since, as can be confirmed,  poverty  F a z  , a1 + t1∗∗ , gu = F a z  , a2 + t2∗∗ , gu = 0.1256 . Briefly, policymakers A and B have a quarrel—and a substantial one at that—on their hands. While quarrels over the choice of the poverty line z have been numerous, quarrels over the choice of g have been relatively muted, with the implicit consensus favouring policymaker A’s approach to the problem. Yet, both choices have implications not only for poverty comparisons but also for the proper targeting of scarce resources in poverty alleviation programmes. There would thus appear to be a case for an explicit statement of the precise choice of g that is made and for a justification of that choice. The issue assumes a particular salience in the context of societies characterized by stratification arising from the historically cumulated maldistribution of

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burdens and benefits across identifiable subgroups of the population. For analysts concerned with the measurement of poverty and anti-poverty policy based on such measurement, the issue resolves itself into not just a problem in logic but into a larger problem in social ethics.

8.8 Concluding Observations In much of mainstream economic theorizing, the only ‘marker’ of identity is income. This is quite clearly evident in, for example, standard approaches to the measurement of poverty. The point is made explicit in Sen’s (1976) seminal paper dealing with the derivation of an ordinal measure of poverty, in which he draws specific attention to an assumption which is at the welfare basis of many poverty measures, and which he calls the ‘monotonic welfare’ axiom. According to this axiom, given any income vector x and any pair of individuals j and k with incomes x j and x k , respectively, if x j > x k , then W j (x) > W k (x), where W j (respectively, W k ) is the welfare level of individual j (respectively, individual k). It should be emphasized that Sen is himself sceptical of the universal validity of this axiom and employs it largely in the spirit of assembling material for a characterization theorem. In a richer framework of welfare, the latter would presumably be a function of arguments other than just income. Specifically, room would have to be made for the notion, as Akerlof and Kranton (2000: 718) put it, that ‘identity is based on social categories…’, and the fact that income classes do not exhaust social categories. If a person’s identity, and the welfare she experiences, depends not only on her income but also, for example, on the colour of her skin, then it is entirely conceivable that, given an income vector x and an n-tuple s describing each individual’s skin colour, one can have a pair of individuals j and k such that x j > x k , j is black and k is white, and V j (x, s) < V k (x, s), where V i (i = 1,…,n) stands for person i’s welfare level, and each person’s welfare level is assumed to be increasing in her/his income, other things equal. In terms of the standard symmetry axiom, aggregate welfare and poverty levels should remain unchanged if j and k were to swap their incomes; however, in terms of the welfare index V, one can easily see that black j would be rendered worse off and white k would be rendered better off if j and k were to swap incomes. Similarly, the standard transfer axiom would endorse a permissible progressive income transfer from j to k; however, such a transfer would only serve to widen the welfare gap (when welfare is measured by V ) between the two individuals. It is clear then that allowing for a plurality of groups in society does have non-trivial implications for the measurement of society-wide deprivation, a point that is emphasized by Thurow (1981: 179, 180, 182): Is the correct economic strategy to resist group welfare measures and group redistribution programmes wherever possible? Or do groups have a role to play in economic justice? … [I]t is not possible for society to determine whether it is or is not an equal opportunity society without collecting and analyzing economic data on groups … Individuals have to be judged based on group data … A concern for groups is unavoidable.

8.8 Concluding Observations

97

This note has been concerned to explore an aspect of the analytics of poverty measurement as a specific application of the exercise of complicating mainstream accounts of the economy by allowing for the pervasive reality of the stratification of society into groups. In the process, it points to two issues that could be salient in a consideration of how to accommodate subgroup poverty in the aggregation exercise of measuring income deprivation. First, it suggests the desirability of the poverty index being a variable, rather than trivial or constant, function of the precise grouping that is employed in partitioning a population into subgroups. By entering the grouping explicitly as an argument in the poverty function, the domain of the function is informationally expanded in a way that enriches a group-sensitive assessment of poverty. Second, it suggests that if it is considered desirable to directly incorporate into the measurement of poverty considerations relating to the inter-group distribution of poverty, then certain conventional axioms of poverty measurement may have to be modified, via restrictions on their domains of applicability, in order to avoid problems of internal consistency in the aggregation exercise.

References Akerlof GA, Kranton RE (2000) Economics and Identity. Quart J Econ CXV:715–753 Amiel Y, Cowell F (1994) Monotonicity, dominance and the Pareto principle. Econ Lett 45:447–450 Anand S, Sen A (1995) Gender inequality in human development: theories and measurement. Occasional paper, 19, Human Development Report Office. UNDP, New York (Cited in UNDP, 1995) Atkinson AK (1987) On the measurement of poverty. Econometrica 55:749–764 Bourguignon F, Fields GS (1990) Poverty measures and anti-poverty policy. Rech Economiques Louvain 56 Foster J, Shorrocks AF (1991) Subgroup consistent poverty indices. Econometrica 54(3):687–709 Foster J, Greer J, Thorbecke E (1984) A class of decomposable poverty indices. Econometrica 52:761–765 Gangopadhyay S, Subramanian S (1992) Optimal budgetary intervention in poverty alleviation programmes. In: Subramanian S (ed) Themes in development economics: essays in honour of Malcolm Adiseshiah. Oxford University Press, Delhi Jayaraj D, Subramanian S (1999) Poverty and discrimination: measurement, and evidence from rural India. In: Harriss-White B, Subramanian S (eds) Illfare in India: essays on India’s social sector in honour of S. Guhan. Sage Publications, New Delhi Keen M (1992) Needs and targeting. Econ J 102:67–79 Loury GC (2000) Racial justice. Lecture 3 of the DuBois Lectures, Harvard University, 17 April 2000. Available at: http://www.bu.edu/irsd/files/DuBois_3.pdf Majumdar M (1999) Exclusion in education: Indian states in comparative perspective. In: HarrissWhite B, Subramanian S (eds) Illfare in India: essays on India’s social sector in honour of S. Guhan. Sage Publications, New Delhi Majumdar M, Subramanian S (2001) Capability failure and group disparities. J Dev Stud 37:104– 140 Sen AK (1970) Collective choice and social welfare. Holden-Day, San Francisco Sen AK (1976) Poverty: an ordinal approach to measurement. Econometrica 44:219–231 Shorrocks AF (1988) Aggregation issues in inequality measurement. In: Eichhorn W (ed) Measurement in economics: theory and applications in economic indices. Physica Verlag, Heidelberg

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Stewart F (2002) Horizontal inequalities: a neglected dimension of development. QEHWPS 81: Queen Elizabeth House Working Papers Series Subramanian S (2003) Aspects of global deprivation and disparity: a child’s guide to some simpleminded arithmetic. In: Carlucci F, Marzano F (eds) Poverty, growth and welfare in the world economy in the 21st Century. Peter Lang AG, Berne Subramanian S, Majumdar M (2002) On measuring deprivation adjusted for group disparities. Soc Choice Welfare 19:265–280 Thurow LC (1981) The zero-sum society: distribution and the possibilities for economic change. Penguin Books Limited, Harmondsworth, Middlesex United Nations Development Programme (UNDP) (1999) Human development report 1999. Oxford University Press, New York

Chapter 9

Reckoning Inter-group Poverty Differentials in the Measurement of Aggregate Poverty

Abstract In a heterogeneous population which can be partitioned into well-defined subgroups, it is plausible that the extent of measured aggregate poverty should depend upon the distribution of poverty across the subgroups. In particular, a judgement against an unequal inter-group distribution of poverty can be upheld as an intrinsic social virtue. The aggregate measure of poverty, in line with this view, would then lend itself to ‘penal adjustment’ in order to reflect the extent of inter-group disparity in the distribution of poverty that obtains. In the present paper, this approach to poverty measurement is examined with specific reference to the advancement of a diagrammatic aid to analysis called the group poverty profile. The latter is a virtual transplantation, to the present context, of the notion of a deprivation profile that has been explored and analysed by A. F. Shorrocks in a different context (‘Deprivation Profiles and Deprivation Indices’, in S. Jenkins, A. Kapteyn and B. Vaan Praag (eds.): The Distributions of Welfare and Household Production: International Perspectives, Cambridge University Press: London, 1996). Keywords Group poverty profile · Group Lorenz profile

9.1 Introduction This conference is concerned with the themes of efficiency, equity, and institutions. Issues of measurement would presumably constitute one component of these concerns. The present paper deals with a particular aspect of equity, and how it may be measured and incorporated into an assessment of the overall poverty that obtains This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2009) Reckoning Inter-group Poverty Differentials in theMeasurement of Aggregate Poverty, Pacific Economic Review, 14: 46–55, https://doi.org/10.1111/j.1468-0106.2009.00434.x. I would like to thank, without implicating, Tony Shorrocks for the benefit of a discussion on the subject. Thanks are also owed to the hospitality of theWorld Institute for Development Economics Research (WIDER), Helsinki, where an earlier version of this chapter was written in the course of a Visiting Fellowship: this earlier version was published, under the same title, inWIDER’s Research Papers series as Research Paper No. 59/05 (September 2005). Finally, my thanks to the anonymous referees of Pacific Economic Review for helpful comments. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_9

99

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9 Reckoning Inter-group Poverty Differentials in the Measurement …

in a society. The particular aspect in question is that of inter-group disparity in the distribution of poverty. Group inequality is an issue of fundamental relevance to the character and performance of institutions. One of the clearest manifestations of this proposition can be found in debates relating to the inherent and instrumental (in terms of efficiency-promoting) advantages of such mechanisms of social justice as ‘compensatory discrimination’ or ‘affirmative action’. These larger debates are greatly aided by the availability of procedures by which inter-group disparities can be measured. While this paper locates the measurement problem in the assessment of poverty, the general approach advanced in it can find specific application with respect to any decomposable real-valued measure of well-being or ‘ill-being’.

9.2 Motivation Departures from the conventional assumption of ‘homogeneity’ as to the constitution of a population can have non-trivial implications for how we view and measure poverty. An aspect of social reality one frequently encounters is that populations are, indeed, heterogeneous. An appreciation of this fact could pave the way for measuring aggregate poverty in such a way that group-related disparities in the distribution of poverty are explicitly taken into account and penalized. Poverty measurement which is so motivated is compatible with the view that inter-group inequality in the distribution of poverty is a dis-valued outcome—for intrinsic reasons of the desirability of equality as a social virtue. In the present paper, this ‘intrinsic’ case for reckoning inter-group differentials in the measurement of aggregate poverty is investigated. Related approaches to social evaluation have been examined by Anand and Sen (1995) in the context of deriving a ‘gender-adjusted human development index’, and variations on this theme have been explored by Jayaraj and Subramanian (1999), Majumdar and Subramanian (2001), Subramanian and Majumdar (2002), and Subramanian (2006). The Subramanian– Majumdar paper provides an axiomatic justification for a specific ‘group inequalityadjusted’ aggregate index of deprivation. A similar index is derived in the present paper, and a diagrammatic link to this index, in the form of a graph called the ‘group poverty profile’, is advanced. This analytical device is directly inspired by Shorrocks’ (1995, 1996) ‘poverty gap/deprivation profile’, although the latter was not conceived to deal specifically with the issue of inter-group poverty differentials. The group poverty profile, it will be contended, could prove to be a useful instrument for comparing alternative regimes of the inter-group distribution of poverty. In addressing these issues, this paper has been organized as follows. The next section deals with the preliminaries of concepts and definitions. The section following examines the ‘intrinsic’ case for inter-group equality, with an emphasis on the graphical device called the group poverty profile. The final section concludes.

9.3 Concepts and Definitions

101

9.3 Concepts and Definitions N is the set of positive integers, R the real line, and R++ the positive real line. For every n ∈ N, X n stands for the set of non-negative n-vectors x = (x1 , . . . xi , . . . , xn ), and xi is the income of individual i in a community of n individuals. Define the set X ≡ ∪n∈N X n . The poverty line is a positive income level z such that all persons with income less than z are certified as being poor. For all x ∈ X, M(x) will stand for the set of all individuals whose incomes are represented in the income distribution x, and Q(x) will stand for the set of poor individuals. Next—see Jayaraj and Subramanian (1999), and Subramanian (2006)—for every n ∈ N, let G n be the set of all possible partitions of the set M = {1,…,n}, and let G be the set ∪n∈N G n . Every g ∈ G is some partition of the population, induced by some appropriate grouping, for example, on the basis of place of birth, religion, caste, gender, etc.; the elements of g—denoted by the running index j—will be taken to be subgroups of the population. It is obvious that, for every g ∈ G and n ∈ N, 1 ≤ #g ≤ n. The atomistic grouping g a induces the finest partition {{1},…,{n}} of {1,…,n}, and the universal grouping g u induces the grossest partition {{1,…,n}} of {1,…,n}: g a and g u define the two polar cases of grouping. For all (x, g) ∈ X × G, the pair (x, g) will be said to be a compatible pair if and only if g partitions a population whose size is the same as the dimensionality of x. We now define a poverty index formally. A poverty index is a mapping P: R++ × X × G → R such that, for every z ∈ R++ , and every compatible pair (x, g) ∈ X × G, P(z, x, g) specifies a real number which is intended to reflect the extent of poverty associated with the regime (z, x, g). Certain axioms which are routinely employed in the measurement of poverty are now quickly and non-technically reviewed. Focus (Axiom F) requires the poverty index to be insensitive to increases in non-poor incomes; continuity (Axiom C) requires the poverty index to be continuous on X n for every n ∈ N; normalization (Axiom N) specifies a value of zero for the poverty index when there is no poor person in the community; symmetry (Axiom S) requires the poverty index to be invariant to any interpersonal permutation of incomes; monotonicity (Axiom M) [respectively, weak monotonicity (Axiom WM)] requires the poverty measure to rise [respectively, not decline] with a decline in the income of any poor person; respect for income dominance (Axiom D; see Amiel and Cowell 1994; Subramanian 2006) requires that, ceteris paribus, poverty associated with the vector x be lower than that associated with the vector y whenever x and y are equi-dimensional and the subvector of poor incomes in x dominates the subvector of poor incomes in y; transfer (Axiom T) requires the value of poverty to decline whenever, other things equal, there is a rank-preserving transfer of income from a poor person to a poorer person; decomposability (Axiom D) requires the poverty index to be expressible as a population-share weighted average of subgroup poverty levels; subgroup sensitivity (Axiom SS; see Jayaraj and Subramanian 1999; Subramanian 2006) requires that, other things equal, poverty should decline whenever a pure redistribution of incomes between two groups causes the relatively disadvantaged group to become less poor

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9 Reckoning Inter-group Poverty Differentials in the Measurement …

and the relatively advantaged group to become poorer while maintaining the relative poverty rankings of the two groups. A poverty index P is said to be degenerate with respect to grouping (or just degenerate) if, for every z ∈ R++ , every x ∈ X, and all distinct g, g ∈ G such that (x, g) and (x, g ) are compatible pairs; it is the case that P(z, x, g) = P(z, x, g ); that is, the value of the poverty index is invariant with respect to the grouping employed. For future reference, we shall denote by  the set of poverty indices which are symmetric, weakly monotonic, decomposable, normalized to lie in the interval [0,1], and degenerate.

9.4 Poverty Aggregation When Inter-group Inequality is Intrinsically Dis-Valued 9.4.1 Properties of Poverty Measures When the Distribution of Subgroup Poverty Matters It can be shown that a poverty index which is required to simultaneously satisfy the sets of requirements constituted by {symmetry, monotonicity, subgroup sensitivity}, or {respect for income dominance, transfer, subgroup sensitivity}, could run into existence problems, on which see Subramanian 2006 (Propositions 4.1 and 4.2). In assessing these impossibility results, the author points to the problematic outcome of insisting on the universal validity of certain axioms irrespective of the context in which they are invoked. A reasonable way out might be to restrict the applicability of a given axiom to a domain that is relatively non-controversial. Thus, in the case of the symmetry axiom, one may require its brief to extend only to the extent that within any subgroup, swapping incomes across members of the subgroup should not alter the level of subgroup poverty, and that in a between-group context, the level of aggregate poverty should not vary with the way in which the subgroups are labelled. Similarly, with the transfer axiom, its sway could be limited to interpersonal income redistributions within a subgroup, while in a between-group context one could require that for a given level of poverty averaged across the subgroups, aggregate poverty should increase with an increase in inequality in the inter-group distribution of poverty. Poverty indices which satisfy these restricted versions of symmetry and transfer also possess the virtue of a certain sort of ‘flexibility’, in terms of which, for example, an interpersonal transfer across members of different subgroups which reduces inter-group disparity in the distribution of poverty may or may not reduce aggregate poverty, depending on how regressive the transfer is: a more rather than less regressive transfer could be partial to the transfer axiom at the expense of the subgroup sensitivity axiom and the other way around with a less rather than more regressive transfer. Examples of such poverty indices are available in a poverty-related version of the Anand–Sen ‘gender-adjusted human development index’, and in the measure

9.4 Poverty Aggregation When Inter-group Inequality …

103

advanced by Jayaraj and Subramanian (1999), and discussed in Subramanian and Majumdar (2002). These poverty measures are presented below.

9.4.2 Two Poverty Measures Which Are Sensitive to the Inter-group Distribution of Poverty First, let (x, g) be a compatible pair belonging to X × G, and let z be any poverty line. Let π be any poverty index belonging to the set  of measures which are symmetric, weakly monotonic, decomposable, normalized to lie in the interval [0,1], and degenerate. We shall assume that g partitions the population into K exclusive and exhaustive subgroups. π will stand for the non-increasingly ordered vector of subgroup poverty levels (π1 , . . . , π j , . . . , π K ), viz. π is ordered such that π j ≥ π j+1 for all j = 1,…,K − 1. We let t j stand for the population share of the jth poorest j group, and T j ≡ k=1 tk for the cumulative proportion of the population belonging to groups whose poverty levels are greater than or equal to that of the jth group. We shall take it that T0 ≡ 0. Recall that since π is a decomposable index, it can  be written as a population-share weighted sum of subgroup poverty levels: π ≡ Kj=1 t j π j . If all groups are of the same size, then it is easy to see that all subgroup poverty levels (the π j ) are accorded the same weight (1/K) by the poverty index π . To secure an Egalitarian tilt, one could think of a system of weights in which a higher weight is accorded to a subgroup with greater poverty. This is the basic idea underlying the construction of ‘group-inequality-sensitive’ indices of aggregate poverty. A specific version of the Anand–Sen index (suitably adapted to the present context) is the index P A , and the poverty measure advanced in Subramanian and Majumdar (2002) is the index P B , which—given z, x, and g—can be written as follows: P A (z, x, g) =

K 

[t j π 2j ]1/2 ;

(9.1)

j=1

and  P (z, x, g) = B

Let

C∗

≡

 K 1 [(K − 1 − j)t j + 1 − T j−1 ]π j K − 1 j=1

[(1/π 2 )

K  j=1

t j π 2j ] − 1

be

the

squared

(9.2)

coefficient

of

variation of the π j , and I ∗ ≡   in the inter-group distribution K 1 j=1 (K + 1 − j)t j + 1 − T j−1 π j − 1 be an alternative measure π(K +1) of inequality for this distribution (introduced and discussed in Subramanian and Majumdar 2002). Subramanian and Majumdar (2002; p. 268) point out that

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‘[b]ecause of the rank-order weighting formula employed in [the expression for I*], one can discern that [I*] bears a close family resemblance to the familiar Gini coefficient of inequality…’. Elsewhere in the same paper (p. 274), they also say that ‘…[I*] is the [group] analogue of the Gini coefficient of interpersonal inequality’. This latter statement is somewhat misleading: the group version of the interpersonal Gini coefficient—call it G*—is, as we shall see later in this paper, different from the index I*, although under an atomistic grouping of the population, both I* and G* will simply be the standard Gini coefficient of inequality in the interpersonal distribution of the variable under analysis. Given (9.1) and (9.2), and the expressions for C* and I*, respectively, it follows that P A = π(1 + C ∗ )1/2 ;

(9.1 )

  

K +1 ∗ I P =π 1+ K −1

(9.2 )

Lim K →∞ P B = π(1 + I ∗ )

(9.2 )

and B

Note also that

The indices P A and P B are essentially the average level of poverty (π ) enhanced by a factor incorporating the extent of inequality in the distribution of the groupspecific levels of poverty: in this sense, their construction is a reflection of an intrinsic preference for inter-group equality. As we have already seen, concerns for group identity and inter-group (as distinct from interpersonal) equality could sit uneasily with the requirements of canonical axioms like symmetry and transfer. These issues have been discussed in some detail in Subramanian (2006) and will therefore not be reviewed here. Rather, an alternative way of deriving an index which is closely related to the index P B will be discussed. This entails the use of a graphical device called the group poverty profile, in presenting which I shall very closely follow Shorrocks’ (1995, 1996) construction of the poverty gap profile. (See also, in this connection, Jenkins and Lambert 1997).

9.4.3 The Group Poverty Profile and the Group Poverty Lorenz Profile Given the ordered vector π of subgroup poverty levels (π1 , . . . , π j , . . . , π K ), the group poverty profile is obtained as a plot of the points {(T j , D j )} j∈{0,1,...,K } , where T0 = D0 ≡ 0 and, for every j = 1,…,K:

9.4 Poverty Aggregation When Inter-group Inequality …

D j (π ; T j ) =

j 

tk πk

105

(9.3)

k=1

The GPP is thus constructed by first arranging the subgroups in the order of poorest to least poor in terms of the values of the π j ; the population-share weighted poverty levels are then cumulated across the subgroups and plotted against the cumulative population shares of the subgroups; the graph obtained by connecting the plotted points by straight lines yields a ‘piece-wise linear’ version of the GPP. Since the maximum value the π j can take is unity, it is clear that the ‘worst-case picture’ is obtained when π j = 1 for all j, in which situation the GPP will be the diagonal of the unit square, and can be called the ‘line of maximal poverty’ (in all of which we are, with appropriate adaptation to the context, closely following Shorrocks 1996). In general, it can be seen that the GPP can be drawn as a non-decreasing, concave curve which lies beneath the diagonal of the unit square. The highest point on the K tk πk , or π , recalling GPP is obviously its final point (T K , DK ) whose height is k=1 that π is a decomposable index. For illustrative purposes, a typical GPP is drawn in Fig. 9.1, where K is taken, for specificity, to be 4. Notice from Fig. 9.1 that if there were no inequality in the inter-group distribution of poverty levels, that is, if it were the case that π j = π for all j, then the GPP would be the straight broken line connecting the points 0 and π in the figure. The actual GPP lies above the broken line, and it is natural to attribute the space enclosed by the two curves to the fact of an unequal distribution of subgroup poverty levels. Indeed,

Fig. 9.1 A group poverty profile

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9 Reckoning Inter-group Poverty Differentials in the Measurement …

an inverted image of the broken line and the GPP suggests something like a Lorenz curve drawn beneath the line of equality, and it is to a construction, precisely, of the group poverty Lorenz profile (GPLP) that we now turn. The GPLP is obtained from plotting the points {(1 − TK − j , L j )} j∈{0,1,...,K } , where T0 = L 0 ≡ 0 and, for every j = 1,…,K: L j (π ; 1 − TK − j ) = (1/π )

K 

tk πk .

(9.4)

k=K − j+1

The graph is drawn by first ranking the subgroups in non-decreasing order of their poverty levels, plotting the cumulative subgroup shares in total poverty against their cumulative population shares, and connecting the plotted points by straight lines to yield a ‘piece-wise linear’ GPLP. Figure 9.2 features a typical GPLP drawn within the unit square, for the special case in which K = 4. Continuing to follow the lead afforded in Shorrocks (1996), one can see that the group poverty profile and the group poverty Lorenz profile are linked by the following relationship: D j (π; T j ) = π [1 − L K − j (π ; 1 − T j )], j = 0, 1, . . . , K .

(9.5)

From (9.3), (9.4), and (9.5), one can see that the lower the GPP or the higher the GPLP is, the further away will the GPP be from the line of maximal poverty, and so

Fig. 9.2 A group poverty Lorenz profile

9.4 Poverty Aggregation When Inter-group Inequality …

107

the ‘better’ may we judge the overall poverty situation to be. This paves the way for comparing alternative group poverty distributions in terms of a poverty dominance relationship based on the GPP. Specifically, for any given poverty line and grouping which partitions the population into K subgroups, we let π (as before) stand for the non-increasingly ordered vector of subgroup poverty levels (π1 , . . . , π K ) and t = (t1 , . . . , t K ) for the corresponding vector of subgroup population shares. Given π and t, a poverty situation s is defined as a K-tuple of pairs of subgroup poverty level and subgroup population share: s = ((π1 , t1 ), . . . , (π K , t K )). For any two poverty situations s1 = ((π1,1 t11 ), . . . , (π K1 , t K1 )) and s2 = ((π1,2 t12 ), . . . , (π K2 , t K2 )), we shall say that s1 poverty-wise dominates s2 , written s1  P s2 , whenever it is the case that the GPP for s1 lies somewhere below and nowhere above the GPP for s2 . (Of course, when subgroup sizes are different in the two poverty situations under comparison, the contributions of population size variations and subgroup poverty variations to the difference in aggregate poverty between the two situations would need to be unscrambled.) In general, the GPP (much like the Lorenz curve employed in standard distributional analysis) is a useful visual aid to group-related poverty analysis and leads naturally to the construction of a strict partial ordering of poverty, namely the binary relation  P . It could find particularly fruitful application in the overtime assessment of, say, the spatial distribution of poverty.

9.4.4 Aggregate Poverty Adjusted for Its Unequal Inter-group Distribution The GPP also paves the way for deriving a measure of aggregate poverty, in much the same way as the Lorenz curve paves the way for deriving the Gini coefficient of inequality. Specifically—and still closely tracking Shorrocks (1995, 1996)—it appears to be natural to obtain a normalized measure of poverty—call it PC —in terms of the area beneath the GPP expressed as a proportion of the area beneath the line of maximal poverty (see Fig. 9.1 again). The area beneath the line of maximal poverty (call this area A) is just one-half, while the area beneath the GPP (call this area B), obtained as a sum of the areas of a number can—with some manip  of trapezoids,   ulation—be seen to be given by: Area B = 2π + Kj=1 t 2j π j − 2 Kj=1 t j T j π j /2. Then, the poverty index PC (=Area B/Area A) will be given by P = 2π + C

K  j=1

t 2j π j

−2

K 

t j Tj π j .

(9.6)

j=1

Turning next to Fig. 9.2, and noting that the Gini coefficient of inequality G* in the distribution of subgroup poverty levels is just one minus twice the area beneath the GPLP, it is easy to verify that

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9 Reckoning Inter-group Poverty Differentials in the Measurement …

G∗ = 1 +

 K j=1

t 2j π j − 2

K j=1

 t j T j π j /π

(9.7)

Making the appropriate substitution from (9.7) into (9.6) yields: P C = π(1 + G ∗ ).

(9.8)

Comparing (9.8) with (9.2 ) and (9.1 ) indicates that PA , PB , and PC are all poverty measures of a type, where aggregate poverty is expressed as the level of poverty averaged across subgroups and then enhanced by a factor that captures the extent of inequality in the inter-group distribution of poverty. Suppose the grouping of the population we resorted to was the ‘atomistic grouping’, in terms of which each person is regarded as constituting a group by herself/himself, so that a typical group—{i}—is the one constituted by person i, for every i ∈ M. Suppose further that x i is the income of the ith poorest person and that πi is the poverty level of this person, which is defined by: πi = max[0, 1− xi /z], that is, a person’s deprivation is measured by the proportionate shortfall of her income from the poverty line if she happens to be poor, and is taken to be zero otherwise. Under these circumstances, the GPP will be precisely what Shorrocks (1995, 1996) calls the poverty gap profile, and the index of aggregate poverty corresponding to PA in (9.7) will be an index  which is given by  = H I (1 + G P ), where H is the familiar headcount ratio (or proportion of the population in poverty), I is the income-gap ratio (or proportionate shortfall of the average income of the poor from the poverty line), and GP is the Gini coefficient of inequality in the distribution of the poverty gap values. , as it happens, is precisely the poverty index derived by Shorrocks (1996), which he shows to be closely related to Sen’s (1976) poverty index, and even more proximately so to Thon’s (1979) index, of which  is the replication-invariant ‘asymptotic’ version. Of the index , Shorrocks (1996; p. 251) says: ‘The index obtained by taking the area below the poverty gap profile relative to that below the “line of maximum poverty” has a particularly appealing interpretation. Although not decomposable across population subgroups, it is an ideal poverty index in all other respects’. That there is a way of reckoning subgroup poverty in the measurement of aggregate poverty which is, substantially, a generalization of a particularly instructive approach to poverty measurement as advanced by Shorrocks is, I believe, both interesting and useful. Finally, it is straightforward to note that when a population is regarded as being entirely homogeneous, the appropriate grouping which reflects this judgement is the ‘universal grouping’ g u , in which case the value of P A will be just π .

9.5 Concluding Observations

109

9.5 Concluding Observations Horizontal, or inter-group, inequalities in the distribution of poverty are a standard feature of many stratified societies. Intrinsic considerations of group fairness would dictate a concern for incorporating inequalities in the group-wise distribution of poverty directly into the aggregation exercise of poverty measurement. Extant approaches to such aggregation procedures have been reviewed in this paper, and in the process, a generalization of Shorrocks’ illuminating approach to poverty measurement, involving the derivation of ‘deprivation indices’ from ‘deprivation profiles’, has been provided. The standardly ‘atomistic’ formulation of measurement concerns has been shown to emerge as a special case of a more general framework of group-inclusive analysis. Three specific products of such a ‘group-inclusive’ analysis have been: (a) the poverty index P C which ‘adjusts’ the average level of poverty in a heterogeneous community for the extent of inter-group inequality in the distribution of poverty; (b) the notion of a ‘group poverty profile’; and (c) the notion of a ‘group poverty Lorenz profile’, the latter two of which, together, facilitate ‘dominance comparisons’ of alternative regimes of the inter-group distribution of aggregate poverty. It is hoped that these ideas can find useful application in poverty measurement exercises which are informed by a concern for equality across groups, in addition to equality across individuals.

References Amiel Y, Cowell F (1994) Monotonicity, dominance and the Pareto principle. Econ Lett 45:447–450 Anand S, Sen, A (1995) Gender inequality in human development: theories and measurement. In: Occasional paper, 19, Human Development Report Office. UNDP, New York Cited in UNDP Jayaraj D, Subramanian, S (1999) Poverty and discrimination: measurement, and evidence from Rural India. In: Harriss-White B, Subramanian S (eds) Illfare in India: essays on India’s social sector in Honour of S. Guhan. Sage Publishers, Delhi Jenkins SP, Lambert PJ (1997) Three I’s of poverty curves, with an analysis of UK poverty trends. Oxford Econ Papers 49: 317–327. Kakwani NC (1993) Performance in living standard: an international comparison. J Develop Econ 41(2):307–336 Majumdar M, Subramanian S (2001) Capability failure and group disparities: some evidence from India for the 1980s. J Develop Stud 37(5):104–140 Sen AK (1976) Poverty: an ordinal approach to measurement. Econometrica 44(2):219–231 Shorrocks AF (1995) Revisiting the Sen poverty index. Econometrica 63(5):1225–1230 Shorrocks AF (1996) Deprivation profiles and deprivation indices. In: Jenkins S, Kapteyn A, Vaan Praag B (eds) The distribution of welfare and household production: international perspectives. Cambridge University Press, London Subramanian S (2006a) Social groups and economic poverty: a problem in measurement. In: McGillivray M (ed) Inequality, poverty, and wellbeing. Palgrave Macmillan, London Subramanian S, Majumdar M (2002) On measuring deprivation adjusted for group disparities. Soc Choice Welfare 19(2):265–280 United Nations Development Programme (UNDP) (1995) Human development report 1995. Oxford University Press, New York

Chapter 10

Poverty Measurement in the Presence of a ‘Group-Affiliation’ Externality

Abstract This paper considers the implications for poverty measurement of the observed fact that any individual’s level of deprivation is a function not only of his own income, but of the general level of prosperity of the group to which he is affiliated. Individual deprivation functions are specialized to a form which reflects this ‘groupaffiliation’ externality, and the resulting poverty measure is studied with respect to its properties and its implications for poverty rankings. Mainstream approaches to measuring deprivation tend to neglect group-related externalities in favour of a certain thorough-going ‘individualism’. This paper is a preliminary attempt at filling this gap. Keywords Groups · Group affiliation · Externality · Poverty axioms

10.1 Introduction Social reality is mediated by two widely recognized facts: (a) that populations are seldom homogeneous; and (b) that deprivation at an individual level is a function of the deprivation status of the group to which the individual is affiliated. An acknowledgement of the first fact is compatible with measuring aggregate poverty in a society in such a way that inequalities in the inter-group distribution of poverty are seen as enhancing poverty and are directly incorporated into the aggregate measure so as to reflect this judgement (see, for example, Subramanian 2006b). The second fact could pave the way for measuring poverty in such a way that individual deprivation is seen This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2009) Poverty Measurement in the Presence of a ‘Group-Affiliation’ Externality, Journal of Human Development and Capabilities, 10:1, 63–76,https://doi.org/10.1080/14649880802675168. The author is grateful to two anonymous referees of the Journal of Human Development for constructive suggestions, to Professor Prasanta Pattanaik for a very helpful discussion of the subject, to D. Jayaraj for earlier collaborative work on related themes, and to Subroto Mukherjee for comments on an earlier version of the chapter. This chapter is based on a part of a longer one titled ‘Reckoning Inter-Group Poverty Differentials in the Measurement of Aggregate Poverty’, held at Brasilia (Brazil) in August 2005. The usual caveat applies. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_10

111

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to depend on group deprivation, so that ‘group-affiliation externalities’ are explicitly reckoned in the assessment of overall poverty. In either case, it turns out that inequality in the distribution of poverty across groups is an undesirable outcome— for intrinsic reasons of valuing equality as a social virtue, in the first case, and for instrumental reasons arising from the link between externality and equity, in the second case. In the present paper, the ‘instrumental’ case for reckoning inter-group differentials in the measurement of aggregate poverty is investigated. The paper explores a simple idea on the connection between individual and group poverty and examines some straightforward implications of the idea for the measurement of aggregate poverty. The idea in question is this: in principle, how poor a person is, is dependent not only on how deprived s/he is, considered as an atomistic unit, but also on the general level of prosperity of the group to which s/he is affiliated. A good deal of the literature on poverty measurement rests on the view of man as economic man, i.e. as an entity wrenched loose from his group moorings. This paper considers some consequences of taking a view of man as social animal, i.e. as an entity whose fortunes are tied, in smaller or larger measure, to the fact of his particular group affiliation. Within this setting, certain conventional approaches to, and outcomes of, poverty measurement can be seen to emerge as special cases of a more general framework that allows for intra-group externality in the determination of individual deprivation. The issue of how outcomes at the individual level are influenced by the fact of group affiliation is well brought out, at a general level, in Loury (2000; pp. 233, 243): Economic analysis begins with a depersonalized agent who acts more or less independently to make the best of the opportunities at hand… [But] [e]ach individual is socially situated, and one’s location within the network of social affiliation substantially affects one’s access to various resources… There is one view of society in which we are atomistic individuals, pursuing our paths to the best of our abilities… But this is a false, or at least incomplete, view of how society works. The fact is that we are all embedded in a complex web of associations, networks, and contacts. We live in families, we belong to communities, and we are members of collectivities of one kind or another. We are influenced by these associations from the day we are born. Our development – what and who we are and become – is nourished by these associations.

The thrust of the preceding considerations, when applied to the context of poverty, resolves itself into a straightforward principle which constitutes the underlying motivation for the present paper and can be stated in terms of the following axiom: Axiom of Group-Mediated Deprivation (Axiom GMD). Ceteris paribus, of two poor persons having the same income, the person who belongs to the group with a lower mean income is more deprived. Before proceeding further, it may be useful to clarify the intended meaning of Axiom GMD. Imagine two situations 1 and 2, in both of which we have four individuals. In situation 1, the four individuals are labelled a, b, c, and d, respectively, while in situation 2, the individuals are labelled r, s, t, and u, respectively. The incomes of a, b, c, and d are, respectively, 20, 30, 40, and 50, while the incomes of r, s, t, and u are, respectively, again 20, 30, 40, and 50. The poverty line in both situations is identical, at 45. It will be assumed that, in situation 1, individuals a, b, and c are members of

10.1 Introduction

113

a ‘backward caste’ (BC) group, while individual d is a member of a ‘forward caste’ (FC) group. In situation 2, persons r and s are members of the BC group, while persons t and u are members of the FC group. Consider, for specificity, individuals c and t. Both have the same income (40). In much of standard poverty analysis, the extent of deprivation of individuals c and t would be judged to be identical. However, Axiom GMD states that person c is more deprived than person t, because c (a backward caste person) belongs to a group with a lower average income (of 30), then t (a forward caste person), who belongs to a group with an average income of 45. A referee of this paper has objected that in this example, the ceteris paribus clause in the statement of Axiom GMD has been violated, because the subgroup means in situations 1 and 2 are not the same. But then, it is precisely this difference in subgroup means which—quite explicitly—drives Axiom GMD! The difference in subgroup means, that is, is not some independent and autonomous source of dissimilarity between situations 1 and 2, but rather simply one particular consequentially constitutive aspect of the recognition that differences in group affiliation do matter in any overall assessment of deprivation. The axiom underlines the point that when other things which are routinely considered to be ‘relevant’ are held constant, the fact of differential group affiliation among individuals does make a difference to the assessment of individual deprivation. Specifically, in much of mainstream analysis, differences in subgroup means are not acknowledged as differences which are in any way any more relevant to a comparative assessment of overall poverty than differences, say, in the shapes of the noses of the individuals involved in the comparisons. Thus, for example, while the Foster– Greer–Thorbecke family of poverty indices (Foster et al. 1984) is explicitly concerned with issues of subgroup monotonicity and subgroup decomposability, these poverty indices would judge the extent of measured poverty in situations 1 and 2 to be identical. The axiom of group-mediated deprivation marks a small and obvious, but nevertheless significant, departure from this ‘standard’ mode of assessing aggregate poverty. The idea underlying Axiom GMD is that a person’s ability to transform income into what Sen (1985) calls functionings is linked to the fortunes of the group to which the person is affiliated. This type of intra-group externality could occur in a number of ways. It is often the case that the extent to which a poor individual is a beneficiary of ‘informal social security’, or of credit facilities, or of supplementary nutrition, or of assistance in the event of sickness or disability, or of scholarships in educational institutions, is an increasing function of the average level of income sufficiency of the group to which the person is affiliated. If a poor person’s access to resources varies directly with the respect and consideration with which s/he is treated by society at large, then such respect and consideration, and therefore personal access to resources, might be expected to be positively correlated with the mean prosperity of the group to which the person belongs. Moreover, a person’s own sense of advantage is often mediated by the economic status of the group of which s/he is a member: even in the absence of differentials in income at the interpersonal level, one can expect interpersonal differentials in the level of achieved functionings arising from groupmediated individual assertiveness, or diffidence, as the case may be. It is important

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10 Poverty Measurement in the Presence …

to note that these sorts of externalities arising from group affiliation are tangible externalities, and not merely ‘psychic’ ones, as reflected in ‘feel good’ or ‘feel bad’ mental affects triggered by considerations of group identity. The failure to see any merit in Axiom GMD (or some appropriate variant of it) is reflected in the arguments (such as they are) that have been advanced by large sections of the upper caste elite in the—often violent—debates which have attended the issue of caste-based reservations in education and employment in India. An extreme version of the forward caste position (which is much more the norm than the exception in this constituency) asserts that reservation on the basis of caste ought to be replaced by reservation according to some purely economic criterion. In this view, there is nothing to distinguish a poor Brahmin from a poor scheduled caste person. This amounts to a comprehensive denial of any form of group-mediated deprivation: in particular, there is no concession to the possibility that a poor Brahmin may be less deprived than a poor scheduled caste person simply because the former belongs to a group that has had millennia of superior advantage in terms of social, educational, and economic status. It is this sort of blunt insensitivity to the link between individual deprivation and group affiliation which Axiom GMD seeks to rectify (in however rudimentary a fashion). In much of what follows, illustrative operational content is given to Axiom GMD. The idea is not to claim any particular social realism for the specific formulations pressed into service; rather, it is to draw out certain elementary conceptual implications of the axiom for poverty comparisons and poverty axiomatics.

10.2 Concepts and Definitions Let M be the set of positive integers, R the real line, and R++ the positive real line. For every n ∈ M, let Xn be the set of non-negative n-vectors x = (x 1 ,…,x i ,…,x n ), where x i is the income of individual i in a community of n individuals. Define the set X ≡ ∪ n∈M Xn . Let the poverty line be denoted by z, where z is a positive and finite income level such that all persons with income less than z are identified as being poor. For all x ∈ X, N(x) will stand for the set of all individuals whose incomes are represented in the income distribution x, and Q(x) will stand for the set of poor individuals. Next—and drawing on Jayaraj and Subramanian (1999) and Subramanian (2006b)—for every n ∈ M, let Gn be the set of all possible partitions of the set N = {1,…,n}, and define the set G ≡ ∪ n∈M Gn . Every g ∈ G is some partition of the population, induced by some appropriate grouping, for example, on the basis of race, caste, gender, etc.; the elements of g—denoted by the running index j—will be taken to be subgroups of the population. Clearly, for every g ∈ G and n ∈ M, it will be the case that 1 ≤ #g ≤ n. Two polar cases of grouping are the atomistic grouping ga which induces the finest partition {{1},…,{n}} of {1,…,n} and the universal grouping gu which induces the grossest partition {{1,…,n}} of {1,…,n}. For all (x, g) ∈ XxG, the pair (x, g) will be said to be a compatible pair if and only if g partitions a population whose size is the same as the dimensionality of x. Given

10.2 Concepts and Definitions

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any compatible pair (x, g) ∈ XxG, xj will stand for subgroup j’s income vector, and μj ≡ μ(xj ) will stand for the mean income of subgroup j, for every j ∈ g. Further, for all compatible (x,g) ∈ XxG, xi will stand for the income vector of the subgroup of g to which person i belongs, and μi ≡ μ(xi ) will denote the average income of the subgroup to which i belongs, for every i ∈ N(x). We now define a poverty index formally. A poverty index is a mapping P: R++ xXxG → R such that, for every z ∈ R++ , and every compatible pair (x, g) ∈ XxG, P(z, x, g) specifies a real number which is intended to represent the extent of poverty associated with the regime (z, x, g). Certain standard axioms invoked in the measurement of poverty are now swiftly and informally reviewed. Focus (Axiom F) requires the poverty index to be invariant with respect to increases in non-poor incomes; continuity (Axiom C) requires the poverty index to be continuous on Xn for every n ∈ M; normalization (axiom N) requires the poverty index to attain a lower bound of zero when there is no poor person in the community; symmetry (Axiom S) requires the poverty index to be invariant to any interpersonal permutation of incomes; monotonicity (Axiom M) requires poverty to increase with a decline in any poor person’s income; transfer (Axiom T ) requires that the poverty index registers a decline in value whenever, other things equal, there is a rank-preserving transfer of income from a poor person to a poorer person; and decomposability (Axiom D) requires the poverty index to be expressible as a population-share weighted average of subgroup poverty levels.

10.3 An ‘Externality-Adjusted’ Poverty Measure 10.3.1 Accommodating Group-Affiliation Externalities: One Specific Approach Consider any (z, x, g) ∈ R++ xXxG, where (x, g) is a compatible pair. For any person i, let d i represent i’s deprivation function. It is reasonable to suggest that, for all i ∈ {1,…,n}, d i = 0 if x i ≥ z; and d i = d i (x i ,μi ) > 0 if x i < z, with d i declining in each of x i and μi . That is, a person whose income is not lower than the poverty line is taken to suffer no deprivation at all; for a person with a below-poverty line income, her deprivation is taken to be a diminishing function of both her own income and the average income of the group to which she is affiliated—and this captures the ‘group externality effect’ postulated by the axiom of group-mediated deprivation. Additionally, and with an eye to normalization, we may suppose that d i (0, 0) = 1: the poverty of a completely poor person belonging to a completely poor group is unity. A functional form for d i which satisfies these requirements is given by di (xi , μi ) = max[(z − xi )/(z + μi ), 0].

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The particular specialization resorted to above is not sought to be uniquely characterized: it is employed only as a convenient illustrative device. I shall now define an ‘externality-adjusted’ poverty index, P*, as a simple average of the individual-specific deprivation functions: P ∗ (z, x, g) [ = (1/n)



di (xi , μi )] = (1/n)

i∈N



[(z − xi )/(z + μi )]

(10.1)

i∈Q

where N is the set of all individuals and Q the set of poor individuals. (10.1). can also be written as ⎤ ⎡   ⎣ {(z − xi )/(z + μ j )}⎦ P ∗ (z, x, g) = (1/n) j∈g

(10.2)

i∈Q j

where Q j is the set of poor persons belonging to subgroup j. Notice now that if is a shorthand for P*(z, xj , gu ), viz. if P j∗ is the poverty level of the jth subgroup when the partitioning of the subgroup’s population is the universal one, then P j∗

P j∗ = (1/n j )



{(z − xi )/(z + μ j )},

(10.3)

i∈Q j

where nj is the dimensionality of xj ; whence—in view of (10.2): P ∗ (z, x, g) = (1/n)

 j∈g

n j P j∗ =



φ j P j∗ ,

(10.4)

j∈g

where φ j ≡ nj /n is the population share of the jth subgroup. The two polar cases of grouping we have considered earlier are of special interest. When the grouping is atomistic, that is, each person is considered to constitute a group by himself or herself, then clearly μi = x i for all i, whence, in view of (10.1), we have    [(z − xi )/(z + xi )]. P ∗ z, x, ga = (1/n)

(10.5)

i∈Q

Equation (10.5) is the sort of ‘individualistic’ poverty index most widely employed in the poverty measurement literature: it turns out to be a special case of the more general formulation (10.1). A matter of some interest is that this poverty index can be independently derived as a certain kind of normalized distance function, in the following sense. Given a non-decreasingly ordered income vector x = (x 1 ,…,x q , x q+1 ,…, x n ), where x q < z and x q+1 ≥ z, define the censored version xc of x as the vector obtained by replacing all the non-poor incomes in x by the poverty line income z (see Takayama 1979), so that xc = (x 1 ,…,x q ,z,…,z). Let z be the n-vector

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(z,…,z) and 0 the n-vector (0,…,0). Then—see Subramanian (2006a)—z can be interpreted as the income distribution with the smallest mean which is compatible with a complete absence of poverty, and 0 as the income distribution representing total poverty. Let δ(z, xc ) represent the ‘vector distance’ between z and xc (the shortfall of the ‘actual’ situation from the ‘no-poverty’ situation), and δ(z,0) represent the vector distance between z and 0 (the shortfall of the ‘complete poverty’ situation from the ‘no-poverty’ situation). Then, in some intuitively straightforward sense, we could take the ratio of the two distances, call it rc ≡ δc(z, xc )/δc(z, 0), to be a normalized measure of poverty. It remains to specify the form of the distance function δ. One candidate is the Camberra distance function δ C (see Wilson and Martinez 1997) nwhich calculates the distance between any two n-vectors a and b as [|ai − bi |/(ai + bi )]. This leads to δ C (a, b) = i=1   r C ≡ δ C z, xc /δ C (z, 0) ⎤  ⎡ n    {(z − xi )/(z + xi )}⎦ (z/z)] = (1/n) [(z − xi )/(z + xi )]. =⎣ i∈Q

i=1

i∈Q

(10.6) From (10.5) and (10.6), we note that the measure P*(z, x, ga ) is just the normalized Camberra distance ratio r C —which confers a simple and handy interpretation on the poverty index. (Of course, this ‘distance function’ interpretation holds only for the atomistic partitioning of the population.) At the other polar extreme, when g = gu , it can be seen from (10.1) that    [(z − xi )/(z + μ)], P ∗ z, x, gu = (1/n)

(10.7)

i∈Q

where μ is the mean of the distribution x. This corresponds to the case where there is only one group—that constituted by the grand coalition of individuals: the community, here, is regarded as being entirely homogeneous. (10.5) and (10.7), it may be reiterated, are special cases of (10.1).

10.3.2 Partitioning the Population on the Basis of Multiple Identities An issue of some significance relates to the basis on which the population is to be partitioned into subgroups. One of the referees of this paper has raised an interesting query regarding the ability of the poverty index P* to accommodate multiple group identities for any given individual. The specific comment is: ‘Individuals are restricted to being members of one unique group. However, we know that in reality, one can typically be associated with multiple groups’. This is not only true but also

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relevant for the sorts of group-based analysis that one may typically be interested in, and so it is important to explicitly address the issue of how multiple identities may be reflected in the grouping one resorts to. One fairly straightforward and simple way of addressing the problem is as follows. In poverty analysis, it appears to be meaningful to partition the population on the basis of multiple categories such as, for example, gender, caste, and sector of residence. A person can be a man or a woman, of ‘backward’ or of ‘forward caste’ affiliation, from a rural area or an urban area. Consider the sets A1 ≡ {female, male}, A2 ≡ {backward caste, forward caste}, and A3 ≡ {rural, urban}. If A is the Cartesian product of the sets A1 , A2 , and A3 , then A = {(female, backward caste, rural), (female, backward caste, urban), (female, forward caste, rural), (female, forward caste, urban), (male, backward caste, rural), (male, backward caste, urban), (male, forward caste, rural), (male, forward caste, urban)}. The set A consists of eight population groupings on the basis of multiple identities for each individual comprehending the individual’s gender, caste, and sector of residence. Among these eight subgroups of the population, we have four pairs of gender-variants (that is, four pairs of subgroups which differ, in each pair, only with respect to the constituents’ gender), four pairs of caste-variants, and four pairs of sector-of-residence-variants. If in every pair of gender-variants we find that the group consisting of females is more impoverished than the group consisting of males, then this is a strong pointer to uniform gender disadvantage arising from being a woman. Similarly with the other attributes. This method of accommodating multiple individual identities has been employed by Majumdar and Subramanian (2002) in studying group-related disparities in the distribution of well-being in India. In general, suppose there are H categories A1 ,…,AH in terms of which the population can be partitioned. (The categories, as just discussed, can be in terms of gender, caste, etc.). Suppose, further, that the kth category Ak possesses mk attributes. (e.g. if the kth category is ‘caste’, its attributes could be ‘scheduled caste’, ‘backward caste’, and ‘forward caste’). The set of possible subgroups will then be given by the Cartesian product A1 × A2 ×· · · ×AH , and the resulting number of subgroups, S, will be m1 × m2 ×· · · × mH . This formulation alerts us to a potential paradox. If, from an excessive zeal for capturing the social reality of multiple identities for a person—which corresponds to what is sometimes called the exercise of determining ‘subject identity’—we were to keep increasing the number of categories H and also the number of attributes within each category, then #S could well converge on n, the number of all persons in the society. In such a situation, we might well wind up with the ‘atomistic’ partitioning of the population, though the whole idea, in the first place, was to effect some departure from the ‘individualistic’ model of social assessment! Between the extreme assumptions of complete homogeneity and complete heterogeneity is a middle ground that is informed by an awareness of what constitutes a socially and politically salient partitioning of the population into a finite set of contextually meaningful subgroups. There is a certain similarity between the exercise of specifying a poverty line and that of specifying a partitioning of a population appropriate to poverty analysis: in both exercises, there is a case for avoiding

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the absurdities of postulating extremes in either direction and for being guided by a ‘feel’ for the socio-economic reality of the situation that is being investigated.

10.3.3 An Application to a Non-income Dimension: Measuring Literacy Basu and Foster (1998) have shown how the conventional headcount measure of literacy could be a misleading indicator of a population’s ‘effective’ literacy status, simply because of the failure of this measure to reckon the intra-household externality accruing to illiterate members of a household from the literacy of their literate cohorts. The externality-related approach to measuring income poverty outlined in this paper has an intimate implication for the Basu–Foster thesis, which has been dealt with in Subramanian (2007), and is briefly sketched in what follows. A ‘literacy distribution’ x can be taken to be a vector detailing each individual’s literacy status, and each element xi of the vector is a member of the set {0,1}, depending on whether the individual in question is illiterate or literate. Corresponding to a poverty line of z in income space, one could postulate a ‘poverty line’ of unity in literacy space: a person is literacy-wise impoverished if her literacy status is less than one (or, equivalently, if her literacy status is zero, since this status is a binary-valued variable). Suppose there are K households in the population and that each household j is treated as a separate ‘group’. Let R i stand for the proportion of literates in the household to which person i belongs, and let R j denote the proportion of literates in the jth household. Now hark back to Eqs. (10.1), (10.2), (10.3), and (10.4). If φ j is the population share of the jth household, if we replace z by unity, if we take xi to be zero for an illiterate person, if we replace μi by R i , and if we replace μ j by R j , then, a little bit of routine manipulation will yield a society-wide ‘effective’, or ‘externality-adjusted’, headcount ratio of illiteracy—analogously to the expression for P∗ in (10.4)—given by I∗ =

K 

φ j [(1 − R j )/(1+R j )].

(10.8)

j=1

I ∗ in (10.8) is, precisely, the illiteracy measure H* advanced and discussed in Subramanian (2007). From (10.8), it is easy to infer that, for any given number of literates in a society, the ‘externality-adjusted’ illiteracy rate I ∗ is minimized through an equal inter-household distribution of the literates. Hence, Basu and Foster’s instrumental justification, based on an argument of efficiency-promoting externality, for planned literacy programmes which aim at equity in the inter-household distribution of literates. Briefly, the phenomenon of intra-group externality is a pervasive one and is fruitfully applied in the measurement of deprivation in more than one context and in more than one dimension.

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10.4 Some Implications of ‘Grouping’ for Poverty Rankings, Anti-poverty Policy, and Poverty Axiomatics 10.4.1 Poverty Rankings The particular manner in which we choose to partition the population can have nontrivial implications for poverty comparisons. This proposition can be illustrated by means of a simple numerical example. Consider two 5-vectors of income, given, respectively, by x = (10, 20, 30, 50, 70) and y = (10, 20, 30, 45, 76). Let the poverty line z be 40. Consider also three alternative groupings of the population: the atomistic grouping ga , the universal grouping gu , and a grouping gr by religion which, let us say, divides the population into two religious groups 1 and 2, respectively, and precipitates the subgroup income vectors x1 = (10,50), x2 = (20, 30, 70), y1 = (10, 45), and y2 = (20, 30, 76). Routine computation, employing Eqs. (10.5), (10.7) and (10.1), respectively, will yield the following results:     P ∗ z, x, ga = P ∗ z, y, ga = 0.2659;     P ∗ z, x, gu = 0.1579 > P ∗ z, y, gu = 0.1575; and     P ∗ z, x, gr = 0.1607 < P ∗ z, y, gr = 0.1621. For an atomistic grouping of the population, the extent of poverty, as measured by the index P*, is the same for both vectors x and y; for the universal grouping, P* suggests that poverty is greater in x than in y, and for a grouping according to religion, P* certifies that there is more poverty in y than in x. These pair-wise rank reversals indicate that it clearly cannot be a matter of indifference how we choose to partition the population. In particular, and almost in a spirit of absent-mindedness, the only partitioning which has generally been held to be relevant is the atomistic one—to the point that it has conventionally not even been considered to be important to specify the particular partitioning invoked as an argument in the poverty function. However, the fact of intra-group externality induces a heterogeneity between groups that compels the need for a context-based attentiveness to the social dimension of economic measurement.

10.4.2 Anti-poverty Policy That the issue just discussed could also have practical implications for the targeting of anti-poverty budgetary allocations is also of significant relevance: treatments of the problem can be found in Jayaraj and Subramanian (1999), Subramanian (2006b) and most especially Dasgupta and Kanbur (2005). In what follows, I provide a simple numerical example of how optimal budgetary intervention in poverty alleviation can

10.4 Some Implications of ‘Grouping’ for Poverty Rankings …

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be variably responsive to the particular grouping of the population we resort to. The form of anti-poverty policy considered here is that of a direct income transfer, and it is assumed that perfect targeting is feasible. Imagine that we have a four-person world and that the income distribution is represented by the non-decreasingly ordered vector x = (10, 20, 50, 70). The poverty line, z, is pitched at 40. Let g be a partitioning of the population according, say, to caste; and suppose this partitioning precipitates two subgroups, a backward caste (subgroup 1) and a forward caste (subgroup 2). We assume that the persons with incomes 10 and 50 belong to subgroup 1 and the remaining individuals belong to subgroup 2. The backward and forward caste subgroup income vectors can then be written, respectively, as x1 = (10, 50), with μ1 = 30, and x2 = (20, 70), with μ2 = 45. Suppose a budget T = 20 is available for distribution among the poor. How should this budget be allocated so as to minimize poverty? Suppose t i (i = 1, 2, 3, 4) is the part of the budget that is allocated to the ith poorest person. The objective is to determine the optimal schedule of transfers {t1∗ , t2∗ , t3∗ , t4∗ }. It is assumed that the sum of transfers cannot exceed the budget constraint T, that all transfers are non-negative, and that no transfer exceeds the poverty gap of a poor person. We find that the optimal anti-poverty allocation depends on the precise partitioning of the population we resort to. (Since the primary interest is in the motivational aspect of the problem, I desist from furnishing tedious derivations of the relevant optimal rules.) Suppose, first, that we partition the population atomistically (that is, the grouping resorted to is ga ). Then, it turns out that the optimal schedule of transfers is given by {t1∗ = 15, t2∗ = 5, t3∗ = t4∗ = 0}. This is just the well-known ‘lexicographic maxi-min solution’, which requires that the incomes of the poorest are raised to a common level at which the budget is exhausted. When, however, the population is partitioned according to the grouping g , it turns out that the optimal solution awards a larger transfer to person 1 and a smaller transfer to person 2, as compared to the solution under the atomistic grouping. In this case, the optimal transfer schedule is given by {t1∗ = 15.83, t2∗ = 4.17, t3∗ = t4∗ = 0}: person 1 has the double disadvantage of having a lower income and of belonging to a group with a lower average income than person 2, and this is reflected in the incremental bias in favour of 1 reflected by the optimal anti-poverty budgetary policy when the population is partitioned not atomistically but in terms of caste.

10.4.3 Poverty Axiomatics What are some of the properties of the poverty index P*, in relation to the ‘standard’ axioms of poverty measurement reviewed in Sect. 10.2? It is fairly easy to see that P* satisfies the monotonicity and normalization axioms, and writing the index in the form of Eq. (10.4) suggests that it is also decomposable. These may well be the limits of P*’s ‘successes’. The focus axiom is violated by P*: the numerical example reviewed in the preceding subsection indicates that the vector y has been derived from

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the vector x by changes to the non-poor incomes which have left the numbers of individuals in poverty unchanged; yet for both the universal grouping and the grouping according to religion, the value of P* is not the same for x and y. The symmetry axiom could also be a casualty. If two persons with distinct incomes and belonging to different groups were to swap their incomes, then the group-specific means would change, and—see Eq. (10.3)—the value of P* could also change. It is not hard to perceive that P* could also fall foul of the transfer axiom. A progressive transfer of income from a poor person belonging to a relatively badly-off group to a poorer person belonging to a relatively well-off group could change group-specific means in such a way as to actually cause the value of P* to rise, in opposition to what the transfer axiom demands. These properties are discussed more elaborately in Jayaraj and Subramanian (1999) and Subramanian (2006b), and they are asserted here for the record (that is, because they happen to be true propositions and not because they are intricately non-obvious, which of course they are not). The point to note is that these ‘axiomatic failures’ of an index such as P* are not necessarily adverse reflections on P*: they may simply be a reflection of the inappropriateness of certain canonical normative properties of poverty measures for contexts which demand that we take the notion of groups and social heterogeneity seriously. ‘Compensatory discrimination’, ‘affirmative action’, and similar principles of group-mediated justice would be impossible to defend if we insisted on swearing context-independent allegiance to axioms like focus, symmetry, and transfer.

10.5 Concluding Observations Externalities arising from group affiliation are an integral and obvious aspect of everyday social existence, yet with few exceptions (Dasgupta and Kanbur 2005 is an important example), they would appear to have been largely neglected in the economics of measuring deprivation. In this paper, an attempt has been made to incorporate the social embeddedness of individuals into an exercise in measuring income poverty, as has been done by Basu and Foster (1998) in the context of measuring literacy. The concerns of this paper have been primarily methodological, and accordingly, the points it addresses have been sought to be made illustratively rather than definitively. A principal reason for this, no doubt, is the difficulty of actually measuring external effects in any precise and unambiguous way—which could well be the reason for their general neglect in the literature. Allowing for groupmediated deprivation enables one to see that how one partitions the population has implications for deprivation comparisons, for the appeal of normative properties of deprivation indices that have been customarily held to be self-evidently desirable, for practical stratagems of budgetary allocation towards redress of deprivation, and for an instrumental defence of inter-group equality and ‘reverse’ discrimination. The standardly ‘atomistic’ formulations of measurement concerns have been shown to emerge as special cases of a more general framework of group-inclusive analysis. As has been stated earlier, it is perhaps understandable that measurement exercises are

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largely confined to that which is more, rather than less, tractably measured. But it is also time for a reappraisal when these severely practical considerations have become overwhelmingly successful in preventing conceptual and normative aspects of social reality from infecting economic theorizing and economic measurement. The present paper is a very preliminary effort at such a reappraisal.

References Basu K, Foster JE (1998) On measuring literacy. Econom J 108:1733–1749 Dasgupta I, Kanbur R (2005) Community and anti-poverty targeting. J Econom Inequality, 3:281– 302 Foster J, Greer J, Thorbecke E (1984) A class of decomposable poverty indices. Econometrica, 52(3):761–765 Jayaraj D, Subramanian S (1999) Poverty and discrimination: measurement, and evidence from rural India. In: Harriss-White B, Subramanian S (eds) Illfare in India: essays on India’s social sector in Honour of S. Guhan. Sage Publishers, New Delhi Loury G (2000) Social exclusion and ethnic groups: the challenge to economics. Institute for Economic Development—Boston University (Discussion Papers) 106 Sen AK (1985) Commodities and capabilities. North Holland, Amsterdam and New York Subramanian S (2006a) A re-scaled version of the foster-Greer-Thorbecke poverty indices based on an association with the minkowski distance function. In: Subramanian S (eds) Rights, deprivation, and disparity: essays in concepts and measurement. Oxford University Press, Delhi Subramanian S (2006b) Social groups and economic poverty: a problem in measurement. In: McGillivray M (ed) Inequality, poverty and well-being. Palgrave-Macmillan, London Subramanian S (2007) Externality and literacy: a note, forthcoming in J Dev Stud Takayama N (1979) Poverty, income inequality, and their measures: professor sen’s axiomatic approach reconsidered. Econometrica 47(3):747–749 Wilson DR, Martinez TR (1997) Improved Heterogeneous distance functions. J Artif Intell Res 6:1–34

Chapter 11

Revisiting the Normalization Axiom in Poverty Measurement

Abstract The ‘normalization’ axiom associated with Sen’s poverty index–and this, indeed, holds for most extant measures of poverty—entails an uncomfortable implication when we adopt a strong, or inclusive, definition of the poor. This paper suggests that we may not always be at liberty to adopt a weak definition. The available alternative then is to change the form of the poverty measure. Accordingly, a modification of his normalization axiom which leads to a variant of Sen’s index, together with a variant also of the Foster–Greer–Thorbecke poverty measures, is advanced and discussed. The derivation of the new normalization axiom benefits from Basu’s decomposition of the Sen axiom. Keywords Normalization · Sen index of poverty · Foster–Greer–Thorbecke Poverty Measure

11.1 Introduction: Classifying the Population by Poverty Status The present paper could be seen as a specific example of a difficulty that can arise from a ‘strong’ (or ‘inclusive’) classification of the relevant population (poor or nonpoor) by poverty status. It sometimes happens that what appears to be a minor or innocuous difference—such as whether to include or exclude those on the poverty line from one’s definition of the poor or the non-poor—can actually have rather material consequences for the outcomes of measurement. To quote Donaldson and Weymark (1986; pp. 668, 670, 687): We consider two definitions of the poor. In the weak definition, the poor consists of all individuals with incomes strictly less than the poverty line. The strong definition of the poor This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2009) Revisiting the Normalization Axiom in Poverty Measurement, Finnish Economic Papers, 22:2, 89– 98. The author is indebted for helpful and constructive suggestions, far exceeding what is compatible with strict duty, to two anonymous referees of Finnish Economic Papers; for patient and considerate advice from its editor, to Andreas Wagener; and for valuable comments, to Kaushik Basu. The usual caveat applies. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_11

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expands the group of poor people to include anyone with an income equal to the poverty line as well. With the weak definition of the poor, all of the axioms we consider are compatible. However, with the strong definition of the poor, a number of impossibilities occur … This seemingly trivial difference in definitions of the poor has important implications … [A]n issue that might seem to be of minor importance, namely the classification of individuals at the poverty line, has significant implications.

This paper explores one such implication, in the context of an axiom called the normalization axiom (to be hereafter referred to as Axiom N) employed by Sen (1976, 1979) in the derivation of his real-valued index of poverty. The normalization axiom states the following. Let H be the headcount ratio (or proportion of the population in poverty), and I the income-gap ratio (or proportionate shortfall of the average income of the poor from the poverty line). Then, Axiom N demands that, in the special case in which all the poor have the same income, the poverty index should be equal to the product of the headcount ratio and the income-gap ratio. If the poor are defined to include all individuals with incomes not exceeding a threshold ‘poverty line’ level of income, then this multiplicative combination of H and I entailed by Axiom N can be problematic, in a specific sense, which is explicated in what follows. It should be clarified, at the very outset, that whether or not Axiom N could be ‘problematic’ would depend on how one chooses to define the populations of the poor and the non-poor. There is one scheme of classification (the ‘exclusive’ scheme, to be discussed later) under which the issue simply ‘dissolves’, and another scheme (the ‘inclusive’ scheme) under which there is a genuine difficulty that needs to be addressed. It must be stated clearly that for those who are persuaded of the unique rightness of the first of these classificatory schemes, this paper must appear to deal with a non-issue. The concerns of the paper, that is, would make any sense only if it is conceded that the alternative (inclusive) classificatory scheme cannot be straightforwardly rejected as unreasonable. To see what is involved, it is useful to consider some preliminary definitions of terms and concepts. Let me motivate the discussion in a slightly oblique fashion, by recalling the work of Watts (1968), who proposed a unified approach to the measurement of poverty, of ‘affluence’ (or, perhaps more appropriately, ‘non-poorness’), and of a welfarerelated combination of both. I focus specifically on Watts because he was the first writer to advance a conjoint treatment of the measurement of poverty, of affluence, and of welfare—by seeing the affluence index as a reflected version of the poverty index, and the welfare index as a simple sum of the poverty and affluence indices. The specificity with respect to Watts, however, need not obscure a certain wider and more general domain of applicability of the concerns under review. In principle, given any measure of poverty, one could conceive of a measure of non-poorness as a sort of mirror-image of the poverty measure. This—with appropriate modifications, of course—has indeed been advocated, with respect both to the Sen index of poverty (on which see Sen 1988), and the Foster–Greer–Thorbecke class of poverty measures (on which see Peichi et al. 2008). The central issue involved here is that, if it is legitimate to measure both poverty and non-poorness, then either the poor or the non-poor population may have to be classified in terms of an inclusive definition. This is true, whether or not any particular poverty index was conceived of in conjunction with

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a complementary ‘non-poorness’ index, and so applies, inter alia, to the Sen index and the Foster–Greer–Thorbecke indices, although the motivating discussion, which follows, will be conducted with reference to the Watts formulation. Let {1, ..., i, ..., n} ≡ N be the set of individuals constituting society, let xi (i = 1, ..., n) be the income of person i, and let z be the threshold, ‘poverty line’ level of income. Define the following three mutually exclusive and completely exhaustive subsets of N : S1 ≡ {i ∈ N |xi < z} ; S2 ≡ {i ∈ N |xi = z} ; and S3 ≡ {i ∈ N |xi > z} The Watts measures of poverty and ‘non-poorness’ (or ‘affluence’, as Watts (1968) loge (z/xi ) if called it) could be seen to be given, respectively, by W P = (1/n) i∈S1  S1 is non-empty and zero otherwise, and W N P = (1/n) i∈S3 loge (z/xi ) if S3 is non-empty and zero otherwise. (Watts also went on to define an overall measure of welfare W W given by summing the measures of poverty and of affluence [W W = W P +W N P ]1 : this is mentioned here only for purposes of completeness and will have no further role to play in this paper.) There is nothing inherently problematic about partitioning the population in this manner, and—as will become apparent shortly— the problem dealt with in this paper would not arise with the three-fold classificatory scheme (consisting of the sets S1 , S2 , and S3 ) just defined. The unexceptionableness of this classificatory scheme comes through particularly strongly at a certain level of abstraction. However, suppose we were to take an ‘ordinary language’ view of the matter, and to adopt some common-sense convention of ‘naming’ the sets S1 , S2 , and S3 , so as to render the ‘meanings’ of the indices W P and W N P transparent. Then, it appears both reasonable and likely that S1 would be defined as the set of the poor and S3 as the set of the non-poor2 —in which case, S2 would have be defined as a set whose typical member is an individual i of whom it is true that s/he is not poor and also not non-poor. This falls foul of a semantic principle, the Principle of Contradiction (POC), which is a dual of another semantic principle, the Bivalence Principle. The former principle demands of any proposition p that it cannot be such that both p and (not-p) are true. If the present scheme of classification by poverty status is seen to be unsatisfactory for the reason just discussed, then one would have to resort to an alternative partitioning of the population—one which can, as it turns out, have adverse implications for Sen’s normalization axiom.   If W P > W N P , then W W will be positive. By normal convention, one would be inclined to treat welfare as positive when the extent of non-poorness exceeds the extent of poverty: to accommodate this, we could take the aggregate welfare measure to be given by (−)W W rather than by W W . 2 For instance, in targeted poverty alleviation schemes, people on the poverty line must either be included within or excluded from the ambit of programme benefits: if they are included, it is presumably because they are regarded as being poor, and if they are excluded, it is presumably because they are regarded as being non-poor. 1

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Under this alternative classificatory system, the population would have to be divided into two mutually exclusive and completely exhaustive subsets—those of the poor and the non-poor. Let Q be the set of poor individuals in N , and Q the set of non-poor individuals. (‘Affluent’ is a convenient, if not quite accurate, substitute for the more apposite term ‘non-poor’: the two are sometimes used interchangeably, but it does less violence to the use of language to employ the term ‘non-poor’, when that is what is meant.) Clearly, the set Q is the complement of the set Q in the setN , and the individuals belonging to Q and Q , respectively, are separated by the poverty line z. But separated how? If the non-poor individuals are defined ‘inclusively’, to signify those with incomes not less than z (that is, Q is defined such that Q ≡ {i ∈ N | xi ≥ z} ), then Q must necessarily be defined ‘exclusively’ (that is, such that Q ≡ {i ∈ N | xi ≥ z} ); on the other hand, if Q is defined ‘exclusively’ (that is, such that Q ≡ {i ∈ N xi ≥ z} , then Q must necessarily be defined ‘inclusively’ (that is, such that Q ≡ {i ∈ N | xi ≤ z} ). The Watts measures of poverty and non-poorness would now be given, respectively, by W p = (1/n) i∈Q loge (z/xi )  if Q is non-empty and zero otherwise, and W N P = (1/n) i∈Q loge (z/xi ) W NP if 















Q is non-empty and zero otherwise. Suppose Q to have been defined inclusively. Imagine two situations, labelled Situation 1 and Situation 2, respectively. In Situation 1, there is exactly one individual with an income of z, while each of the remaining n − 1 individuals has an income of z/2 each. In Situation 2, all n individuals have an income of z each. It is easy to see that, in Situation 1, W P = [(n − 1)/n] loge 2, while in Situation 2, W P = 0: the poverty level in Situation 1 is finite, while there is no poverty in Situation 2, which seems to be a reasonable enough judgment, considering that a finite number of persons in Situation 1 are poor while no-one in Situation 2 is poor. Next, consider the non-poorness measure W N P . It is easy to see that W N P = 0 in both Situations 1 and 2: this must certainly be judged odd, since there is only one non-poor person (with an income of z) in Situation 1, while all n persons out of n, with the same income of z each, are non-poor in Situation 2. The oddness of this judgment can, of course, be easily rectified, by switching from the inclusive to the exclusive definition of the non-poor: in this case, the set Qˆ would be empty in both Situations 1 and 2, and W N P would—not unreasonably—be zero in both situations. The difficulty with this ‘easy rectification’, however, is that with Q now being defined exclusively, Q would necessarily have to be defined inclusively. Now imagine two situations, labelled Situation 3 and Situation 4, respectively, such that, in Situation 3, there is exactly one person with an income of z and there are n − 1 persons with an income of 3z/2 each, while in Situation 4 (as in Situation 2), each of the n persons has an income of z each. Under the inclusive definition of Q, it is easy to verify that W P = 0 in each of Situations 3 and 4—which is an odd judgment given that in Situation 3 there is only one person (with an income of z) who is poor, while all n persons out of n, with the same income of z each, are poor in Situation 4. Briefly, a form of resolution of the problem which entails switching from the inclusive to the exclusive definition is not very satisfactory, because it effectively 



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tells an analyst that she may measure poverty or non-poorness, but not both, without allowing odd judgments to mediate the exercise. A more promising approach to resolution may, instead, require a re-consideration of the measures of poverty (or non-poorness) themselves. Let me re-iterate, at this point, that the concerns of this paper will be seen to have any validity only if some merit is conceded to the notion that it may be problematic to define both the poor and the non-poor populations exclusively, for reasons discussed in the preceding paragraphs. If this concession is made, then one can examine— without loss of generality—the consequences for measurement of defining the poor population inclusively, as is done in what follows.

11.2 The Sen Normalization Under an Inclusive Definition of the Poor An asymptotic expression for Sen’s index of poverty (that is, in a situation where the numbers of the poor are ‘large’)—Sen (1976)—is given by P s = H [I + (1 − I )G P ],

(11.1)

where G P is the Gini coefficient of inequality in the distribution of income among the poor. Notice that if the set of poor individuals is defined so as to include those with incomes equal to the poverty line, then when I is zero because all the poor have the same income, equal to the poverty line, clearly G P is zero, and hence—in view of (11.1)—P s is zero. This is an undesirable property of the poverty index: for any given population and any given poverty line, any two equal distributions of income among the poor in which all the poor receive the poverty line income must be judged to have the same extent of poverty, despite the fact that only one person out of one hundred may be poor under the first distribution, while one hundred persons out of one hundred may be poor under the second. That is, Sen’s poverty index does not allow for monotonicity in H when I is zero by virtue of all poor individuals sharing the poverty line income. This is ensured by the multiplicative form entailed by the normalization axiom, which demands that when there is no inequality in the distribution of poor incomes, poverty is given by the product of H and I . Indeed, this difficulty is a feature also of the Foster et al. (1984) family of poverty indices, given by Pα = (1/n)

q 

[(z − xi )/z]α , α ≥ 0,

(11.2)

i=1

where z is the poverty line, xi is the income of the ith person in the set of poor persons, q is the number of poor persons, n is the total population, and α is a measure of ‘poverty aversion’. For given n and z, and for all α > 0, it is easy to see from (11.2)

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that all distributions in which xi = z∀i = 1, ..., q, will be certified by Pα to reflect zero poverty, irrespective of the value of q in relation to n. As we have noted earlier, it is, of course, true that a simple way out of the problem just discussed would be to exclude individuals on the poverty line from the definition of the poor. But—and this was also noted earlier—exclusive definitions of the poor or of the non-poor are, from one point of view, compatible with ‘satisfactory’ measurement of only either poverty or non-poorness, but not of both. This is not a very appealing way out of the problem, if it is conceded that it is, after all, legitimate to be interested in comparing distributions not only in terms of poverty but also in terms of ‘non-poorness’. Under these circumstances, there may be a case for taking another look at the normalization axiom itself. Sen’s normalization, entailing a multiplicative combination of H and I is, as we have seen, not entirely satisfactory. Takayama (1979) proposed a normalization axiom in which—when poor incomes are equally distributed—the extent of poverty is identified with the minimum of the headcount ratio and the income-gap ratio. For a given population and a given poverty line, Takayama’s normalization also fails, when all poor incomes coincide with the poverty line, to allow for monotonicity of the poverty measure in the headcount ratio. This is true also for the normalization axiom proposed by Pattanaik and Sengupta (1995), which demands that the poverty measure should (a) be zero when all poor incomes are equal to the poverty line, and (b) coincide with the headcount ratio when all poor incomes are equal to zero. In this paper, I propose a normalization axiom (N*) in which the extent of poverty is equated to one-half the sum of the headcount ratio and the income-gap ratio. Axiom N* is sought to be rationalized within an axiomatic framework. In the process, some aspects of Basu’s (1985) rationalization of Sen’s normalization axiom N are reviewed, and the relevance and usefulness of Basu’s arguments for justifying Axiom N* are noted. Axiom N* can be used (in conjunction with the other axioms proposed by Sen for justifying his own poverty index P s ) to derive a variant P s∗ of P s . P s∗ is free of a particular difficulty which is a feature of P s , namely the failure of the poverty index to be a monotonically increasing function of the headcount ratio, in the special case in which the strong definition of the poor is adopted and all the poor have an income equal to the poverty line. We have noted that this difficulty is also a feature of the Foster et al. (1984) Pα family of poverty indices. A minor change in the specification of the functional form of the poverty index will rectify this difficulty: a family of poverty indices Pα∗ which are variants of the indices in the Pα family are presented in this paper. The new normalization axiom, and the corresponding variants of Sen’s index and the Foster–Greer–Thorbecke indices, however, are secured at a possible price, namely the violation of certain other canonically valued axioms of poverty measurement. These difficulties are discussed in the paper. Overall, the essay points to the troublesome possibility that seemingly rather small issues can result in rather large complications for the measurement of poverty.

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131

11.3 Axiom N* and Its Rationalization I start with some basic definitions, drawing on Basu (1985). Let Xn ≡ {x ∈ Rn+ | xi ≤ xi+1 , i = 1, ....., n − 1}, where Rn+ is the non-negative orthant of n-dimensional real n space. Every x belonging to X ≡ ∪∞ n=1 X is then a description of an n-person ordered distribution of (non-negative) incomes. z is the poverty line, which is a level of income such that any individual whose income does not exceed z is certified to be poor. (This, to recall, is the ‘strong’ definition of the poor). For every x ∈ X, let S(x) be the set of poor individuals in x. A poverty measure is a function P : X → [0, 1] such that, for all x ∈ X, 0 ≤ P(x) ≤ 1. The headcount ratio H and the income-gap ratio I are defined as follows: ∀x ∈ X, H (x) = #S(x)/#x. ∀x ∈ X, I (x) = 0 if S(x) = {}, and I (x) =

 i∈S(x)

(z − xi )/(#S(x) · z), otherwise.

Let X∗ (⊂ X) ≡ {x ∈ X|xi = xi−1 ∀i ∈ S(x)\{1}} . It will be assumed that there exists a function f such that ∀x ∈ X∗ , P(x) = f (H (x), I (x))

(11.3)

It is useful, first, to consider an approach to rationalizing Sen’s normalization axiom, of the type adopted by Basu (1985). Sen’s normalization axiom can be stated as follows. Axiom N. ∀x ∈ X∗ , P(x) = f (H (x), I (x)) = H (x) · I (x). As pointed out by Basu, H (X∗ ) ≡ Q ∩ [0, 1] where Q is the set of all rational numbers (note that since #S(x) and #x are both integers, the headcount ratio must be a rational number), while I(X*) ≡ [0,1]. Given this, Basu takes the function f to be a mapping from ( Q ∩ [0, 1]) × [0, 1] to [0, 1]. Properly speaking, however, the domain of the function f is a strict subset of (Q ∩ [0, 1]) × [0, 1]. For notice that when H is zero, I must also be zero: combinations of H and I for which H is zero and I is finite do not exist. Let {0} be the point (0, 0) in two-dimensional real space. Define the set M ≡ {0} ∪ (Q ∩ (0, 1]) × [0, 1]. Then, we can correctly assert that f : M → [0, 1].

(11.4)

Basu now proposes the following axioms to justify Sen’s normalization axiom. Axiom 1(a). f (1, 1) = 1. Axiom 1(b).Lim H →0 f (H, I ) = Lim I →0 f (H, I ) = 0. Axiom 2. ∀H1 , H2 , H3 , H4 ∈ Q ∩ [0, 1] and ∀I ∈ [0, 1], [H1 − H2 > (=)H3 − H4 ] → [ f (H1 , I ) − f (H2 , I ) > (=) f (H3 , I ) − f (H4 , I )]

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11 Revisiting the Normalization Axiom in Poverty Measurement

Axiom 3. ∀I1 , I2 , I3 , I4 ∈ [0, 1] and ∀H ∈ Q ∩ [0, 1], [I1 − I2 > (=)I3 − I4 ] → [ f (H, I1 ) − f (H, I2 ) > (=) f (H, I3 ) − f (H, I4 )]. Basu establishes that Axioms 1–3 are equivalent to Axiom N. The preceding brief review paves the way for a set of axioms which uniquely imply the following variant of Sen’s normalization axiom: Axiom N*. ∀x ∈ X∗ , P(x) = f (H (x), I (x)) = [H (x) + I (x)]/2. Towards rationalizing Axiom N*, I begin with a definition: every m = (m1 , m2 ) ∈ M is a poverty regime, defined by an ordered pair of headcount ratio and income-gap ratio. (M, to recall, is the set {0} ∪ (Q ∩ (0, 1]) × [0, 1]). The following axioms are restrictions on the function f : M → [0, 1]. Axiom I (Boundary Axiom). 1. 2.

f (0, 0) = 0; f (1, 1) = 1.

That is, when there are no poor individuals, the extent of poverty is zero, while when everyone is poor and receives no income, the extent of poverty is unity. Axiom II (Proportional Monotonicity Axiom).     1. ∀m, m ∈ M such that m 2 = m 2 , f (m) − f m = k1 m 1 − m 1 , where k1 ∈ R++ ;     2. ∀m, m ∈ M such that m 1 = m 1 , f (m) − f m = k2 m 2 − m 2 , where k2 ∈ R++ . That is, if two poverty regimes differ from each other only with respect to the headcount ratio (respectively, income-gap ratio), then the difference in the extent of poverty in the two regimes varies directly with the difference in the headcount ratios (respectively, income-gap ratios) in the two regimes. Notice that proportional monotonicity implies a stronger condition than monotonicity which would require only that, other things equal, a regime with a higher headcount ratio (respectively, income-gap ratio) should reflect greater poverty. Axiom III (Symmetry Axiom). . ∀m, m ∈ M such that m 1 = m 2 and m 2 =  

m 1 , f (m) = f m . That is, the extent of poverty remains unchanged when one poverty regime is derived from another by a simple interchange of the headcount ratio and the incomegap ratio. (Loosely, the headcount ratio and the income-gap ratio are of ‘equal importance’ in determining the extent of poverty.) It must be admitted that the judgment embodied in Axiom III is somewhat arbitrary. For example, consider the poverty regimes (0.1, 1) and (1, 0.1): in the first regime, ten per cent of the population are in extreme deprivation while in the second regime, the burden of deprivation is more evenly spread among the entire population; it could well be held that a small incidence of acute deprivation is worse than a large incidence of relatively mild deprivation. For the purposes of this note, however, we shall stick with Axiom III; and it might be noted that Axiom N also satisfies the symmetry axiom. The following proposition is now true.

11.3 Axiom N* and Its Rationalization

133

Theorem 11 Let f : M → [0, 1]. Then, ∀m ∈ M, f (m) = (m 1 + m 2 )/2 if and only if f satisfies Axioms I, II and III. Proof (a). Sufficiency: Straightforward. (b) Necessity: Consider the poverty regime (m 1 , 0) ∈ M. By Axiom II(a), f (m 1 , 0) − f (0, 0) = k1 m 1 , k1 > 0 or f (m 1 , 0) = k1 m 1 ,

(11.5)

since f (0,0) = 0 by Axiom I(a). Consider the regime (m 1 , m 2 ) ∈ M. By Axiom II(b), f (m 1 , m 2 ) − f (m 1 , 0) = k2 m 2 , k2 > 0, whence f (m 1 , m 2 ) = k1 m 1 + k2 m 2 ,

(11.6)

since f (m 1 , 0) = k1 m 1 by (11.5). Setting m 1 = m 2 = 1 in (11.6), and invoking Axiom 1(b), we have: k1 + k2 = 1.

(11.7)

Let. (s, t) and (t, s) be two poverty regimes, with s = t. By virtue of (11.6), we have: f (s, t) = k1 s + k2 t,

(11.8)

f (t, s) = k1 t + k2 s.

(11.9)

and

By Axiom III, f (s, t) = f (t, s). This, in conjunction with (11.8) and (11.9), and recalling that s = t by assumption, leads to k1 − k2 = 0

(11.10)

From (11.7) and (11.10), we obtain: k1 = k2 = 1/2.

(11.11)

Substituting for k1 and k2 from (11.11) into (11.6) yields: ∀m ∈ M, f (m) = (m 1 + m 2 )/2, as desired. (Q.E.D.) An alternative justification of the normalization axiom N* can be obtained by retaining Axioms I and III and replacing Axiom II by the following axiom proposed (and discussed) in Basu (1985): Axiom II (Difference Preservation Axiom).

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1.

11 Revisiting the Normalization Axiom in Poverty Measurement

∀m1 , m2 , m3 , m4 ∈ M such that m 11 = m 21 = m 31 = m 41 ,              4 m 2 − m 32 > (=) m 22 − m 12 → f m4 − f m3 > (=) f m2 − f m1

2.

∀m1 , m2 , m3 , m4 ∈ M such that m 12 = m 22 = m 32 = m 42 ,              4 m 1 − m 31 > (=) m 21 − m 11 → f m4 − f m3 > (=) f m2 − f m1 The following proposition is now true.

Theorem 11 Let f : M → [0, 1]. Then, ∀m ∈ M, f (m) = (m 1 + m 2 )/2if and only if f satisfies Axioms I, II and III. The proof of Theorem 2 is here omitted: the result can be proved very much along the lines of the proof which Basu (1985) uses in order to justify Axiom N (The proof is available with the author on request).

11.4 Variants of the Sen and the Foster–Greer–Thorbecke Poverty Measures If we preserve the axiomatic structure employed by Sen (1976) in the derivation of his poverty index, but replace his normalization axiom N by the normalization axiom N* advanced in this note, and write the general form of the poverty function as P = A(n, q, z) + B(n, q, z)

q 

(z − xi )(q + 1 − i),

i=1

(where A and B are normalizing parameters and z, x i , q, and n are as already defined in Sect. 2 of this paper), then it can be shown that the poverty index implied by this axiomatic structure is given (for ‘large’ values of q) by  ∗ P S = H + I + (1 − I )G P /2

(11.12)

A precise statement of the proposition, and its proof, are here omitted, since the required result can be routinely derived by mechanical application of the stratagem of proof employed by Sen (1976) to derive his poverty index PS . (The theorem and its proof are available, on request, with the author). Notice now, by comparing (11.12) with (11.1), that when all poor incomes coin∗ cide with the poverty line, P S depends only on H—which is as it should be: unlike ∗ in the case of the Sen index, P S does not vanish whenever I becomes zero owing ∗ to all the poor having the poverty line income. While P S is obviously a very simple S refinement of P , the refinement could conceivably be of some practical significance:

11.4 Variants of the Sen and the Foster–Greer–Thorbecke …

135

∗

specifically, it is important to note that P S can reverse the poverty ranking of distributions by PS . An actual empirical example should help to highlight this. Subramanian (1988) furnishes some estimates of rural poverty in the Indian State of Tamil Nadu, and for the years 1961–62 and 1970–71 (here represented by the subscripts 1 and 2, respectively), he furnishes the following poverty-related information in terms of the triples (H 1 = 0.4434, I 1 = 0.2991, G 1P = 0.1539) and (H 2 = 0.4967, I 2 = 0.2682, G 2P = 0.1271).3 Assuming ‘large’ numbers of the poor, it is easy to verify that P1S ∗ ∗ = 0.1804 > P2S = 0.1794, but P1S = 0.4252 < P2S = 0.4290. A similar refinement can be effected in the context of the Foster–Greer–Thorbecke (FGT)-related family of poverty indices Pα∗ , given by: Pα∗ = (1/2n)

q   1 + ((z − xi )/z)α , α ≥ 0.

(11.13)

i=1

Three distinguished values of α are: α = 0, α = 1, and α = 2 (see Foster et al. 1984). By setting α = 0, 1, and 2, respectively, in (11.2), the FGT indices  for these values of α can be seen to be given by: P0 = H , P1 = H I , and P2 = H I 2 + (1 − I )2 C 2P , where C 2P is the square of the coefficient of variation in the distribution of income ∗ among the poor. For the same values of α, the values of the  indices Pα (see (11.13)) are given by: P0∗ = H , P1∗ = H (1 + I )/2, and P2∗ = H 1 + I 2 + (1 − I )2 C 2P /2. In general, for α greater than zero, Pα∗ , unlike Pα , does not vanish when I is zero by virtue of all poor persons having the income z: it is an increasing function of the headcount ratio H for every value of the income-gap ratio I in the interval [0, 1]—which is an attractive property not found in the Pα family of indices (α positive) when I is zero. Finally, for α > 0, Pα∗ and Pα can rank distributions in opposing ways. Again, an empirical illustration is afforded by poverty statistics for Finland. From estimates provided by Riihela et al. (2008), we can directly obtain or deduce,4 for two specific years, 2001 and 2004, the values of H, I and C 2P , which are, respectively, 0.0160, 0.2625, and 0.0917 in 2001, and 0.0165, 0.2364, and 0.0705 in 2004. It is easily verified that P2 has declined from 0.0019 in 2001 to 0.0016 in 2004, but P2∗ has increased from 0.00895 in 2001 to 0.00905 in 2004. The story, however, does not quite end on a happy note. As pointed out by an anonymous referee, the measure P S∗ , like the measure P s , can violate the strong transfer axiom (which requires the poverty index to register an increase in value whenever a regressive transfer of income from any poor person to any other person occurs), while the measure Pα∗ —unlike the measure Pα —can violate both strong transfer and restricted continuity (that is, continuity in poor incomes). The violation of strong transfer by Pα∗ can be illustrated by means of an example supplied by the referee. Imagine two ordered five-person income distributions given by x = 3

The data can be found in Table 2 of Subramanian (1988). The poverty line is taken to be a consumption expenditure level of Rupees 15 per person per month at 1960–61 prices. 4 The authors provide data on H , H I and P , from which the values of H, I and C 2 can be 2 P easily inferred. The estimates reproduced here relate to consumption poverty, and the poverty line employed is a relative one, pitched at 40 per cent of the median consumption level.

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11 Revisiting the Normalization Axiom in Poverty Measurement

(5, 5, 10, 15, 27) and y = (1, 5, 10, 15, 31), and suppose the poverty line to be given by z = 30. It is clear that y has been derived from x by a regressive (or upward) transfer of four units of income from the poorest individual to the richest one—a transfer which has also pushed the richest individual from below the poverty line to above it. It is easy to verify, for example, that P2 (x) = .4187 < P2 (y) = 0.4647, but P2∗ (x) = .7093 > P2∗ (y) = .6323: the transfer has reduced the valued of the poverty index P2∗ . That Pα∗ can also fail continuity is reflected q  at the poverty line in the following: recalling that Pα∗ = (1/2n) i=1 1 + ((z − xi )/z)α , α ≥ 0, note that if xi = z∀i ∈ N , then Pα∗ = 1/2, α ≥ 0; if now each person’s income were to rise by an infinitesimal amount ε, then Pα∗ would plummet discontinuously to zero. Briefly, if strong transfer and continuity are regarded as indispensable properties for a poverty index, then, in terms of these properties, it would appear that P S∗ is no better than P s , while Pα∗ is actually worse than Pα . But are properties like strong transfer and continuity indeed unambiguously indispensable? This is questionable, as the following three lengthy quotations indicate. The first two—relating to the desirability of the continuity axiom—are due to Atkinson (1987; p. 754) and Donaldson and Weymark (1986; p. 674); and the third—relating to the desirability of the strong transfer axiom—is again due to Donaldson and Weymark (1986; p. 674): Here, there is room for difference of opinion. On the one hand, there are those who agree with Watts that there is a continuous gradation as one crosses the poverty line. On the other hand, there are people who see poverty as an either/or condition. A minimum income may be seen as a basic right, in which case the headcount [the archetypically discontinuous poverty function] may be quite acceptable as a measure of the number deprived of that right. Given the difficulties involved in measuring incomes accurately, it seems reasonable to require a poverty index to vary continuously with income. On the other hand, the use of a poverty line to sharply demarcate the rich from the poor suggests, but does not require, that a poverty index might be discontinuous at the poverty line. Thus, the a priori case for continuity is somewhat uneasy. Transfer principles are normally justified by the ethical appeal that similar principles have had in the measurement of inequality…With the strong transfer principles the arguments are less clearcut, since the number of poor people can change as a result of the transfer. An upward transfer may lower the number of poor people by pushing the richer poor person over the poverty line. At the same time, it increases the shortfall from the poverty line of the more destitute. These effects work in opposite directions, the first reducing poverty and the second increasing it (given monotonicity). It is true that inequality is increased by such a transfer, but it is not at all obvious that poverty is, at least if the poverty line represents a demarcation representing “absolute deprivation in terms of a person’s capabilities” (Sen 1983). Thus it may be desirable to permit poverty to decrease because of such a transfer in some cases, and [strong upward transfer] must be rejected. Sen (1976) himself suggested that [strong upward transfer] was a reasonable property of a poverty index, but subsequently advocated [weak transfer] instead. The strong transfer principles have been a source of controversy ever since.

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11.5 Concluding Observations This paper amounts to one more confirmation of the proposition that the measurement of poverty can be a complicated enterprise. It suggests that one may not always be free to indifferently adopt a strong or weak definition of the poor. Whether or not this suggestion is a convincing one, the paper proceeds, in a positive spirit, to consider one specific and troublesome implication of adopting a strong, or inclusive, definition. This implication, for a number of extant measures of poverty, is that in the special case in which all the poor individuals in a society have the poverty line income, measured poverty fails to be a monotonically increasing function of the headcount ratio. The paper advances alternative versions of the Sen and Foster–Greer–Thorbecke poverty measures as a means of avoiding this problem. In the case of the Sen index, the problem is traced to Sen’s normalization axiom, and an alternative normalization axiom is advanced and rationalized. In the process of conducting this exercise, the paper also reviews and draws on Basu’s decomposition of Sen’s normalization axiom. The variants of the Sen and Foster–Greer–Thorbecke indices may circumvent the specific problem which motivated their quest, but the latter set of variants, unlike the corresponding originals, fall foul of certain axioms that have been proposed in the literature, such as the strong transfer property and the continuity property. These properties are not, it can be argued, self-evidently and completely compelling. Clearly, something must give, and it must be left to the analyst to effect the trade-offs on the basis of her values, priorities, and perceptions. It is hoped that the problem of choice which is entailed is at least not a wholly trivial one.

References Atkinson AB (1987) On the measurement of poverty. Econometrica 55(4):749–764 Basu K (1985) Poverty measurement: a decomposition of the normalization axiom. Econometrica 53(6):1439–1443 Donaldson D, Weymark JA (1986) Properties of fixed population poverty indices. Int Econ Rev 27(3):667–688 Foster J, Greer J, Thorbecke E (1984) A class of decomposable poverty indices. Econometrica 52(3):761–765 Pattanaik PK, Sengupta M (1995) An alternative axiomatization of Sen’s poverty measure. Rev Income Wealth 41(1):73–80 Peichi A, Schaefer T, Scheicher C (2008) Measuring richness and poverty: a micro data application to Europe and Germany. IZA DP No. 3790, Discussion Paper Series of the Study of Labour (IZA), Bonn, Germany Riihela M, Sullstrom R, Tuomala M (2008) Economic poverty in Finland 1974–2004. Finn Econ Pap 21(1):57–77 Sen AK (1976) Poverty: an ordinal approach to measurement. Econometrica 44(2):219–231 Sen AK (1979) Issues in the measurement of poverty. Scandinavian J EcoNom 81(2):285–307 Sen AK (1983) Poor, relatively speaking. Oxf Econ Pap 35(2):153–169 Sen PK (1988) The harmonic Gini coefficient and affluence indices. Math Soc Sci 16(1):65–76

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Subramanian S (1988) Poverty and inequality. In Madras Institute of Development Studies (ed) Tamil Nadu economy: performance and issues. Oxford and IBH Publishing Co. Pvt. Ltd., New Delhi Takayama N (1979) Poverty, inequality, and their measures: professor sen’s axiomatic approach reconsidered. Econometrica 47(3):747–759 Watts H (1968) An economic definition of poverty. In: Moynihan DP (ed) On understanding poverty. Basic Books, New York

Chapter 12

The Focus Axiom and Poverty: On the Coexistence of Precise Language and Ambiguous Meaning in Economic Measurement …"That is not what I meant at all. That is not it, at all.” -T. S. Eliot: The Lovesong of J. Alfred Prufrock ‘The question is,’ said Alice, ‘whether you can make words mean so many different things.’ ‘The question is,’ said Humpty Dumpty, ‘which is to be master – that’s all.’ -Lewis Carroll: Through the Looking Glass

Abstract Despite the formal rigour that attends social and economic measurement, the substantive meaning of particular measures could be compromised in the absence of a clear and coherent conceptualization of the phenomenon being measured. A case in point is afforded by the status of a ‘focus axiom’ in the measurement of poverty. ‘Focus’ requires that a measure of poverty ought to be sensitive only to changes in the income distribution of the poor population of any society. In practice, most poverty indices advanced in the literature satisfy an ‘income focus’ but not a ‘population focus’ axiom. This, it is argued in the present paper, makes for an incoherent underlying conception of poverty. The paper provides examples of poverty measures which either satisfy both income and population focus or violate both, or which effectively do not recognize a clear dichotomization of a population into its poor and non-poor components, and suggests that such measures possess a virtue of consistency, and coherent meaning, lacking in most extant measures of poverty available in the literature. Keywords Poverty measure · constituency principle · Income focus · Population focus · Comprehensive focus

This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2012). The Focus Axiom and Poverty: On the Co-existence of Precise Language and Ambiguous Meaning in Economic Measurement. Economics: The Open-Access, Open-Assessment E-Journal, 6 (2012-8): 1–21. I am indebted to Manimay Sengupta, an anonymous referee of Economics E-Journal, and the Editor in charge of this paper, Satya Chakravarty, for very helpful comments. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_12

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12.1 Introduction What do we mean when we say we are ‘measuring poverty’? It would be easy enough to respond by suggesting that what we mean is reflected precisely in what we say— and that would be the end of that, if the response were an accurate one. The difficulty is that the literature on the measurement of poverty affords reason to doubt the accuracy of the response. In particular, it is not at all obvious that the measurement of poverty, as it is largely practised today, is informed by a clear and coherent notion of what it is that is really being measured. This want of coherence, it will be argued in this note, can be traced to an inconsistent—because it is partial—deference to a widely employed axiom in the measurement of poverty. The axiom in question is the so-called focus axiom (see Sen 1981). The focus axiom (potentially) comes into play whenever poverty measurement involves the so-called identification exercise, namely the stipulation of a ‘poverty line’ level of income whose job it is to separate the poor section of a population from what is perceived to be that population’s definitely non-poor component. The axiom is a particular application, in the context of poverty, of a more general principle in population ethics which Broome (1996) calls the ‘constituency principle’. This latter principle can be stated, in loose terms, as the requirement that in comparing the goodness of alternative states of the world, one should confine attention to how good the states are from the point of view of some identified constituency of individuals who alone are judged to be the relevant and interested parties to the outcome of the comparison exercise. For instance, in comparing the goodness of alternative histories of the world, it could be claimed that which history is better ought to be determined exclusively by which history is better for the constituency yielded by the intersection of the populations that exist in both histories. As applied to poverty comparisons, the constituency principle/focus axiom would demand that in determining the relative poverty status of two societies, we ought to confine our concern to, or ‘focus’ our attention on, the condition of the poor constituency in the two societies. In practice, the constituency principle in poverty measurement has resolved itself into what one may call an ‘income focus axiom’, which is the requirement that, other things equal, any increase in the income of a non-poor person ought not to affect one’s assessment of measured poverty. A properly thoroughgoing appreciation of the constituency principle ought to extend the scope of the principle also to an appropriately formulated ‘population focus axiom’, which is the requirement that, other things equal, any increase in the size of the non-poor population ought not to affect one’s assessment of measured poverty. With few exceptions (see, among others, Subramanian 2002; Paxton 2003; Chakravarty et al. 2006; Hassoun 2010; Hassoun and Subramanian 2011), the population focus axiom has received little attention in the poverty measurement literature. Effectively, extant measurement protocols, as reflected in a number of real-valued measures of poverty in current use, seem to suggest that income focus must be respected though population focus may (indeed, must) be violated. This partial and inconsistent deployment of the constituency principle in measurement exercises which nevertheless are governed

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by rigorous axiom systems is ultimately a reflection of some confusion on precisely what one means when one claims to be engaged in measuring poverty. Here is an attempt, through the employment of an analogy, to uncover the nature of the confusion. Suppose we are interested in measuring the ‘blueness’ of a purple mixture of given quantities of blue and red paint. It appears to me that there are at least two different notions of the measurement of ‘blueness’ that one could have in mind: 1. 2.

The first notion would relate to the question of ‘how blue’ is the purple. The second notion would relate to the question of ‘what is the quantity of blue’ in the purple.

We can tell these two notions apart with the aid of the following test. Suppose, first, that we employed a darker shade (without altering the amount) of red paint in the purple. Suppose, next, that we employed a greater quantity of (the same shade of) red. In either case, the described changes to the purple mixture would certainly lead to the judgement that the purple has become ‘less blue’, though neither change is compatible with the notion that ‘the quantity of blue’ in the purple has changed. We can think of the blue as the ‘poor’ and the red as the ‘non-poor’. Employing a darker shade of red is analogous to increasing the income(s) of the non-poor, while employing a greater quantity of red is analogous to increasing the size of the nonpoor population. If measured poverty is to be invariant with respect to increases in non-poor incomes or populations—as a properly exhaustive concession to a focus axiom would demand—then we are effectively subscribing to a measure of poverty that seeks to assess the ‘quantity of poverty in a society’. If, however, we do not find the focus axiom to be persuasive, then we are effectively subscribing to a measure of poverty that seeks to assess ‘how poor’ a society is. The distinction, in terms of population focus, can be clarified with the help of yet another analogy. Imagine a very small cup of coffee with two spoons of sugar in it and a very large cup with three spoons of sugar in it. It would be natural to judge that the first cup of coffee is more sugary than the second cup, even though it is also true that there is a smaller quantity of sugar in the first cup than in the second. The distinction made above between ‘how poor a society is’ and ‘the quantity of poverty there is in a society’ echoes an invitation, in Hassoun (2010; p. 8), to recognize just such a difference: ‘…it may be important to clearly distinguish between a population’s poverty and how much poverty there is in a population’. This strikes me as being a key insight, but while Hassoun seems to regard ‘how much poverty there is in a population’ to be the only proper object of poverty measurement, my own inclination would be to allow both notions of poverty alluded to above to be quantified by a measure of poverty, provided there is a clear declaration of which conception of poverty it is that is being measured. It is perhaps understandable that one’s intuitions on how compelling a focus axiom is could be shifting and uncertain. An unstructured view of the matter, which is informed largely by immediate apprehension, is compatible with positions both for and against a focus axiom, an issue that is discussed with the help of some illustrations in the Appendix. The examples reviewed in the Appendix suggest that

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there is reason to believe that both views of the intended meaning of a measure of poverty (‘how poor is a society?’ [which flows from denying any version of a focus axiom] and ‘what is the quantity of poverty in a society?’ [which flows from deferring to both income focus and population focus]) are valid ones, although, of course, it would be a great help for the practitioner to explicitly specify which view she espouses. It is important to add here that the ‘how poor is a society?’ view of poverty does not derive only from denying the appeal of focus (while accepting a clear demarcation between the poor and the non-poor segments of a population). It can also arise in a situation where ‘focus’ is an irrelevant concept. This would happen if the ‘identification exercise’ is not seen to be an integral or essential part of the poverty measurement exercise. In such a situation, no recourse is had to the specification of a ‘realistic’ poverty line intended to serve the purpose of certifying individuals with incomes in excess of the line as being wholly and unambiguously non-poor. As we shall see later, ‘fuzzy’ views of poverty, in which (effectively) every individual in a society is seen as being more or less poor, are compatible with this perspective on poverty. There are two implications to the issue of whether or not one subscribes completely and consistently to a constituency principle in the measurement of poverty. The first implication is that partial acknowledgement of a constituency principle (as reflected, for instance, in acceptance of the income focus axiom and denial of the population focus axiom) is unreasonable and inconsistent—and also a feature of most poverty indices in use: if the inconsistency is sought to be rectified by requiring respect for the population focus axiom as well, then this, in conjunction with other commonly accepted properties of poverty measures, can result in logically incoherent aggregation. The second implication is that in either of the events of a comprehensive acceptance or comprehensive rejection of a constituency principle, or in the event of a bypassing of the identification exercise, poverty indices would have to be specified differently from the way in which they have thus far been specified in the bulk of the poverty measurement literature. These issues, which are both simple and basic, but arguably also of importance for the cause of meaningful measurement, are reviewed in the rest of this essay.

12.2 Focus and the Possibility of Coherent Aggregation An income distribution is an n-vector x = (x1 , . . . , xi , . . . , xn ), where xi (≥ 0) is the (finite) income of individual i in an n-person society, and n is any positive integer. The set of all n-vectors of income is Xn , and the set of all possible income vectors is the set X ≡ ∪n Xn . We shall let D ⊆ X stand for the comparison set of income distributions, viz. the set of conceivable income vectors whose poverty ranking we seek. The poverty line is a positive level of income z such that persons with incomes less than z are designated poor and the rest non-poor. For all x ∈ D and z ∈ R++ (where R++ is the set of positive real numbers), n(x) is the dimensionality of x, N (x) is the set of all people represented in x, Q(x; z) is the set of poor people represented

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in x, and xzP is the subvector of poor incomes in x. It will be convenient to see D as a collection of ordered income vectors, that is, for every x ∈ D, the individuals in N (x) will be indexed so as to ensure that x i ≤ x i+1 for all i ∈ N (x) − {n(x)}. For every x ∈ D and i ∈ N (x), the rank order of the ith poorest person in the vector x is defined as ri (x) ≡ n(x) + 1 − i (with income ties taken to be broken arbitrarily). If R is the set of reals, then a poverty measure is a function P : D × R++ → R such that, for all x ∈ D and z ∈ R++ , P assigns a real number which is supposed to be a measure of the poverty associated with the regime (x; z). We define D∗ to be the set of all vectors in which each person’s income is zero: D∗ ≡ {x ∈ D|xi = 0∀i = 1, . . . , n(x)}. In everything that follows, we shall assume the poverty index P to be anonymous, that is to say, invariant to interpersonal permutations of income. The income focus (IF) axiom requires that for all x, y ∈ D and z ∈ R++ , if n(x) = n(y) and xzP = yzP , then P(x; z) = P(y; z). The population focus (PF) axiom requires that for all x, y ∈ D and z ∈ R++ , if y = (x, x) for any x ≥ z, then P(x; z) = P(y; z). The maximality (M) axiom requires that for all x, y ∈ D and z ∈ R++ , if x ∈ D∗ and y ∈ / D∗ , then P(x; z) ≥ P(y; z). The poverty growth (PG) axiom requires that for all x, y ∈ D and z ∈ R++ , if xzP = (x, . . . , x) for any x satisfying 0 ≤ x < z, N (x)\Q(x) = , and y = (x, x), then P(y; z) > P(x; z). Axiom IF requires measured poverty to be invariant with respect to increases in non-poor incomes, while Axiom PF—see also Paxton’s (2003) poverty noninvariance axiom—requires measured poverty to be unchanging with respect to increases in the non-poor population. It may be mentioned here that Axiom PF is diametrically opposed in spirit to what Kundu and Smith (1983) call the ‘non-poverty growth axiom’, which is the requirement that poverty should decline with an increase in the non-poor population. Population focus, clearly, takes a ‘quantity of poverty in a society’ view of a poverty measure, while the non-poverty growth property takes a ‘how poor is a society’ view of a poverty measure. As stated earlier, it may well be hard to display dogmatism, one way or the other, with respect to the appeal of a focus axiom. In any event, it is not a concern of this paper to argue the substantive merits of one position or the other. What, however, can be said is that it does seem to be inconsistent to see merit in one of the two focus axioms but not in the other. Inasmuch as virtually all known poverty indices satisfy Axiom IF, there appears to be a strong case for votaries of IF to require a poverty measure to also fulfil Axiom PF. The difficulty is that Axiom PF, in conjunction with Axioms M and PG, both of which are standard features of most known poverty indices, results in incoherence. Before examining this problem, let us take quick stock of the maximality and poverty growth axioms. The maximality axiom simply requires that poverty is never worse than when every person in the population has zero income. This property is satisfied by most known poverty indices and is compatible with a normalization axiom due to Pattanaik and Sengupta (1995) which requires, in part, that the poverty measure should simply be the headcount ratio when the entire population has zero income: as the authors

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point out, this accords with a standardization procedure in which the upper bound on the measure is defined by unity—the case of ‘extreme’ poverty where every person is maximally poor (i.e. has zero income). (Indeed, a weaker version of the maximality axiom can be obtained by simply normalizing the poverty index to lie in the interval [0,1] and requiring that, for all x ∈ D∗ and z ∈ R++ , P(x; z) = 1. )1 Finally, the poverty growth axiom requires that in a situation where there is at least one non-poor person and where all the poor have the same income, the addition of another person to the population with this income should cause measured poverty to rise. This is a weakened version (see Subramanian 2002 and Hassoun and Subramanian 2011) of a similar axiom, with the same name, proposed by Kundu and Smith (1983). Notice that the poverty growth axiom would make sense whether we understood a poverty measure to signify ‘how poor’ a society is or ‘what the quantity of poverty’ in the society is. Population focus, clearly, takes a ‘quantity view’ of poverty. Maximality, on the other hand, clearly adopts a ‘how poor’ view of poverty. The combination of these clashing views on what constitutes a ‘measure of poverty’ must inevitably result in contradiction, as suggested by the following simple proposition. Proposition There exists no poverty measure P : D × R++ → R satisfying the maximality (M), population focus (PF), and poverty growth (PG) axioms. Proof Let z be the poverty line, and let x be a level of income such that x ≥ z. Consider the income vectors a = (0, . . . 0), b = (a, x), and c = (b, 0). By Axiom PG, P(c; z) > P(b; z), and by Axiom PF, P(b; z) = P(a; z), whence P(c; z) > P(a; z) which, however, contradicts P(a; z) ≥ P(c; z), as dictated by Axiom M (Q. E. D.). What the proposition above suggests—and similar impossibility results can be found in Subramanian (2002) and Hassoun and Subramanian (2011)—is the following. If we wish to defer to a constituency principle, we should do so in its entirety; that is, we must accept the population focus, and not only the income focus, axiom. When, in the cause of consistency, this latter requirement is explicitly imposed on a poverty measure, then we find that its combination with other properties that are a standard feature of known poverty indices of the type that satisfy income focus leads to impossibility. Virtually, all available real-valued measures of poverty in the literature satisfy income focus, but they also incorporate the headcount ratio, which—as pointed out in Hassoun (2010)—violates Axiom PF (notice that the headcount ratio declines with a rise in the non-poor population). This inconsistent attitude towards focus is troublesome and leads to the problem of ‘incoherent aggregation’ alluded to earlier.

1

This was pointed out to the author, in personal communication, by Manimay Sengupta.

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12.3 Where Focus is Comprehensively Respected: Examples of Measures Which Assess the ‘Quantity of Poverty’ The preceding observations suggest that extant measures of poverty are essentially confused about what view of poverty is actually sought to be captured by its measurement. If a ‘focus’ view—one that upholds a ‘quantity of poverty’ interpretation—is favoured, then it will not suffice to defer to income focus alone: population focus must also be deferred to. An implication of such comprehensive deference to a focus principle is, as mentioned earlier, that one cannot continue to advance the sorts of poverty measures that abound in the literature. This problem is, however, susceptible of a ready solution. Specifically, one particularly simple means to the end of satisfying both income and population focus would be to take any of the many widely employed poverty measures in current use—measures which incorporate the headcount ratio– and simply multiply them by the size of the total population (see Subramanian 2000; Hassoun 2010). For instance, if P S (x; z) is the Sen (1976a) measure of poverty for an income vector x when the poverty line is z, then the simplest way of deriving from this measure one which also satisfies population focus is to advance the cause of the measure 

S

P (x; z) ≡ n(x)P S (x; z).

(12.1)

The proposal, that is, is simply to replace—in the composition of any ‘standard’ poverty index which satisfies income focus—the headcount ratio by the aggregate headcount. It must be added here that one well-known poverty index that adopts a consistent stance towards focus by satisfying both the income focus and the population focus axioms is the so-called income-gap (or IG) ratio which, for any income vector x and poverty line z, and given that μzP (x) is the mean income of the poor population, is defined by   P IG (x; z) ≡ z − μzP (x) /z.

(12.2)

P IG is not a generally favoured index of poverty because it takes no account of the prevalence—as measured by any headcount—of the poor. Nevertheless, it bears remarking that P IG is one of the few known indices in the poverty measurement literature that manifests a consistent response to the notion of focus.

12.4 Where Focus is Comprehensively Violated: Examples of Measures Which Assess the ‘Poorness’ of a Society One is not, of course, obliged to see any merit in any version of a focus axiom: one is, after all, free to take a view of poverty which measures ‘how poor’ a society is,

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rather than ‘how much poverty there is’ in the society. A consistent response to the merits of a focus axiom, that is, resides as much in accepting both income focus and population focus as in rejecting both. This raises the question of the sort of poverty measure one might expect to derive when one comprehensively discards focus, even while accepting the view that a section of a population can be unambiguously nonpoor. As it happens, there is at least one known index in the poverty measurement literature, due to Anand (1977), which rejects both population and income focus. For any income vector x ∈ D and poverty line z, and given that μ(x) is the mean income of the distribution, Anand’s modification of the Sen index of poverty (see also Thon 1979) is yielded by  P A (x; z) ≡

 z P S (x; z). μ(x)

(12.3)

Notice that like any other ‘standard’ poverty measure that incorporates the headcount ratio, the index P A violates population focus; in addition, it violates income focus too: an increase in the income of a non-poor person would change the value of μ and therefore of the poverty measure. Despite the consistent stance displayed by P A towards the notion of focus, it is ironical that its appeal as a measure of poverty (as distinct from its appeal as a measure of ‘the difficulty of alleviating poverty’) has been questioned (see, e.g., Sen 1981). It could be argued that this criticism misses the point that P A is a consistent measure of poverty which, by denying any merit to any version of a focus axiom, asserts the validity of a view of poverty that reflects the notion of ‘how poor a society is’. Indeed, and by contrast, it is not quite clear precisely what view of a measure of poverty an index such as P S upholds, since it endorses one focus axiom (income) and violates the other (population). A second example of a poverty measure which violates both focus axioms is one due to the present author (Subramanian 2009a). This measure of poverty seeks to incorporate the possibility that an individual’s deprivation status is determined not only by her own income but also by the average income of the social or ethnic or geographic (or other appropriately relevant) group that she is affiliated with. The idea is to capture some element of ‘horizontal’ (inter-group) inequality, in addition to the more familiar phenomenon of ‘vertical’ (interpersonal) inequality, in the measure of poverty, by postulating an externality arising from group affiliation such that, other things equal, a poor person’s poverty status is seen to be a declining function of the average level of prosperity of the group to which she belongs. Suppose the population to be partitioned into a set of mutually exclusive and exhaustive subgroups (such as on racial lines), that x is the given income distribution, and that μi (x) is the average income of the group to which  personi belongs. Then, person i’s deprivation z−xi (which is just a specialization of the status can be written as: di = max z+μ i ,0 requirement that a poor person’s poverty status be a declining function of both her own income and of her group’s average income). One can now define a ‘groupaffiliation externality’-adjusted measure of poverty, P G (where the superscript G is a reminder of the ‘group’-mediated deprivation which the measure incorporates), as a

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simple average of the values of the individual deprivation functions di : for all x ∈ D and z ∈ R++ ,    z − xi 1 . P (x; z) = i∈Q(x) z + μi (x) n(x) 

G

(12.4)

(Note that, properly speaking, the precise ‘grouping’, or partitioning of the population into groups, ought to be entered as an argument in the poverty function, but this has been omitted in Expression (12.4) in order to lighten the notational burden.) Motivationally, the distinctive feature of the poverty measure P G is that it takes account of the inter-group distribution of poverty: specifically, for any given x and associ ated distribution μi (x) of the mean incomes of the groups to which individuals belong, poverty is minimized when μi (x) = μ(x)∀i = 1, . . . , n(x). Of interest in the context of the present paper’s concerns is that P G , like most conventional poverty indices, violates population focus; additionally, and like the index P A , it also violates income focus: changes in non-poor incomes will typically change group means and therefore—given (4)—the value of the poverty index.

12.5 Where Focus is Irrelevant: Further Examples of Measures Which Assess the ‘Poorness’ of a Society Yet, another means of addressing the question of how to measure poverty when focus is not deferred to is to adopt a thoroughgoingly fuzzy approach, in one specific sense, to reckoning poverty. In this approach, one is under no compulsion to truncate a distribution at some specified poverty line and focus one’s attention only on the income distribution below the poverty line. The idea, rather, is to allow for the possibility that everybody in a society is more or less poor (which is compatible, as we shall see, with specifying a ‘pseudo-poverty line’, that is, a line pitched high enough that no-one’s income is likely to exceed it): in effect, one can, under this procedure, avoid engaging in the messy ‘identification exercise’ of specifying a ‘realistic’ and satisfactory poverty line. In particular, a vague approach to reckoning poverty is compatible with postulating a ‘fuzzy membership function’, which assigns a ‘poverty status’ in the interval [0, 1] to each individual income in a distribution. On fuzzy poverty measurement, the reader is referred to, among others, Kundu and Smith (1983), Shorrocks and Subramanian (1994), Chiappero-Martinetti (2000), Qizilbash (2003), and Subramanian (2009b) (It should be noted that the fuzzy approach described in this paper corresponds to what Qizilbash (2003) calls the ‘degree’ approach, and which he distinguishes from ‘epistemic’ and ‘supervaluationist’ approaches to accounting for vague predicates. In particular, my concern is not to claim any particular and superior merit for the fuzzy approach considered here, but rather to present illustrative examples of how one variant of such an approach might effectively deny a focus axiom any role in the measurement of poverty).

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Before we deal with membership functions, some further investment in notation is required. With this in mind (see Subramanian 2009b), let the highest level of income  in any of the vectors contained in D be designated by x: ¯ that is, x¯ ≡ maxx∈D xn(x) . It is useful now to define a sort of ‘pseudo-poverty line’ as a very large, finite level of income Z which is ‘sufficiently larger than x’ ¯ to ensure that every person in each of the distributions contained in D would have to be regarded as being (more or less) poor. The precise value assigned to Z is not a matter of any great significance: as we shall see, Z will simply be employed as a ‘device’, dictated by considerations of arithmetical convenience, for deriving a fuzzy poverty index. Fuzzy poverty membership functions can be—to borrow the terminology even if not the exact context of Barrientos (2010)—‘relational’ or ‘non-relational’ (corresponding, respectively, to ‘Egalitarian’ and ‘Prioritarian’ social valuations of income, in the sense of Parfit 1997, though it could be needlessly misleading to employ these latter terms in the present context). ‘Relational’ membership functions assign poverty status to an income after locating that income within the overall distribution of income, that is to say, in relation to other incomes, in a distinctly ‘menu-dependent’ way. ‘Non-relational’ membership functions, on the other hand, assign the same poverty status to any given level of income irrespective of what relation the income level in question bears to other incomes in the income distribution, that is to say, in a distinctly ‘menu-independent’ way. In either case, poverty status may be expected to be a non-increasing function of income (a simple and appealing monotonicity requirement) and to be bounded from below by zero (no poverty) and from above by unity (complete poverty), which is a simple zero–one normalization. Formally, ‘relational’ membership functions are drawn from the set M R ≡ {m : R+ × D → [0, 1]|m(x; x) is continuous and non-increasing in x ∀x ∈ D, m(0; x) = 1 ∀x ∈ D, and Limx→Z m(x; x) = 0 ∀x ∈ D}. ‘Non-relational’ membership functions are drawn from the set MNR ≡ {m : R+ → [0, 1]|m(x) is continuous and non-increasing, m(0) = 1, and Limx→Z m(x) = 0} (R+ is the non-negative real line). Here is an example of a ‘relational’ membership function (employed in Subra ∈D manian 2009b). Given an ordered income vector x = x1 , . . . , xi , . . . , xn(x)   Z −xi  ri (x) 1 , where, and some Z, consider the membership function m (xi ; x) ≡ Z n(x) to recall, ri (x) ≡ n(x) + 1 − i is the rank order of the ith poorest person in the vector x. The function m 1 (xi ; x) assigns to the ith poorest person in the vector x a poverty status that depends on i’s income and also—in a ‘relational’ way—on her rank order in the vector. Notice that m 1 declines as we move up the income ladder, that m 1 (xi ; x) = 1 when xi = 0, and that m 1 (xi ; x) goes to zero as xi goes to Z, so m 1 satisfies the ‘monotonicity’ and ‘normalization’ properties alluded to earlier. A fuzzy poverty index can be written as a simple average of the poverty statuses, as reflected in the values of their membership functions, of all the individuals in a society. Such a poverty index, corresponding to the fuzzy membership function m 1 , will then be given, for any x ∈ D and Z, by

12.5 Where Focus is Irrelevant: Further Examples of Measures …

P 1 (x; Z ) = [1/n(x)]

i∈N (x)

m 1 (xi ; x) = [1/n(x)]

149



i∈N (x)

Z − xi Z



 ri (x) . n(x) (12.5)

Consider Sen’s (1976b) welfare index, or ‘distributionally adjusted measure of real national income’, which is given, for any x ∈ D, by W S (x) = μ(x)[1 − G(x)],

(12.6)

where G(x) is the Gini coefficient of inequality in the distribution of incomes in x. It has been shown in Subramanian (2009b) that if n(x) is large enough to allow [n(x) + 1]/[n(x)] to be approximated to 1, then the fuzzy poverty measure P 1 can be approximated to the expression 1/2 − (1/2Z )μ(x)[1 − G(x)]. That is to say, for any x ∈ D and Z, the following is true for sufficiently ‘large’ values of n(x): P 1 (x; Z ) = 1/2 − (1/2Z )W S (x).

(12.7)

As pointed out in Subramanian (2009b), the fuzzy poverty index P 1 is simply a re-cardinalization of the ‘crisp’ welfare index W S : the one is just a negative affine transform of the other, and this enables us to see that the particular value attached to Z (so long as Z satisfies x¯ 1, and t > 0, [I (x) < I (λx)] and [I (x) > I (x + t)], where t ≡ (t, ..., t) and n(x) = n(t).4 To sum up: the case for intermediate inequality measures asserts itself from the (not unreasonable) recognition of the possibly ‘extreme’ values underlying scaleand translation-invariant inequality measures. Among the best known of intermediate inequality measures are those due to Kolm (1976) and Bossert and Pfingsten (1990). We shall not review these measures here because, as pointed out by Zheng (2007), neither of these measures satisfies even the weak requirement of unit-consistency, and this is problematic for an inequality measure. As it happens, two approaches to an intermediate view of inequality which are, additionally, compatible with unit-consistency are available from pursuing directions of enquiry suggested by Moyes (1987) and del Rio and Alonso-Villar (2010). In what follows, we present ingredients of a simple account of these two approaches to inequality measurement. We follow this up with a discussion of yet another intermediate measure of inequality, the one due to Krtscha (1994) which, despite its drawbacks, is nevertheless a useful index to employ.

3

Kolm was himself well aware of this. As he puts it (Kolm 1976a; p. 419): ‘The topic was an equal increase in all incomes rather than an equal decrease in them. But it is the first point which is relevant in our progressive societies. Anyway, all that is said here is that it is no less legitimate to attach the inequality between two incomes to their difference than to their ratio. One view must not be judged from the other’s prejudice. The term “leftist” used below must not be taken too literally: the measure corresponds to this view of society in very important cases, not in all imaginable ones’. 4 Symmetrically, of course, we would also require of an intermediate inequality measure I that for all x ∈ X, λ ∈ (0, 1) and t < 0, [I (x) > I (λx)] and [I (x) < I (x + t)], where t ≡ (t, ..., t) and n(x) = n(t).

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163

13.3 The Lorenz Curve and the Gini Coefficient: Beyond Pure Relativism While the formal basis of the distinction between relative and absolute conceptions of inequality has been dealt with in the preceding section, we shall consider here a more specific application of the issue, by focusing on the distinction between the relative and the absolute Lorenz curve, as also between the relative and the absolute Gini coefficient of inequality. The short but pointed paper of Moyes (1987) is of considerable significance in this context; the reader is also referred to Yoshida (2005). Given a non-decreasingly ordered income/wealth n-vector x = (x1 , ...xi , ..., xn ) with mean μ, where xi is the income/wealth of the ith poorest person, the relative (respectively, absolute) Lorenz curve for the income/wealth distribution under review, is defined by L R (x; i/n) = (1/n)

i 

(x j /μ), i = 1, ..., n;

j=1

and L A (x; i/n) = (1/n)

i 

(x j − μ), i = 1, ..., n.

j=1

(We shall take it that L R (x; 0) = L A (x; 0) = 0.)  The relative Lorenz curve is obtained by connecting the points (0, 0), ..., (i/n, ij=1 (x j /nμ), ..., (1, 1). For a typical, unequal distribution, the relative Lorenz curve is a non-decreasing convex curve beginning at the point (0, 0) and terminating at the point (1, 1) of the unit square. For a perfectly equal distribution, the relative Lorenz curve coincides with the diagonal of the unit square. Figure 13.1 a presents a typical relative Lorenz curve. The absolute Lorenz curve is obtained by connecting the points i  (x j − μ)/n), ..., (1, 0). For a typical, unequal distribution, (0, 0), ..., (i/n, j=1

the absolute Lorenz curve is a double-valued function beginning at the point (0,0), which initially falls and then rises, before terminating at the point (1, 0) in the fourth quadrant. For a perfectly equal distribution, the absolute Lorenz curve coincides with the abscissa, while for a perfectly concentrated distribution, it is the straight line from the origin to the point (−)μ on the ordinate for all values on the abscissa less than 1 and becomes zero discontinuously at 1. Figure 13.1b presents a typical absolute Lorenz curve. It may be noted that the relative and absolute Lorenz curves reflect, respectively, two alternative traditional ways of comprehending the dissimilarity between two quantities—the first as a ratio of the two quantities and the second as the difference between them: specifically, if μi is the average income/wealth of the poorest i recipients, then a typical point on the relative Lorenz curve will be given by

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Fig. 13.1 a Relative Lorenz Curve. b Absolute Lorenz Curve

(i/n, (i/n)(μi /μ)), while a typical point on the absolute Lorenz curve will be given by (i/n, (i/n)(μi − μ)). The relative Gini coefficient of inequality is twice the area enclosed by the relative Lorenz curve and the diagonal of the unit square, while the absolute Gini is twice the area enclosed within the absolute Lorenz curve, which, in turn, is just the relative Gini times the mean μ of the distribution. The expressions for the relative and absolute Gini coefficients have been provided in Eqs. (13.1) and (13.3), respectively, of Sect. 13.2. For those unaccustomed to the absolute Gini coefficient, it may appear to be an ‘extreme’ measure of inequality. However, as the philosopher P. G. Wodehouse has often pointed out, ‘everything depends on the point of view’: in particular, the relative Gini is no less ‘extreme’ for the absolutist than the absolute Gini is for the relativist!

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165

A middle path between the two versions of the Gini is afforded by a measure GI which one may call the ‘Intermediate Gini’ index, and which is obtained, quite simply, as a product of the two versions: for all x ∈ X, G I (x) = G A (x)G R (x)[= μ(x){G R (x)}2 ].

(13.5)

The version of the intermediate Gini in (13.5) has been proposed in Subramanian and Jayaraj (2013)5 and employed by the authors in the estimation of inequality in the over-time distribution of consumption expenditure in India. The ‘product rule’ is, of course, arbitrary, but not unreasonable inasmuch as the product of any relative inequality measure with an absolute inequality measure must yield an intermediate measure. This follows from the fact that when all incomes are uniformly scaled up, the relative component of the intermediate measure will remain unchanged, while the absolute component will increase, causing the product to increase, and when all incomes are raised by the same absolute amount, the absolute component of the intermediate measure will remain unchanged, while the relative component will decline, causing the product also to decline. That G I is an intermediate measure follows from the fact that a uniform scaling up of all incomes will leave G R unchanged and increase the value of G A (thereby causing G A G R to rise), while a uniform increment to all incomes will leave G A unchanged and reduce the value of G R (thereby causing G A G R to decline). (Indeed, this result generalizes to the proposition that any inequality measure which is a product of a relative measure and an absolute measure must be an intermediate measure.) A nice feature of the intermediate measure G I is that it is unit-consistent: this is evident from the fact that, for all x ∈ X and all λ ∈ R++ , G I (λx) = G R (λx)G A (λx) = μ(λx){G R (λx)}2 = λμ(x){G R (x)}2 ≡ λG I (x), and similarly, G I (λy) = λG I (y), whence [G I (x) ≥ G I (y)] → [G I (λx) ≥ G I (λy)], as required for unit consistency to be satisfied. A final point to note about absolute and intermediate indices of inequality is that, by virtue of being mean-dependent, the means of distributions across different regimes of comparison must be expressed in comparable units. This means that in time-series comparisons means must be deflated to a common base year through employment of an appropriate price index; in cross-section comparisons, means must be expressed in terms of an appropriate common currency that will reflect a proper ‘real’ exchange rate, and in time-series-cum-cross-section comparisons, one must have recourse to both an appropriate inter-temporal price deflator and an appropriate cross-regional exchange deflator. The choice of appropriate deflators is seldom unproblematic; and to the extent that scale-invariant inequality measures do not require us to employ deflators, such measures dispense with a potentially troublesome complexity. This is sometimes held out as a justification for exclusive reliance on relative inequality measures. It is important to note though that while relative measures avoid a specific complexity that is a feature of mean-dependent 5

For an alternative intermediate version of the Gini coefficient, the reader is referred to Atkinson and Brandolini (2004).

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measures, the complexity is not a painlessly avoidable one. To turn one’s back on an arguably conceptually sound procedure simply because it presents some practical difficulties in implementation is not a particularly compelling approach to adopt. In any event, the problem of identifying completely satisfactory price indices or real exchange rates is one that is commonly encountered in other branches of economic inquiry as well, such as real-income comparisons, welfare comparisons, and poverty comparisons.6

13.4 A ‘Linear Invariance’ Approach to Intermediate Inequality Measures Suppose we begin with some initial distribution of income which is subjected to an increase in its mean. If the incremental income is distributed equally, then an absolute measure of inequality would certify that inequality has remained unchanged. If the incremental income is distributed in the same proportion as obtains for the existing distribution, then a relative measure of inequality will certify that inequality has remained unchanged. An intermediate measure will preserve inequality when the incremental income is distributed equally in part and, in part, according to the pattern of distribution in the initial vector of incomes. A ‘linear invariance’ approach to intermediate inequality measures will require the path of inequality invariance to be linear, by which is simply meant that the division of emphasis between relative and absolute orientations should remain constant with successive increments of mean income. This is an avenue of enquiry that has been explored by, among others, del Rio and Ruiz-Castillo (2000), del Rio and Alonso-Villar (2008), and Azpitarte and Alonso-Villar (2011). Del Rio and Alonso-Villar (2008) actually advance an intermediate measure of inequality which is compatible with linear invariance (or what the authors call ‘ray invariance’). We shall not ourselves discuss this measure, but rather employ the ray-invariance approach in order to compare inequality from an intermediate perspective along lines suggested in del Rio and Ruiz-Castillo (2001), Azpitarte and Alonso-Villar (2011), and, especially, Bosmans et al. (2011). Suppose x and y to be two distributions of income with means μ(x) and μ(y), respectively, with μ(y) > μ(x). Employing x as the reference distribution, let ya be a distribution obtained by adding to each person’s income in x an equal division of the 6

Indeed, Kolm (1976a; p.419) anticipates the issue when he says: ‘When all incomes xi are multiplied by the same number, whereas [a relative inequality measure]… does not change, [an absolute inequality measure]…is multiplied by this number. Therefore, if we study variations of [an absolute inequality measure]… over time in an inflationary country, we must call xi the real incomes, discounted for inflation; or if we make international comparisons of [an absolute inequality measure]…, we must use the correct exchange rates. This need not be done if we use [a relative measure]. But these problems are exactly the same ones which are traditionally encountered in the comparisons of national or per capita incomes, and they can be given the same traditional solutions. Anyway, convenience could not be an alibi for endorsing injustice’. Moyes (1987) strongly endorses this view of Kolm’s.

13.4 A ‘Linear Invariance’ Approach to Intermediate …

167

increase in aggregate income associated with the passage from x to y, viz. ya = x+m where m ≡ (μ(y) − μ(x), ..., μ(y) − μ(x)) and n(m) = n(x). Similarly, let yr be a distribution obtained by simply scaling the initial distribution x up bya factor equal μ(y) r to the ratio of the y and x distributions’ respective means, viz. y = μ(x) x. Let yˆ

be a convex combination of ya and yr (ˆy = βya + (1 − β)yr , 0 ≤ β ≤ 1) such that yˆ is judged to be inequality-wise indistinguishable from x. Here, the parameter β encompasses our judgment on the division of emphasis between absolute and relative orientations such that, with this division, the resulting distribution yˆ is pronounced to have the same extent of (intermediate) inequality as the initial, reference distribution x. The value of β will reflect how ‘right-’ or ‘left-’ leaning our intermediate stance is. If β is unity, we are wholly ‘leftist’; if β is zero, we are wholly ‘rightist’; as β increases from zero to one-half, we become progressive less ‘right-leaning’; as β increases from one-half to unity, we become progressively more ‘left-leaning’; and when β is exactly one-half, we may be said to be ‘properly’ centrist. In Fig. 13.2, the point P represents some initial (unequal) distribution. The straight line from the origin through P is a locus of points for which income is distributed between the two individuals in the same proportion as at the point P. Call this the ‘relative line’ (RL). A 45° line drawn with P as the origin is a locus of points for

Fig. 13.2 Representation of a linear inequality invariance curve for a ‘Properly Centrist’ orientation

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13 Assessing Inequality in the Presence of Growth: An Expository Essay

which income is equally distributed. Call this the ‘absolute line’ (AL). Suppose we judge that the inequality level at P is preserved, for successive increments of income, by dividing the income into two halves, with one-half distributed between the two individuals in the same proportion as in P and the other half by splitting it evenly. Then, the straight line which bisects the angle between the ‘absolute line’ and the ‘relative line’ previously described is a locus of points for which the incremental income is divided between the two individuals, after the fashion of a ‘compromise’ between absolutist and relativist orientations, in the manner just indicated. In terms of Fig. 13.2, the ‘compromise’ point for the first incremental rupee is the point Q. Continuing to treat P as the initial, reference distribution, if we were to repeat the operation just described, we would arrive at R as the compromise point corresponding to the second incremental rupee. With P remaining the reference distribution and once more traversing the steps just described, we would arrive at the point S which is the compromise point corresponding to the third incremental rupee. The ‘compromise’, or iso-inequality, path is, then, the straight line connecting the points P, Q, R, S…. If the two individuals’ incomes are interchanged, then the point P would become the new initial, reference distribution (a point symmetric about the 45° line to the point P), and Q  , R  , S  ,… are the ‘compromise’ points corresponding to each incremental rupee that is divided, starting with the initial distribution at P . The paths described by connecting the points P, Q, R, S,… and P  , Q  , R  , S  are, as can be seen from Fig. 13.2, straight lines which are symmetric about the 45° line. The path of inequality invariance is thus a linear curve (drawn, in Fig. 13.2, for a ‘properly centrist’ orientation, entailing that β = 1/2). If distribution y lies to the left of the iso-inequality line traced by the points P, Q, R, S,…, then y is more unequal than x; if y lies on the iso-inequality line, then y is exactly as unequal as x; if y lies to the right of the iso-inequality line, then y is less unequal than x.7 In general, and given an n-vector of incomes (n ≥ 2), how may we employ the ray-invariance apparatus to compare inequalities? Suppose, for specificity, that we choose to be ‘properly’ centrist. Suppose also that an initial reference distribution x with mean μ(x) evolves into another distribution y with a higher mean income μ(y). How may we compare inequality in x with inequality in y in terms of our ‘properly centrist’ orientation? Note first that in terms of the stated orientation, we should hold μ(y) x), yˆ to be inequality-wise indistinguishable from x, when yˆ = (0.5)(x + m + μ(x) with m ≡ (μ(y) − μ(x), ..., μ(y) − μ(x)) and n(m) = n(x). Now consider any conventional, widely used relative measure of inequality—say the relative Gini coefficientG R . Call G R (ˆy) the warranted Gini, that is, the value of the Gini coefficient for the distribution yˆ , which has been judged to be inequalitywise indifferent to the reference distribution x, and call G R (y) the actual Gini, that is, the value of the Gini coefficient for the actual distribution y which is the subject of comparison with x. If the warranted Gini is greater than the actual Gini, then we 7

This would also be true for any other distribution z. Del Rio and Alonso-Villar (2011) are careful to point out that in order to obtain a consistent inequality ranking of x, y, z,…, the iso-inequality curve employed must be the one drawn for a unique reference distribution, say x: one cannot, for instance, seek a ranking of x, y, and z by employing x as the reference distribution for ranking x and y, and x and z, while employing y as the reference distribution for ranking y and z.

13.4 A ‘Linear Invariance’ Approach to Intermediate …

169

can infer that inequality, according to our ‘properly centrist’ values, has declined in the transition from x to y, and we should infer just the contrary if the warranted Gini is less than the actual Gini. In this exercise of comparison, we have fixed the value of β (at one-half, as it happens; though it could be any other number between zero and one which captures our division of emphasis between the absolutist and relativist conceptions of inequality). An alternative approach might be to find out that value of β—call it β∗—for which the warranted Gini just equals the actual Gini. Suppose β∗ turns out to be 0.20. Then, the warranted Gini would exceed the actual Gini for all values of β less than 0.20, and we would pronounce inequality to have declined in the passage from x to y only provided our notion of intermediate inequality sits comfortably with the relatively low value for β of 0.20, that is, only if we are rather seriously ‘right-leaning’. The ray-invariance approach to reckoning intermediate inequality can thus be employed to carry out comparisons of inequality in terms of familiar inequality measures such as the relative Gini. Finally, it may be noted that the ray-invariance procedure has the great advantage of satisfying the property of unit-consistency.

13.5 The Krtscha Approach to Intermediate Inequality Measures Are there other intermediate unit-consistent measures one could invoke? The Krtscha measure, it turns out, is one such attractive intermediate measure, which is given—see Krtscha (1994)—for all x ∈ X, by  n(x) 

K (x) = (1/n(x)μ(x))

 (xi − μ(x))

2

.

(13.6)

i=1

We strongly believe there is a case for a wider acceptance of the Krtscha index in routine applied work, and accordingly, in this section, we dwell at some length on an elaboration of the index.8 Various aspects of Krtscha’s (1994) proposed intermediate inequality measure are discussed in the form of a series of points in what follows. (a) Suppose, starting from a certain initial distribution of incomes, a certain additional amount of income is to be distributed: what would be a reasonable construction of an ‘intermediate’ approach to this division problem? To fix ideas, it is helpful to work, as Krtscha does, with a two-person distribution. Of the additional amount to be distributed, consider the division of the first incremental rupee. A ‘fair compromise’ division of this rupee, Krtscha suggests, would consist in distributing one-half of the rupee equally between the two individuals and the other half in the same 8

Examples of significant empirical work whose conceptual basis is also clearly explained would include Atkinson and Brandolini (2004) and Bosmans et al. (2014) on disparities in the global distribution of income, and Del Rio and Ruiz-Castillo (2000, 2001) on inequality in the Spanish distribution of income.

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13 Assessing Inequality in the Presence of Growth: An Expository Essay

Fig. 13.3 Parabolic inequality invariance (or fair compromise) curve under the Krtscha formulation

proportion as obtains with the initial distribution: note that an absolute conception of inequality would view inequality as remaining unaltered by an equal split of the incremental rupee, while a relative conception would favour the notion that inequality remains unaltered when the incremental rupee is distributed according to the proportions dictated by the existing distribution. Starting with the initial distribution at the point A in Fig. 13.3, the straight line from the origin through A is a locus of points for which income is distributed between the two individuals in the same proportion as at the point A. A 45° line drawn with A as the origin is a locus of points for which income is equally distributed. The straight line which bisects the angle between these two lines is then a locus of points for which the incremental income is divided between the two individuals in the ‘fair compromise’ just described. In terms of Fig. 13.3, the ‘fair compromise’ point for the first incremental rupee is the point B. Treating B as the new initial distribution, if we were to repeat the operation just described, we would arrive at C as the ‘fair compromise’ point corresponding to the second incremental rupee. Employing C as the new initial distribution and again going through the paces just described would fetch us at point D, which is the ‘fair compromise’ point corresponding to the third incremental rupee. One can proceed in this fashion, incremental rupee by incremental rupee, until the budget that was needed to be divided between the two individuals is exhausted. The ‘fair

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171

compromise’ path is then the curve connecting the points A, B, C, D,9 …. Notice that a flip-flop of incomes between the two individuals would lead to A , in Fig. 13.3, being the initial distribution (which is a point symmetric about the 45° line to the point A), and B  , C  , D  , . . . are the ‘fair compromise’ points corresponding to each incremental rupee that is divided, starting with the initial distribution at A . The path described by connecting the points A, B, C, D,… and A , B  , C  , D  , . . . is, as can be seen from Fig. 13.3, a parabola which is symmetric about the 45o line. The path of inequality invariance is thus a nonlinear curve. We shall return to the parabolic shape of this curve at a later stage. (b) Krtscha asks: for a two-person income distribution, what real-valued measure of inequality which is symmetric, normalized, and transfer-preferring would trace the inequality invariance path just described? Without going into the details of his derivation, we shall simply note here that Krtscha’s answer resides in an inequality ¯ where measure which is any increasing function of the quantity (x1 − x2 )2 /2 x, x1 and x2 are the incomes of persons 1 and 2, respectively, and x¯ is the mean of the two-person distribution (x1 , x2 ). For a general, n-person distribution x, it turns out—as we have already seen in Eq. (13.3)—that Krtscha’s measure is given by 

n(x)  (xi − μ(x))2 . (More accurately, we have a family of K (x) = (1/n(x)μ(x)) i=1

measures, given by the set of all increasing functions of K, but it is convenient to simply take K to be the Krtscha measure.) (c) The fact that K is a compromise measure between a relative and an absolute index of inequality is clearly reflected in the fact that, given the expressions for Krtscha’s measure, the coefficient of variation (CV ), and the standard deviation (SD) furnished earlier in Eqs. (13.6), (13.2), and (13.4), respectively, it is a very simple matter to verify that, for all x ∈ X, K (x) = C V (x) · S D(x) : the Krtscha measure is just a product of two very well-known indices of inequality—the coefficient of variation, which is a relative measure, and the standard deviation, which is an absolute measure. (We may well be mistaken, but we have not seen this simple point established elsewhere.) (d) This is a digression, but not an entirely irrelevant one. Urbanization is often measured in terms of the proportion of a population living in urban areas (a relative conception), though the size of a city as measured by the numbers of people living in it (an absolute conception) could also matter in an overall assessment of the extent of urbanization. In the case of poverty measures too, presumably both the headcount ratio H (the proportion of the population below the poverty line) and the aggregate headcount A (the absolute numbers of the poor) ought to matter in an overall headcount view of poverty. This problem was noted by Arriaga (1970) in the context of urbanization, and he came up with a measure of urbanization which is just a product of the proportion of the urbanized population and the numbers of the urbanized 9

It should be noted, in Fig. 13.3, that whether we move directly from the first incremental unit of income to the third unit, or via the second unit, the iso-inequality curve will pass through the point C: this is the property of ‘path-independence’, by which judgements on equal inequalities do not depend on the particular route taken from one distribution to another. This is an important feature of Krtscha’s proposed ‘fair compromise’ procedure.

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13 Assessing Inequality in the Presence of Growth: An Expository Essay

population. A ‘composite’ headcount view of poverty would, similarly, be given by the product HA of the headcount ratio and the aggregate headcount. The ‘compromise’ involved is precisely analogous to the Krtscha compromise of multiplying the coefficient of variation by the standard deviation. In general, there is a commonality between the neutrality-to-units-of-measurement issue and the neutrality-topopulation-replications issue in the measurement of poverty which invites remarking (a consideration of this problem is available in Hassoun and Subramanian 2011). (e) It is interesting to ask how the Krtscha measure reflects the division of emphasis between relative and absolute conceptions of inequality. It turns out that along the isoinequality curve, the share of the poorer person’s income in a two-person distribution keeps increasing. This is demonstrated with some simple formalities in the Appendix to this paper. The Appendix shows that, for any given initial average level of income, the greater the increase in income, the more ‘left-leaning’ is the resultant distribution required to be for inequality to be preserved. This latter result is what accounts for the parabolic shape of the ‘iso-inequality curve’: with each successive increment in income, the ‘equally unequal’ distribution moves closer to the ‘equal division rule’. Thus, other things equal, under the Krtscha formulation, starting with any initial distribution, as total income increases, for inequality to remain unchanged the required orientation is never to be ‘right-leaning’, and to become progressively more ‘left-leaning’ as income increases by larger amounts. (f) The Krtscha index is subgroup decomposable (see Zheng 2007): for any partitioning of the population into a set of mutually exclusive and exhaustive subgroups, the measure can be expressed as an exact sum of a ‘between-group’ inequality component and a ‘within-group’ inequality component, the latter being a weighted sum of subgroup inequality levels. More formally, suppose the population is partitioned into S(≥ 2) mutually exclusive and completely exhaustive subgroups, and that xs is the income vector, π(xs ) is the population share, σ (xs ) is the income share, a and = 1, ..., S). Define the withinI (xs ) is the inequality level of the sth subgroup (s S group component of inequality as: IW (x) ≡ s=1 ws (x)I (xs ), where ws (x) is a weight on the sth subgroup’s inequality level, with the weight depending on π(xs ) or σ (xs ) or some combination of the two. Define the between-group component of inequality I B (x) as the value of the inequality index obtained by replacing each person’s income within any subgroup s by the mean income μ(xs ) of that subgroup, so that, if xs0 ≡ (μ(xs ), ..., μ(xs )) and n(xs0 ) = n(xs ) for all (s = 1, ..., S), then I B (x) ≡ I (x10 , ..., xs0 ). For all x ∈ X, I (x) is a subgroup decomposable inequality measure if and only if I (x) = IW (x) + I B (x). For the Krtscha intermediate measure of inequality, it can be verified—see Subramanian (2011)—that, for all x ∈ X: K (x) = K W (x) + K B (x),

(13.7)

where K W (x) ≡

S  s=1

σ (xs )K (xs );

(13.7a)

13.5 The Krtscha Approach to Intermediate Inequality Measures

173

and K B (x) =

 S 

 n(xs )μ (xs )/n(x)μ(x) − μ(x). 2

(13.7b)

s=1

Subgroup decomposability is of particular salience when we are considering the inter-group inclusiveness of growth: the property enables us—for whatever meaningful socio-economic partitioning of the population we may be interested in—to work out the trend in each of the within-group and between-group components of inequality, and the relative rates at which the two components have changed over time. (Note also that the proportionate contributions of the within- and betweengroup components of the Krtscha measure of inequality, for any x ∈ X, are K (x) = K W (x)/K (x) and c BK (x) = K B (x)/K (x), with given, respectively, by cW K K cW (x) + c B (x) = 1). (g) The Krtscha index is also unit-consistent.10 To see this, note that for all x ∈ X and λ ∈ R++ , K (λx) = C V (λx)· SV (λx) = C V (x)·λS D(x) = λC V (x)· SV (x) ≡ λK (x), whence for all x, y ∈ X and λ ∈ R++ , [K (x) ≥ K (y)] → [K (λx) ≥ K (λy)], as required for unit-consistency to be satisfied. (h) In short, the Krtscha measure of inequality exhibits a number of properties which can be argued to be desirable: it is symmetric, normalized, transfer-preferring, replication-invariant, subgroup decomposable, avoids the extreme value judgments of scale- and translation-invariant inequality measures, and is unit-consistent. It is also easily comprehended as the product of two well-known inequality measures, one of which is relative (the coefficient of variation), and the other absolute (the standard deviation). Having said this, it must be clarified that it is no part of one’s intention to suggest that the Krtscha index is an unqualifiedly blemishless index. Among other things (and arising from the fact that it is a variance-based indicator), it does not satisfy Kolm’s ‘principle of diminishing transfers’, which is a ‘transfer-sensitivity’ property that demands a greater sensitivity, in an inequality measure, to transfers at the lower than at the upper end of a distribution: the Krtscha index displays no such differential sensitivity to transfers. In the end, one supposes that trade-offs are unavoidable, and it might be justified to suggest that, despite its imperfections, the Krtscha measure is at least an eligible contender in a reasonable shortlist of intermediate measures of inequality. 10

The properties of subgroup decomposability and unit-consistency are uniquely a feature of a measure derived from allocation of one-half the incremental income to distribution according to existing shares and the other half to equal distribution. The ‘half-and-half’ division is necessary to secure these properties, which are thus special to the Krtscha measure. One generalization of Krtscha which preserves these properties, due to Zheng (2007), relates to the value of the exponential parameter in the expression for the Krtscha measure (see Eq. (13.3)). An alternative route to generalization is based on the notion that the ‘half-and-half division’ is only one special, ‘centrist’, case of a general ‘alpha-and-one-minus-alpha division’, whereby one can have an entire spectrum of ‘left-to-right’ intermediate values of a parameter α which assumes values in the continuum (0,1). This is the route to generalization of Krtscha proposed by Yoshida (2005). (The generalization, however, does not accommodate subgroup decomposability and unit-consistency.).

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13 Assessing Inequality in the Presence of Growth: An Expository Essay

13.6 Summary and Conclusions In assessing the inclusiveness or otherwise of the growth process in an economy, one would imagine that a reasonable means to the end would entail tracking the over-time trend of inequality in the distribution of income in the economy. This, on the face of it, is a fairly straightforward exercise, but what tends to complicate it is the fact that there are alternative ways of reckoning inequality—through inequality measures that may be relative (or ‘rightist’) or absolute (or ‘leftist’) or in-between (or ‘intermediate/centrist’). The predominant practice in the profession is to rely on a wholly relative conception of inequality. This could bias one’s findings heavily on the side of conservatism, just as a swing to wholly absolute measures could bias one’s findings heavily on the side of liberalism. Given problems of both logical coherence and normative appeal attending the use of either exclusively relative or exclusively absolute measures of inequality, there is a case for resorting to the employment of intermediate measures. The purpose of this paper has been twofold. The first objective has been to present a relatively easily accessible exposition of the complexities of relating judgments on the distribution of a pie to the size of the pie. In the process, the properties and uses of two ‘reasonable’ intermediate measures of inequality—the so-called Intermediate Gini Coefficient and the Krtscha Measure—have been discussed: both indices satisfy a certain crucial property of ‘unit-consistency’, and one of them—the Krtscha measure—virtually uniquely among intermediate measures (Zheng 2007), also satisfies the requirement of ‘subgroup decomposability’, which is of particular value in assessing the intergroup inclusiveness of growth. The first purpose of this essay has therefore been essentially in the cause of exposition, explanation, and clarification—with little, if anything, of original content in it. The second purpose has been essentially in the cause of persuasion, namely to point to the importance of adopting a more catholic approach to the measurement of inequality in applied work, which would require relaxing the near strangle-hold which relative measures of inequality would appear to have on standard professional practice in the matter. This should pave the way for the use also of absolute measures and—more significantly—of intermediate or centrist measures that have the virtue of not biasing one’s findings on inclusiveness owing to the ‘extreme’ value judgments inherent in the normative orientation of both relative and absolute measures. ‘Inclusiveness’ is a vital ingredient of policy, narrowly, and of politics, broadly: a reasonably clear understanding of the values underlying the measurement of the phenomenon is therefore a prime requirement in its empirical appraisal. The present paper has been an effort in addressing this requirement.

Appendix

175

Appendix On the Parabolic Shape of the Krtscha Iso-Inequality Curve Let x = (x1 , x2 ) be an unequal, increasingly ordered two-person distribution, with x1 + x2 ≡ X , and let y = (y1 , y2 ), with y1 + y2 ≡ Y , be a distribution derived from x through an increase of income  from X to Y, so that Y − X ≡  > 0. Imagine now that  is comprised of some K finite and equal increments, so that the rise in total income from X to Y is seen to be composed of a sequence of transitions from one distribution to another. The transitions are supposed to preserve inequality and are of the following nature: from x to x1 ≡ (x11 , x21 ) (with x11 + x21 ≡ X 1 = X + /K ); from x1 to x2 ≡ (x12 , x22 ) (with x12 + x22 ≡ X 2 = X 1 + /K );… and from x K −1 to x K ≡ (x1K , x2K ) (with x1K + x2K ≡ Y = X K −1 +/K ). Note further that each income vector is supposed to be derived from the preceding income vector by distributing one-half of the incremental total income /K in the proportions as they obtain in the preceding vector and the other one-half equally between the two persons. The vectors x, x1 , x2 ,..., xK−1 , xK are points on the Krtscha iso-inequality curve. Given how x1 is derived from x, we can write           1  1 1 x1 1 + = (4K X x1 + 2x1 + X ). x1 = x1 + 2 X K 2 2K 4K X It follows then that the difference between the income share of the poorer person in distribution x1 and his income share in distribution x is given by x1 x1 = δ1 ≡ 1 − X1 X



 x 1 1 (4K X x1 + 2x1 + X ) − . 4K X (K X + ) X

(A1)

It can be verified that the right hand side of Equation (A2) simplifies to the following expression:  δ1 =

  (x2 − x1 ) > 0 since, by definition, (x2 > x1 ). 4X (K X + )

(A2)

Defining δ2 , ..., δ K analogously to δ1 , we can prove, exactly along the lines just demonstrated, that δk > 0 for all k = 2, ..., K . That is to say, in order to preserve the level of inequality, the income share of the poor person has to keep increasing with every successive increment of income. This is the same thing as saying that the Krtscha iso-inequality locus will, in the limit, veer towards the ‘equal division’ outcome as income increases, hence the parabolic shape of the locus, as featured in Fig. 13.3 in the text.

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References Amiel Y, Cowell FA (1999) Thinking about inequality. Distributional analysis research programme: London School of Economics (PWD). Available at: http://darp.lse.ac.uk/papersdb/amiel_cowell_ tai.pdf. (Published 1999: Cambridge University Press: Cambridge.) Arriaga EE (1970) A new approach to the measurement of urbanization. Economic Development and Cultural Change, 18(2):206–218 Atkinson AB (1970) On the measurement of inequality. J Econ Theory 2(3):244–263 Atkinson AB Brandolini A Global world inequality: absolute, relative or intermediate? Paper prepared for the 28th general conference of the international association for research in income and wealth, Cork, Ireland, August 22–28, 2004. Available at: http://www.iariw.org/papers/2004/ brand.pdf Azpitarte F, Alsonso-Villar O (2011) Ray-invariant intermediate inequality measures: a lorenz dominance criterion. ECINEQ working paper, 2011–26 Bosmans K, Decancq K, Decoster A (2011) The evolution of global inequality: absolute, relative and intermediate views, Center for Economic Studies Discussion Paper Series (DPS) 11.03, Katholieke Universiyeit Leuven Bosmans K, Decancq K, Decoster A (2014) The relativity of decreasing inequality between countries. Economica 81(322):276–292 Bossert W, Pfingsten A (1990) Intermediate inequality: concepts, indices and welfare implications. Math Soc Sci 19(2):117–134 Chakravarty SR, Tyagarupananda S (1998) The sugroup decomposable absolute indices of inequality. In: Chakravarty SR, Coondoo D, Mukherjee R (eds) Quantitative economics: theory and practice, essays in honour of professor N. Bhattacharya. Allied Publishers Limited, New Delhi Chakravarty SR, Tyagarupananda S (2009) The subgroup decomposable intermediate indices of inequality. Span Econ Rev 11(2):83–97 Del Rio C, Alonso-Villar O (2008) New Unit-Consistent Intermediate Inequality Indices. Econom Theo 42(3):505–521 Del Rio C, Alonso-Villar O (2010) New unit-consistent intermediate inequality indices. Econom Theo 2(3):505–521 Del Rio C, Alonso-Villar O (2011) Rankings of income distributions: a note on intermediate inequality indices. Universidade de, Vigo. Available at: http://webs.urigo.es/crio/REI.pdf Del Rio C, Ruiz-Castillo J (2000) Intermediate inequality and welfare. Soc Choice Welf 17(2):223– 239 Del Rio C, Ruiz-Castillo J (2001) Intermediate inequality and welfare: the case of Spain. Rev Income Wealth. 47(2):221–237 Jayaraj D, Subramanian S (2012) On the interpersonal inclusiveness of India’s consumption expenditure growth. Econ Polit Wkly XLVII(45):56–66 Jayaraj D, Subramanian S (2013) On the inter-group inclusiveness of India’s consumption expenditure growth. Econ Polit Wkly 10:65–70 Jenkins SP, Jäntti M (2005) Methods for summarizing and comparing wealth distributions, ISER working paper 2005–05.University of Essex, Institute for Social and Economic Research, Colchester Kolm SC (1969) The optimal production of social justice. Chapter 7, In Margolis J. Guitton H (eds.) Public Economics: An Analysis of Public Production and Consumption and their Relations to the Private Sectors pp 145–200. London: Macmillan Kolm SC (1976a) Unequal inequalities I. J Econ Theory. 12(3):416–454 Kolm SC (1976b) Unequal inequalities II. J Econ Theory. 13(1):82–111 Krtscha M (1994) A new compromise measure of inequality. In: Eichhorn W (eds) Models and measurement of welfare and inequality. Springer, Heidelberg Moyes P (1987) A new concept of lorenz domination. Econ Lett 23(2):203–207

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Ravallion M (2003) The debate on globalization, poverty and inequality: why measurement matters. Int Aff 79(4):739–753 Ravallion M, Chen S (2002) Measuring pro-poor growth’. Econ Lett 78(1):93–99 Shorrocks AF (1988) Aggregation issues in inequality measurement. In: Eichhorn W (eds) Measurement in economics: theory and applications in economic indices. Physica Verlag, Heidelberg Subramanian S (2011) ‘Inequality measurement with subgroup decomposability and levelsensitivity’, economics: the open-access. Open-Assessment E-Journal. 5:2011–2019. https://doi. org/10.5018/economics-ejournal.ja.2011-9 Subramanian S, Jayaraj D (2013) The evolution of consumption and wealth inequality in India: a quantitative assessment. J Globalization Develop (Published Online: 11/29/2013) Yoshida T (2005) Social welfare rankings of income distributions: a new parametric concept of intermediate inequality. Soc Choice Welf 24(3):557–574 Zheng B (2007) Unit-consistent decomposable inequality measures. Economica 74(293):97–111 Zoli C (2012) Characterizing inequality equivalence criteria’, Department of Economics (University of Verona) Working Paper 32. 2012. Available at: http://leonardo3.dse.univr.it/home/workingpa pers/IneqEquivRevision20wp20201220VR.pdf.

Chapter 14

Revisiting an Old Theme in the Measurement of Inequality and Poverty

Abstract This essay subjects to criticism the dominant convention in the inequalitymeasurement and poverty-measurement literature of employing wholly ‘relative’ indices, and advocates, instead, the routine use of ‘centrist’ measures. In the process, the paper revisits some old debates on the logical adequacy and normative appeal of measures of inequality and poverty that are either wholly relative or wholly absolute. The implication of these issues for the diagnosis of magnitudes and trends in inequality and poverty is illustrated by means of a couple of simple empirical examples drawn from Indian data. Keywords Poverty · Inequality · Relative indices · Absolute indices · Centrist indices

14.1 Introduction Precisely how we choose to quantitatively assess the phenomena of inequality and poverty must necessarily serve as an important guide to our diagnosis of the gravity of these phenomena in the society under review and therefore to the nature and urgency of the public policy measures that are initiated to address these problems. This proposition is starkly in evidence in certain old, but unfortunately somewhat neglected, debates on the merits of relative and absolute indices of inequality and poverty. This paper offers a compact treatment of these debates which have been explored more thoroughly and elaborately elsewhere by the present author (Subramanian 2018). Most extant measures of poverty and inequality are ones which are normalized with respect to both the mean income and population size, that is, they are incomerelative and population-relative measures. It will be maintained in this paper that relative measures, however, are as arbitrary and unreasonable, in their way, as are wholly absolute measures. This would pave the way for more ‘moderate’ intermediate This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2020) Revisiting an Old Theme in the Measurement of Inequality and Poverty,in: Saleth R., Galab S., Revathi E. (eds) Issues and Challenges of Inclusive Development, Springer: Singapore, https://doi. org/10.1007/978-981-15-2229-1_13. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_14

179

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14 Revisiting an Old Theme in the Measurement of Inequality …

measures that mitigate the problems of logical coherence and ethical appeal which tend to afflict comprehensively relative and comprehensively absolute measures. The paper illustrates, by means of a couple of simple empirical examples, how our view of inequality and poverty is a variable function of how we choose to measure these phenomena.

14.2 On Intermediate Measures of Inequality and Poverty 14.2.1 Inequality Two problems confronted by distributional analysts are what might be called the problem of variable size and the problem of variable populations. The first problem poses the question of how to compare welfare, inequality, and poverty between distributions of the same population but different mean incomes. The second problem poses the question of how to compare welfare, inequality, and poverty between two distributions with the same mean income but different population sizes. The convention has been to locate the answers in two well-known ‘Invariance’ properties. The property of scale invariance says that when all incomes in a distribution are raised or lowered equi-proportionately, inequality must be deemed to remain the same. (In poverty comparisons, scale invariance would require poverty to remain unchanged when the poverty line and all incomes are changed in the same proportion.) The property of replication invariance states that when each income level in a distribution is replicated k times over (where k is any positive integer), inequality must be deemed to remain the same. (Replication invariance is defined analogously for poverty comparisons). As one can easily see, scale invariance will certify that inequality is unchanged if the ratio of each person’s income to the mean income remains unchanged. Replication invariance certifies that inequality remains unchanged if the relative frequency of each income in a distribution remains unchanged. Scale-invariant inequality measures are thus wholly ‘income-relative’ measures, while replication-invariant measures are wholly ‘population-relative’ measures. The bulk of the theoretical and applied research in measurement favours a relative view of inequality and poverty. A very commonly employed relative measure of inequality is the (relative) Gini coefficient, GR . Scale invariance is an unexceptionable property, on the face of it. However, as far back as the 1920s, Hugh Dalton expressed reservations about the unqualified appeal of relative inequality measures, as did Serge-Christophe Kolm, in the mid 1970s (see Dalton 1920, Kolm 1976a, b). This is because while an equi-proportionate increase in all incomes will leave relative inequality unchanged, it will, however, increase the absolute difference between incomes. Consider the two-person ordered income distributions x = (10, 20) and y = (20,40). It is easy to see that y is derived from x by doubling each person’s income: a relative measure of inequality will remain unchanged in going from x to y. However, the absolute difference between the two

14.2 On Intermediate Measures of Inequality and Poverty

181

persons’ incomes rises from 10 in distribution x to 20 in distribution y. From this absolute perspective, inequality must be deemed to have increased. This immediately presents the case for a rival to the scale invariance property, a property which Kolm called translation invariance, and which requires that inequality should remain unchanged with an equal addition to (or subtraction from) each person’s income. In this view, it is not equi-proportionate changes in income, but rather equal absolute changes, under which measured inequality should remain unchanged. Patrick Moyes (1987) advanced an absolute version GA1 of the Gini coefficient, which is given simply by the product of the mean income m and the relative Gini: GA1 = mGR . Kolm has suggested that in the presence of income growth, relative inequality measures tend to display ‘rightist’ values, while absolute measures display ‘leftist’ values; and this characterization is switched around in the presence of income contraction. For notice that in moving from x = (10, 20) to y = (20, 40), a relative measure takes no account of the increase in the absolute difference between the two persons’ incomes, which is certainly not reflective of a ‘radical’ perspective on inequality. By the same token, in moving from y = (20, 40) to z = (0, 20), an absolute measure takes no account of the fact that the share of the poorer person’s income in total income has declined from one-third to zero—which again is certainly not reflective of a ‘radical’ perspective on inequality. Briefly, neither a wholly relative nor a wholly absolute conception of inequality is entirely satisfactory. This paves the way for what Kolm called an ‘intermediate’ measure: in the context of the problem of variable size, an income-intermediate inequality measure would be one which satisfies the property of displaying an increase in value when all incomes in a distribution are raised by the same proportion and a reduction in value when all incomes in a distribution are raised by the same absolute amount. Such an income-intermediate version GI1 of the Gini coefficient would be given by the geometric mean of the incomerelative and the income-absolute measures, parameterized by the quantity α ∈ [0, 1]: G I 1 (α) = (G R )α (G A1 )1−α = m 1−α G R where α is a measure of ‘pro-absoluteness’. When α is exactly √ one-half, we have a ‘properly centrist’ income-intermediate Gini measure G ∗I 1 ≡ mG R . We turn now to an analogous consideration of the problem of variable populations. Given an n-person distribution x, suppose r is the number of individuals such that each of these individuals has at least one other person earning a higher income than herself. These are people who might be thought of as having a ‘complaint’ (Temkin 1993) about inequality. Clearly, the maximum possible number of complainants is n − 1. The proportion of complainants in x, then, is r/(n − 1). Suppose now that y is derived from x through a k-fold replication of the population at each income level in x. Then, the number of complainants in y will rise to kr, while the proportion of complainants will remain constant at r/(n − 1) (= kr/k(n − 1)). A relative view of inequality is concerned only with the proportion of complainants, while an absolute view would take account of the number of complainants. Such an absolute view would renounce the replication invariance property in favour of one which we might call replication scaling (Subramanian 2002). Replication scaling demands that a kfold increase of the population at each income level should lead to a k-fold increase in

182

14 Revisiting an Old Theme in the Measurement of Inequality …

measured inequality. A population-relative inequality measure is one which satisfies replication invariance, while a population-absolute measure is one which satisfies replication scaling. A population-absolute version GA2 of the relative Gini coefficient is given simply by the product of the total population n and the relative Gini: GA2 = nGR . If we wish to avoid the ‘extreme’ values of both absolute and relative measures, then we would favour a population-intermediate measure, namely a measure which increases, but less than proportionately, with a k-fold increase in the population at each income level. Such a population-intermediate version GI2 of the Gini coefficient would be given by the geometric mean of the population-relative and the population-absolute measures, parameterized by the quantity β ∈ [0, 1]: G I 2 (β) = (G R )β (G A2 )1−β = n 1−β G R , where β is a measure of ‘pro-absoluteness’. When β is exactly one-half, we have a ‘properly centrist’ population-intermediate √ Gini measure G ∗I 2 ≡ nG R . A comprehensively absolute version of the relative Gini coefficient, that is, a version GA which is both income-absolute and population-absolute, woud be given simply by the product of aggregate income mn and the relative Gini: GA = mnGR . And a comprehensively intermediate version of the Gini GI , namely one which is both income-intermediate and population-intermediate, would be given by the geometric mean of the comprehensively relative and the comprehensively absolute measures, parameterized by the quantity γ ∈ [0, 1] : G I (γ ) = (G R )γ (G A )1−γ = (nm)1−γ G R , where γ is a measure of ‘pro-absoluteness’. When γ is exactly one-half, we have a ‘properly centrist’ income-intermediate Gini measure: G ∗I ≡

√

nmG R

(14.1)

For all the reasons discussed earlier, there is a strong case for employing a comprehensively centrist measure of inequality such as G ∗I in Eq. (14.1). The dominant tradition in applied work is to employ the comprehensively relative Gini coefficient. It is on the strength of the time trend of this latter measure of inequality in the distribution of consumption expenditure (especially in rural India) that many commentators have inferred, effectively, that economic inequality in the country is not a seriously threatening issue (see, for example, Ahluwalia 2011, Bhalla 2011). The charts comparing the trends in the relative and comprehensively intermediate Gini coefficients, featured in Fig. 14.1a–d, separately for rural and urban India from 1987–88 to 2011–12, and based on data on the distribution of consumption expenditure in various rounds of the National Sample Survey, speak for themselves.

14.2.2 Poverty A widely employed family of relative poverty indices is the PηR family due to Foster– Greer and Thorbecke (1984), where, if z is the poverty line, x i is the income of the ith poorest person in a community of n individuals of whom q are poor (i.e. have incomes lower than the poverty line), and η(≥ 0) is a parameter reflecting aversion

14.2 On Intermediate Measures of Inequality and Poverty

183

Fig. 14.1 a Relative Gini for consumption distribution: rural India 1987–88 to 2011–12. b Centrist Gini for consumption distribution: rural India 1987–88 to 2011–12. c Relative Gini for consumption distribution: urban India 1987–88 to 2011–12. d Centrist Gini for consumption distribution: urban India 1987–88 to 2011–12

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14 Revisiting an Old Theme in the Measurement of Inequality …

Fig. 14.1 (continued)

to inequality in the distribution of poor incomes, then PηR = (1/nz η )

q  i=1

(z − xi )η , η ≥ 0.

(14.2)

14.2 On Intermediate Measures of Inequality and Poverty

185

As is well-known, P0R is just the headcount ratio of poverty, P1R is the per capita income-gap ratio (or the product of the headcount ratio and the proportionate shortfall of the average income of the poor from the poverty line), and P2R (the ‘squared poverty gap’ index) additionally incorporates information on the squared coefficient of variation in the distribution of poor incomes, that is, is sensitive to inequality among the poor. The comprehensively absolute counterparts of the relative Foster– Greer–Thorbecke poverty indices are obtained by simply desisting from normalizing the indices with respect to population size (n) and the poverty line (z) and are given by PηA =

q 

(z − xi )η ≡ nz η PηR , η ≥ 0.

(14.3)

i=1

It is easy to see from Eq. (14.3) that if P0R is the headcount ratio, then P0A is the aggregate headcount. The aggregate headcount, unlike the headcount ratio, violates what one might call a ‘Likelihood Principle’, namely the principle that a poverty measure should convey some information on the probability of encountering a poor person in any community. The headcount ratio, unlike the aggregate headcount, violates a ‘population focus principle’, namely, that a poverty measure should not be sensitive to increases in the non-poor population. In general, both comprehensively relative poverty measures (that is, measures that are relative with respect to both income and population) and comprehensively absolute poverty measures (that is, measures which are absolute with respect to both income and population) are predicated on ‘extreme values’, and the case for ‘intermediate’ poverty measures is as persuasive as is the case for intermediate inequality measures. The comprehensively intermediate family of Foster–Greer–Thorbecke (FGT) poverty indices is given by the geometric mean of the class of relative FGT measures and absolute FGT measures, in terms of a parameter δ ∈ [0, 1] and given by PηI (δ) = (PηR )δ (PηA )1−δ = (nz η )1−δ PηR , δ ∈ [0, 1].

(14.4)

As δ in Eq. (14.4) increases from 0 to 1, the poverty measure becomes less and less absolute and more and more relative. A ‘properly centrist’ intermediate measure ∗ PηI is one which gives equal weight to both the absolute and the relative conceptions of poverty and is realized when δ in Eq. (14.4) is set at one-half: ∗

PηI =

√

nz η PηR .

(14.5)

While the overwhelmingly popular convention in the measurement literature is to employ purely relative poverty measures, it is our contention that properly centrist ∗ measures such as PηI mitigate the extreme outcomes to which the values underlying relative and absolute measures are prone. (In this connection, the reader is referred to the works of, among others, Zheng 2007 and Subramanian 2018).

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14 Revisiting an Old Theme in the Measurement of Inequality …

Here is an empirical example, involving urban poverty estimates for India based on National Sample Survey data on the distribution of consumption expenditure in 2004–05 and 2011–12, of how our diagnosis of money-metric poverty can change when we relax some of the customary assumptions underlying the ‘identification’ and ‘aggregation’ exercises of standard poverty measurement (to the extent that such measurement is meaningful). In Table 14.1a, we present information on the headcount ratio, the per capita income-gap ratio, and the squared poverty gap index— each in its customarily purely relative form—for a poverty line that is unvarying in real terms over the two years involved in the poverty comparison: following on the Tendulkar Committee’s (Planning Commission 2009) recommendation, the poverty line is pegged at Rs. 505.27 at 2001 prices (the price deflator employed being the Consumer Price Index for Industrial Workers (CPIIW)). By this reckoning, poverty in 2011–12 is just between a third and a half of poverty in 2004–05, depending on which relative poverty measured is employed. In Table 14.1b, we defer to the view that the poverty line should be continuously adapted and augmented with time, such as has been advocated by commentators like Peter Townsend (1979). One way of doing this is to allow the poverty line of Rs. 505.27 in 2004–05 to increase at the arbitrary, but modest, compound rate of growth of one percent per annum, so that, in 2011–12, the line becomes Rs. 563.71 at 2001 prices. Further, we relax the norm Table 14.1 a Relative poverty for a fixed poverty line in urban India: 2004–05 and 2011–12 Year

Poverty line at 2001 prices (Rupees)

P0R

P1R

P2R

2004–05

505.27

0.2674

0.0634

0.0204

2011–12

505.27

0.1344

0.0252

0.0071

50.26

39.48

34.80

Terminal year * poverty as a percentage of base year poverty (%)

*

b Centrist poverty for a variable poverty line in urban india: 2004–05 and 2011–12 ∗

∗

∗

Year

Poverty line at 2001 prices (Rupees)

P0I (Millions of Persons)

P1I (Millions of Rupees)

P2I (Millions of Rupees)

2004–05

505.27

3.68

25.47

184.22

2011–12

563.71

3.67

17.70

114.93

99.73

69.49

62.39

Terminal year * poverty as a percentage of base year poverty (%)

*

Source Estimates based on the figures in Tables 1 and 2 of Subramanian and Lalvani (2018), themselves computed from the 61st and 68th Rounds of the National Sample Survey on distribution of consumption expenditure

14.2 On Intermediate Measures of Inequality and Poverty

187

of relativity in the aggregation exercise to allow for properly centrist measures. In such an event, the poverty level in 2011–12 as a proportion of its level in 2004–05 rises from about a third to about three-fifths for the FGT-2 index, from about twofifths to about seven-tenths for the FGT-1 index, and from about one-half to about one hundred per cent for the FGT-0 index: the decline in poverty rates becomes altogether less dramatic!

14.3 Summary and Conclusion Our response to the problems of disparity and deprivation is inevitably determined by our perception of the magnitudes of, and trends in, these phenomena. Our perception, in turn, is inevitably determined by the precise protocols of measurement we choose to employ in order to assess the quantitative significance of the phenomena in question. One must be a ‘measurement-nihilist’ to deny the truth of this proposition and does not have to be a ‘measurement-fetishist’ in order to affirm it. This is particularly in evidence in certain old debates on whether inequality and poverty are best measured in relative, in absolute, or in some intermediate form. The debates can be traced back to the pioneering work of Hugh Dalton in the 1920s and their revival by SergeChristophe Kolm in the 1970s. Despite the profound importance of the issues of logical and ethical appeal involved in the debates, they have tended, unfortunately, to be largely neglected in the measurement literature, in favour of wholly relative measures of inequality and poverty. When we correct for this bias, we find that the problems of both inequality and poverty in India are more severe than results based on conventional measurement procedures will allow. Measurement is far from being the only matter of concern when we deal with issues of disparity and deprivation. Equally, however, it is very far from being a matter of inconsequential concern.

References Ahluwalia MS (2011) Prospects and policy challenges in the twelfth plan. Econ Polit Wkly XLV I(21):88–105 Bhalla SS (2011) Inclusion and growth in India: some facts, some conclusions. LSE Asia Research Centre Working Paper 39 Dalton H (1920) The measurement of the inequality of incomes. Econ J 30(119):348–361 Foster JE, Greer J, Thorbecke E (1984) A class of decomposable poverty measures. Econometrica 52(3):761–766 Kolm SCh (1976a) Unequal inequalities I. J Econ Theor 12(3):416–454 Kolm SCh (1976b) Unequal inequalities II. J Econ Theor 13(1):82–111 Planning Commission (2009) Report of the expert group to review the methodology for estimation of poverty. Government of India, New Delhi

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Srinivasan TN (2017) Planning, poverty and political economy of reforms: a tribute to Suresh D. Tendulkar. In Krishna KL, Pandit V, Sundaram K, Dua P (eds) Perspectives on Economic Policy and Development in India: In Honour of Suresh Tendulkar, Springer: Delhi Subramanian S (2002) Counting the poor: an elementary difficulty in the measurement of poverty. Econ Philos 18(2):277–285 Subramanian S (2018) On comprehensively intermediate measures of inequality and poverty, with an illustrative application to global data. J Global Dev. https://doi.org/10.1515/jgd-2017-0027 Subramanian S, Lalvani M (2018) Poverty, growth, inequality: some general and India-specific considerations. Indian Growth and Development Review 11(2):136–151 Temkin L (1993) Inequality. Oxford University Press, Clarendon Townsend P (1979) The development of research on poverty. In: Department of health and social security: social security research: the definition and measurement of poverty. HMSO, London Zheng B (2007) Unit-consistent poverty indices. Econ Theor 31(1):113–142

Chapter 15

Inequality Measurement with Subgroup Decomposability and Level-Sensitivity

Abstract Subgroup Decomposability is a very useful property in an inequality measure, and level-sensitivity, which requires a given level of inequality to acquire a greater significance the poorer a population is, is a distributionally appealing axiom for an inequality index to satisfy. In this paper, which is largely in the nature of a recollection of important results on the characterization of subgroup decomposable inequality measures, the mutual compatibility of subgroup decomposability and level-sensitivity is examined, with specific reference to a classification of inequality measures into relative, absolute, centrist, and unit-consistent types. Arguably, the most appealing combination of properties for a symmetric, continuous, normalized, transfer-preferring and replication-invariant (S-C-N-T-R) inequality measure to satisfy is that of subgroup decomposability, centrism, unit-consistency and levelsensitivity. The existence of such an inequality index is (as far as this author is aware) yet to be established. However, it can be shown, as is done in this paper, that there does exist an S-C-N-T-R measure satisfying the (plausibly) next-best combination of properties—those of decomposability, centrism, unit-consistency and level-neutrality. Keywords Subgroup decomposability · Level-sensitivity · Absolute inequality measure · Relative inequality measure · Centrist inequality measure · Unit-consistency

15.1 Introduction In the axiomatic approach to the measurement of inequality, a number of desirable properties of inequality indices have been advanced. In this article, we consider two specific properties—those of ‘decomposability’ and ‘level-sensitivity’—and check This chapter draws heavily and directly on a previously published paper of the author’s, the content from which is re-used here with the permission of the copyright holder: S. Subramanian (2011): ‘Inequality Measurement with Subgroup Decomposability and Level-Sensitivity’, Economics: The Open-Access, Open-Assessment E-Journal, Vol. 5, 2011-9. https://doi.org/10.5018/economics-ejo urnal.ja.2011-9. I acknowledge helpful comments from an Associate Editor and an anonymous referee of the journal. © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 S. Subramanian, Social Values and Social Indicators, Themes in Economics, https://doi.org/10.1007/978-981-16-0428-7_15

189

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for their mutual compatibility in the presence of other specified properties. The points made in this essay draw on a number of important results which have already been established in the literature: it is then mainly a matter of putting these results together in order to present a set of observations on the prospects of simultaneously meeting the requirements of decomposability and level-sensitivity. The outcome is arguably useful, insofar as taxonomies (in this case of inequality measures) are generally useful; the outcome is also inarguably dependent on a great deal of important prior work that has been done on the subject of decomposable inequality measures. Subgroup decomposability (see Bourguignon 1979; Cowell 1980; Cowell and Kuga 1981; Shorrocks 1980, 1984, 1988) is the property that an inequality measure be expressible as an exact sum of a ‘between-group component’ (obtained by imagining that each person in any subgroup receives the subgroup’s mean income) and a ‘withingroup component’ (obtained as a weighted sum of subgroup inequality levels, the weights depending on the subgroups’ income shares or population shares or some combination of the two shares). Level-sensitivity can be thought of as a group-related Egalitarian requirement that arises when a population is partitioned into non-overlapping income groups of the same size: it postulates that in this circumstance, and other things remaining the same, a given increase in subgroup inequality should cause overall inequality to rise by more the poorer (in terms of subgroup mean income) the subgroup is. This property has a strong affinity to a concern expressed in an early contribution by Amartya Sen (1973), and relating to the question of how our view on inequality ought to vary with the general level of a society’s prosperity. As observed by Sen (1973: 36): Can it be asserted that our judgment of the extent of inequality will not vary according to whether the people involved are generally poor or generally rich? Some have taken the view that our concern with inequality increases as a society gets prosperous since the society can ‘afford’ to be inequality-conscious. Others have asserted that the poorer an economy, the more ‘disastrous’ the consequences of inequality, so that inequality measures should be sharper for low average income. This is a fairly complex question and is bedeviled by a mixture of positive and normative considerations. The view that for poorer economies inequality measures must be themselves sharper can be contrasted with the view that greater importance must be attached to any given inequality measure if the economy is poorer. The former incorporates the value in question into the measure of inequality itself, while the latter brings it in through the evaluation of the relative importance of a given measure at different levels of average income.

It is the former of the two views asserted by Sen at the conclusion of the preceding quote that is upheld by the level-sensitivity axiom. In this essay, we examine the mutual compatibility of subgroup decomposability and level-sensitivity for certain broad classes of inequality measures, taxonomized according to their invariance to multiplicative or additive transformations of an income distribution. In terms of this classification, inequality measures can be relative or absolute (see Blackorby and Donaldson 1980). A relative inequality measure is ‘scale invariant’, while an absolute inequality measure is ‘translation invariant’. Scale invariance is the property that the value of an inequality measure should remain

15.1 Introduction

191

unchanged if all persons’ incomes were to be uniformly multiplied by any positive scalar, while translation invariance requires such constancy in the value of an inequality measure when all persons’ incomes are increased (or decreased) by the addition (or subtraction) of a fixed amount. The invariance requirements just considered have both purely ‘analytical’ and ‘normative’ implications. At the analytical level, scale invariance ensures that the value of an inequality index does not change with the units in which income is measured, while translation invariance violates this property of neutrality with respect to the units of measurement. From this ‘analytical’ perspective, scale invariance would appear to possess an attractive advantage over translation invariance. However, from a ‘normative’ perspective, scale invariance can be seen to uphold a ‘rightwing’ view of inequality and translation invariance to uphold a ‘left-wing’ view, as pointed out by Serge-Christophe Kolm (1976a, 1976b). Notice that, given a twoperson ordered income distribution x = (1, 100), a doubling of each person’s income would lead to the distribution y = (2, 200): a scale-invariant index would uphold the (typically right-wing) judgment that the extent of inequality is the same in both distributions, despite the fact that out of the additional total income of 101 units in y vis-à-vis x, 100 units of income have gone to the richer person and only 1 unit to the poorer person. In contrast, if z were to be derived from x by the addition of 100 units of income to each person, so that z = (101, 200), a translation-invariant index would uphold the (typically left-wing) judgment that the extent of inequality is the same in both distributions, despite the fact that in the transition from x to z, the poorer person’s income has risen by a factor of 10,000% and the richer person’s income by a factor of just 100%. One can see now that one can have inequality measures which are a ‘compromise’ between absolute and relative measures. The compromise we effect would depend on whether we take a purely analytical or a normative view of the two classes of measures. Under a purely analytical interpretation, a compromise class of measures would be unit-consistent measures (Zheng 2007), namely inequality measures which satisfy the requirement that the inequality-ranking of distributions is invariant with respect to the choice of units in which income is measured. As it happens, all rightwing measures and some left-wing measures are unit-consistent. A different type of compromise is the normative one between right-wing and left-wing measures, which leads to a class of centrist or intermediate measures (see, for example, Zheng 2007): an intermediate measure is one which satisfies the property that (i) a uniform scalingup of every individual’s income should increase inequality and (ii) the addition of any given income to every person’s income should reduce inequality. It should be noted that the two types of compromise we have just considered are mutually independent: unit-consistent inequality measures are not necessarily centrist measures, and similarly centrist inequality measures are not necessarily unit-consistent. In examining subgroup decomposability and level-sensitivity of inequality measures for a classification of measures according to their disposition toward distributional values and unit-consistency, this article proceeds as follows. The following section introduces concepts and notation. This is followed by a section which advances a set of observations on subgroup decomposability and level-sensitivity

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for alternative types of inequality measures. The final section offers a summary and conclusions.

15.2 Basic Concepts N is the set of positive integers, and R is the real line. For every n ∈ N , Xn is the set of positive n-vectors x = (x1 , . . . , xi , . . . , xn ), and each x is to be interpreted as an income vector whose typical element xi is the income of individual i in a community of n individuals. X is the set ∪n∈N Xn , and an inequality index is a mapping I : X → R such that, for every x ∈ X, I (x) is a real number which is supposed to indicate the amount of inequality associated with the distribution x. For every income vector x ∈ X, N (x) is the set of individuals represented in x, and n(x) ≡ #N (x) is the dimensionality of x, while μ(x) is the mean income of x. If a population is partitioned into K (≥ 1) subgroups {1, . . . , j, . . . , K }, then x j is the  is the set of individuals represented in x j , income vector of the jth subgroup, N x j   n x j is the dimensionality of x j , μ(x j ) is the mean income of x j , and I (x j ) is the extent of inequality associated with x j ( j = 1, . . . , K ). Wherethere    is no ambiguity, we shall also write I for I (x), n for n(x), μ for μ(x), I j for I x j , n j for n x j , μ j for μ(x j ), and so on. Let I* be the set of inequality measures such that a typical member of this set, I : X → R, satisfies the following properties: Symmetry (Axiom S), which is the requirement that for all x ∈ X , I (x) = I (x) where  is any appropriately dimensioned permutation matrix (so measured inequality is impervious to the personal identities of individuals);   Normalization (Axiom N), which is the requirement that for all x ∈ X, I x0 = 0, where x0 is the vector obtained from x by setting xi0 = μ(x)∀i = 1, . . . , n(x) (so that inequality is taken to be zero when all incomes are equalized); Continuity (Axiom C), which is the requirement that I be continuous on Xn for all n ∈ N (so that ‘similar income distributions have similar inequality values’); Strict Schur-Covexity (Axiom SSC), which is the requirement that for all x ∈ X, I (x) > I (xB) where B is any appropriately dimensioned bistochastic matrix which is not a permutation matrix (so that any movement toward equalization of the incomes in a distribution causes measured inequality to decline); Replication Invariance (RI), which is the requirement that for all x, y ∈ X, I (x) = I (y) whenever y is a q-replication of x, that is, y = (x, . . . , x), n(y) = qn(x), and q is any positive integer greater than 1 (so that inequality values depend only on the relative, not the absolute, frequency distribution of incomes); and Differentiability (D), which is the requirement that for all x ∈ X, I should have continuous first and second partial derivatives (∂ I /∂ xi ∀i ∈ N (x) and ∂ 2 I /∂ xi2 ∀i ∈ N (x), respectively) in each income in the vector.

15.2 Basic Concepts

193

Some basic definitions relating to relative and absolute inequality measures, and ‘compromise’ versions of these, are now provided (Note that relative inequality measures are also referred to as ‘right-wing’ measures, and absolute inequality measures as ‘left-wing’ measures). Definition 15 (Relative Inequality Measure) An inequality measure I : X → R is relative if and only if it is scale invariant, that is, if and only if, for all x ∈ X, I (x) = I (λ x) for any λ ∈ R++ . Definition 15 (Absolute Inequality Measure) An inequality measure I : X → R is absolute if and only if it is translation invariant, that is, if and only if, for all x ∈ X, I (x) = I (x + t) where t = (t, . . . , t) for any t ∈ R and n(t) = n(x). Definition 15 (Unit-Consistent Inequality Measure) An inequality measure I : X → R is unit-consistent if and only if, for all x, y ∈ X, I (x) < I (y) implies I (λ x) < I (λ y) for any λ ∈ R++ (see Zheng 2007). Definition 15 (Centrist Inequality Measure) An inequality measure I : X → R is centrist if and only if, for all x ∈ X, (i) I (x) < I (λ x) for any λ > 1 and (ii) I (x) > I (x + t) where t = (t, . . . , t) for any t ∈ R++ and n(t) = n(x) (see Zheng 2007). Definition 15 (Bossert–Pfingsten Restriction) A centrist inequality measure I : X → R will be said to obey the Bossert–Pfingsten restriction (see Bossert and Pfingsten 1990) if and only if, for all x ∈ X, I (x) = I (x + a[π x + (1 − π )t]), where a ∈ R, π ∈ [0, 1], and t = (t, . . . , t) for any t ∈ R++ and n(t) = n(x). [The restriction stated above provides a particular operationalization of the notion of a centrist inequality measure by specifying a plausible condition under which the measure should remain unchanged for some combination of a uniform scale increase and a uniform incremental increase in all incomes of a distribution]. Next, the notion of ‘level-sensitivity’ is defined. Level-sensitivity essentially demands that, when a population is partitioned into equi-dimensional nonoverlapping income groups, then, other things equal, a given increase in subgroup inequality should cause aggregate inequality to rise by more the poorer (in terms of mean income) the subgroup is. More formally: Level-Sensitivity (Axiom LS). An inequality measure I : X → R is level-sensitive if and only if, for all x,y, z ∈ X, if x = (x1 , . . . , x K ), y = (y1 , . . . , y K ), z = (z1 , .. . , z K ), n xj = n x j+1 (= n(x)/K )∀ j = 1, . . . , K − 1, n x j  = n y j = n z j ∀  j ∈ {1, . . . , K }, μ z ∀ j ∈ {1, . . . , K }, μx j  = μy j = j  μ x j < μ x j+1 ∀ j = 1, . . . , K − 1, I x j = I x j+1 = I (say) ∀ j = 1, . . . , K − 1, and I x j = I y j ∀ j ∈ {1, . . . , K }\{s} and I x j = I z j ∀ j ∈ {1, . . . , K }\{t} for some subgroups s and t such that I (ys ) = I (zt ) = I + ,  ∈ R++ . and s < t (so that μ(xs ) < μ(xt )), then:

194

15 Inequality Measurement with Subgroup Decomposability …

I (y) > I (z) > I (x). The level-sensitivity axiom is kindred in spirit to a property of poverty measures which Nanak Kakwani (1980) has called Monotonicity-Sensitivity, namely the requirement that ‘if (P)i represents the increase in the poverty measure due to a small reduction in the income of the ith poor, then (P)i > (P) j for j > i [given that incomes are arranged in non-descending order]’ (Kakwani 1980: 438). Indeed, Kakwani (1993) has upheld the attractiveness of a similar level-sensitivity property in related settings involving the use of generalized indicators of living standards. Finally, we state the axiom of subgroup decomposability: Subgroup Decomposability (Axiom SD). For all x ∈ X, I (x) is a subgroup decomposable inequality measure if and only if I (x) = IW (x) + I B (x), where IW (x) is the within-group component of inequality, defined as: K      IW (x) = w xj I xj , j=1

    level, with w x j where w x j is a weight attached to subgroup j’s inequality  depending only on subgroup j’s population share π x j or income share σ x j or both; and of inequality, defined as: I B (x) is the   between-group component 0 0 0 I B (x) = I x1 , . . . , x j , . . . , x K , where for all x ∈ X, x0 is the vector obtained by setting xi0 = μ(x)∀i = 1, . . . , n(x). Notice that if a population is partitioned into non-overlapping income groups, then as long as the group-specific inequality levels and population shares remain unchanged, it is reasonable—even if the group-specific income shares should change—to expect the within-group component of a decomposable inequality index to also remain unchanged. Decomposability subjected to this reasonable restriction can be called ‘proper decomposability’, and it is easy to see that proper   decomposability implies the requirement that the group-specific weights w x j appearing in the definition of subgroup decomposability should depend only on the subgroup population shares (and, in particular, not at all on the subgroup income shares): Proper Subgroup Decomposability (Axiom  PSD). Axiom PSD is derived from Axiom SD by replacingthephrase ‘where w x j is a weight attached to subgroup  j’s depending only on subgroup j’s population share π xj inequality level, with w x j     or income share σ x j or both’ with thephrase ‘where w x j is a weight attached to subgroup  inequality level, with w x j depending only on subgroup j’s population  j’s share π x j ’. A number of important results relating to the characterization of subgroup decomposable inequality indices have been established in the literature. Some of these results are summarized in what follows: Result 1 (Shorrocks 1980). For all x ∈ X, a relative inequality measure I belongs to the set I* and satisfies subgroup decomposability if and only if it is a positive multiple of a member of the following class Ic (x) of generalized entropy measures:

15.2 Basic Concepts

195

n(x)   1 (xi /μ(x))c − 1 , c ∈ R, c = 0, 1; n(x)c(c − 1) i=1 

 n(x)

1 xi xi = ln , c = 1; n(x) i=1 μ(x) μ(x)  n(x)

1 μ(x) = ln , c = 0. n(x) i=1 xi

Ic (x) =

Result 2 (Chakravarty and Tyagarupananda 1998; Bosmans and Cowell 2010). For all x ∈ X, an absolute inequality measure I belongs to the set I* and satisfies subgroup decomposability if and only if it is a continuous and strictly increasing function of the following class Ib (x) of measures:  1  b(x1 −μ(x)) e − 1 , b ∈ R, , b = 0; n(x) i=1 n(x)

Ib (x) =

1 [xi − μ(x)]2 , b = 0. n(x) i=1 n(x)

=

Result 3 (Chakravarty 2000). For all x ∈ X, an absolute inequality measure I belongs to the set I* and satisfies proper subgroup decomposability if and only if it is a positive multiple of the variance, given by: 1 V (x) = (xi − μ(x))2 . n(x) i=1 n(x)

Result 4 (Chakravarty and Tyagarupananda 2009). For all x ∈ X, a centrist inequality measure I belongs to the set I*, obeys the Bossert–Pfingsten restriction [namely the requirement that I (x) = I (x + a[π x + (1 − π )t]), where a ∈ R, π ∈ [0, 1], and t = (t, . . . , t) for any t ∈ R++ and n(t) = n(x)], and is subgroup decomposable, if and only if it is a member of the following class I c (x) of transformed generalized entropy measures:

  1 {(xi + v)/(μ(x) + v)}c − 1 , c ∈ R, c = 0, 1; n(x)c(c − 1) i=1 

 n(x)

1 xi + v xi + v = ln , c = 1; n(x) i=1 μ(x) + v μ(x) + v

Iˆc (x) =

n(x)

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15 Inequality Measurement with Subgroup Decomposability …

=

 n(x)

μ(x) + v 1 ln , c = 0, n(x) i=1 xi + v

where v = (1 − π )/π and c depends on both a and π . Result 5 (Zheng 2007). For all x ∈ X, a unit-consistent inequality measure I belongs to the set I* and satisfies subgroup decomposability if and only if it is a positive multiple of a member of the following class Ic (x) of measures:   1 x c − μc (x) , c, d ∈ R, c = 0, 1; Ic (x) = c(c − 1)n(x)μd (x) i=1 i 

 n(x)

1 xi xi = ln , c = 1, d ∈ R; n(x)μd−1 (x) i=1 μ(x) μ(x)  n(x)

1 μ(x) = ln , c = 0, d ∈ R. n(x)μd (x) i=1 xi n(x)

[In the interests of formal accuracy, it should be pointed out that in the BosmansCowell 2010 version of Result 2, the axioms of normalization and differentiability are dispensed with, and Result 3 (Chakravarty 2000) does not really invoke the replication invariance property.] Result 3 relates to the characterization of a properly subgroup decomposable inequality measure which is absolute, while Results 1, 2, 4, and 5 relate to the characterization of subgroup decomposable measures which are, respectively, relative, absolute, centrist, and unit-consistent. How do these measures fare in relation to level-sensitivity? This issue is examined in the following section.

15.3 Some Observations on Subgroup Decomposability and Level-Sensitivity While both subgroup decomposability and level-sensitivity appear to be attractive properties of an inequality index, it may not always be possible for an inequality measure to satisfy both properties. We illustrate this proposition by considering the Gini coefficient G of inequality which, though it is not a subgroup decomposable (nor even subgroup consistent) measure, does lend itself to decomposability in the special case in which the population is partitioned into non-overlapping income groups (see Anand 1983). Specifically, it can be shown that if a population is divided into, say, K non-overlapping    income groups of the same size, so  with n x j  = n(xk )∀ j, k ∈ {1, . . . , K } [so that , . . . , x , . . . , x that x = x 1 j K   π x j = π (xk ) = 1/K ∀ j, k ∈ {1, . . . , K } ], then one can write: G(x) = G B (x) + G W (x), where

15.3 Some Observations on Subgroup Decomposability and Level-Sensitivity

G B (x) = 1 + G W (x) =

1 K

1 K

K 

−

2 K

K 

197

(K + 1 − j)σ j ; and

j=1

σjG j.

j=1

Of interest is the fact that in the expression for the within-group component of aggregate inequality, the weight on the jth subgroup’s inequality level is σ j /K : if the groups are indexed in ascending order of mean income, then it is clear that when G j = G k ∀ j, k ∈ {1, . . . , K }, a given increase in inequality will raise aggregate inequality by more the richer (in terms of mean income) the subgroup is, since the weight on G j , σ j /K , is an increasing function of j: this precisely reverses what the axiom of level-sensitivity demands. What can be said at a more general level about subgroup decomposability and level-sensitivity? A first and immediately obvious conclusion that emerges from a consideration of the concepts and definitions discussed in the preceding section is that there is a mutual incompatibility between the properties of proper decomposability and level-sensitivity of an inequality measure. This follows from noting that when a population is partitioned into non-overlapping income groups of equal size, any properly decomposable inequality measure I belonging to the set I* will (by definition) have a within-group inequality component which is a weighted sum of subgroup inequality levels where the weights depend only on the subgroup population shares—which must all be equal since the subgroups are of equal size: a given increase in subgroup inequality will therefore cause overall inequality to rise by the same extent, irrespective of the average level of prosperity of the subgroup. The outcome is that level-sensitivity is a casualty. This leads to our first observation: Observation 1. There exists no properly decomposable inequality measure I ∈ I ∗ which is level-sensitive. Observation 1 suggests that if level-sensitivity is a desired normative property of an inequality index, then insistence on proper decomposability may have to be sacrificed. Indeed, the following observation, it can be shown, is true: Observation 2. There exists a relative inequality measure I ∈ I ∗ which satisfies both subgroup decomposability and level-sensitivity. To see this, recall from Result 1 that the only relative inequality measures in I* which satisfy subgroup decomposability are positive multiples of members of the class of generalized entropy indices Ic . As a matter of convention, the only members of Ic in common circulation are restricted to the case in which the parameter c assumes non-negative values: specifically, c = 1 and c = 0 correspond to the two well-known Theil indices T1 and T2 , respectively, while c = 2 yields one-half the squared coefficient of variation C. None of these three indices is level-sensitive: it is well-known that the weight on the inequality level of subgroup j in the within-group component of inequality is σ j for T1 , π j for T2 , and π j σ j2 for C. The implications for level-sensitivity are plain: T1 and C are level-insensitive, while T2 is level-neutral. The picture, however, becomes promising when we consider negative values for the

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parameter c. Specifically, if we set c = −1, then we obtain an inequality measure— n(x)  μ(x)−xi  call it I−1 —given by: I−1 (x) = (1/n(x)) i=1 xi +xi . The index I−1 is, as it happens, closely related to a member of the Atkinson (1970) family of measures,  n(x) λ 1/λ xi /μ(x), λ < 1, λ = 0. When λ = −1, given by: Aλ (x) = 1 − (1/n(x)) i=1 it is easily verified that  I−1 is a strictly increasing transform of A−1 : specifically, I−1 = A−1 /2(1 − A−1 ) . (It may also be noted, in passing, that the inequality measure I−1 is quite similar in formulation to one advanced by Jayaraj and Subramanian 2006, which can be derived as a normalized Canberra distance function, and is given by n(x)  μ(x)−xi  the expression ICanberra (x) = (1/n(x)) i=1 μ(x)+xi : this latter index, however, is not decomposable.) What is relevant to note is that the decomposition  of I−1 (x) is  njμ K − 1 defined by:I−1 (x) = I−1 B (x) + I−1 W (x), where I−1B (x) = (1/2) j=1 nμ j  and I−1W (x) = Kj=1 w j I−1 j , with w j = π j /σ j , j = 1, . . . , K . Since the weight on subgroup inequality is a declining function of the subgroup income share, I−1 will satisfy the level-sensitivity requirement. But what if our distributional values were left-wing rather than right-wing? Observation 3 below addresses this question. Observation 3. There exists an absolute inequality measure I ∈ I ∗ which satisfies both subgroup decomposability and level-sensitivity. Result 2 enables us to see the truth of Observation 3. The class of indices Ib (I) = n(x)   1  eb(xi −μ(x)) − 1 , b ∈ R, b = 0 is the class of exponential inequality indices, n(x) i=1

and is ordinally equivalent to the Kolm (1976) class of measures. Chakravarty (2000) has established that a subgroup decomposition of Ib yields a within group component in which the weight on the inequality value for the jth subgroup is given by w j = π j eb(μ j −μ) ; if the population is partitioned into K non-overlapping income groups of the same size, then w j = (1/K )ebh j , where h j ≡ μ j − μ, j = 1, . . . , K , so dw that dh jj = (b/K )ebh j < 0 for b < 0. That is to say, Ib is level-sensitive whenever b is negative. Thus, the exponential inequality measures, for negative values of the parameter b, are both subgroup decomposable and level-sensitive. Notice now that since all relative inequality indices are also unit-consistent, we are assured by Observation 2 that there exists a unit-consistent relative inequality measure belonging to the set I* which is also level-sensitive. Unfortunately, we have no such assurance regarding absolute inequality measures from Observation 3, since absolute measures may or may not be unit-consistent. Result 2 confines our attention to those absolute indices which are either exponential indices or the variance. Zheng (2007) points out that the family of exponential indices is not unit-consistent. The variance, however, is a unit-consistent measure, but Result 4 (Chakravarty 2000) asserts that the only absolute inequality measure in the set I* which is properly decomposable is the variance; and from Observation 1 we know that no properly decomposable index belonging to the set I* is level-sensitive. This leads to the following negative observation:

15.3 Some Observations on Subgroup Decomposability and Level-Sensitivity

199

Observation 4. There exists no absolute unit-consistent inequality measure I ∈ I ∗ which is level-sensitive. Observation 4 is a harsh verdict for those who would value both subgroup decomposability and level-sensitivity but whose distributional judgments favour only leftwing inequality indices. For those who are happy to settle for centrist measures, the present state of knowledge may be inadequate to arrive at a definitive conclusion on the prospects of meeting the requirements of both subgroup decomposability and level-sensitivity, as reflected in the following observation. Observation 5. Since (to the best of this author’s awareness) there is no characterization available of unit-consistent, centrist inequality measures which are subgroup decomposable, it is not known if there exists a unit-consistent and centrist measure which is both decomposable and level-sensitive. It may be added that the available evidence on this question is not encouraging. Result 4 (Chakravarty and Tyagarupananda 2009) presents a class I c of centrist inequality measures belonging to the set I* which are decomposable, but, as pointed out by Zheng (2007), none of these indices is unit-consistent. Result 5 (Zheng, 2007) presents a class Ic of unit-consistent inequality measures belonging to the set I* which are decomposable. Two classes of centrist measures which are subsets of the

Ic class are the following ones (see Zheng 2007):

  1 xi2 − μ2 (x) , 0 < d < 2. d n(x)μ (x) i=1 n(x)

I ∗ (x) =

  1 xic − μc (x) , 0 < c < 2, c = 1. n(x)c(c − 1) i=1 n(x)

I ∗∗ (x) =

I ∗ is what Zheng (2007) refers to as a generalization of the Krtscha (1994) measure. It can be verified that the subgroup decompositions of the families of indices I ∗ and I ∗ ∗ yield the following outcomes: I ∗ (x) = I B ∗ (x) + IW ∗ (x), where I B∗ = with w j = π 1−d σ jd , j = 1, . . . , K ; and j

K

n j μ2j nμd

j=1

1 I ∗ ∗ (x) = I B∗ ∗ (x) + IW∗ ∗ (x), where I B∗ ∗ = c(c−1) K ∗∗ j=1 w j I j , with w j = π j , j = 1, . . . , K .

− μ2−d and IW∗ =

K j=1

w j I j∗ ,

  K 1 n

 c c and IW∗ ∗ = n μ − μ j j=1 j

An examination of the weights on subgroup inequality levels in the within-group component of inequality suggests that I ∗ is level-insensitive, while I ∗ ∗ is levelneutral. Briefly, the decomposable and centrist measures proposed by Chakravarty and Tyagarupananda are not unit-consistent, while the decomposable and centrist measures proposed by Zheng are unit-consistent but not level-sensitive. Whether there exist decomposable, centrist, unit-consistent and level-sensitive inequality measures is an open question.

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15.4 Summary and Conclusion This article has been mainly a quick review of a set of important results on the characterization of decomposable inequality measures, classified into relative, absolute, centrist, and unit-consistent indices, and an examination of the mutual compatibility of the properties of subgroup decomposability and level-sensitivity. For inequality measurement to be coherent, it appears that inequality measures must be unitconsistent. For inequality measurement to be informed by non-extreme distributional values, it also seems to be desirable that inequality measures be centrist. Thus, in the interests of both coherence and normative appeal, there would appear to be a strong case to confine attention to the set of unit-consistent and centrist measures. Decomposability is an extremely convenient property for an inequality index to possess, though it is not clear that this property is imbued with any particularly striking normative values (except in so far as what the philosopher Derek Parfit 1997 has called ‘Prioritarianism’ is compatible with the strong separability underlying additively decomposable inequality indices). Level-sensitivity is a fairly compelling property of an inequality measure, requiring as it does that inequality be regarded as a more severe problem the poorer the population experiencing it is. Level-neutrality is a weaker requirement, demanding only that inequality should be regarded as a problem whose severity does not diminish as a population becomes poorer. In an ‘ideal’ situation, one may wish to have inequality measures which are centrist, unit-consistent, subgroup decomposable, and level-sensitive. Whether such measures exist is still (as far as the present author is aware) an open question. What can, however, be asserted is that there does exist a symmetric, normalized, continuous, differentiable, strictly Schurconvex, and replication-invariant measure which is unit-consistent, centrist, subgroup decomposable, and level-neutral. This is the index,  c or rather family of indices (see n(x) 1 c x − μ (x) , 0 < c < 2, c = 1. Zheng 2007), given by I ∗∗ (x) = n(x)c(c−1) i i=1

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