HANDBOOI( OFINCOME INEQUALITY MEASUREMENT

RECENT ECONOMIC THOUGHT SERIES

Editors:

Warren J. SamuelsMichigan State UniversityEast Lansing, Michigan, USA

Other books in the series:

William Darity, Jr.University ofNorth CarolinaChapel Hill, North Carolina, USA

Davis, John B.: THE STATE OF THE INTERPRETATION OF KEYNESWells, Paul: POST-KEYNESIAN ECONOMIC THEORYHoover, Kevin D.: MACROECONOMETRICS:

DEVELOPMENTS, TENSIONS AND PROSPECTSKendrick, John W.: THE NEW SYSTEMS OF NATURAL ACCOUNTSGroenewegen, John: TRANSACTION COST ECONOMICS AND BEYONDKing, J.E.: AN ALTERNATIVE MACROECONOMIC THEORYSchofield, Nonnan: COLLECTIVE DECISION-MAKING: SOCIAL CHOICE

AND POLITICAL ECONOMYMenchik, Paul L.: HOUSEHOLD AND FAMILY ECONOMICSGupta, Kanhaya L.: EXPERIENCES WITH FINANCIAL LIBERALIZATIONCohen, Avi J., Hagemann, Harald, and Smithin, John:

MONEY FINANCIAL INSTITUTIONS AND MACROECONOMICSMason, P.L. and Williams, R.M.:

RACE, MARKETS, AND SOCIAL OUTCOMESGupta, Satya Dev: THE POLITICAL ECONOMY OF GLOBALIZATIONFisher, R.C.: INTERGOVERNMENTAL FISCAL RELATIONSMariussen, A. and Wheelock, J.: HOUSEHOLDS, WORK AND ECONOMIC

CHANGE: A COMPARATIVE INSTITUTIONAL PERSPECTIVEGupta, Satya Dev: GLOBALIZATION, GROWTH AND SUSTAINABILITYGupta, Satya Dev: DYNAMICS OF GLOBALIZATION AND DEVELOPMENTMedema, Steven G.: COASEAN ECONOMICS: LAW AND ECONOMICS AND

THE NEW INSTITUTIONAL ECONOMICSPeoples, James: REGULATORY REFORM AND LABOR MARKETSDennis, Ken: RATIONALITY IN ECONOMICS: ALTERNATIVE

PERSPECTIVESAhiakpor, James C.W.: KEYNES AND THE CLASSICS RECONSIDEREDWolfson, Murray: THE POLITICAL ECONOMY OF WAR AND PEACEJain, A.K.: ECONOMICS OF CORRUPTIONWheelock, J. and Vail, J.: WORK AND IDLENESS: THE POLITICAL

ECONOMY OF FULL EMPLOYMENTDean, James M. and Watennan, A. M.C.: RELITION AND ECONOMICS:

NORMATIVE SOCIAL THEORYGupta, Kanhaya: FOREIGN AID: NEW PERSPECTIVESMacDonald, R. and Stein, J.: EQUILIBRIUM EXCHANGE RATESChilcote, Ronald M.: THE POLITICAL ECONOMY OF IMPERIALISM:

CRITICAL APPRAISALS

HANDBOOI( OF INCOME INEQUALITY MEASUREMENT

edited by

J acques Silber Bar-Dan University, Israel

with a foreword by Amartya Sen

1IIt...

" SPRINGER SCIENCE+BUSINESS MEDIA, LLC

Library of Congress Cataloging-in-Publication Data

Handbook of income inequality measurement / edited by Jacques Silber ; with a foreword by Amartya Sen.

P. cm. -- (Recent economic thought series ; 71) Includes index. ISBN 978-94-010-5897-1 ISBN 978-94-011-4413-1 (eBook) DOI 10.1007/978-94-011-4413-1 1. Income distribution--Statistical methods.

II. Series. HBS23.H361999 339.2--dc21

1. Silber , Jacques.

99-28390 CIP

Copyright © 1999 by Springer Science+Business Media New York Second Printing 2001.

Origina11y published by Kluwer Academic Publishers in 1999 Softcover reprint of the hardcover 1 st edition 1999

AlI rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanica1, photo-copying, recording, or otherwise, without the prior written permission of the publisher,Springer Science+Business Media, LLC.

Printed on acid-free pa per.

This printing is a digital duplication of the original edition.

"Who is rich? He who rejoices in his portion."

Chapters of the Fathers 4:4

I.

2.

3.

4.

5.

6.

7.

CONTENTS

List of Contributors

Acknowledgements

ForewordAmartyaSen

Introduction: Thirty Years of Intensive Research on Income InequalityJacques Silber

The Rational Foundations of Income Inequality MeasurementSerge-Christophe KolmComment: Nripesh Podder

Linking the Functional and Personal Distributions of IncomeCamiloDagumComment: Michel Martinez and Christian Morrison

Income Inequality Measurement: The Normative ApproachCharles Blackorby, Walter Bossert and David DonaldsonComment: John A. Weymark

Measuring Inequality: The Axiomatic ApproachSatya R. ChakravartyComment: Udo Ebert

The Mathematical Foundations of Inequality AnalysisMichel Le BretonComment: Wolfgang Eichborn

Stochastic Dominance and the Lorenz CurvePatrick MoyesComment: Shlomo Yitzhaki

The Measurement of Income Inequality: The Subjective ApproachYoramAmielComment: Bernard M. S. Van Praag

xi

xv

xvii

19

101

133

163

187

199

227

viii

8. Income Inequality Measurement: The Statistical Approach 245Giovanni M. GiorgiComment: Maria-Pia Victoria-Feser

9. Estimation oflnequality Indices 269Frank A. CowellComment: Achille Lemmi

10. Parametric Approximations of the Lorenz Curve 291Hang K. Ryu and Daniel 1. SlottjeComment: Joan Esteban

II. Tests of Significance for Lorenz Partial Orders 315John A. Bishop and John P. FormbyComment: Joseph L. Gastwirth and Tapan K. Nayak

12. How Do Income Sources Affect Income Inequality? 341Robert I. LermanComment: Lea Achdut

13. Inequality Decomposition by Population Subgroups and the Analysisof Interdistributional Inequality 363Joseph Deutsch and Jacques SilberComment: Cecilia A. Conrad

14. Equivalence Scales and Inequality 405Frank A. Cowell and Magda Mercader-PratsComment: Yves Fluckiger

15. Multidimensioned Approaches to Welfare Analysis 437Esfandiar MaasoumiComment: Francois Bourguignon

16. Redistributional Effects of Progressive Income Taxes 485Peter J. LambertComment: Andreas Pfingsten

17. Lifetime versus Annual Income Distribution 513JohnCreedyComment: Timothy Smeeding

18. Horizontal Inequity Measurement: A Basic Reassessment 535Stephen P. Jenkins and Peter 1. LambertComment: Robert D. Plotnick

19.

20.

The Measurement of Income Mobility: An Introduction to the LiteratureGary S. Fields and Efe A. OkComment: Valentino Dardanoni

Inequality, Welfare and Poverty: Three Interrelated PhenomenaNanak KakwaniComment: Martin Ravallion

ix

557

599

Epilogue: Reflections on Income Inequality MeasurementAnthony B. Atkinson

Index

635

639

List of Contributors

Lea Achdut, Economic and Social Research Institute, The Labor Federation of Israel,Jerusalem

Yoram Arniel, Ruppin Institute, Israel

Anthony B. Atkinson, Warden, Nuffield College, Oxford University, UK

John A. Bishop, East Carolina University, North Carolina, USA

Charles Blackorby, University of British Columbia, Canada and GREQAM, Universited'Aix-Marseille, France

Walter Bossert, University of Waterloo, Canada

Fran90is Bourguignon, DELTA (ENS), Paris, France

Satya Chakravarty, Indian Statistical Institute, Calcutta, India

Cecilia A. Conrad, Pomona College, California, USA

Frank A.Cowell, London School of Economics and Political Science, UK

John Creedy, University of Melbourne, Australia

Camilo Dagum, University of Bologna, Italy

Valentino Dardanoni, Universita di Palermo, Italy

Joseph Deutsch, Bar-Han University, Israel

David Donaldson, University of British Columbia, Canada

Udo Ebert, University of Oldenburg, Germany

Wolfgang Eichhorn, Universitat Karlsruhe, Germany

Joan Esteban, Institut d'Analisi Economica, Barcelona, Spain

Gary S. Fields, Cornell University, New York, USA

Yves Fliickiger, Universite de Geneve, Switzerland

John P. Formby, University of Alabama, USA

xii

Joseph L. Gastwirth, George Washington University, Washington, DC, USA

Giovanni M. Giorgi, "La Sapienza" University, Rome, Italy

Stephen P. Jenkins, Institute for Social and Economic Research, University of Essex, UK

Nanak Kakwani, University of New South Wales, Sydney, Australia

Serge-Christophe Kolm, EHESS, Paris, France

Peter J. Lambert, University of York, UK

Michel Le Breton, Universite d' Aix-Marseille 2 and Institut Universitaire de France, France.

Achille Lemmi, CRIDIRE, Universita degli Studi di Siena, Italy

Robert I. Lerman, The Urban Institute and American University, Washington, DC, USA

Esfandiar Maasoumi, Southern Methodist University, Texas, USA

Michel Martinez, DELTA (E.N.S), Paris and Thema, Universite de Cergy-Pontoise, France

Magda Mercader-Prats, Universitat Autonoma de Barcelona, Spain

Christian Morrisson, Universite de Paris I, France

Patrick Moyes, CNRS, IDEP and GRAPE, Universite Montesquieu, Bordeaux, France

Efe A. Ok, New York University, USA

Tapan K. Nayak, George Washington University, Washington, DC, USA

Andreas Pfingsten, Institut fur Kreditwesen, Universitiit Miinster, Germany

Robert D. Plotnick, University of Washington, USA

Nripesh Podder, The University of New South Wales, Australia

Martin Ravallion, The World Bank, Washington, DC, USA

Hang K. Ryu, Chung Ang University, Seoul, South Korea

Amartya Sen, Master, Trinity College, Cambridge University, UK

Jacques Silber, Bar-Dan University, Israel

Daniel J. Slottje, Southern Methodist University, Texas, USA

Timothy Smeeding, University of Syracuse, New York, USA

xiii

Maria-Pia Victoria-Feser, Universite de Geneve, Switzerland

Bernard M.S. von Praag, Foundation for Economic Research, The University of Amsterdam,The Netherlands

John A. Weymark, University of British Columbia, Canada

Shlomo Yitzhaki, The Hebrew University, Israel

ACKNOWLEDGEMENTS

In preparing this volume I have accumulated many debts and therefore owe many thanks.

My colleague Warren Young came up with the idea of including this book in the series

Recent Economic Thought published by Kluwer Academic Publishers. I have to thank also

William Darity and Warren Samuels, the editors of the series, for encouraging me to publish

this Handbook and Zachary Rolnik, Vice President and Managing Director of Kluwer

Academic Publishers, for giving me editorial advice and being patient with additional delays

I asked for. I am specially grateful to Peter Lambert who agreed, in addition to his

contribution with Stephen Jenkins on horizontal inequity, to write an additional chapter on

tax progressivity, once I found that the author I had contacted originally would not be able

to keep his commitment. Frank Cowell several times gave me useful advice as did Dan

Slottje. I am thankful to Amartya Sen for inviting me to come to Trinity College. The

discussion I had with him on the various contributions presented in this book as well as on

Income Inequality Measurement was indeed a very rewarding experience. Various letters

from Serge-Christophe Kolm and quite a few conversations with him on the phone were also

very helpful to me. I should indeed thank all the authors, whether of a chapter or of a

comment, for having done their utmost in preparing their contribution, despite the many

commitments most of them have, and for having been patient during the long "gestation

period" of this book. The University of Geneva was very kind in offering me the

opportunity to spend a sabbatical year in 1997-1998 and Yves Fliickiger deserves special

credit for making there all the arrangements I needed to be able to complete this book. Jacob

Rosenberg, Chair of the Department of Economics at Bar-Han University, and Pnina Manor,

Administrative Vice-Chair, were very helpful in agreeing that the secretaries would devote

part of their time to the preparation of this book. Special thanks are also due to various

people who helped with the preparation of the manuscript: Suzanne St. Clair of Kluwer

Academic Publishers who gave us very useful advice, Howard Kaplon whose expertise on

WordPerfect saved us many times, Karen Horovitz who drew most of the graphs and Helen

Ovadia at Bar-Han University who did not mind retyping several chapters, no matter how

xvi Acknowledgements

much mathematics they included. My deepest gratitude goes however to Gilda Kurtzman

without whose help this book would never have been completed. She typed many chapters

and gave me constant editorial assistance, whether in improving the style of several chapters

or in planning the [mal presentation of this book. Her perfectionism was an excellent

complement to my tendency to be very flexible. Finally lowe special thanks to my wife

Fanny, my children and grandchildren, who agreed to my leaving Israel for one year for a

sabbatical so that I could complete some unfinished work, among which was the editing of

this book. To all the individuals I thanked and those I should have thanked but forgot, I can

just say that I hope this book deserves their help.

FOREWORD

Amartya Sen

"Equality," I spoke the wordAs if a wedding vowAh, but I was so much older then,I am younger than that now.

Thus sang Bob Dylan in 1964. Approbation of equality varies not only with our age (though

it is not absolutely clear in which direction the values may shift over one's life time), but

also with the spirit of the times. The 1960s were good years for singing in praise of

equality. The spirit of the present times would probably be better reflected by melodies in

admiration of the Federal Reserve System.

And yet the technical literature on the evaluation and measurement of economic

inequality has grown remarkably over the last three decades. Even as actual economic

policies (especially in North America and Europe) have tended to move towards focusing on

virtues other than the avoidance of economic inequality, the professional literature on

assessing and gauging economic inequality has taken quite a jump forward. A great many

different problems have been addressed and effectively sorted out, and new problems

continue to be posed and analyzed.

The Contents: A Review

Jacques Silber has done a great service to the subject by producing this collection of

admirably helpful and illuminating papers on different aspects of the measurement of income

inequality. The reach of this collection is quite remarkable. Along with a thorough

overview from the editor himself, the major areas in this complex field have been carefully

examined and accessibly discussed.

The historical background to the relation between personal and functional distributions

of incomes is well explored by Camilo Dagum, and the statistical approach to systematizing

xviii Foreword

the assessment -- and ranking -- of inequalities is critically surveyed by Giovanni Giorgi.

Michel Le Breton provides a cogent and concise account of the mathematical foundations of

inequality analysis. grounded on identifying partial orderings that fulfil some very basic

requirements of inequality comparison (such as the rule that a pure transfer of income from

a richer to a poorer person must -- other things given -- reduce income inequality). This

approach has led to a rich literature on stochastic dominance, which is dependably surveyed

and sCl"Qtinized by Patrick Moyes, who also takes the reader through the distinctions between

first, second and third order dominances and their respective implications and roles.

The pioneering works of Serge Kolm and A.B. Atkinson, which did much to initiate

the contemporary literature on the evaluation of inequality, had linked the approach of

dominance to their normative -- and welfare-economic -- implications. They have, in their

respective pieces in this volume, commented (in rather different styles) on the connection

between descriptive statistics and normative evaluation. Aside from these interesting remarks

by the pioneers themselves, we get in this collection a remarkably illuminating account, by

Charles Blackorby, Walter Bossert and David Donaldson, of the entire normative approach

to the measurement of income inequality, which includes an analysis of dominance reasoning

as well as other -- more particular -- normative priorities.

The subject of inequality measurement has often taken a strongly axiomatic form,

involving normative demands as well as requirements of descriptive perspicacity. Satya

Chakravarty takes the readers deftly through the substance of axiomatic reasoning and its

many applications in this subject. While the axioms can be judged in different ways (for

example, through invoking abstract reasoning on principles of justice, or through appeals to

elementary statistical requirements), the relation between the invoked axioms and people's

subjective perceptions of inequality can be of great interest, and this is the subject matter of

Yoram Arniel's illuminating paper.

While the chosen axioms are standardly related to the incomes of particular individuals

(often involving a small number of people), practical inequality analysis is typically applied

to aggregate statistics over large groups. The statistical approach -- presented at a general

level with historical background by Giovanni Giorgi -- receives forceful scrutiny from Frank

Amartya Sen xix

Cowell, who focuses on estimation problems in general. The statistical approach to

comparisons of Lorenz curves -- central to the use of dominance reasoning -- both requires

methods of approximation in deriving Lorenz curves and a reading of the significance of

partial orderings that reflect Lorenz comparisons. While Hang Ryu and Daniel Slottje

introduce the readers to the first problem, John Bishop and John Formby provide a useful

understanding of the latter task.

Inequality analysis lends itself to seeking decomposability in various forms, and this

difficult literature is helpfully presented and critically assessed by Joseph Deutsch and

Jacques Silber. The requirements of decomposability links closely to the important question

addressed by Robert Lerman as to how income sources may affect income inequality.

If decomposability is a challenging problem, so is the use of "equivalence scales" which

"correct" actual incomes to arrive at effective incomes through adjustments that reflect

variations of non-income characteristics of individuals or of families. That problem receives

attention from Frank Cowell and Magda Mercader-Prats, who searchingly investigate a

particular class of approaches to tackling this problem. The different characteristics that may

be relevant to inequality analysis can also be simultaneously considered in a multidimensional

framework, and the large literature on multidimensional analysis of inequality is extensively

examined by Esfandiar Maasoumi.

The next set of papers takes the readers through specific policy issues, in particular

income redistribution and progressive taxation (Peter Lambert), the linkages between

inequality in life-time incomes and that over a short period such as a year (John Creedy), the

relevance and demands of horizontal inequity (Stephen Jenkins and Peter Lambert),

measurement of income mobility, taking note of profiles of inequalities (Gary Fields and Efe

Ok), and the complex relationship between assessments of inequality, poverty and welfare

(Nanak Kakwani). The respective problems are well addressed and, in different ways, these

contributions help to bridge the gap between theoretical analysis and practical reason.

Even this extremely brief recounting of subject matters covered in this volume brings

out how extensive and comprehensive that coverage is. Also, in addition to the far-reaching

usefulness of the papers themselves, the readers can benefit from the comments and

xx Foreword

criticisms on each contribution from well-chosen commentators, who are often themselves

leaders in their respective fields (Lea Achdut, Francois Bourguignon, Cecilia Conrad,

Valentino Dardanoni, Udo Ebert, Wolfgang Eichhorn, Joan Esteban, Yves Fliickiger, Joseph

Gastwirth, Achille Lemmi, Michel Martinez, Christian Morrisson, Tapan Nayak, Andreas

Pfingsten, Robert Plotnick, Nripesh Podder, Martin Ravallion, Timothy Smeeding, Bernard

van Praag, Maria-Pia Victoria-Feser, John Weymark and SWomo Yitzhaki). Jacques Silber

deserves much credit for producing this book with insightful imagination as well as energy

and determination.

Having confmed myself, so far, to describing this book, let me now take the liberty to

point briefly to a few unsettled issues related to the foundations of measuring and assessing

economic inequality. These are not meant, in any way, as criticism of this collection, but

as possible ideas for future work.

Economic Inequality and Non-income Influences

First, income inequality can diverge not only from welfare inequality, but also from

economic inequality broadly defined, since many non-income factors may influence the

economic limits of our freedom to achieve what we want to achieve. This divergence is

typically not disputed by those who nevertheless choose to concentrate on income inequality

only, for one reason or another. The reason often is tractability of income data compared

with the difficulty of getting and using some non-income determinants of feasible actions and

choices. But insofar as this limited focus leaves our understanding of economic inequality

seriously incomplete, we have to think about ways of bridging this gap.

One route is that of "correction" of income data in the light of non-income information,

so that the income distribution measures are applied to equivalent incomes (or effective

incomes, or corrected incomes). The paper by Cowell and Mercader-Prats explores that

route, and shows its effectiveness in dealing with certain types of non-income considerations

related particularly to "household differences." But as they point out, many of the basic

issues have remained unaddressed.

Amartya Sen xxi

A second route is to see economic inequality in explicitly multidimensional tenns. In

that general fonnat, non-income factors can be accommodated along with income (or its

detenninants). Esfandiar Maasoumi's paper discusses the nature of this approach and its

reach and merits. This approach can be very inclusively used, and there are different ways

of arriving at scalar measures of inequality from distributional infonnation in a

multidimensional space.

In contrast with these two approaches -- well surveyed in this collection -- a third

approach takes the fonn of evaluating the overall advantage that a person enjoys, based on

the different detenninants as they apply to her (these detenninants are components of the

multidimensional infonnation). We can then compare the respective measures of advantage

of different persons to arrive at an evaluation of overall inequality. One particular avenue

that has been explored is to see the implications of the various detenninants of quality of life,

in the fonn of valuable functionings (in largely Aristotelian lines), and then to construct

indices of valuable functioning combinations (using agreed valuations and broad consensus

on particular trade-offs). There is a growing literature on this alternative approach, with

various different techniques of evaluation. l It is useful, in this context, to appreciate the

presence of different types of non-income factors that influence the individual's economic

prosperity, well-being and freedom. Various types of contingencies lead to variations in the

"conversion" of incomes into levels of living. Four important sources of variation can be

readily identified.

(1) Personal heterogeneities: Human beings have disparate physical characteristics connected

with disability, illness, age, or gender, making their needs diverse.

(2) Environmental diversities: Conversion of income into freedom and achievement can be

influenced by environmental conditions, including climatic circumstances, such as

extremities of temperatures, or risks of flooding or cyclones, or the effects of man­

made pollution.

(3) Variations in social climate: What people can achieve with a given level of income may

greatly vary with various social conditions, including public health care and

epidemiology, educational arrangements, the prevalence or absence of crime and

xxii Foreword

violence, and various social attitudes that have recently received attention as parts of

the so-called "social capital."

(4) Differences in relational perspectives: Income needed for meeting the same basic

functioning may vary with the established conventions in a community or a group, and

this may depend inter alia on the general standards of prosperity in that society,

including relative incomes. For example, as Adam Smith had noted, to be able to

"appear in public without shame" may require higher standards of clothing and other

visible consumption in a richer society than in a poorer one. Personal resources

needed for taking part in the life of the community -- or even to achieve self-respect

-- may in general vary with the lives of others in that community. This is primarily

an inter-societal variation, rather than an inter-individual variation within a given

society, but the two concerns are inter-linked, especially in the context of such issues

as "decomposability" (on which more presently).

Each of the three broad approaches outlined earlier can be tried out as ways of handling

the multiplicity of determining factors. Indeed, in dealing with problems of such difficulty

as well as importance, there is a good case for exploring each approach without giving up

the others. The "corrected income" approach has the merit of linking more closely to our

intuitions derived from the literature on income inequality, well reflected in this volume. On

the other hand, it is an indirect approach and to that extent, it is less transparent than dealing

directly with the ends that income and non-income means help to promote. It is also worth

mentioning that insofar as inequality measures are inescapably concerned with measurement

of "distances", we must take note of the fact that a small distance in the income space

(including the equivalent income space) could correspond to quite a large distance in the

space of achievements. For example, if a small fall in income makes a poor person a victim

of starvation -- and possibly of death -- then that small distance in the income space stands

for a very large distance in the space that ultimately matters. The metrics of inequality in

direct and indirect spaces are not congruent.

Amartya Sen xxiii

Is Income Inequality an Internal Characteristic of Income Distribution?

Closely related to this problem is a second issue, to wit, whether measurement of income

inequality can be taken to depend only on the "internal" characteristics of income

distribution. If ~ is an income distribution vector over a given population in a particular

economic situation, should we expect that the extent of income inequality [ in that situation

must be independent of everything else, that is: [ = [(JJ?

An elementary intuition might well suggest that this must be the case. How can the

same income distribution vector have two different measures of income inequality, varying

with non-income characteristics? Indeed, once the problem of measurement of income

inequality is separated out from that of measuring economic inequality more broadly defined,

it would seem to be plausible enough to insist that the level of income inequality must depend

only on income information and on nothing else. And indeed all the measures of "descriptive

statistics" (such as the Gini coefficient, coefficient of variation, etc.) satisfy the functional

requirement [ = [(y).

And yet the significance of income gaps may depend on the nature of the non-income

features, and this cannot but affect normative evaluations of inequality. How much

normative importance to attach to any income gap must depend on what results follow from

the income gap, in particular how they relate to differences in those variables (such as

aggregate utility or social welfare) in terms of which the significance of income inequality

is ultimately judged. This dependence is well acknowledged in a formula like that of

Atkinson, in which inequality is measured by the loss of the income equivalent of the sum­

total of u(yJ as a result of the presence of unequal incomes. The significance of the income

gaps depends on the concavity of the u(yJ function. Income inequality is, in this account,

not a purely internal characteristic of income distribution.

It must be noted that this problem is different from -- though related to -- the

distinction between income inequality and inequality of other variables (such as utility or

freedom or capability). For example, the Atkinson formula is not measuring the inequality

of the levels of u(yJ, but measuring the inequality specifically of incomes by looking at the

loss of the income equivalent of the aggregate value of the vector of u(yJ. Intuitively, this

xxiv Foreword

is an attempt to relate inequality measurement to an exercise of social welfare evaluation (and

not specifically to inequality in any other space).

Decomposition and Non-income Characteristics

A third problem -- closely related to the first two -- concerns the requirement of

decomposability. Decomposability has been seen to be an attractive property of an inequality

measure, and as the paper of Deutsch and Silber brings out, this can impose a strong

requirement on permissible inequality measures, with quite far-reaching implications. But

is the demand of decomposability sensible?

It can be argued that it is not a sensible demand when imposed on income inequality

measures if non-income characteristics have a significant role. As Adam Smith had noted,

the consumption that must count as a "necessity" has to depend on the standards of

consumption established in a community within which a person operates. The ability to take

part of the life of the community may, thus, depend on a person's relative income vis-a-vis

those of others in that community, and in judging the significance of income gaps this

consideration may have to be taken into account. This poses no problem if we partition the

total population into communities within which people interact, so that the level of overall

income inequality for the total population can be seen as a function of a weighted sum of

within-group inequalities (for those community groups) and a measure of between-group

inequality (among those community groups). This way of splitting up the total population

in the decomposition exercise is, thus, just fine.

But decomposability as a requirement has to work for every partition of the total

population -- not just for the one chosen partition. Suppose we partition the population on

some other basis, such as the first initials of the persons' surnames. Again, decomposability

would require that the overall inequality be exactly made of a weighted sum of within-group

inequalities (now for these surname-based groups) and a measure of between-group inequality

(among those surname-based groups). But since the relevance of relativities cut across the

surname-based groups, this would be a most unreasonable requirement, and it is easy to show

that for these other ways of partitioning -- other than that of the relevant communities within

Amartya Sen xxv

which people socially interact -- the implications of decomposability would be entirely

unreasonable and worth violating.

Can we not impose the requirements of decomposability for some partitions, but not

others? We certainly can, but then the measurement of inequality ceases to be dependent

only on income vectors, and depends also on which individual belongs to which group or

community. This is where this discussion relates to the previous point that measurement of

inequality may not be seen as only an internal characteristic of the income distribution

vectors. Other information is relevant even in evaluating income inequality,

It also follows that choosing "decomposable" measures (such as the Theil index of

entropy or the coefficient of variation) over "non-decomposable" ones (such as the Gini

coefficient) on grounds of the merits of general decomposability is not as sound a reasoning

as is frequently claimed. Measures such as entropy or the coefficient variation manage to

pass the decomposability test by making the evaluation of each person's income completely

independent of the income of everyone else (thereby going firmly against Adam Smith's

arguments), so that no matter which partitioning is chosen, the splitting up formula would

work. Despite the attractiveness of doing decomposability analysis for some partitions (such

as localities, racial groups, other communities), in the demanding universal form -- applied

to all partitions -- decomposability is surely an over-kill.

Theory of Measurement and Practical Reason

Given the remarkable achievements of this Handbook of income inequality measurement, I

have chosen to comment here on some issues that link the exercise of assessing income

inequality to more general concerns about social welfare and also to the inequality of other

variables such as well-being or freedom. In particular, I have focused on some interrelated

implications of the significance of non-income factors even when our primary purpose is

specifically to evaluate and measure income inequality. The literature on income inequality

has not, I believe, fully come to grips with these interdependences, and there is scope for

going beyond the literature surveyed here to broaden the coverage of the exercise.

xxvi Foreword

In his comments on these papers, Tony Atkinson has also drawn attention to a number

of other extensions that could be sensibly pursued. I think there are good grounds to take

note particularly of Atkinson's complaint that "the recent theoretical developments have so

far had relatively little impact on empirical practice."

Many people....continue to use the same conventional summary measures of inequality;rarely does one find tests of significance for observed differences; policy decisions aremade about the treatment of different family types without regard to the sensitivity ofthe measured distributional consequences to the choice of equivalence scale; and studiesof poverty typically present only the headcount. (p. 633)

It is fitting that a handbook which has done so much to make the existing literature

available to general readers should also include suggestions for the direction in which we

might proceed in the future. For example, Atkinson's emphasis on the importance and

impact of "readily available software" illustrates the kind of things that can be done to make

greater use of the impressive theoretical literature that has emerged. I would like to

supplement these suggestions for application by arguing for some extra work on the theory

of measurement as well, for example in integrating the role of non-income characteristics

more fully into the evaluation of economic inequality in general and income inequality in

particular.

It is a measure of the success of a volume of this kind that it not only consolidates what

has been already achieved, but also inspires us to think about what more needs to be done.

The ideas that have been explored in this impressive handbook call not just for understanding

and appreciation, but also for constructive responses of various kinds. There is a future as

well as a past.

Notes

I. See the references given in Foster and Sen (1997).

Reference

Foster, James E. and Amartya Sen. 1997. "On Economic Inequality after a Quarter Century." In A. Sen, OnEconomic Inequality, Expanded Edition. Oxford: Oxford University Press.