Occupational and Residential Segregation, Volume 17

OCCUPATIONAL AND RESIDENTIAL SEGREGATION RESEARCH ON ECONOMIC INEQUALITY Series Editor: John Bishop RESEARCH ON ECO...

Author: Jacques Silber | Yves Fluckiger | Sean F. Reardon

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OCCUPATIONAL AND RESIDENTIAL SEGREGATION

RESEARCH ON ECONOMIC INEQUALITY Series Editor: John Bishop

RESEARCH ON ECONOMIC INEQUALITY VOLUME 17

OCCUPATIONAL AND RESIDENTIAL SEGREGATION EDITED BY

YVES FLU¨CKIGER University of Geneva, Switzerland

SEAN F. REARDON Stanford University, USA

JACQUES SILBER Bar-Ilan University, Israel

United Kingdom – North America – Japan India – Malaysia – China

Emerald Group Publishing Limited Howard House, Wagon Lane, Bingley BD16 1WA, UK First edition 2009 Copyright r 2009 Emerald Group Publishing Limited Reprints and permission service Contact: [email protected] No part of this book may be reproduced, stored in a retrieval system, transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without either the prior written permission of the publisher or a licence permitting restricted copying issued in the UK by The Copyright Licensing Agency and in the USA by The Copyright Clearance Center. No responsibility is accepted for the accuracy of information contained in the text, illustrations or advertisements. The opinions expressed in these chapters are not necessarily those of the Editor or the publisher. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-1-84855-786-4 ISSN: 1049-2585 (Series)

Awarded in recognition of Emerald’s production department’s adherence to quality systems and processes when preparing scholarly journals for print

CONTENTS LIST OF CONTRIBUTORS

ix

INTRODUCTION

xi

PART I: INFORMATION THEORY AND THE MEASUREMENT OF SEGREGATION THE INFORMATION THEORY OF SEGREGATION: UNITING SEGREGATION AND INEQUALITY IN A COMMON FRAMEWORK Paul A. Jargowsky and Jeongdai Kim THE INVARIANCE PROPERTIES OF THE MUTUAL INFORMATION INDEX OF MULTIGROUP SEGREGATION Ricardo Mora and Javier Ruiz-Castillo

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PART II: THE GINI INDEX AND THE MEASUREMENT OF SEGREGATION THE GINI COEFFICIENT AND SEGREGATION ON A CONTINUOUS VARIABLE Jeongdai Kim and Paul A. Jargowsky GENERALIZED GINI OCCUPATIONAL SEGREGATION INDICES Satya R. Chakravarty, Conchita D’Ambrosio and Jacques Silber v

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71

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CONTENTS

PART III: MEASURING SEGREGATION WITH ORDERED CATEGORIES OCCUPATIONAL SEGREGATION WITH ECONOMIC DISADVANTAGE: AN INVESTIGATION OF DECOMPOSABLE INDEXES Robert Hutchens

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COMMENT ON ROBERT HUTCHENS, ‘‘OCCUPATIONAL SEGREGATION WITH ECONOMIC DISADVANTAGE: AN INVESTIGATION OF DECOMPOSABLE INDEXES’’ Paul A. Jargowsky

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A RESPONSE TO PAUL JARGOWSKY’S COMMENT Robert Hutchens

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MEASURES OF ORDINAL SEGREGATION Sean F. Reardon

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PART IV: EXPLORING CHANGES IN SEGREGATION EXPLORING CHANGES IN THE SPATIAL PATTERN OF INCOME SEGREGATION DURING THE 1990s Casey J. Dawkins

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ANALYZING CHANGES IN OCCUPATIONAL SEGREGATION: THE CASE OF SWITZERLAND (1970–2000) Joseph Deutsch, Yves Flu¨ckiger and Jacques Silber

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Contents

PART V: WAGE INEQUALITY AND SEGREGATION WAGE INEQUALITY AND SEGREGATION BETWEEN NATIVE AND IMMIGRANT WORKERS IN SWITZERLAND: EVIDENCE USING MATCHED EMPLOYEE–EMPLOYER DATA Tobias Mu¨ller and Jose´ Ramirez

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GENDER SEGREGATION IN THE WORKPLACE AND WAGE GAPS: EVIDENCE FROM URBAN MEXICO 1994–2004 Sebastiań Calońico and Hugo Nõpo

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LIST OF CONTRIBUTORS Sebastiań Calońico

Department of Economics, University of Michigan, MI, USA

Satya R. Chakravarty

Indian Statistical Institute, Economic Research Unit, Kolkata, India

Conchita D’Ambrosio

Universita` di Milano-Bicocca, DIW Berlin and Econpubblica, Universita` Bocconi, Milano, Italy

Casey J. Dawkins

Department of Urban Affairs and Planning, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA

Joseph Deutsch

Department of Economics, Bar-Ilan University, Ramat-Gan, Israel

Yves Flu¨ckiger

De´partement d’eć- onomie politique, Universite´ de Gene`ve, Geneva, Switzerland

Robert Hutchens

School of Industrial and Labour Relations, Cornell University, New York, USA

Jeongdai Kim

EcoMotto Energy, Plano, TX, USA

Paul A. Jargowsky

School of Economic, Political, and Policy Sciences, University of Texas at Dallas, TX, USA

Ricardo Mora

Departamento de Economı´ a, Universidad Carlos III de Madrid, Spain

Tobias Mu¨ller

Department of Econometrics, University of Geneva, Switzerland ix

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LIST OF CONTRIBUTORS

Hugo Nõpo

Inter-American Development Bank, Education Division, Bogota, Colombia

Jose´ Ramirez

Geneva School of Business Administration, University of Applied Sciences, Western Switzerland, Carouge-Geneva, Switzerland

Sean F. Reardon

School of Education, Institute for Research on Education Policy and Practice, Stanford University, Stanford, CA, USA

Javier Ruiz-Castillo

Departamento de Economı´ a, Universidad Carlos III de Madrid, Spain

Jacques Silber

Department of Economics, Bar-Ilan University, Israel

INTRODUCTION Although the measurement of segregation by gender or ethnic group in the labor force has long been of interest to both sociologists and economists, the sociology and economics literatures on this topic have evolved in different ways and remained largely separate. This has also been the case to some extent with research on the measurement of residential segregation. Although much of the segregation measurement literature is in sociology and geography, economists have contributed to this field as well, particularly in the development of measures of residential income segregation. Again, however, the economics literature has remained largely separate from that in geography and sociology. Because of this separation, Yves Flu¨ckiger and Jacques Silber took the initiative to organize on July 1–6, 2007, an international conference that took place in Monte Verita`, in Ascona, Switzerland, devoted to the topic ‘‘New Frontiers in the Field of Segregation Measurement and Analysis.’’ The idea was to bring together specialists from several fields of social sciences, all of them working on subjects related to segregation measurement. The present volume gathers some of the papers that were presented at this conference as well as a few papers by authors who did not participate in this event. All the chapters in this volume have been reviewed by referees. The volume is divided into five parts. Part I (Chapters 1 and 2) is devoted to ‘‘Information Theory and Segregation Measurement,’’ Part II (Chapters 3 and 4) to ‘‘The Gini Index and the Measurement of Segregation,’’ Part III (Chapters 5 to 8) to ‘‘Measuring Segregation with Ordered Categories,’’ Part IV (Chapters 9 and 10) to ‘‘Exploring Changes in Segregation,’’ and Part V (Chapters 11 and 12) to ‘‘Wage Inequality and Segregation.’’ Let us now review into more details each of these 12 chapters. In the first chapter, Paul Jargowsky and Jeongdai Kim propose an Information Theory of Segregation. In this chapter, entitled ‘‘The Information Theory of Segregation: Uniting Segregation and Inequality in a Common Framework,’’ they argue that all measures of segregation and inequality may be united within a single conceptual framework. For these authors accepting this framework implies that all measures of inequality can also be used to measure segregation and that all measures of segregation are fundamentally based on measures of inequality. They thus show that all xi

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INTRODUCTION

common measures of inequality can be used to form measures of segregation and that the resulting measures can be applied to binary, polytomous, and continuous variables. As a consequence they develop several new measures, including a Gini Segregation Index for continuous variables and a version of the Index of Dissimilarity suitable for measuring economic segregation. They also emphasize that segregation measures may easily be adapted to handle persons of mixed race, and describe what they call the Non-Exclusive Index of Dissimilarity and the Non-Exclusive Entropy Index of Segregation. In the second chapter, Ricardo Mora and Javier Ruiz-Castillo examine ‘‘The Invariance Properties of the Mutual Information Index of Multigroup Segregation.’’ They start by arguing that in the context of educational segregation by ethnic group, for example, a rigorous pairwise segregation comparison over time or across space should be invariant to changes in the ethnic composition of the population, assuming the distribution of each ethnic group over the schools remains constant (they call this case the first type of invariance), as well as to the variation in the size distribution of schools, assuming the ethnic composition of each school remains constant (they call this case the second type of invariance). They first note that both properties have strong implications and that there are reasons to defend that the overall segregation index need not satisfy either one. They then present two decompositions of the M index into three terms in pairwise comparisons over time or across space, which isolate the effects of the changes in the marginal distributions. In the first decomposition, the first term captures the effects on the index M of changes in the ethnic groups’ marginal distribution. The second term captures the effects of the changes in the entropy, or concentration, of the schools. The last term captures changes in the M index when there are no changes in the groups’ marginal distributions and no changes in the concentration of schools. In the second decomposition, the first term captures the effects on the index M of changes in the organizational units’ marginal distribution. The second term captures the effects of the changes in the concentration of the groups, while the last term captures changes in the M index otherwise. Finally, they indicate that these decompositions can be used to reach results similar to those derived by Deutsch et al. in Chapter 10 of the present volume. In Chapter 3, Jeongdai Kim and Paul Jargowsky, look at ‘‘The Gini Coefficient and Segregation on a Continuous Variable.’’ They argue that the Gini coefficient, a very popular measure of income inequality, has also been used as a segregation measure, but only in the case of binary variables like race or gender. The purpose of this chapter is to develop a general version of the Gini Segregation Index that can accommodate either continuous or

Introduction

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binary variables, and discuss its relationship to existing measures. The authors also provide a graphical illustration. Finally, using the Public Use Microdata Sample for 25 U.S. metropolitan areas from the 2000 Census, they illustrate the calculation of the index and show that it is highly correlated with an existing measure of economic segregation. In Chapter 4, Satya Chakravarty, Conchita D’Ambrosio, and Jacques Silber describe what they call ‘‘Generalized Gini Occupational Segregation Indices.’’ They derive axiomatically a class of numerical indices of integration (equality) in the distribution of different types of workers across occupations. The associated segregation (inequality) indices parallel one form of multidimensional generalized Gini inequality indices. The authors also compare this new index with the other Gini-related segregation indices. Finally, using U.S. occupational data, they provide a numerical illustration of the family of indices they derived. Chapter 5, written by Robert Hutchens, is entitled ‘‘Occupational Segregation with Economic Disadvantage: An Investigation of Decomposable Indexes.’’ It starts by noting that measures of segregation usually ignore economic disadvantage. For example, when measuring occupational segregation of men and women, one usually examines variation in gender ratios across occupations without worrying about whether women tend to be concentrated in poorly compensated occupations. Similarly, when measuring residential segregation, one often considers the ethnic or racial mix of a neighborhood without worrying about whether minorities tend to be concentrated in more impoverished neighborhoods. The chapter derives numerical indexes of segregation that use the average level of compensation in an occupation as a cardinal measure of economic status. It develops methods for decomposing changes in these indexes into analytically meaningful components. It concludes with an analysis of occupational segregation of men and women in the United States between 1970 and 2000. In Chapter 6, Paul Jargowsky, commenting on Chapter 5, argues that Hutchens develops a class of indices that combine two dissimilar types of information: segregation by occupation and inequality between men and women in a characteristic of those occupations, namely income. Jargowsky would rather label this measure occupational disadvantage since it combines disadvantage from segregation per se and inequality on an economic or status dimension. In Chapter 7, Hutchens replies to Jargowsky, asserting that Jargowsky should have more precisely defined the difference he makes between occupational disadvantage and segregation. Hutchens believes that this field of inquiry would greatly benefit from a more intensive use of the mathematical language.

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INTRODUCTION

Chapter 8, written by Sean Reardon, addresses similar concerns using a different framework. In this chapter, ‘‘Measures of Ordinal Segregation,’’ Reardon argues that traditional methods of measuring segregation focus on the measurement of segregation among groups defined by nominal categorical variables (e.g., race and gender) among organizational units also defined by nominal categorical units (e.g., schools and neighborhoods). These measures are inappropriate when either the groups or the organizational units are defined by ordered categories. Reardon develops methods appropriate to measuring segregation among ordered categories, such as occupational categories or levels of educational attainment. The new approach described in this chapter combines the appealing features of defining segregation as a ratio of variation within categories to variation in a population with the computational ease of defining ordinal segregation measures as averages of binary segregation measures among pairs of categories defined by an ordered variable. Chapter 9, written by Casey Dawkins, is devoted to ‘‘Exploring Changes in the Spatial Pattern of Income Segregation During the 1990s.’’ It starts by noting that evidence suggests that during the 1990s, many U.S. metropolitan areas saw fundamental changes in the spatial distribution of household income. Following two decades of increasing economic segregation, many metropolitan neighborhoods saw declines in economic segregation, particularly those neighborhoods located within central cities and rural areas. To explore these trends the author extends earlier work based on the use of the Spatial Ordering Index. On the basis of U.S. Census data, he calculates economic segregation indices for a sample of 205 U.S. metropolitan areas in 1990 and 2000 and decomposes changes in the indices into portions attributable to changes in the spatial distribution of households and portions capturing changes in the spatial distribution of aggregate income. He also examines regional variations in the decompositions. The results of this empirical investigation suggest that changes in the spatial distribution of households and changes in the spatial distribution of income each influenced metropolitan economic segregation in different ways during the 1990s. Furthermore, the spatial dynamics of income segregation exhibited significant regional heterogeneity. In Chapter 10, which is entitled ‘‘Analyzing Changes in Occupational Segregation: The Case of Switzerland (1970–2000),’’ Joseph Deutsch, Yves Flu¨ckiger, and Jacques Silber look at the changes that took place in occupational segregation in Switzerland during the period 1970–2000. The chapter starts by using correspondence analysis to detect changes in occupational segregation by gender and nationality. It then combines a

Introduction

xv

known algorithm on the adjustment of a frequency table when the expected margins of this table are known with the so-called Shapley decomposition procedure in order to derive a clear breakdown of the variation over time in occupational segregation (change in gross segregation) into a component measuring changes in net segregation and another one corresponding to changes in the margins. The latter is then itself broken down into two components, one measuring variations in the occupational structure and the other changes in the shares of the subpopulations (e.g., the genders) in the labor force. This new decomposition may easily be extended to the cases where more than two categories are distinguished or when there are more than two dimensions. The results of the empirical illustration, based on Swiss Census data for 1970 and 2000, show that in several instances, variations in gross and net segregation worked in opposite directions. In Chapter 11 Tobias Mu¨ller and Jose´ Ramirez examine ‘‘Wage Inequality and Segregation Between Native and Immigrant Workers in Switzerland: Evidence Using Matched Employee–Employer Data.’’ More precisely, they look at wage inequality and segregation between immigrants and natives in the workplace, taking into account the interaction between skill level and immigration status (work permit). They first calculate exposure rates in order to analyze segregation at the firm level along these two dimensions. Then they analyze the role of segregation in the explanation of wage inequality between different skill–nationality groups. Their empirical investigation is based on the use of data from the Swiss Wage Structure Survey 2002, which records individual wages among a very large sample of establishments in all industries, covering approximately 42,000 firms and 1 million workers. The authors conclude that interfirm segregation is particularly pronounced for unskilled foreign workers and for recently arrived, highly skilled foreigners. The former earn lower wages than equally skilled Swiss workers and the latter are paid higher wages than highly skilled Swiss workers. In both cases, interfirm segregation accounts for almost the entire wage differential. Finally Chapter 12, written by Sebastiań Calońico and Hugo Nõpo, looks at ‘‘Gender Segregation in the Workplace and Wage Gaps: Evidence from Urban Mexico 1994–2004.’’ This chapter analyzes the evolution of gender segregation in the workplace in Mexico between 1994 and 2004, using a matching comparisons technique to explore the role of individual and family characteristics in determining gender segregation and wage gaps. The results suggest that the complete elimination of vertical segregation would reduce the observed gender wage gaps by 5 percentage points, while the elimination of occupational segregation would have increased gender wage gaps by approximately 6 percentage points. The results also indicate that the role of

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INTRODUCTION

occupational segregation in wage gaps has been increasing in magnitude during the period of analysis, while the role of vertical segregation on the determination of wage gaps has been decreasing. We cannot conclude this introduction without some words of thanks. The Swiss Federal Institute of Technology in Zurich, which owns the Centro Stefano Franscini in Monte Verita` where the conference took place, was kind enough to put the Centro at our disposal for close to a week and provide an important part of the financial resources that were necessary to organize such a conference. Financial support by the Socie´te´ acade´mique, the Fonds geńe´ral of the University of Geneva, the Faculty of Social and Economic Sciences, as well as the Department of Economics (University of Geneva) is also gratefully acknowledged. We are also very thankful to Francine Gennai, who was in charge of the technical planning of this conference and was extremely helpful to everyone during the whole conference. Finally we would like to thank John A. Bishop, the editor of the series Research on Economic Inequality, for having given us the opportunity to edit this volume. Yves Flu¨ckiger Sean F. Reardon Jacques Silber Editors

PART I INFORMATION THEORY AND THE MEASUREMENT OF SEGREGATION

THE INFORMATION THEORY OF SEGREGATION: UNITING SEGREGATION AND INEQUALITY IN A COMMON FRAMEWORK Paul A. Jargowsky and Jeongdai Kim ABSTRACT Purpose – We propose the Information Theory of Segregation, which holds that all measures of segregation and of inequality are united within a single conceptual framework. Accepting this framework implies that all measures of inequality can also be used to measure segregation and that all measures of segregation are fundamentally based on measures of inequality. Methodology – We state several propositions that follow from the information theory perspective, and show mathematically that many common measures of inequality and segregation satisfy the propositions. Findings – We show that all common measures of inequality can be used to form measures of segregation and that the resulting measures can be applied to binary, polytomous, and continuous variables. Further, we develop several new measures, including a Gini Segregation Index (GS) for continuous variables and Income Dissimilarity Index (ID), a version of the Index of Dissimilarity suitable for measuring economic segregation. Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 3–31 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017004

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PAUL A. JARGOWSKY AND JEONGDAI KIM

We show that segregation measures can easily be adapted to handle persons of mixed race, and describe the Non-Exclusive Index of Dissimilarity (NED) and the Non-Exclusive Entropy Index of Segregation (NEH). We also develop a correction for structural constraints on the value of segregation measures, comparable to capacity constraints in a communications channel, which prevent them from reaching their theoretical maximum or minimum value. Originality – Placing inequality and segregation measures in a common framework is useful for several reasons. It highlights a common mathematical structure shared by many different segregation measures, and it suggests certain useful variants of these measures that have not been recognized previously.

INTRODUCTION Shannon’s landmark paper ‘‘The Mathematical Theory of Communication’’ (1948) examined the problems of transmitting information over limited and potentially noisy channels. The paper, which has been cited over 20,000 times, launched the field of information theory and is considered foundational in several disciplines, including communications, digital computing, and cryptography. Previous work in the measurement of segregation has drawn on information theory measures and concepts, but only to a limited extent. We propose a much more direct and fundamental connection between segregation and information theory, one that unites measures of segregation and inequality in a single framework. Specifically, we argue that the question of segregation is analogous to the problem of communicating information over a noisy channel. We propose an Information Theory of Segregation that defines segregation in terms of the ratio of inequality among individuals to inequality between groups of individuals, for example, neighborhoods. We state four specific propositions concerning inequality and segregation measures that follow from this approach, and discuss the implications of these propositions. As result of the application of information theory to the question of segregation, we are able to develop several important results. First, all inequality measures can be used to form segregation measures. Second, most commonly used segregation measures can be expressed as the ratio of two inequality measures. Third, we show that several segregation measures that cannot be expressed this way are not measures of segregation at all, but rather measures of neighborhood

The Information Theory of Segregation

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inequality. Fourth, by understanding their common mathematical structure, we can see that segregation measures that were previously used only in the binary case have analogues that are applicable to continuous variables, such as income. Fifth, insights generated by this approach show how racial segregation measures can be applied to situations involving mixed-race individuals, such as Tiger Woods or Barack Obama. Our goal in this paper is to provide a conceptual foundation for segregation measures based on information theory. Not all measures of segregation that have been proposed are consistent with the framework developed here. Researchers are obviously free to adopt a different or broader conceptualization of the meaning of segregation. However, the clarity and new insights generated by the information theory approach are strong arguments in its favor.

INFORMATION THEORY, SEGREGATION, AND INEQUALITY Shannon wrote, ‘‘the fundamental problem of communication is that of reproducing at one point y a message selected at another point’’ (1948, p. 379). A communication system consists of an information source, a destination, and a possibly noisy transmission process over a medium that has limited capacity, such as a telephone cable. Defining the information content of the source was the crucial first step to Shannon’s analysis. He characterized the information value of a source as a function of the number of potential messages, or different states of the system, that the source could produce. The technical problem in the field of communication is whether the intended recipient of a message can distinguish it from the many possible messages that the sender could have sent. Communication is a problem only if the receiver is uncertain about what message the source intends. The more potential messages, that is the greater the information content of the source, the greater the uncertainty about the intended message. Therefore, the information content of the source is at a minimum if there is only one possible message, and increases as the number of possible messages and therefore the uncertainty about any specific message increases. All of this may seem far removed from the issues of inequality and segregation, yet we argue that this perspective is directly applicable. We conceptualize the source as information about a population of individuals, specifically the values of a particular variable that characterizes the

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individuals. The information content of the source is the variation between individuals. If individuals are identical, then choosing an individual at random will produce the same message every time. In Shannon’s terms, the information of the source is zero because there is no uncertainty about the message. If there is a great deal of variation among the individuals, then there are many possible messages that can be produced (many different ‘‘states of the system’’) and the information value of the source is high. The communication process is analogous to the sorting into groups. For example, the groups can be a two-dimensional grid of neighborhoods. The message received is contained in a set of values, one for each group, which characterizes the individuals in the group. Usually, but not necessarily, that value is the mean of the variable for the individuals in that group. The message also contains the number of individuals in each group. After the sorting takes place, the information content of the message received depends on the resulting distribution of neighborhood values of the outcome variable. If all the groups have the same mean, the sorting process destroys all of the information about the underlying distribution of the variable at the level of individuals. In certain circumstances, all of the information about the underlying distribution might be preserved. Imagine a Martian attempting to learn about income inequality in a certain city. The Martian goes to the Census Bureau, which gives him a vector of mean incomes for census tracts and the populations of those tracts but no information on individuals. If all the neighborhood means are identical, the Martian can say nothing about the underlying distribution of income. However, if each person lives in a neighborhood with persons of identical income, then all the information from the original income distribution is preserved, including the total number of individuals at each income level. The former situation represents a state of integration, the latter represents extreme segregation.1 The Information Theory of Segregation, therefore, holds that segregation and inequality have a very specific relationship. Inequality is information about variation in an outcome of interest. Segregation is the degree of preservation of that information after a grouping process. Integration is the destruction of information about inequality by grouping. Segregation should therefore be measured by comparing the information about inequality available in the group summary data to the information about inequality in the individual-level data. Information may be measured in many different ways, but the same measure of inequality should be applied to the group-level data as applied to the individual-level data, so that the degree of ‘‘equivocation’’


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in the signal is due to the grouping process and not differences in weighting or measurement. Although there are specific mathematical relationships implied by this perspective on segregation, as discussed in detail below, the point is entirely conceptual. The Information Theory of Segregation should be adopted if the perspective it offers proves compelling and useful. We argue that an information theory perspective on segregation clarifies the relationships between many measures of inequality and measures of segregation and that it solves problems in the measurement of segregation, such as how to handle multiracial individuals or how to apply several traditional segregation measures to continuous variables.

NOTATION AND ASSUMPTIONS For the purpose of illustration and to set notation, we assume a finite population composed of N individuals. They could be people or households or corporations or trees in the forest, but for the purpose of this analysis we assume they are indivisible. The individuals have a characteristic of interest, which is the value of a variable Yi, where i indexes individuals from 1 to N. If two subscripts are needed, we use h and i to index individuals. Each individual is a member of a group of some kind, and there are j=1 to M mutually exclusive and collectively exhaustive groups. If two subscripts are needed, we use j and k to index the groups. The population sizes of the groups may be equal or uneven, but in any case they are denoted by nj or nk. Empty groups are not allowed, and at least one group must have more than one individual, implying that MoN. By construction, the sum of nj equals N. The groups are typically spatial units, such as neighborhoods, but the grouping can be any clearly defined categorization, such as social class, caste, or occupations. In some cases, the outcome variable Y is continuous, such as income. The overall mean of Y is denoted by m, and the mean for group j is mj. In other cases, the variable is binary, such as gender. For a dichotomous variable, the overall mean is P and the group mean is pj. For nominal variables with more than two categories, such as a race or religion, we index the categories with r ranging from 1 to R, and prj is the proportion of category r in group j. With few exceptions, subscripts denote indexes and superscripts (other than exponents) denote qualifiers; for example, Bi means the value of B for the ith individual, but Bi is a function B calculated on the basis of individual-level data.

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Measures of inequality examine the distribution of Y among the N individuals with no regard to the M groups of which they are members. Inequality exists if and only if some members of the population differ in terms of Y, which is equivalent to saying that some members have values of Y not equal to the central tendency of the distribution. A measure of inequality is a single statistic that represents the degree of inequality across the whole distribution, allowing comparisons between one distribution and another or between the same distribution at two points in time. Many inequality measures focus on income percentiles, that is, deviations from the median, such as the interquartile range or the ratio of the 90th to the 10th percentile. Other measures are based on deviations from the mean, including the variance, the variance of log Y, and the Theil Index. The Gini Index is based on the mean difference between individuals in the population. These measures differ mainly in how they weight deviations from the central tendency (Firebaugh, 1999, p. 1608). In contrast, measures of segregation provide a summary measure of the differential distribution of the N individuals into the M groups according to the characteristic Y. Y is often race, most often defined as a dichotomous variable of white versus black. Segregation measures abound for the binary case, but the most common is the Index of Dissimilarity. The relationships among these measures, their properties, and their comparative advantages have been reviewed extensively (Duncan & Duncan, 1955; Frankel & Volij, 2007; Hutchens, 2001; James & Taeuber, 1985; Massey & Denton, 1988; White, 1986; Winship, 1977). Many have been extended to the polytomous case (James, 1986; Morgan, 1975; Reardon & Firebaugh, 2002; Sakoda, 1981). A smaller set of measures and procedures has been used to measure segregation when the characteristic Y is continuous, such as household income (Hardman & Ioannides, 2004; Ioannides, 2004; Jargowsky, 1996; Massey & Eggers, 1990). Shannon (1948) defined the entropy score (E) as the information value of a given state of a system, but the argument here is independent of a particular measure of information.2 In our application, the information of interest is the diversity or variation of Y over a population of individuals. The information content of the population is zero if and only if all members of the population are identical, in which case we are certain of the outcome of choosing a person at random from the population. Given a set of possible outcomes for a discrete variable or a range of possible outcomes for a continuous variable, information is maximized when all possible values of Y are equally likely (Shannon, 1948), that is, a uniform distribution or equal probabilities for each category in a binary or polytomous variable.


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Segregation and inequality are closely related, as has long been recognized. If there is no inequality (in the sense of variation between individuals), then segregation is a moot point – segregation by what? Every member of the population is indistinguishable. On the other hand, if inequality exists in income, race, or some other characteristic, we can ask about the extent to which different values of the outcome are clustered in space or sorted into certain identifiable groups. Hutchens (2001, p. 15), for example, states that segregation is a form of inequality: ‘‘segregation assesses inequality in the distribution of people across groups.’’ We go beyond the notion that segregation is related to inequality among groups to defining segregation as the retention of information about inequality when comparing the group information to the individual-level information. Several authors have noted parallels and borrowed techniques from inequality research to study segregation, notably Hutchens (2001, 2004), James and Taeuber (1985), and Reardon and Firebaugh (2002). However, to our knowledge, prior researchers did not articulate a fundamental relationship between these two concepts. Several propositions follow from thinking about segregation as the transmission of information about inequality over, or through, a grouping process. The first three propositions are closely related and concern the relationship between measures of inequality and measures of segregation. The fourth proposition concerns capacity constraints on the grouping process that render the theoretical maximum and minimum value of segregation unattainable. We attempt to demonstrate these propositions by showing how they apply to several commonly used measures of segregation and inequality. Several new measures and relationships among existing measures are demonstrated.

IMPLICATIONS OF AN INFORMATION THEORY APPROACH TO SEGREGATION In this section, we briefly outline four propositions that follow from adopting an information theory approach to the conceptualization and measurement of segregation. In the following sections, we provide support for the propositions and illustrate their implications by examining common measures of inequality and segregation and showing their interrelationships. Proposition 1. For any inequality measure F, there is a corresponding segregation measure LF, formed by the ratio of the inequality measured at the group level to the same inequality measure measured at the individual

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level. Thus, LF ¼

Fj Fi

(1)

Segregation measures formed in this way possess the inequality ratio property. Proposition 2. Conversely, any measure of segregation LF can be expressed as the ratio of two underlying inequality measures F j and Fi. In other words, we argue that all ‘‘valid’’ measures of segregation have the inequality ratio property. Obviously, researchers are free to define a multitude of measures and call them segregation measures, whether or not they have the inequality ratio property. However, if one accepts the conceptual framework we have described relating to segregation and inequality through information theory, Proposition 2 must hold. We argue that measures that do not possess the inequality ratio measure are actually hybrid measures, combining an element of segregation with other concerns. Taken together, Propositions 1 and 2 unify the concepts of inequality and segregation and imply a one-to-one theoretical and technical relationship between measures of inequality and segregation. Proposition 3. A surprising outcome of viewing segregation through information theory is the finding that all measures of inequality that can be applied to either binary or continuous variables generate measures of segregation that can be applied to either binary or continuous variables. Thus, measures such as the Gini and Atkinson measures of segregation that have been applied only in the binary case have forms that can be used for continuous variables like income. Moreover, the Index of Dissimilarity, which is normally limited to binary or polytomous variables, may be used in the analysis of continuous variables as well. Proposition 4. Segregation measures have a theoretical minimum of perfect integration and a theoretical maximum of perfect segregation. However, in many cases the theoretical limits may not be attainable for reasons that have nothing to do with segregation. Shannon described an analogous problem regarding communication over a noisy channel or a channel with a capacity constraint. To address this, we define two functions, SortMax and SortMin. The SortMax is the maximum value of

11


F j that can be produced given the set of group sizes {n1, n2, y , nj, y , nM} and the set of individual values of Y {Y1, Y2, y, Yi, y , YN} and is denoted Fjmax . Likewise, SortMin is the minimum value of neighborhood inequality and is denoted Fjmin . To account for structural maxima and minima unrelated to sorting, a segregation statistic can be normalized as follows: Ln ¼

j F j Fmin j j Fmax Fmin

(2)

DISCUSSION AND EXAMPLES Another way of stating the inequality ratio property is that the degree of segregation is measured by the ratio of information from the grouped data, given by F j, to the information from the individual data, Fi. The information from the grouped data cannot be greater than the information from the individual data; hence, the theoretical maximum of the inequality measure is 1, regardless of the scale of the inequality measure. This is analogous to Shannon’s Theorem 7, stating that the information value of the output must be less than or equal to the information value of the source. The theoretical minimum information at the group level is zero, if and only if all the groups are equivalent in terms of the first moment of Y. For example, the groups may have equal means, if Y is a continuous variable, or equal proportions, if Y is categorical (binary, ordinal, or nominal). If the neighborhoods are Y-equivalent in this sense, then F j is zero by the definition of information. Thus, the minimum potential value of LF is zero. We assume only that the amount of information at the individual level is greater than zero. If there is no variation at the individual level, then there is no inequality and the question of segregation is meaningless. We illustrate Proposition 1 by discussing the correlation ratio and the application of the Gini Index to the measurement of income segregation. In both cases, the segregation measure is formed by a ratio of corresponding group and individual inequality measures. We then illustrate Proposition 2 by showing that several common segregation measures – the Index of Dissimilarity and the Entropy Index – are also effectively ratios of underlying inequality measures, a fact not previously recognized.

12


Variance Ratio Measures The correlation ratio, also referred to as eta2 or the variance ratio, has been used as a measure of segregation for both dichotomous and continuous variables (Bell, 1954; Farley, 1977; Schnare, 1980; Zoloth, 1976). Reardon and Firebaugh (2002) developed a version for multiple groups. If the j groups are census tracts, the correlation ratio ‘‘may be thought of as a one-way analysis of variance model in which the overall variance y is divided into within census tract and between census tract variances’’ (Farley, 1977, p. 503). In an application to economic segregation, Jargowsky (1996) proposed the Neighborhood Sorting Index (NSI) as a measure of economic segregation. Individuals in this case are households and the groups are neighborhoods proxied by census tracts. The square of the NSI is the ratio of the between-neighborhood variance in household income to the overall variance of the income distribution:

NSI2 ¼

s2j s2i

M P

¼

ðnj =NÞðmj mÞ2

j¼1 N P

(3) ð1=NÞðY i mÞ2

i¼1

However, another way to look at this measure is simply the ratio of the variance of the j distribution to the variance of the i distribution.3 The denominator is clearly an inequality measure, whereas the numerator is a measure of the inequality of the grouped data, weighted by tract population. The denominator can also be considered to be weighted, with each individual having a weight of 1/N (Firebaugh, 1999, p. 1607). Thus, the NSI and similar segregation measures are consistent with Proposition 1, because they are formed by a measure of inequality at the group level to the same measure of inequality at the individual level. The Gini Segregation Coefficient The Gini coefficient is perhaps the best known measure of inequality (Gini, 1912, 1921). Gini coefficients are routinely computed for all nations in the world that collect adequate data (Milanovic, 2002). Several authors have employed the Gini as a measure of segregation, but its use has been limited to dichotomous groups (Silber, 1989; James & Taeuber, 1985; Massey & Denton, 1988). Reardon and Firebaugh (2002) extended the Gini to polytomous categorical variables and Dawkins (2004) has proposed a spatial version.


13

The Gini Index can be computed by a number of different formulas and procedures (Pyatt, 1976; Silber, 1989; Yao, 1999). For our purposes, we employ the mean difference formulation (Gini, 1912, 1921) of the Gini coefficient: Gi ¼

N X N 1 X jY h Y i j 2N 2 m h¼1 i¼1

(4)

The Gini measure can just as easily be applied to the smaller set of values comprising the group or neighborhood means: Gj ¼

M X M 1 X nj nk jmj mk j 2N 2 m j¼1 k¼1

(5)

where j and k both index groups. Kim and Jargowsky (2009, this volume) develop this index geometrically and show that the grouped (i.e., neighborhood-level) Gini Index is always less than or equal to the Gini computed from the individual level. The group-level Gini Index has been used to measure the spatial distribution of health-care providers (Brown, 1994; Schwartz, Newhouse, Bennett, & Williams, 1980). Hutchens considered and rejected a grouped version of the Gini as a measure of occupational segregation (2004). The group-level Gini, however, simply measures the inequality of the groups without conveying any information on how the distribution of Y changes when individuals are sorted into groups. G j reaches a maximum of Gi when each individual lives in a neighborhood (or is a member of a group) with a mean income identical to their own. When any group contains individuals who differ in terms of Y, G j must be less than Gi. As an example, let Y be income and consider a city with 100 neighborhoods and 200 persons, 100 of whom have an income of $9,999 and the remaining 100 have incomes of $10,001. Clearly, there is little inequality. There is complete neighborhood equality if each neighborhood pairs a ‘‘rich’’ and ‘‘poor’’ person. There is complete segregation if each neighborhood contains only one income level or the other. In both, however, the group-level Gini will still report a value close to zero because the neighborhoods are quite similar despite the complete segregation of the rich from the poor. The important point is that G j is not a pure measure of segregation. The maximum value of G j can never exceed Gi. The maximum possible degree of inequality between the M groups is limited by the amount of inequality in

14


the original income distribution of the N individual units. The grouping process may destroy some of the inequality information available in the individual distribution. Segregation is a question of organization, not the degree of disparity, a distinction that does not arise in the case of nominal variables. Segregation may interact with inequality to produce worse outcomes on a variety of important variables, but segregation and inequality are conceptually distinct. The problem is solved by dividing the group-level Gini by the individual-level Gini, canceling the common terms: M P M P

nj nk jmj mk j Gj j¼1 k¼1 L ¼ i¼ N N PP G jY h Y i j G

(6)

h¼1 i¼1

We call this the Gini Segregation Index (GS). Returning to the example of the previous paragraph, LG produces 0 when the rich and poor are paired and 1 if they are completely segregated. As a check on the logic of Eq. (6), we simplify the GS as we have developed it to the binary case to see if it is consistent with the Gini in use in racial segregation studies. Let Yi be an indicator variable equal to 1 for black and 0 for white. The mean of binary variable Y becomes P, the percentage black, and the mean of Y in neighborhood j becomes pj, the percentage black in neighborhood j. Then, the special case of the GS for binary variables is: M P M P

LGð0=1Þ ¼

M P M P

nj nk jpj pk j

j¼1 k¼1 N P N P

¼ jY h Y i j

nj nk jpj pk j

j¼1 k¼1 2

2N Pð1 PÞ

(7)

h¼1 i¼1

This is a standard formula for the GS in the binary case (James & Taeuber, 1985; White, 1986).4 Thus, the Gini that has been applied to racial segregation is a properly formed segregation measure according to Proposition 1. The procedure for deriving the binary version of the Gini Segregation measure involved dividing by the maximum possible value of Gini (Jahn, Schmid, & Schrag, 1947) to scale the measure to a maximum value of 1. The maximum value for Gini was fairly clear in the binary case, but harder to intuit in the case of a continuous variable. The Information Theory of Segregation, however, clarifies the correct denominator. Thus, there is no reason that the use of Gini as a segregation statistic needs to be limited to dichotomous


15

variables. Kim and Jargowsky (2009, this volume) show an application of the GS for the continuous variable income and compare it to the NSI. It is not enough, however, to show that some segregation measures can be formed as the ratio of an inequality measure applied at the individual and group levels. Rather, to demonstrate that the information theory perspective on segregation is correct and useful, we have to argue that this structure is common to all or at least to most commonly accepted measures of segregation. We turn now to two existing measures – the Index of Dissimilarity (D) and the Theil Entropy Index of Segregation (H) – and show that they are consistent with the Information Theory of Segregation, even though they have not been developed or described this way in past research.

The Index of Dissimilarity Dissimilarity measures the unevenness of the distribution of a characteristic across neighborhoods. In a familiar application, we have two groups, whites and blacks. The total number of whites and blacks is W and B, respectively, while wj and bj represent the number of each group in a specific neighborhood. Then, the Index of Dissimilarity is written as: M 1X bj wj (8) D¼ 2 j¼1 B W D ranges between 0 and 1, and the specific value tells the proportion of one group or the other that would have to move to obtain an even distribution. If D is a ratio of two inequality measures, we need a version of D that measures neighborhood racial inequality (Dj) and a version that measures racial inequality at the individual level (Di). Rather than thinking about whites and blacks as two separate groups, we can express the measure in terms of the variable Y, equal to 1 if a person is black and 0 otherwise. The overall mean of Y is the percentage black, denoted P. The mean of Y for each neighborhood is the neighborhood proportion black, pj, and nj is the neighborhood’s total residents. Converting Eq. (8) to this notation shows that D is actually a measure of neighborhood inequality in the percentage black: M M X nj pj nj ð1 pj Þ 1X 1 ¼ nj jpj Pj ¼ D j D¼ 2 j¼1 NP Nð1 PÞ 2NPð1 PÞ j¼1

(9)

16


At the individual level, by assumption, the proportion white or black for any individual is either 0 or 100%, and the number of individuals in each individual is 1, so the equation simplifies to: N 1X Yi ð1 Y i Þ i D ¼ 2 i¼1 NP Nð1 PÞ " # X 0 1 X 1 0 1 ¼ þ 2 Y ¼1 NP Nð1 PÞ NP Nð1 PÞ Y ¼0 i

i

1 ¼ ð 1 þ 1Þ ¼ 1 2

ð10Þ

In other words, to revert to the traditional interpretation of D, 100% of whites or 100% of blacks would have to switch from one race to the other to have a perfectly even distribution of race across the population. Returning to the proposition, the argument above demonstrates that the segregation measure based on the Index of Dissimilarity follows the pattern established by NSI, Gini, and so forth: LD ¼

Dj Dj ¼ Dj ¼ 1 Di

(11)

In other words, the common Index of Dissimilarity is both a measure of segregation of the characteristic race across neighborhoods and a measure of neighborhood inequality of mean racial composition because individual inequality in race is equal to 1.5

Extension of Dissimilarity to Multirace Individuals Di is equal to 1, as argued in the previous section, but only because researchers typically characterize all persons as one group or another. For example, many studies employing D examine white versus nonwhite, or examine white versus black ignoring all others. It does not need to be so. In a situation where a significant number of persons are mixed race, the standard dissimilarity measure fails to reflect reality. Armed with an understanding of the nature of segregation as a ratio of inequality measures, we can easily incorporate mixed-race persons. Define Yi as the proportion black for each person. Most persons would be 0 or 1, but others could be 0.5, 0.25, 0.125, or any value between 0 and 1. Then the Non-Exclusive


Index of Dissimilarity (NED) is: M nj m M n ð1 mj Þ P P j j nj jmj mj NEDj j¼1 Nm Nð1 mÞ j¼1 LNED ¼ ¼ ¼ N Y N P P NEDi i ð1 Y i Þ jY i mj Nm Nð1 mÞ i¼1 i¼1

17

(12)

In most racial applications in the US, there are probably too few persons categorized as multiracial for this refinement to matter. However, explicit acknowledgment of mixed-race individuals is becoming more common over time, and the new measure could prove useful in studying ethnic and racial segregation in the presence of mixed marriages. The NED could also be applied in contexts other than race and ethnicity. For example, if one were examining the segregation of persons who possessed two genetic markers, many individuals may have both genetic markers. In the Introduction, we discussed a Martian attempting to learn about income inequality in a city. The Martian is given the population of the neighborhoods and the mean income of each neighborhood. We argued that if economic segregation were zero, all the neighborhood means would be the same and the Martian would have no information about the underlying distribution of income. In other words, all the information at the individual level would be destroyed by the grouping process. The argument seems to break down in the case of a binary variable.6 Imagine a city comprised of three-fourth whites and one-fourth blacks. Further, assume perfect racial integration, so that every neighborhood is one-fourth black. It seems that if the Martian knows the neighborhood racial proportions and the population of each neighborhood, he knows everything about the individual distribution of race. That conclusion rests on an assumption: that each person belongs to one and only one racial group. The Martian, unfamiliar with one-drop rule and other Earthly conventions about race, will not assume that everyone has to be either white or black. If every neighborhood is one-fourth black, the Martian will not know whether every neighborhood is composed of 75% white persons and 25% black persons or, alternatively, composed of 100% of persons who are three-fourths white and one-fourth black. In the former case, segregation at the individual level is total; in the latter, there is an even distribution of white and black across individuals in the population. Just as in the income inequality case, if all neighborhoods have the same percentage black, all the information about individual-level racial inequality has been destroyed. We are not used to thinking about race in this way, but an

18


increasing recognition of mixed-race individuals in our culture and in our data may require that researchers adopt a more nuanced perspective in future research.

The Index of Dissimilarity and Continuous Variables The fact that some variations of D are really inequality measures suggests other possible applications. For example, economic segregation can be thought of as the uneven distribution of people and money. Starting with the standard D formula, let income (Yi) replace one racial group and let persons replace the other. The individual is the unit of analysis, and there is one person per individual, so the Income Dissimilarity Index (ID) is given by: N N 1X Y i 1 1 X i ¼ jY i mj (13) ID ¼ 2 i¼1 T N 2Nm i¼1 in which T is the total income in the system, and the identity T=Nm is used to simplify the equation. IDi measures the uneven distribution of money over people, and the figure it generates tells us what proportion of the total dollars would have to be reallocated to achieve an even distribution of money over people. Unlike Di in the racial case, IDi will rarely, if ever, be 1. The ID at the neighborhood level is: IDj ¼

M M 1X 1 X nj mj nj ¼ nj jmj mj N 2 j¼1 T 2Nm j¼1

(14)

The resulting figure indicates the proportion of dollars that would have to switch neighborhoods to achieve equal mean incomes across neighborhoods. As in the case of the neighborhood-level Gini, however, this version of D is not a pure measure of segregation. The maximum value it can achieve is based on the underlying inequality of the income distribution. We start with an income distribution in which each individual has a fixed income. You can transfer money from one neighborhood to another by swapping a richer person for a poorer one. The most segregated pattern that can be achieved is limited by the inequality at the individual level, so that if one person has all the money and lives alone, we achieve a near-total segregation of people and money. If everyone has nearly identical incomes, even the most segregated arrangement of people still results in a fairly even distribution of people and money across neighborhoods.

19


The creation of a measure of economic segregation based on the Index of Income Dissimilarity is achieved as usual by dividing the group-level inequality by the corresponding individual-level inequality: M P

L

ID

nj jmj mj IDj j¼1 ¼ ¼ N P IDi jY i mj

(15)

i¼1

The formula is identical to that of the NED above, but the meaning of Yi is different. In the non-exclusive case, Yi varies between 0 and 1, whereas for ID, Yi is the continuous variable income. The meanings of m and mj vary accordingly. ID looks very similar to the square of the NSI (Jargowsky, 1996): M P

NSI2 ¼

nj ðmj mÞ2

j¼1 N P

(16) ðY i mÞ

2

i¼1

However, in the absence of individual-level data, the ID measure is likely to be less sensitive to assumptions that have to be made about the distribution of incomes in the highest income bracket to estimate the parameters of the individual income distribution (Jargowsky, 1996). The Entropy Index of Segregation Shannon (1948) borrowed the entropy score from thermodynamics, and applied it as a measure of the information content or diversity of a source. Theil and Finezza (1971; see also Theil, 1972) adapted entropy to the measure of segregation in school systems, defining the Entropy Index as the weighted average difference between the entropy of schools and the overall entropy of the school system divided by the overall entropy. The measure has been described, again in the education context, as ‘‘a measure of how diverse individual schools are, on average, compared with the diversity of their metropolitan area school enrollment as a whole’’ (Reardon, Yun, & Eitle, 2000, p. 353). Given the prevalence and desirable properties of this measure, an important test of the Information Theory of Segregation is whether H satisfies the inequality ratio property, as required by Proposition 2. The most common application of H is distinct racial groups, with the number of groups given by r=1 to R. Diversity in the population is

20


measured by Shannon’s entropy score: E¼

R X

pr ln pr

(17)

r¼1

where pr is the proportion of the category r in the overall population. A corresponding entropy score Ej is calculated for each group j: Ej ¼

R X

pjr ln pjr

(18)

r¼1

If there is complete integration, every group has an exactly proportional share of each racial group; then Ej is equal to E for each group j. In a case of maximal segregation, pjr is 1 for one category r and 0 for all other r for all j; hence, Ej is 0 for all j. Moreover, the weighted average of the diversity of the groups can never exceed the diversity of the population, because, as Shannon showed, the grouping process cannot create more information than exists in the population. Thus, the weighted average group entropy is divided by E to create a variable with a maximum of 1: M P

H¼

M P

ðnj =NÞðE E i Þ

j¼1

E

¼

nj ðE E i Þ

j¼1

NE

(19)

Written in this form, H does not seem to satisfy the inequality ratio property. In fact, H does have the structure of a ratio of inequality at the group level to inequality at the individual level. To see this, let E represent the diversity in the population, and let Ei and Ej represent the diversity of individuals and groups, respectively. Then, the inequality in diversity at the group level is the weighted mean of (EjE), and the inequality in diversity at the individual level is the mean of the expression (EiE). From Proposition 2, the Entropy Index of Segregation must have the form: M P j

M P

ðnj =NÞðE j EÞ

H j¼1 L ¼ i¼ N P H H

i¼1

¼ ð1=NÞðE i EÞ

nj ðE j EÞ

j¼1 N P

(20) ðE i EÞ

i¼1

Each individual, at least in applications of H to date, is categorized in one and only one category of the nominal variable Y, so the diversity of any

21


given individual is zero. In such cases, we can simplify the expression: M P j

M P

nj ðE j EÞ

H j¼1 L ¼ i¼ N P H H

¼ ð0 EÞ

M P

nj ðE j EÞ

j¼1

NE

¼

nj ðE E j Þ

j¼1

NE

¼H

(21)

i¼1

proving that the formulation based on Proposition 2 is equivalent to one in common use.7 H is therefore a special case of a broader measure, LH, that could easily incorporate mixed-race persons or other categorizations that allow one individual to belong to multiple groups. In the presence of mixed-race (multi-category) persons, Ei would not equal zero in all cases and the more general version of the formula would need to be applied. Letting pir represent the proportion of each person that is category r, we define the Non-Exclusive Entropy Index of Segregation (NEH) as: R M M P P P n ðE EÞ n ðp ln p p ln p Þ j j j jr jr r r NEHj j¼1 j¼1 r¼1 ¼ ¼ LNEH ¼ (22) N N P R P P NEHi ðE i EÞ ðpir ln pir pr ln pr Þ i¼1

i¼1 r¼1

It is also interesting to note the similarity of form between this expression and the other measures. Of course, in the usual case where individuals are assigned to one and only one category, the first term in the denominator is either (1)ln(1)=0 or (0)ln(0) that is typically defined to be zero in segregation studies (Reardon et al., 2000). A related point is that the Theil inequality measure, often applied to income data (Theil, 1972), can also form an income segregation measure. By Proposition 1, if Yi is income, we can form the Theil Income Segregation (TI) measure as follows: M P

nj ðmj =mÞ lnðmj =mÞ TI j¼1 ¼ i¼ N P TI ðY i =mÞ lnðY i =mÞ j

TI

L

(23)

i¼1

This measure has not been identified previously, but follows naturally from the Information Theory of Segregation. Like the measures discussed

22


previously, the Theil measure can also be used to measure segregation with respect to binary, polytomous, and continuous variables.

The Exposure Index The Exposure Index, also known as the Interaction Index (I ), is often described as an alternative to the Index of Dissimilarity (D) as a measure of segregation (Bell, 1954; Lieberson & Carter, 1982; Schnare, 1980; Zoloth, 1976). Unlike D, the Exposure Index is asymmetric and is sensitive to the underlying population proportion, which can be viewed as an advantage or a disadvantage (White, 1986). On the one hand, many have argued that composition invariance is a desirable property of a segregation measure (James & Taeuber, 1985). On the other hand, I reflects actual differences in the probability of residential contact between two groups (Lieberson & Carter, 1982). Assume a simple, two-group situation. Let W equal total whites in the metropolitan area and B be the total blacks. N is the total population. wj, bj, and nj are, respectively, the corresponding parcel figures for neighborhood j. Then, the white exposure to blacks and the black exposure to whites are given by: M w b P j j wI b ¼ nj j¼1 W (24) M b P wj j bI w ¼ nj j¼1 B The white exposure to blacks is simply the average proportion black in the neighborhood parcels, weighted by the population of whites. It may be interpreted as the percentage black ‘‘experienced by the average white.’’ The second measure has a parallel interpretation. Restating these equations in terms of pj, the neighborhood percentage black, and P, the overall percentage black, reveals their underlying structure: M M nj ð1 p Þ nj p X P 1 j j ¼ nj pj ð1 pj Þ wI b ¼ nj Nð1 PÞ i¼1 j¼1 Nð1 PÞ (25) M M nj p P 1 X j nj ð1 pj Þ ¼ nj pj ð1 pj Þ bI w ¼ nj NP i¼1 j¼1 NP

23


Two things are apparent from this reformulation. The first is that the two measures are equal only if P=0.5. The second is that both are the variance in group-level percent black, multiplied by different constants. Interaction indices are therefore measures of group-level inequality. The individual-level analogue of the exposure of whites to blacks is simply: N X wi B B i I ¼ ¼ ¼P (26) w b W N N i¼1 In other words, the percentage black in the metropolitan area experienced by the average white is simply the metropolitan area percentage black. Thus, the properly formed segregation measure based on the Interaction Index is: M P ðwj =WÞðbj =nj Þ j j¼1 wI b L wIb ¼ (27) ¼ i B=N wI b This is equivalent to the normalized Index of Interaction (Bell, 1954). When normalized in this way, the measure is no longer asymmetric, and it is exactly equivalent to eta2 (White, 1986). To see this, substitute pj for the percentage black in neighborhood j and P for the overall proportion black. Then, canceling and rearranging terms, the Exposure Index of Segregation measure becomes: M P I

L ¼

ðPNÞ=N M P

¼

ððð1 pj Þnj Þ=ðð1 PÞNÞÞððpj nj Þ=nj Þ

j¼1

ðnj =NÞpj ð1 pj Þ

j¼1

Pð1 PÞ

¼

s2j s2i

ð28Þ

This is nothing more than the correlation ratio as applied to a binary variable. The numerator is the variance of the group proportions, that is the mean of the race variable if it is defined as a binary indicator variable. The relationships presented here are well known. What is new is that the normalization of the Exposure Index, resulting in the correlation ratio, is motivated by the information theory notion that a proper segregation measure must be expressed as the ratio of a group inequality measure to the corresponding individual inequality measure. Further, this perspective clearly supports the position that, without normalization, the Exposure Index is not a pure measure of segregation, despite its many advocates.

24


Other Measures We have shown that several common inequality measures can be used to measure segregation by applying the Information Theory of Segregation. In fact, any inequality measure can be employed in this way, simply by applying the measure as usual to the Yi and then applying the measure to the Yj. For example, the Atkinson family of inequality measures will produce a corresponding family of Atkinson Segregation measures (AS): " #1=ð1Þ M P 1 ð1=MÞ ðmj =mÞ1 j j¼1 A (29) LAðÞ ¼ i ¼ 1=ð1Þ N A P 1 1 ð1=NÞ ðY i =mÞ i¼1

where e is the inequality aversion parameter, and can be used by researchers to make the measure more sensitive to segregation at different levels of the income distribution. There is an Atkinson segregation statistic that has been described for the binary case, but it has seldom been used (James & Taeuber, 1985; White, 1986). For a binary variable, regardless of the shape parameter e, the individual-level Atkinson inequality measure is 1. So, once again, the binary variable formula is shown to be a special case of the continuous variable case and the Atkinson Segregation statistic is equally applicable to binary and continuous variables. Segregation measures based on percentile ranks are possible as well. For example the, the Interquartile Segregation (IQS) measure is: LIQR ¼

j j IQRj Q3 Q1 ¼ IQRi Qi3 Qi1

(30)

While this measure would be insensitive to income extremes, it has the virtue of being very easy to calculate; often researchers may have group data but only summary statistics on the individual-level data, including the quartiles. Of course, any available quantiles could be used, such as the 90th and 10th percentiles. Clearly, the number of measures of segregation can be greatly expanded by understanding the structural relationship between inequality and segregation. The choice between these measures should be based on their properties, judgments about the implicit weighting, and the data demands of each measure in relation to the available data. It is worth noting that a number of segregation measures that have been proposed may not possess the inequality ratio property.


25

Massey and Denton (1988) define five dimensions of segregation. Two (evenness and isolation) have already been addressed in the discussion of the Index of Dissimilarity and the Exposure Index. The other three are spatial concentration, centralization, and clustering. These are perfectly valid and interesting concepts, but we think that each is a hybrid measure that combines segregation, in the sense defined here, with additional information. Echenique and Fryer (2005) propose a Spectral Segregation Index, which aggregates information about the social networks of members of different groups. While this may be a very desirable measure for many purposes, from our perspective it combines information about the sorting of individuals into groups with information about interactions between and among members of those groups.

Theoretical versus Structural Limits Propositions 1 and 2 show that segregation is a measure of the information loss in the measured level of inequality when moving from individuals to groups. Segregation is the result of a process of allocating individuallevel values across spatial units. In theory, segregation measures should vary from 1, representing no information loss, to 0, representing total information loss. However, with a finite population, these extrema may be unachievable through sorting alone. Shannon’s ‘‘fundamental theorem for a discrete channel with noise’’ is exactly on point. It states that if a channel has a capacity C and the source has a total quantity of information H, and H is greater than C, then ‘‘there exists a coding scheme such that the output of the source can be transmitted over the channel with an arbitrarily small frequency of errors.’’ This is true even on a noisy channel with random errors, a fact that makes modern communications possible. However, ‘‘there is no method of encoding which gives an equivocation less than H–C’’ (1948, Theorem 11). In the case of segregation, the channel through which the information about the underlying individual inequality is transmitted is the grouping structure, which in effect limits the information that can be transmitted. Examples will clarify the point. If the variable Y is income, and if the number of unique values of income is greater than the number of groups M, then there is no possible way to allocate individuals to groups without the destruction of information from the individual level, even if individuals are assigned to groups explicitly by income rank. At least two distinct

26


incomes will have to be averaged within one group, losing information. Thus, F joFi and L will be less than 1, solely for structural reasons, because of the choice of the number of parcels. One might think that this is a problem only with continuous variables, but it can happen for binary variables as well. Suppose there are seven whites and seven blacks in a city with two neighborhoods. In theory, perfect segregation could be produced. However, every household must live in a housing unit, which are durable and fixed in the short run. If one neighborhood has eight housing units and the other has six, there is no way to achieve perfect segregation. Again, the maximum level of neighborhood inequality will fall short of the individual inequality for reasons other than sorting of units. The minimum level of neighborhood inequality may fall short of zero. The clearest example is the ‘‘Bill Gates problem.’’ If Bill Gates is one of the members of the population in question, there is no way to create neighborhoods with equivalent mean incomes. Whichever neighborhood contains Bill Gates and his money will be far richer than any other neighborhood. One could achieve neighborhood equality only by redistributing his income. While that might be a good idea for a variety of reasons, it is not a sorting process and it would change the underlying income distribution. In the binary context, with fixed housing stock, neighborhoods with even numbers of housing units cannot be made to have identical proportions with neighborhoods with odd numbers of housing units. As a result of these considerations, two different metropolitan areas (or the same metropolitan area at two different points in time), both of which have been segregated to the maximum possible extent given Yi and nj, may differ in how close the measured level of neighborhood inequality may approach the individual level of inequality. To address this issue, we define two functions, SortMax and SortMin. SortMax is the maximum value of F j that can be produced given the set of group sizes {n1, n2, y , nj, y ,nM} and the set of individual values of Y j . Likewise, SortMin is {Y1, Y2, y , Yi, y , YN} and is denoted Fmax j . the minimum value of neighborhood inequality, and is denoted Fmin To account for these structural maxima and minima, Proposition 4 is that a segregation statistic can be normalized as follows:

Ln ¼

j F j Fmin j j Fmax Fmin

(31)


27

It is obvious that the minimum and maximum values of L, without normalization, are given by: j j Fmin F j Fmax Fi Fi Fi

(32)

Subtracting the first term shifts the scale of the measure to have a zero minimum: ! ! j j j F j Fmin Fmax Fmin (33) 0 Fi Fi Dividing by the final term and simplifying shows that the normalized segregation statistic is properly bounded by zero and one: ! j F j Fmin 0 1 (34) j j Fmax Fmin Note that while it appears that individual-level inequality has dropped out of the definition of the segregation measure, it remains implicitly in the j , which has an upper limit of Fi. Thus, the normalization of form of Fmax Proposition 4 is consistent with Proposition 2. If a measure of segregation is desired that reflects only sorting, rather than neighborhood structure, group size, and so forth, then the normalized measure should be used. However, the SortMax and SortMin functions do not have analytic solutions, and would have to be estimated on a case-bycase basis using simulation, which requires individual-level data. In the case of binary values, the difference is likely to be miniscule. In the case of continuous variables, the values and rank ordering of the larger units (states, countries, metropolitan areas, etc.) could be substantially altered. On the other hand, if the number and size of the neighborhoods are significant aspects of the segregation process, rather than arbitrary administrative groupings, it could be argued that the un-normalized measure is both descriptively and substantively accurate.

CONCLUSION We argue that segregation is fundamentally analogous to the loss of information from an aggregation process. Integration is information loss

28


and segregation is information retention. In the case of signal processing, compression of graphics files, or the estimation of income distribution parameters from grouped data, the question is how to minimize information loss. In the case of segregation, the question is how much information about the underlying distribution remains after the population is sorted in groups, particularly geographic neighborhoods. While these questions have different motivations, they have the same mathematical structure. We argued that the information theory framework requires that a segregation measure must be structured as the ratio of an inequality measure applied to the group data to the same inequality measure applied to the individuals in the population. Of course, any researcher is free to invent any measure and call it segregation. In particular situations, a nonconforming measure may prove useful. However, we suspect that any such measure will not be a pure segregation measure, just as the neighborhood-level Gini reflected both sorting and underlying inequality. Ultimately, what constitutes a segregation measure is a conceptual question, not a mathematical one. Our position is normative in that we think the logic of the information theory analogy is so strong that all researchers should adopt it. (Most have implicitly done so already, by using measures of segregation that are consistent with the framework.) With any theoretical formulation, the test of its value is the extent to which it unites diverse phenomena and proves useful in understanding and resolving problems within a field. A reexamination of a number of segregation and inequality measures in light of the framework revealed several interesting points and useful applications. First, some measures that have been used to measure segregation, like the neighborhood-level Gini and the Exposure Index, are not proper segregation measures at all. In fact, they are neighborhood inequality measures. Second, all the standard measures of segregation can handle binary or continuous variables. The distinction between these two types of measures is largely artificial. The binary formulas in use were correct because the denominator of the correctly specified segregation measure is 1. Based on Proposition 2, the correct denominator for the continuous case is easy to specify. We presented several new measures using this approach, such as the GS and ID. Third, the requirement for nonoverlapping groups in segregation studies is shown to be an artificial limitation that was a function of not understanding the implicit denominator of the Index of Dissimilarity and related measures. We presented two new measures, the NED and the NEH, that can easily accommodate mixed-race individuals in a segregation analysis. Fourth, although the formulas and means of calculation for a number of segregation measures look quite different in their standard forms, they can


29

all be recast in a common mathematical structure. As Firebaugh (1999) observed with respect to inequality measures, measures of segregation differ primarily in the functional form used to aggregate individual or group differences from the overall mean. Fifth, the lower and upper bounds of a segregation measure are theoretically 0 and 1, but with finite groups of fixed size and a given finite distribution of the outcome of interest, the actual bounds will often fall short of those values. Comparisons among different cities or school systems could well be affected by these considerations. A normalization is proposed that provides a pure measure of segregation unaffected by structural limitations.

NOTES 1. The argument also applies to binary variables, as will be demonstrated below. 2. Shannon (1948) actually used H to denote entropy, but in the segregation literature, H is reserved for the Entropy Index of Segregation while E usually denotes the entropy, that is, diversity, score of groups or the system as a whole. 3. Similar measures using other metrics have been proposed, including the ratio of neighborhood and individual variance of log income (Ioannides, 2004) and coefficients of variation (Hardman & Ioannides, 2004). 4. Reardon and Firebaugh (2002) also describe a multigroup version. 5. A parallel argument holds for some other segregation measures. For example, Hutchens (2004) develops the Square Root and describes its properties. This pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P Index ðs1j Þðs2j Þ, also satisfies the Inequality measure, which has the general form 1 M j¼1 Ratio property (Propositions 1 and 2). In the individual-level formulation of the Square Root Index, one of the two terms under the radical would always be zero. Thus, in the ratio of the group-level square root index to the individual-level square root index, the denominator is always equal to 1. The same argument applies to the generalization of the Hutchens measure discussed by Chakravarty and Silber (2007). 6. We are grateful to an anonymous reviewer for raising this point. 7. Since the normalization by E produces a measure consistent with the inequality ratio property, measures that fail to normalize by E are not pure measures of segregation as that concept is defined in this paper. An example is Mutual Information Index (Mora & Castillo, 2009, this volume), which combines information about sorting with information about the overall community diversity.

ACKNOWLEDGMENTS An earlier version of this paper was presented at the conference ‘‘New Frontiers in the Field of Segregation Measurement and Analysis,’’ July 3–8, 2007, in Monte Verita, Switzerland. We are grateful to Nathan Berg, Brian

30


Berry, Marie Chevrier, Euel Elliott, Yves Flu¨ckiger, Christopher Jencks, Murray Leaf, Sean Reardon, Jacques Silber, and the conference participants for helpful comments and discussions.

REFERENCES Bell, W. (1954). A probability model for the measurement of ecological segregation. Social Forces, 32, 357–364. Brown, M. C. (1994). Using Gini-style indexes to evaluate the spatial patterns of health practitioners: Theoretical considerations and an application based on Alberta data. Social Science and Medicine, 38, 1243–1256. Chakravarty, S. R., & Silber, J. (2007). A generalized index of employment segregation. Mathematical Social Sciences, 53, 185–195. Dawkins, C. (2004). Measuring the spatial pattern of residential segregation. Urban Studies, 41, 833–851. Duncan, O. D., & Duncan, B. (1955). A methodological analysis of segregation indexes. American Sociological Review, 20, 210–217. Echenique, F., and Fryer, R. G. (2005). On the measurement of segregation. Unpublished paper. Harvard University. Farley, R. (1977). Residential segregation in urbanized areas of the United States in 1970: An analysis of social class and racial differences. Demography, 14, 497–517. Firebaugh, G. (1999). Empirics of world income inequality. American Journal of Sociology, 104, 1597–1630. Frankel, D. M., & Volij, O. (2007). Measuring segregation. Working Paper #07009. Department of Economics, Iowa State University, Ames, IA. Gini, C. (1912). Variabilita e mutabilita. Reprinted in Pizetti, E., & Salvemini, T. (Eds). (1955). Memorie di metodologia statistica. Rome: Libreria Erendi Virgilio Veschi. Gini, C. (1921). Measurement of inequality of Incomes. The Economic Journal, 31, 124–126. Hardman, A., & Ioannides, Y. M. (2004). Neighbors’ income distribution: Economic segregation and mixing in US urban neighborhoods. Journal of Housing Economics, 13, 368–382. Hutchens, R. (2001). Numerical measures of segregation: Desirable properties and their implications. Mathematical Social Sciences, 42, 13–29. Hutchens, R. (2004). One measure of segregation. International Economic Review, 45, 555–578. Ioannides, Y. M. (2004). Neighborhood income distributions. Journal of Urban Economics, 56, 435–457. Jahn, J., Schmid, C. F., & Schrag, C. (1947). The measurement of ecological segregation. American Sociological Review, 3, 293–303. James, D. R., & Taeuber, K. E. (1985). Measures of segregation. Sociological Methodology, 15, 1–32. James, F. J. (1986). A new generalized ‘exposure-based’ segregation index: Demonstration in Houston and Denver. Sociological Methods and Research, 14, 301–316. Jargowsky, P. A. (1996). Take the money and run: Economic segregation in US metropolitan areas. American Sociological Review, 61, 984–998.


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Kim, J., & Jargowsky, P. A. (2009). The Gini coefficient and segregation on a continuous variable. In: Y. Flu¨ckiger, S. F. Reardon & J. Silber (Eds), Occupational and residential segregation (Vol. 17). UK: Emerald. Lieberson, S., & Carter, D. K. (1982). Temporal changes and urban differences in residential segregation: A reconsideration. American Journal of Sociology, 88, 296–311. Massey, D. S., & Denton, N. A. (1988). The dimensions of racial segregation. Social Forces, 67, 281–315. Massey, D. S., & Eggers, M. L. (1990). The ecology of inequality: Minorities and the concentration of poverty, 1970–1980. American Journal of Sociology, 95, 1153–1188. Milanovic, B. (2002). True world income distribution, 1988 and 1993: First calculation based on household surveys alone. The Economic Journal, 112, 51–92. Mora, R., & Ruiz-Castillo, J. (2009). The invariance properties of the mutual information index of multigroup segregation. In: Y. Flu¨ckiger, S. F. Reardon & J. Silber (Eds), Occupational and residential segregation (Vol. 17). UK: Emerald. Morgan, B. S. (1975). Segregation of socio-economic groups in urban areas: A comparative analysis. Urban Studies, 12, 47–60. Pyatt, G. (1976). On the interpretation and disaggregation of Gini coefficients. The Economic Journal, 86, 243–255. Reardon, S. F., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32, 33–67. Reardon, S. F., Yun, J. T., & Eitle, T. M. (2000). The changing structure of school segregation: Measurement and evidence of multiracial metropolitan-area school segregation, 1989– 1995. Demography, 37, 351–364. Sakoda, J. M. (1981). A generalized index of dissimilarity. Demography, 18, 245–250. Schnare, A. B. (1980). Trends in residential segregation by race: 1960–1970. Journal of Urban Economics, 7, 293–301. Schwartz, W. B., Newhouse, J. P., Bennett, B. W., & Williams, A. P. (1980). The changing geographic distribution of board-certified physicians. New England Journal of Medicine, 303, 1032–1038. Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379–423 (July), 623–656 (Published in October). Silber, J. (1989). Factor components, population subgroups and the computation of the Gini index of inequality. Review of Economics and Statistics, 71, 107–115. Theil, H. (1972). Statistical decomposition analysis. Amsterdam: North Holland. Theil, H., & Finezza, A. J. (1971). Note on measurement of racial integration of schools by means of informational concepts. Journal of Mathematical Sociology, 1, 187–193. White, M. J. (1986). Segregation and diversity measures in population distribution. Population Index, 52, 198–221. Winship, C. (1977). A revaluation of indexes of residential segregation. Social Forces, 55, 1058–1066. Yao, S. (1999). On the decomposition of Gini coefficients by population class and income source: A spreadsheet approach and application. Applied Economics, 31, 1249–1264. Zoloth, B. S. (1976). Alternative measures of school segregation. Land Economics, 52, 278–293.

THE INVARIANCE PROPERTIES OF THE MUTUAL INFORMATION INDEX OF MULTIGROUP SEGREGATION Ricardo Mora and Javier Ruiz-Castillo ABSTRACT In the context of educational segregation by ethnic group, it has been argued that rigorous pairwise segregation comparisons over time or across space should be invariant in two situations: when the ethnic composition of the population changes while the distribution of each ethnic group over the schools remains constant (invariance 1), or when the size distribution of schools changes while the ethnic composition of each school remains constant (invariance 2). This paper makes two contributions to the segregation literature. First, it argues by means of the Mutual Information or M index, which is neither invariant 1 nor 2, that both properties have strong implications, and it provides reasons to defend that the overall segregation index need not satisfy either one. Second, nevertheless, it is shown that in pairwise comparisons this index admits two decompositions into three terms. In the first decomposition, a term is invariant 1 and also satisfies a weak version of invariance 2. In the second decomposition, a term is invariant 2 and also satisfies a weak version of invariance 1. It is shown

Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 33–53 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017005

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34

RICARDO MORA AND JAVIER RUIZ-CASTILLO

that these decompositions can be used to reach the analogous ones obtained in Deutsch et al. (2006).

INTRODUCTION The phenomenon of segregation has preoccupied sociologists and economists during the last 50 years in two different contexts: the occupational segregation by gender in the labor market and the residential or school segregation by ethnic group.1 For concreteness, and given the increasing ethnic diversity in many countries, this paper focuses on school segregation in the multigroup case; that is, when there are more than two ethnic groups. This paper studies two invariance properties of a segregation index, originally discussed in the context of pairwise segregation comparisons over time or across space. Consider for a moment the special but important case of occupational segregation by gender, and assume that segregations in 1950 and 2000 are being compared in a given country. The two questions often asked are the following (see, inter alia, Watts, 1998). First, should the measurement of occupational segregation be independent of the fact that female labor participation has greatly increased over time? Many people would agree that, as long as the male and female distributions over occupations remain constant, the degree of segregation should be the same in the two situations (this is known as composition invariance, or invariance 1). In the multigroup case, the question becomes: should segregation be invariant to changes in the ethnic composition of the population as long as the distribution of each group over the schools remains constant? Second, should occupational segregation be independent from the fact that agricultural and industrial occupations are much more important in 1950 than in 2000, while services occupations carry much more weight in 2000 than in 1950? Many people would agree that, as long as the gender composition of each occupation remains constant, the degree of segregation should be the same in the two situations (this is known as occupational invariance, or invariance 2). In the multigroup case, the question becomes: should segregation be invariant to changes in the size distribution of schools as long as the ethnic composition of each school remains constant? This paper makes two contributions to the literature in the homogeneous case in which the two situations under comparison share a common structure, that is, the same number of demographic groups and organizational units. In the first place, it is argued that both invariance properties have strong implications, and hypothetical examples are given to defend that the overall

Invariance Properties of Mutual Information Index

35

segregation index need not be invariant 1 or invariant 2.2 This position is substantiated by means of the Mutual Information or M index, based on the entropy concept used in information theory that was first introduced in the segregation literature by Theil and Finizza (1971). Although the M index violates both invariance properties, it is very appealing for the following reasons. First, Frankel and Volij (2008a) have characterized the underlying segregation ordering in terms of eight axioms. Second, the M index has been shown to satisfy a number of other desirable properties discussed in the literature, including two additive separability properties into between- and within-group terms for any partition of either the schools or the ethnic groups (see Mora & Ruiz-Castillo, 2008a, 2008c). Third, Mora and RuizCastillo (2008b) show that a test on segregation in an unrestricted statistical model leads to the M index and that differences in segregation over time or across space according to the M index can be tested in an appropriate statistical framework. It may be argued that intertemporal or international comparisons of segregation levels using the M index are ‘‘contaminated’’ by changes in the marginal distributions. However, in its second contribution, this paper presents two decompositions of the M index into three terms in pairwise comparisons over time or across space, which isolate the effects of the changes in the marginal distributions.3 In the first decomposition, the first term captures the effects on the index M of changes in the ethnic groups’ marginal distribution. The second term captures the effects of the changes in the entropy, or concentration, of the schools. The last term captures changes in the M index when there are no changes in the groups’ marginal distributions and no changes in the concentration of schools. Thus, the third term is invariant 1 and also satisfies a weak version of invariance 2. In the second decomposition, the first term captures the effects on the index M of changes in the organizational units’ marginal distribution. The second term captures the effects of the changes in the concentration of the groups, while the last term captures changes in the M index otherwise. Thus, the last term is invariant 2 and also satisfies a weak version of invariance 1. It is shown that these decompositions can be used to reach the analogous ones obtained in Deutsch, Flu¨ckiger, and Silber (2006). The remaining of this paper is organized in four sections. In the first section, the notation, the notions of segregation most often used in the literature, and a variety of invariance axioms are presented. In the second section, the M index is introduced and a number of examples given that illustrate how demanding invariance 1 and invariance 2 really are, and why it may be justified that a segregation index violates them. The third section is

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devoted to the presentation of the two decompositions of the M index, which isolate either an invariance 1 or an invariance 2 term for the purpose of making pairwise segregation comparisons over time or across space. The last section concludes.

NOTATION AND INVARIANCE AXIOMS Notation Let X be a city with G ethnic groups distributed over N schools, indexed by g ¼ 1,y, G and n ¼ 1,y, N, respectively. Denote by T ng the number of students of ethnic group g in school n. The data available in empirical situations can be organized as follows: 2

T 11 6. .. . X ¼6 . 4. 1 TG

1

3 TN 1 .. 7 . 7 5 TN G N

Ethnic groups

2

3 1 6 .. 7 4. 5

(1)

G

Schools

P n n where T g ¼ N n¼1 T g is the number of students of ethnic group g, T ¼ PG PG n g¼1 T g the number of students attending school n, and T ¼ g¼1 T g ¼ PN n n¼1 T is the total number of students in the city. We are often interested in spatial or intertemporal segregation comparisons between heterogeneous situations, that is, between cities with a different set of demographic groups and/or schools. However, in what follows it will be assumed that we have data about two comparable cities that share the same number of ethnic groups and schools. A discussion of the possible empirical strategies to deal with the heterogeneous case is beyond the scope of this paper.4 The information contained in the joint distribution of ethnic groups and schools is usually summarized by means of numerical indices of segregation. Let X be the set of all cities with G ethnic groups and N schools. A segregation index is a real valued function defined in X, S(X), providing the extent of school segregation for any city X. The concept of segregation used in this paper embraces two views. First, the notion advocated by James and Taeuber (1985), according to which


37

segregation is seen as the tendency of ethnic groups to have different distributions across schools. Second, the idea of ‘‘representativeness’’ emphasized by Frankel and Volij (2008a), which asks to what extent schools have different ethnic compositions than the population as a whole.5 As can be seen in Eq. (1), where the rows are ethnic groups and the columns are schools, evenness and representativeness are dual concepts: deviations from evenness (representativeness) correspond to differences in the row (column) percentages. The following observation indicates how close these two views are to each other. Remark 1. If a segregation index S that captures the notion of evenness when applied to the G N array X in X is applied to the N G array Xu, where the roles of demographic groups and schools are reversed, so that 0 T ng ¼ T gn for all n and g, then what will be called the reverse index, S, captures equally well the notion of representativeness (and vice versa). In general, S(X) will be different from S(Xu). Otherwise, that is, when S(X) ¼ S(Xu), we say that the segregation index S is transpose invariant. Relative and Absolute Segregation Views It is customary to ask: should segregation depend on the population size T ? Most answers in the literature have been in the negative; that is, most segregation indices satisfy the following invariance property. Size invariance. (James & Taeuber, 1985; Weak Scale Invariance in 0 Frankel & Volij, 2008a). If X and Xu are two cities in X such that T ng ¼ lT ng for all n and g with lW0, then S(X) ¼ S(Xu). This property resembles the Population Principle in the income inequality literature, according to which replications of the income distribution do not alter income inequality. In our context, size invariance captures a relative view of segregation in which all that matters are the joint, conditional and marginal frequency distributions:6 pgn ¼ T ng =T: proportion of students of group g and school n in the city; : joint distribution of ethnic groups and schools; Pgn ¼ fpgn gG;N PN g¼1; n¼1 pg ¼ n¼1 pgn : proportion of students of group g in the population; : marginal distribution by ethnic groups; Pg ¼ fpg gG P g¼1 p pn ¼ G g¼1 gn : proportion of students attending school n in the population; Pn ¼ fpn gN n¼1 : marginal distribution by school size;

38


pgjn Pgjn pnjg Pnjg

¼ pgn =pn : proportion of students in school n who belong to group g; ¼ fpgjn gG g¼1 : conditional distribution by groups of students at school n; ¼ pgn =pg : proportion of students in group g who attend school n; ¼ fpnjg gN n¼1 : conditional distribution by schools of students in group g.

The Main Invariance Properties As pointed out in the Introduction, in the empirical literature on gender segregation, it has been noticed that both the overall gender composition of employment as well as the distribution of the employed population across occupations typically change over time and/or space. Similar phenomena are present in other segregation contexts. Due to differential fertility or the vagaries of domestic or international migration flows, the relative size of certain ethnic groups may drastically change over time and across space. Similarly, the size distribution of schools in two cities or in a given city in two moments in time may be very different indeed. Consequently, it has been forcefully argued that rigorous segregation comparisons in spatial and time-series studies should be made independent of changes in the marginal distributions, be it the overall gender shares of the employed population and its distribution across occupations in studies of occupational segregation by gender, or the population ethnic composition, Pgd, and the population distribution across schools, Pdn, in studies of school segregation by ethnic group.7 The following two axioms have been proposed to capture these ideas. To motivate the first one, consider a situation in which only the size of one or more ethnic groups varies, so that the marginal distribution Pgd changes, but the allocation of ethnic groups across schools, Pn|g, remains constant. Under these circumstances, it is frequently argued that the segregation level should be unchanged. Invariance 1 (I1) (Composition Invariance in James & Taeuber, 1985; Watts, 1998; Homogeneity in Hutchens, 1991; Scale Independence in 0 Frankel & Volij, 2008b). If X and Xu are two cities in X such that T ng ¼ n lg T g for all n and g with lgW0 for each g, then S(X) ¼ S(Xu). Under I1, differences in segregation cannot be due solely to differences in citywide ethnic composition. In other words, a group’s weight in the segregation index cannot depend only on its size. As a matter of fact, the only relevant magnitudes in the domain of an I1 index are the conditional


39

school distributions by ethnic group, Pn|g; that is, if S1 satisfies I1, then S1(X)6¼S1(Xu) only if the conditional distributions Pn|g and P0njg differ for some group g. This implies that users of segregation indices that satisfy I1 restrict themselves to an evenness notion of segregation. In the important G ¼ 2 case, there is a very close relation between I1 indices and segregation curves, first suggested by Duncan and Duncan (1955). In the context of occupational segregation by gender, a segregation curve represents the cumulative fraction of females (on the ordinate) and the cumulative fraction of males (on the abscissa) with occupations sorted in ascending order according to the ratios (pn|female/pn|male). A segregation curve is said to dominate another if it lies at no point below and at some point above the other. Just as with Lorenz curves, segregation curves provide an incomplete ranking of distributions of employed people across occupations. Hutchens (1991, 2001) showed that a segregation index is consistent with the ranking obtained from segregation curves only if it satisfies I1. Thus, the failure to satisfy I1 implies that a segregation index is not consistent with the ordering provided by segregation curves. Notice, however, that since segregation curves are well defined only when G ¼ 2, this defense of I1 does not carry over to the multigroup case with GW2.8 For the next invariance axiom, consider situations in which the school size distribution, Pdn, changes, while the ethnic mix within each school, Pg|n, remains constant. It has also been argued that under these conditions the segregation index should not change. Invariance 2 (I2) (Occupational Invariance in Watts 1998; Blackburn, Jarman, & Siltanen, 1993, 1995). If X and Xu are two cities in X such that 0 T ng ¼ ln T ng for all n and g with lnW0 for each n, then S(X) ¼ S(Xu). Under I2, differences in segregation should not be due solely to differences in school sizes; consequently, a school’s weight in a segregation index cannot depend only on its size. The only relevant magnitudes in the domain of I2 indices are the school’s ethnic composition, Pg|n; that is, if S2 satisfies I2, then S2(X) 6¼ S2(Xu) only if the conditional distributions Pg|n and P0gjn differ for some school n. This implies that users of segregation indices that satisfy I2 restrict themselves to a representativeness notion of segregation. Remark 2. In view of Remark 1, the reciprocal segregation index S of any I1 index S becomes an I2 index (and vice versa).

40


THE MUTUAL INFORMATION INDEX OF SEGREGATION In information theory, the expression ! G X pgjn n M ¼ pgjn log pg g¼1

(2)

is known as the expected information of the message that transforms the set of proportions Pgd to the set of proportions Pg|n. The value of this expected information is zero whenever the two sets of proportions are identical, and it takes larger and larger positive values when the two sets are more different. In other words, the term Mn measures the extent to which the ethnic distribution of individuals in school n, Pg|n, differs from the overall ethnic distribution, Pgd. Therefore, Mn can be interpreted as an index of (local) segregation in school n when segregation is interpreted as deviation from representativeness.9 Because Mn indicates the existence of local segregation whenever Pg|n differs from Pgd, Mn is obviously not independent from changes in the ethnic distribution of the population, Pgd. Thus, the Mn index is not I1. To justify this choice, consider the following two examples with two demographic groups (black and white), who attend three schools. In example A, a student population of four people, consisting of two black and two white students, is distributed in the three schools as follows: Ethnic groups

1 1 0 A X ¼ 0 1 1 ½ s1 s2 s3

Black White

Schools

Rows represent the distributions of blacks and whites across the three schools. Columns inform about the ethnic mix for each of the three schools. School 1 is entirely black, pblack|s1 ¼ 1, while school 3 is entirely white, pblack|s3 ¼ 0. Since the proportion of black students in the population is pblackd ¼ 0.5, the local indices Ms1 ¼ Ms3 ¼ log(2) ¼ 0.69 indicate large segregation levels in both schools. Since pblack|s2 ¼ 0.5, Ms2 ¼ 0, indicating the absence of segregation in school 2. Consider now example B where, relative to example A, the size of the white population increases by a factor of 100:

1 1 0 XB ¼ 0 100 100 ½ s1 s2 s3 Schools

Ethnic groups

Black White

41


Since now pblackd ¼ 1/101, the event pblack|s1 ¼ 1, becomes much more anomalous than before when pblackd ¼ 0.5. Correspondingly, it is reasonable to have Ms1 ¼ log (101) ¼ 4.62 in school 1, a larger value than before. Similarly, the event pblack|s3 ¼ 0 fits more into the overall pattern than in example A, which causes Ms3 ¼ 0.99 102 in school 3 to be much lower than before. Finally, since pblack|s2 ¼ pblackd ¼ 1/101, indicating that school 2 is characterized as before by the absence of segregation, we have Ms2 ¼ 0. Thus, in spite of the fact that the distributions of both ethnic groups across schools remain constant, the change in the overall black share from pblackd ¼ 0.5 in example A to pblackd ¼ 1/101 in example B drastically alters the way segregation in certain schools is measured by the local index Mn. Of course, all indices of (local) segregation for demographic groups satisfying I1 will indicate that the extent of segregation among blacks and whites remain constant between situations A and B. A weighted average of the Mn indices of local segregation defined in Eq. (2) will constitute an index of segregation for the city as a whole. The selection of the weights is an important issue. One possible option is to give the same weight to each school, thus ensuring that the index satisfies I2. However, we agree with the argument provided by England (1981) in the context of occupational segregation by gender. Suppose that schools that are more segregated grow faster over time. An index that reveals this increase in segregation seems to us preferable over one that adjusts the change out because it resulted from an increase in the relative size of highly segregated schools. Thus, the M index of overall segregation is defined by M¼

N X

pn M n

(3)

n¼1

That is to say, M is the weighted average of information expectations, with weights equal to the number of people in each school.10 Consider the following illustration of the role of such weighting scheme. Relative to example A, in example C the size of school 2 is increased by a factor of 100: 1 100 0 X ¼ 0 100 1 ½ s1 s2 s3 C

Ethnic groups

Black White

Schools

The black share in the three schools and in the population as a whole coincides with those of example A so that the local indices of segregation Mn

42


remain constant in each school. However, the relative sizes of schools 1 and 3 have now drastically decreased with respect to those in example A. Correspondingly, the overall segregation index is now M (XC) ¼ 0.68 103oM(XA). Of course, an index that satisfies I2 would give the same level of segregation in the two examples. Looking at the three examples together, advocates of segregation indices that satisfy I1 but not I2 would recommend that segregation remains constant in examples A and B, but it should typically decrease in example C. Advocates of segregation indices that satisfy I2 but not I1 would recommend that segregation remains constant in examples A and C, but it should typically decrease in example B. Interestingly enough, advocates of segregation indices that satisfy both invariance properties would insist on segregation remaining constant through the three examples.11 As we have seen, the M index favored in this paper would indicate instead that segregation continuously decreases in the sequence from example A to example C. To further illustrate the differences between a segregation index that satisfies one or both invariance properties and the M index, consider the following example D (used for other purposes in Frankel & Volij, 2008b). There are two cities Y1 and Y2 with the following data matrixes: 99 0 Y ¼ 0 1 ½ s1 s2 1

Schools

Ethnic groups

Black ; White

50 Y ¼ 0 ½ s1 2

0 50 s2

Ethnic groups

Black White

Schools

All conditional distributions in each city are equal to

1 0 Pnjg ¼ Pgjn ¼ 0 1 Therefore, a student’s ethnicity is determined by her school in both cities. Thus, according to an index satisfying both invariance properties, segregation is the same in both cities.12 However, as Frankel and Volij (2008b) indicate, because the population ethnic distribution Pgd is (0.99, 0.01) in Y1 and (0.50, 0.50) in Y2, there is initially less uncertainty about a student’s ethnicity in Y1 than in Y2. Correspondingly, the M index reveals that the second city is more segregated, M(Y2) ¼ 0.69, than the first one, M(Y1) ¼ 0.56 101. As long as any aspect of the joint distribution of ethnic group and school changes – including changes solely in the marginal


43

distributions with both conditional distributions remaining constant – the M index reflects a change in the pattern of segregation. Finally, one may wonder how the notions of evenness and representativeness are handled by the M index. For that purpose, notice that the expression N X pnjg (4) pdn log Mg ¼ p dn n¼1 is the expected message that transforms the set of proportions Pdn to the set of proportions Pn|g. The term Mg measures the extent to which the distribution of students in group g across schools differs from the school size distribution for the population as a whole. Therefore, Mg can be interpreted as a (local) index of segregation in ethnic group g when segregation is taken to mean deviation from evenness. It turns out that the weighted average of information expectations defined in Eq. (4), with weights equal to the number of people in each group, coincides with the overall index of segregation for the population as a whole defined in Eq. (3), that is, M¼

G X

pg d M g

(5)

g¼1

Eqs. (3) and (5) show that the M index is transpose invariant (for the twogroup case, this is shown in Mora & Ruiz-Castillo, 2005). Consequently, we may say that it treats evenness and representativeness in a symmetric fashion. According to the alternative motivation in Frankel and Volij (2008a), if the ethnic group and school of a randomly selected student are thought of as random variables, then the M index equals the mutual information of these variables: the reduction in uncertainty about one variable that occurs when one learns the value of the other (Cover & Thomas, 1991). Since mutual information is a symmetric concept, these authors establish that the M index is transpose invariant. Therefore, it can be interpreted both as the average amount of information that a student’s school conveys about her ethnicity as well as what a student’s ethnicity reveals about her school. A final example E, inspired in another one from Frankel and Volij (2008b), will help to understand the difference between the M index and any index that satisfies I1, according to which a change in the segregation scheme in a relatively small ethnic group may lead to a large change in

44


school segregation. Consider two cities Z1 and Z2 with the following data matrixes: 2

89

6 Z 1 ¼ 4 10 1 ½ s1

10

3

7 89 5 1 s2

Ethnic groups 2 3 Black 6 White 7 4 5; Mixed

2

89 10

3

6 7 Z2 ¼ 4 10 89 5 10 1 ½ s1 s2

Schools

Ethnic groups 2 3 Black 6 White 7 4 5 Mixed

Schools

Since the black and white distributions are very similar in the two situations and the mixed group is very small, the weighted M index does not change much: M(Z1) ¼ 0.36 and M(Z2) ¼ 0.37.13 This is not the case for an I1 segregation index for which a group’s weight cannot depend on its size. For example, take the unweighted Atkinson index A defined by:14 ! N G X Y 1=G AðXÞ ¼ 1 ðpnjg Þ n¼1 1

g¼1

2

In this case, A(Z ) ¼ 0.29, while A(Z ) ¼ 0.37. Since a group’s size does not matter, the small mixed-race group receives the same weight, 1/3, in the geometric mean as each of the two larger groups. Because this group’s segregation pattern considerably changes from Z1 to Z2, the overall segregation changes by as much as 27.6%. It is reasonable to search for segregation indices that satisfy either I1 or I2. However, we believe that it is equally reasonable to work with segregation indices that violate both properties. In particular, as the above examples illustrate, there are good a priori reasons to work with the M index where the influence of any index of (local) segregation at the group or the school level in overall segregation depends on the group or the school size.

DECOMPOSITIONS OF PAIRWISE COMPARISONS OF THE MUTUAL INFORMATION INDEX Differences in the index of segregation between any two situations may result from differences in the marginal distributions, Pgd and Pdn, as well as from differences in the conditional distributions, Pn|g and Pg|n. As indicated in the Introduction, there are reasons to argue that pairwise comparisons of segregation should net out the effect of differences in the marginal distributions. Such comparisons can be accomplished in at least two ways.

45


First, if the index is invariant, then the comparison of the index will also be invariant. Second, as advocated, inter alia, by Blau and Hendricks (1979), Jonung (1984), Beller (1985), and Watts (1992, 1998), if the index is not invariant, then observed differences may be decomposed so that one of the terms in the decomposition reflects changes in segregation that are due only to changes in one of the conditional distributions. This is the strategy applied for the M index in the rest of this section. For the sake of concreteness, assume that there is data on a city X in X during two periods, X(t), t ¼ t1, t2. Applying the M index to this data for any t, we have: G N X X pnjg ðtÞ MðXðtÞÞ ¼ pg ðtÞ pnjg ðtÞ log pn ðtÞ g¼1 n¼1 ! N G X X pgjn ðtÞ ¼ pn ðtÞ pgjn ðtÞ log pg ðtÞ n¼1 g¼1 As we know, a change in pdn(t) and/or pgd(t) keeping pg|n(t) and/or pn|g(t) invariant will result in a change in the index; that is, M does not satisfy I1 or I2. However, as will be seen presently, there are ways to decompose pairwise comparisons using the M index in order to isolate either an invariant 1 or an invariant 2 term. In the first place, to identify an invariant 1 term in a decomposition of a pairwise comparison, note that the M index can be written as: ( ) G N X X pg ðtÞ pnjg ðtÞ logðpnjg ðtÞÞ þ T c ðtÞ (6) MðXðtÞÞ ¼ g¼1

n¼1

where N X

1 pn ðtÞ log T ðtÞ ¼ pn ðtÞ n¼1 c

is Theil’s index of concentration for the marginal distribution of students by G schools. For the moment, let Pg pg g¼1 be any G array of real numbers PG 15 such that 0rpgr1 and Given Eq. (6), the intertemporal g¼1 pg ¼ 1. change in overall segregation can be decomposed into the following three terms: MðXðt2 ÞÞ MðXðt1 ÞÞ ¼ DNðPg Þ þ DSCg þ DEGðPg Þ

(7)

46


where DNðPg Þ ¼

G X

pg

g¼1 c

N X pnjg ðt2 Þ logðpnjg ðt2 ÞÞ pnjg ðt1 Þ logðpnjg ðt1 ÞÞ , n¼1

2

DSCg ¼ T ðt Þ T c ðt1 Þ, ( ) G N X X X 1ðt¼t1 Þ ð1Þ ðpg ðtÞ pg Þ pnjg ðtÞ logðpnjg ðtÞÞ DEGðPg Þ ¼ g¼1

t¼t1 ;t2

n¼1

and 1(t ¼ t1) is the indicator function. The three terms in Eq. (7) can be interpreted as follows. First, DSCg isolates segregation changes due to intertemporal changes in the concentration of students across schools. Second, DEG(Pg) isolates segregation changes due to differences between the marginal distributions of students by ethnic groups, Pgd(t), and Pg. Third, DN(Pg) gives the difference in segregation that arises from changes in Pn|g(t) when both the Theil’s index of concentration Tc(t) and the marginal distribution Pgd(t) remain constant, and the latter equals Pg. Notice that DN(Pg) satisfies I1 in the sense that it equals zero if T ng ðt2 Þ ¼ lg T ng ðt1 Þfor all n and g with lgW0. In addition, DN(Pg) satisfies a weaker version of I2 in the following sense. Suppose that the conditional distribution Pn|g(t) remains constant but the Theil’s index of concentration T c(t) changes, then DN(Pg) ¼ 0. In the second place, to identify an invariant 2 term in a decomposition of a pairwise comparison, note that the M index can also be written as: ( ) N G X X pn ðtÞ pgjn ðtÞ logðpgjn ðtÞÞ þ T c ðtÞ (8) MðXðtÞÞ ¼ n¼1

g¼1

where G X

1 T c ðtÞ ¼ pg ðtÞ log p g ðtÞ g¼1

!

is Theil’s index of concentration for the marginal distribution of students n n N by ethnic groups. For the moment, denote PGby Pn fp gn¼1 any N array n of real numbers such that 0rp r1 and g¼1 p ¼ 1. Given Eq. (8), the intertemporal change in overall segregation can be decomposed into the following three terms: MðXðt2 ÞÞ MðXðt1 ÞÞ ¼ DNðPn Þ þ DEGn þ DSCðPn Þ

(9)

47


where DNðPn Þ ¼

N X

pn

n¼1 n

G X

pgjn ðt2 Þ logðpgjn ðt2 ÞÞ pgjn ðt1 Þ logðpgjn ðt1 ÞÞ ,

g¼1 2

DEG ¼ T c ðt Þ T c ðt1 Þ, ( ) N G X X X 1ðt¼t1 Þ n n ð1Þ ðpn ðtÞ p Þ pgjn ðtÞ logðpgjn ðtÞÞ DSCðP Þ ¼ n¼1

t¼t1 ;t2

g¼1

The three terms in Eq. (9) can be interpreted as follows. First, DEGn isolates segregation changes due to changes in the concentration of students across ethnic groups. Second, DSCðPn Þ isolates segregation changes due to differences between the marginal distributions of individuals by schools, Pdn(t), and Pn. Third, DN(Pn) gives the difference in segregation that arises from changes in Pg|n(t) when the Theil’s index of concentration Tc(t) and the marginal distribution of students by ethnic groups Pdn(t) remain constant, and the latter equals Pn. Note that DN(Pn) is I2 in the sense that it equals zero if T ng ðt2 Þ ¼ ln T ng ðt1 Þ for all n and g with lnW0 In addition, DN(Pn) satisfies a weaker version of I1 in the following sense. If the conditional distribution Pg|n(t) remains constant but the Theil’s index of concentration Tc(t) changes, then DN(Pn) ¼ 0. Decompositions (7) and (9) can be computed only after specific values for Pg and Pn are chosen. Consider the decomposition in Eq. (7). A simple choice for Pg would be Pgd(t1), in which case the terms DN(Pg) and DEG(Pg) in the right-hand side of Eq. (7) become: DNðPg ðt1 ÞÞ ¼

G P

pg ðt1 Þ

g¼1

DEGðPg ðt2 ÞÞ ¼

N P

fpnjg ðt2 Þ logðpnjg ðt2 ÞÞ pnjg ðt1 Þ logðpnjg ðt1 ÞÞg;

n¼1 G P g¼1

ðpg ðt2 Þ pg ðt1 ÞÞ

N P

pnjg ðt2 Þ logðpnjg ðt2 ÞÞ

n¼1

The interpretation of DN(Pg(t1)) as a change in segregation when marginals are kept constant requires additional conditions. First, Pgd(t2) ¼ (Pgd(t1)); that is, the marginal distribution of students across ethnic groups in the second period is assumed to be the same as the marginal distribution in the first period. Second, the Theil’s index of concentration for the marginal distribution of students by schools in both periods must be equal, Tc(t2) ¼ Tc(t1). Finally, the conditional distribution by schools of students from group g in the second period must equal the actual one,

48


Pn|g(t2) ¼ (Pn|g(t2)). As Karmel and MacLachlan (1988) argue, this approach will not be generally appropriate because marginal and conditional distributions are not independent of each other: pg ðtÞ ¼

N P

pgjn ðtÞpn ðtÞ

n¼1

pn ðtÞ ¼

G P

pnjg ðtÞpg ðtÞ

g¼1

The additional conditions will almost never be met. Instead, they propose the use of an iterative numerical procedure originally suggested by Deming and Stephan (1940) to identify a joint distribution of schools and ethnic groups for, say, period t1, which maintains the original association structure between schools and ethnic groups but closely approximates the marginal distributions of, say, period t2. Using this transformation, these authors establish that changes in segregation can be decomposed into four terms that capture variation in segregation due to a variation in the relative weights of the schools, a change in the overall proportions of ethnic groups, a net change in segregation, plus an interaction term. More recently, Deutsch et al. (2006) combine the Karmel and MacLachlan approach with the concept of Shapley value to provide an alternative decomposition without any interaction term. These authors accomplish this valuable task by applying this approach to the dissimilarity segregation index first proposed in Duncan and Duncan (1955). Interestingly enough, when this approach is applied to pairwise comparisons of the M index to isolate I1 and I2 terms, it turns out that the decomposition thus achieved can be obtained from Eqs. (7) and (9) for certain specifications of the sets Pg and Pn, respectively. The remainder of this section is devoted to establishing the close connection between the two approaches. First, consider the computation of an I1 term. Following the iterative algorithm described in Deming and Stephan (1940), assume that the matrix X(t1) converges into a new matrix X(t1, t2) such that the conditional distribution Pn|g closely resembles the original conditional distribution in X(t1), while the marginal distributions Pgd and Pdn are equal to those of period t2. Similarly, matrix X(t2) can also be transformed into a new matrix X(t2, t1) such that the conditional distribution Pn|g remains approximately constant but the marginal distributions equal those of period t1. Define 1 1 2 1 1 ½MðXðt ; t ÞÞ MðXðt ÞÞ þ ½MðXðt2 ÞÞ MðXðt1 ; t2 ÞÞ DN g ¼ 2 2

49


and

1 1 2 2 1 ½MðXðt ÞÞ MðXðt ; t ÞÞ þ ½MðXðt1 ; t2 ÞÞ MðXðt1 ÞÞ DM g ¼ 2 2

DNg measures changes in segregation due to changes in the conditional school distribution by ethnic group, Pn|g, while DMg captures the effect on segregation of changes across periods in both marginals, Pgd and Pdn. It is easy to observe that MðXðt2 ÞÞ MðXðt1 ÞÞ ¼ DN g þ DM g

(10)

To establish the relation between DNg and DN(Pg) in Eq. (7), assume that the algorithm used to obtain X(t2, t1) actually delivers a new matrix that exactly replicates the conditional distribution in period t2, that is, pn|g(t2, t1) ¼ pn|g(t2). In this case, we have ( ) G N X X 2 1 1 2 2 MðXðt ; t ÞÞ ¼ pg ðt Þ pnjg ðt Þ logðpnjg ðt ÞÞ þ T c ðt1 Þ g¼1

n¼1

If we now choose Pg ¼ Pgd(t1), then it can be seen that ½MðXðt2 ; t1 ÞÞ MðXðt1 ÞÞ ¼ DNðPg ðt1 ÞÞ Similarly, assuming that the algorithm used to obtain X(t1, t2) actually delivers a new matrix that exactly replicates the conditional distribution in period t1, that is pn|g(t2, t1) ¼ pn|g(t2), and choosing Pg ¼ Pgd(t2), it can be seen that ½MðXðt2 ÞÞ MðXðt1 ; t2 ÞÞ ¼ DNðPg ðt2 ÞÞ Therefore, DN g ¼

1 1 DNðPg ðt1 ÞÞ þ DNðPg ðt2 ÞÞ 2 2

(11)

On the other hand, it can be seen that DM g ¼ DSCg þ DEGg

(12)

where DSCg is defined in Eq. (7) and 1 1 DEGðPg ðt1 ÞÞ þ DEGðPg ðt2 ÞÞ DEGg ¼ 2 2 Eqs. (11) and (12) imply that a decomposition that isolates an I1 term following the approach in Deutsch et al. (2006) can be achieved by applying

50


Eq. (7) for Pg ¼ Pgd(t1) and Pg ¼ Pgd(t2), and then computing the average for each of the terms in the decomposition. Second, consider the computation of an I2 term following the Deutsch et al. approach. Using a similar argument as in the previous case, it can be shown that: MðXðt2 ÞÞ MðXðt1 ÞÞ ¼ DN n þ DEGn þ DSCn where DEGn is defined in Eq. (9) and 1 1 DN n ¼ DNðPn ðt1 ÞÞ þ DNðPn ðt2 ÞÞ 2 2 1 1 DSCðPn ðt1 ÞÞ þ DSCðPn ðt2 ÞÞ DSCn ¼ 2 2

ð13Þ

Eq. (13) shows that a decomposition that isolates an I2 term can be achieved by evaluating decomposition (9) for Pn ¼ Pdn(t1) and Pn ¼ Pdn(t2), and then computing the average for each term in the decomposition.

CONCLUSIONS In the segregation literature it has been advocated that rigorous pairwise segregation comparisons over time or across space in the multigroup case should be invariant in two situations: when the ethnic composition of the population changes while the distribution of each ethnic group over the schools remains constant (invariance 1), or when the size distribution of schools changes while the ethnic composition of each school remains constant (invariance 2). This paper has made two contributions to this literature. First, it has argued that both properties have strong implications and that there are reasons to defend that the overall segregation index need not satisfy either one. The contrast between invariant segregation indices and the Mutual Information segregation index that violates both properties has been illustrated with a number of examples. Second, it has nevertheless been shown that in pairwise comparisons, this index admits two decompositions into three terms. In the first decomposition, a term is invariant 1 and also satisfies a weak version of invariance 2. In the second decomposition, a term is invariant 2 and also satisfies a weak version of invariance 1. It is shown that these decompositions can be used to reach the analogous ones obtained in Deutsch et al. (2006).


51

NOTES 1. For a recent treatise in occupational segregation by gender, see Flu¨ckiger and Silber (1999), and for references to residential and educational segregation, see Reardon and Firebaugh (2002). 2. We are not alone in this position. See, inter alia, Coleman, Hoffer, and Kilgore (1982), Flu¨ckiger and Silber (1999, pp. 84–85), Watts (1992, 1998), and Frankel and Volij (2008a). 3. Using other segregation indices, many authors have defended this strategy before. See, inter alia, Blau and Hendricks (1979), Jonung (1984), Beller (1985), and Watts (1992, 1998). 4. Herranz, Mora, and Ruiz-Castillo (2005) may provide a starting point for the treatment of heterogeneity. Using the M index in the two-group case, these authors propose an algorithm to reduce the number of occupations in two different time periods without significantly reducing the segregation values. 5. These two notions are closely connected with the ‘‘evenness’’ and ‘‘isolation’’ dimensions distinguished in Massey and Denton (1988) in the two-group case. 6. For a study that focuses on translation invariant segregation indices that represent an absolute view of segregation, see Chakravarty and Silber (1992). 7. See, inter alia, James and Taeuber (1985), Charles (1992, 1998), Charles and Grusky (1995, 2004), Grusky and Charles (1998), and Hutchens (1991, 2001, 2004). 8. For another notion of segregation curves in the multigroup case, see AlonsoVillar and Del Rio (2008). 9. In principle, the logarithm could be computed in any base. In the examples discussed, natural logarithms will be used. 10. Notice that, with this aggregation scheme, the citywide segregation in examples A and B above become M(XA) ¼ 0.35 and M(XB) ¼ 0.03, respectively, indicating larger overall segregation in example A than in example B. In contrast, an index that satisfies I1 would give the same level of segregation in the two examples. 11. For a segregation index that satisfies both I1 and I2 in the G ¼ 2 case, see Charles (1992, 1998), Charles and Grusky (1995, 2004), and Grusky and Charles (1998). 12. Of course, indices that satisfy either I1 or I2 will also give the same measurement of segregation in the two cities. 13. Since the ethnic mix in both schools and in the population as a whole does not change much in the two cities Z1 and Z2, it can be conjectured that the extent of segregation according to any I2 index will also be of a similar order of magnitude. 14. Frankel and Volij (2008b) characterize the underlying Atkinson ordering in terms of an ordinal version of the I1 property and five other ordinal axioms. 15. Alternative choices for Pg will be discussed later in text.

ACKNOWLEDGMENTS The authors acknowledge financial support from the Spanish DGI, Grants SEJ2006-05710 and SEJ2004-01959. We wish to thank an anonymous reviewer whose insightful comments helped us improve the paper.

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REFERENCES Alonso-Villar, O., & Del Rio, C. (2008). Local versus overall segregation measures. Working Paper no. 08/02. Departamento de Economı´ a Aplicada, Universidade de Vigo. Beller, A. (1985). Changes in the sex composition of U.S. occupations, 1960–1981. Journal of Human Resources, 20, 235–250. Blackburn, R. M., Jarman, J., & Siltanen, J. (1993). The analysis of occupational gender segregation over time and place: Considerations of measurement and some new evidence. Work, Employment and Society, 7, 335–336. Blackburn, R. M., Siltanen, J., & Jarman, J. (1995). The measurement of occupational gender segregation: Current problems and a new approach. Journal of the Royal Statistical Society A, Part 2, 158, 319–331. Blau, F., & Hendricks, W. (1979). Occupational segregation by sex: Trends and prospects. Journal of Human Resources, 12, 197–210. Chakravarty, S. R., & Silber, J. (1992). Employment segregation indices: An axiomatic characterization. In: W. Eichhorn (Ed.), Models and measurement of welfare and inequality. New York: Springer-Verlag. Charles, M. (1992). Cross-national variation in occupational sex segregation. American Sociological Review, 57, 483–502. Charles, M. (1998). Structure, culture, and sex segregation in Europe. Research in Social Stratification and Mobility, 16, 89–116. Charles, M., & Grusky, D. (1995). Models for describing the underlying structure of sex segregation. American Journal of Sociology, 100, 931–971. Charles, M., & Grusky, D. (2004). Occupational ghettos. Stanford, CA: Stanford University Press. Coleman, J., Hoffer, T., & Kilgore, S. (1982). Achievement and segregation in secondary schools: A further look at public and private school differences. Sociology of Education, 55, 162–182. Cover, T., & Thomas, J. (1991). Elements of information theory. New York: Wiley. Deming, W. E., & Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginals are known. Annals of Mathematical Statistics, 11, 427–444. Deutsch, J, Flu¨ckiger, Y., & Silber, J. (2006). The concept of Shapley decomposition and the study of occupational segregation: Methodological considerations with an application to Swiss data, mimeo. Duncan, O., & Duncan, B. (1955). A methodological analysis of segregation indices. American Sociological Review, 20, 210–217. England, P. (1981). Assessing trends in occupational sex segregation, 1900–1976. In: I. Berg (Ed.), Sociological perspectives on labor markets. New York: Academic Press. Flu¨ckiger, Y., & Silber, J. (1999). The measurement of segregation in the labor force. Heidelberg: Physica-Verlag. Frankel, D., & Volij, O. (2008a). Measuring school segregation, mimeo, September. Frankel, D., & Volij, O. (2008b). Scale invariant measures of segregation, mimeo, June. Grusky, D. B., & Charles, M. (1998). The past, present, and future of sex segregation methodology. Demography, 35, 497–504. Herranz, N., Mora, R., & Ruiz-Castillo, J. (2005). An algorithm to reduce the occupational space in gender segregation studies. Journal of Applied Econometrics, 20, 25–37.


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Hutchens, R. M. (1991). Segregation curves, Lorenz curves and inequality in the distribution of people across occupations. Mathematical Social Sciences, 21, 31–51. Hutchens, R. M. (2001). Numerical measures of segregation: Desirable properties and their implications. Mathematical Social Sciences, 42, 13–29. Hutchens, R. M. (2004). One measure of segregation. International Economic Review, 45, 555–578. James, D. R., & Taeuber, K. E. (1985). Measures of segregation. In: G. Schmid & R. Weitzel (Eds), Sex discrimination and equal opportunity: The labor market and employment policy. London: Gower Publishing Company. Jonung, C. (1984). Patterns of occupational segregation by sex in the labor market. In: N. B. Tuma (Ed.), Sociological methodology. San Francisco: Jossey-Bass. Karmel, T., & MacLachlan, M. (1988). Occupational sex segregation: Increasing or decreasing? Economic Record, 64, 187–195. Massey, D., & Denton, N. (1988). The dimensions of residential segregation. Social Forces, 67, 281–315. Mora, R., & Ruiz-Castillo, J. (2005). Axiomatic properties of an entropy based index of segregation. Working Paper no. 05–62, Economics Series 31, Universidad Carlos III. Mora, R., & Ruiz-Castillo, J. (2008a). A defense of an entropy based index of multigroup segregation. Working Paper no. 07–76, Economics Series 45, Universidad Carlos III. Mora, R., & Ruiz-Castillo, J. (2008b). The statistical properties of an entropy based index of multigroup segregation. Working Paper no. 07–74, Economics Series 43, Universidad Carlos III. Mora, R., & Ruiz-Castillo, J. (2008c). Multigroup and multilevel residential segregation: The U.S. case, 1989–2005. Working Paper no. 08–61, Economics Series 28, Universidad Carlos III. Reardon, S., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32, 33–67. Theil, H., & Finizza, A. J. (1971). A note on the measurement of racial integration of schools by means of information concepts. Journal of Mathematical Sociology, 1, 187–194. Watts, M. (1992). How should occupational segregation be measured? Work, Employment and Society, 6, 475–487. Watts, M. (1998). Occupational gender segregation: Index measurement and econometric modelling. Demography, 35, 489–496.

PART II THE GINI INDEX AND THE MEASUREMENT OF SEGREGATION

THE GINI COEFFICIENT AND SEGREGATION ON A CONTINUOUS VARIABLE Jeongdai Kim and Paul A. Jargowsky ABSTRACT Purpose – The Gini coefficient is a widely used measure of income inequality. It has also been used as a segregation measure, but only in the case of binary variables, for example race or gender. We develop a general version of the Gini Segregation Index (Gs) that can accommodate either continuous or binary variables, and discuss its relationship to existing measures. Methodology – The Gini Index of Segregation is developed graphically and derived mathematically, illustrating the relationship between Gini’s use in segregation and inequality applications. Findings – Using the Public Use Microdata Sample for 25 U.S. metropolitan areas from the 2000 Census, we illustrate the calculation of the index and show that it is highly correlated with an existing measure of economic segregation. Originality – This paper develops and illustrates a measure of segregation for continuous variables, a task for which there are few alternative measures. Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 57–70 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017006

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58

JEONGDAI KIM AND PAUL A. JARGOWSKY

BACKGROUND The Gini coefficient is a common measure of income inequality (Gini, 1912, 1921). It is routinely computed for all nations in the world that collect adequate data (Milanovic, 2002). It has also been employed as a measure of segregation, although its use as a segregation measure has been limited to categorical variables. Most commonly, it has been used and discussed in the context of black/white residential segregation (Duncan & Duncan, 1955; Jahn, Schmid, & Schrag, 1947; James & Taeuber, 1985). It has also been employed in studies of gender segregation in occupations (Deutsch, Flu¨ckiger, & Silber, 1994; Flu¨ckiger & Silber, 1990). Reardon and Firebaugh (2002) described a version of the Gini that is suitable for more than two groups. We develop a modified version of the Gini coefficient that may be used to measure segregation for continuous variables, a task for which there are few alternative measures (White, 1986). The Gini, therefore, can be used to contribute to the study of such topics as economic segregation and the segregation of wealth. As an empirical illustration, we examine economic segregation in 25 U.S. metropolitan areas and compare the results to an existing measure of economic segregation, the Neighborhood Sorting Index (NSI) (Jargowsky, 1996). Segregation refers to an uneven distribution of a characteristic of a population among subgroups within that population (James & Taeuber, 1985; Hutchens, 2004). Typically, but not always, the subgroups are residential neighborhoods. The issue of economic segregation has received less attention in the literature than racial segregation. One of the reasons, we believe, is that there are few measures of segregation for continuous variables such as income. A typical approach for a continuous variable has been to divide the continuous variable into two or more discrete categories, and then compute traditional group-based measures (Massey & Eggers, 1990; Townshend & Walker, 2002). This approach has been criticized for discarding information and for potentially confounding changes in the underlying parameters of the continuous variable’s distribution with changes in segregation (Jargowsky, 1996). The NSI, a version of eta squared, has been used to measure income segregation (Jargowsky, 1996). However, without comparable measures, the performance of NSI has not been examined. This paper aims to provide a valid measure of segregation for continuous variables based on the Gini coefficient. In contrast to measures of segregation, measures of inequality examine the distribution of a characteristic of the primary units within a given group

The Gini Coefficient and Segregation on a Continuous Variable

59

with no regard for the membership of those units to subgroups or neighborhoods. The Gini coefficient has long been the workhorse of income inequality analyses. It has also been employed as a measure of segregation, although in that context, its use has generally been limited to dichotomous groups (Silber, 1989; James & Taeuber, 1985; Massey & Denton, 1988). We develop a modified version of the Gini coefficient that can be used as a segregation measure for continuous variables. We use household income as an example, but the argument is applicable to any continuous variable.

THE GEOMETRY OF THE INDEX Assume that we wish to examine economic segregation among households (indexed by i ¼ 1 to N) sorted into neighborhoods (indexed by j ¼ 1 to M). The Gini coefficient may be computed in various ways, but the logic of measure is closely related to the Lorenz curve. To construct a Lorenz curve, one plots the cumulative percentage of household income as a function of the cumulative percentage of total households, after sorting the households from lowest to highest income (Lorenz, 1905). If all households have the same income, cumulative income and cumulative population will rise in tandem and the Lorenz curve is a 451 line, known as the line of equality. The Gini coefficient is defined as the area between the Lorenz curve and the line of equality, divided by the total area under the line of equality. A corresponding Lorenz curve and Gini coefficient can be constructed for neighborhoods (or any defined subgroups of the population) by sorting the neighborhoods by the mean household income, and then plotting the cumulative percentage of income and the cumulative percentage of population of the neighborhood totals (Theil, 1972, p. 108). In Fig. 1, assume that Li is a Lorenz curve based on individual-level data and Lj is a Lorenz curve based on neighborhood-level data. If every neighborhood contains individuals with identical incomes, Lj will be identical to Li, but if neighborhoods contain households with different levels of income, Lj will lie above Li. Based on the areas under the line of equality identified in Fig. 1, the Gini coefficient of income inequality (Gi) is defined by: Gi ¼

A1 þ A2 A1 þ A2 þ B

(1)

60


100%

Cumulative Income

Lj

Li A1 A2

B

Cumulative Households

Fig. 1.

100%

Lorenz Curves for Individual Households and Neighborhoods.

Gi has a minimum of zero if all households have equal income and approaches 1 as the population increases if all income is possessed by a single household. The Gini coefficient of neighborhood inequality (G j ) is between-group inequality (Pyatt, 1976; Silber, 1989) and is calculated in an analogous manner to Gi: Gj ¼

A1 A1 þ A2 þ B

(2)

G j has a minimum of zero if all neighborhoods have identical mean incomes, so that the area A1 that separates Lj from the line of equality is zero. However, it has a maximum of Gi if no neighborhood contains households with different levels of income, so that the area A2 that separates Li and Lj is zero. This form of Gini has been used to describe unevenness in the spatial distribution of health-care providers (Brown, 1994; Schwartz, Newhouse, Bennett, & Williams, 1980). However, G j is not a pure measure of segregation, because it simply measures the inequality of the neighborhoods,


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which results from an interaction of the sorting of households (segregation) and the underlying degree of income inequality. The upper bound of G j varies depending on the inequality of the individual units. When there is maximal segregation – every household living in neighborhoods with other households of the same income – the Lorenz curve for neighborhoods (Lj) converges to the Lorenz curve for individuals (Li) and the upper bound of G j is determined by Gi. A version of Gini that depends solely on the sorting of households into neighborhoods can be obtained by normalizing G j by Gi. Therefore, we define the Gini coefficient of segregation (Gs) as: Gs ¼

Gj A1 =ðA1 þ A2 þ BÞ A1 ¼ ¼ Gi ðA1 þ A2 Þ=ðA1 þ A2 þ BÞ A1 þ A2

(3)

Maximum segregation, given the existing income distribution, is indicated by Gs being equal to 1, and minimum segregation, by zero. In the case of maximum integration, all neighborhoods have identical mean incomes and the index takes on a value of zero. In the case of maximum segregation, every household lives only with other households of identical incomes, in which case G j equals Gi and therefore Gs equals 1. As a practical matter, if the number of unique income values exceeds the number of neighborhoods, the maximum is effectively less than 1, because at least one neighborhood will combine households of different incomes. Jargowsky and Kim (2009) argue on theoretical grounds that all segregation measures may be expressed as the ratio of an inequality measure computed on grouped data to the same inequality measure based on individual data. Gini fits this pattern. The binary form of Gini coefficient in use in racial segregation studies incorporates this normalization implicitly. Jahn et al. (1947, p. 298), analyzing black/white residential segregation, defined the Gini index as ‘‘the ratio of the Gini index of concentration computed y for census tracts,’’ that is, neighborhoods, which we have called G j, to the value ‘‘the index would obtain if there were ‘complete segregation.’ ’’ They calculated this maximum to be 1-P, the overall proportion black. When used as a segregation measure, the Gini has been often presented in a manner that obscured the distinction between segregation and inequality. The segregation Gini is illustrated, or at least described, in terms of a Lorenz curve with cumulative black on the vertical axis and cumulative white on the horizontal axis (e.g., Duncan & Duncan, 1955). However, this is not a precise analogy to the traditional Gini measure, with cumulative income on

62


the vertical axis and cumulative population on the horizontal axis. The analogy to income in the segregation case is ‘‘black,’’ a binary variable equal to 1 if the person is black and 0 otherwise. Therefore, the graph analogous to the traditional Gini income inequality graph should have cumulative black on the vertical axis and cumulative population on the horizontal axis. Fig. 2 clarifies the relationship between the Gini computed by Jahn et al. (1947) and subsequent researchers and Gs as we have defined it. It shows the Lorenz curve plotted in cumulative black population against cumulative white population, as is common in segregation studies (Panel I), compared to the cumulative black population versus cumulative total population, the more common representation in income inequality studies (Panel II). First, note that the individual-level Lorenz curve (Li) in Panel II is a horizontal

Panel I Cumulative White Population C Cumulative Black Population

NP Ls A

B

Panel II H N(1 -P)

NP

Cumulative Black Population

NP Li Lj F

D E Cumulative Total Population

Fig. 2.

Comparison of Black/White and Black/Total Ratio Frames.


63

line at zero until it reaches 1-P; then it is a 451 line from 1-P to 1. Therefore, the Gini coefficient for binary race at the individual level (Gi) is the area of triangle DEH divided by the area of triangle DFH: DDEH N 2 Pð1 PÞ=2 ¼ ¼1P DDFH N 2 P=2

(4)

This is also the maximum possible value of Gj, obtained if whites live only with whites and blacks live only with blacks, as Jahn et al. noted. They normalized the measure by this value, but only because they understood it to be the maximum level of racial segregation, not because it is the level of inequality at the individual level of a binary variable denoting membership in a racial group. Second, it is obvious that the area of triangle ABC in Panel I is equal to the area of triangle DEH in Panel II. It can be shown that the areas between the Lorenz curve Ls and Lj and their respective lines of equality are also equal. Therefore, Gs is nothing more than G j represented in a black/white ratio space. Dividing by 1-P, the maximum possible segregation for a binary variable is a special case of the more general principle of dividing the neighborhood-level Gini by the individual-level Gini. The conclusion is that Gs is a general measure of segregation that can be used either for binary or for continuous variables.

FORMULATION The Gini coefficient can be expressed as a mean difference in income among all pairs of individuals in a population (Anand, 1983, citing Kendall & Stuart, 1963):

Gi ¼

N X N 1 X jyh yi j 2 2N m h¼1 i¼1

(5)

where yh and yi are the income of the individual households, N the total number of households, and m is mean household income of the population. Literature on the decomposition of Gini provides various computation methods for G j (Mussard, 2004; Pyatt, 1976; Silber, 1989; Yao, 1999). Adapting the formula in Theil (1972, p. 108) to the current notation, the

64


group-level Gini can also be expressed in the form of mean difference as follows: M X M 1 X nj nk jmj mk j (6) Gj ¼ 2N 2 m j¼1 k¼1 where nj and nk are the count of households in specific neighborhoods and mj and mk are the mean household incomes in those neighborhoods. Finally, the Gini Segregation Index (Gs) is the ratio of the two: M P M P

nj nk jmj mk j G j¼1 k¼1 G ¼ i ¼ N P N P G jyh yi j j

s

(7)

h¼1 i¼1

To see the applicability of Eq. (7) to binary cases, m is replaced with the overall percentage black, denoted P, and y equals 1 for black and 0 for white. Since Gi, the denominator of Gs in Eq. (7), reduces to 1P, as demonstrated above, the binary form is: M P M P

nj nk jpj pk j G j j¼1 k¼1 G ¼ i ¼ G 2N 2 Pð1 PÞ s

(8)

We will contrast Gs with the NSI, which is based on the idea that there is a distribution of individual household incomes and of neighborhood mean incomes. To the extent that individuals of different incomes reside in neighborhoods together, the dispersion of neighborhood income will be less than that of household income (Jargowsky, 1996). Thus, the NSI may be expressed as: M P nj ðmj mÞ2 Variance between neighborhoods j¼1 ¼ N (9) NSI2 ¼ P Total variance ð yi mÞ2 i¼1

where mj is the average income of the neighborhood j. NSI attains a minimum of zero when all neighborhood mean incomes are equal to areawide mean income, so there is no variance between neighborhoods. It attains a theoretical maximum of 1 when all persons live only with others who have identical incomes, so that all variance in household income is between neighborhoods. The NSI2 is equivalent to eta squared from a oneway analysis of variance in which the categorical variable is neighborhood.

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EMPIRICAL APPLICATION For the purpose of illustration, we use the 5-percent Public Use Microdata Sample (PUMS) from the 2000 U.S. Census (Ruggles et al., 2004). The data is a 5-percent sample of all U.S. households, a subset of the households that completed the 2000 Census ‘‘long form’’ survey. Two levels of geography are employed: (1) the 25 largest U.S. metropolitan areas and (2) Public Use Microdata Areas (PUMAs), which function as neighborhoods in this analysis.1 We exclude households in nonmetropolitan areas. Table 1 shows the number Table 1. Households, Income, and Sample Characteristics of the 25 Largest Metropolitan Areas, 2000 U.S. Census 5-Percent PUMS. Metropolitan Area

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Total

New York-Northeastern, NJ Los Angeles-Long Beach, CA Chicago-Gary-Lake, IL Philadelphia, PA/NJ Washington, DC/MD/VA Detroit, MI Houston-Brazoria, TX Atlanta, GA Boston, MA Dallas-Fort Worth, TX Phoenix, AZ Minneapolis-St. Paul, MN Riverside-San Bernardino, CA St. Louis, MO/IL Tampa-St. PetersburgClearwater, FL San Diego, CA Baltimore, MD Orange County, CA Pittsburgh-Beaver Valley, PA Seattle-Everett, WA Nassau Co, NY Cleveland, OH Oakland, CA Denver-Boulder-Longmont, CO Miami-Hialeah, FL

Households

Sample Size

PUMS Areas

Mean Household Income ($)

3,482,324 3,135,972 2,937,970 1,913,900 1,777,008 1,692,431 1,462,022 1,460,471 1,320,037 1,225,886 1,133,243 1,096,388 1,033,275 1,011,780 1,007,772

144,070 154,419 122,571 79,438 86,350 66,836 63,582 64,092 62,541 56,568 53,493 38,674 48,449 41,947 49,057

64 67 54 37 33 32 30 32 29 23 22 21 20 19 20

62,237 60,975 68,059 62,560 80,813 63,025 61,519 67,331 74,018 66,613 59,445 67,598 54,767 57,295 50,889

993,065 958,948 936,488 934,238 932,566 916,196 892,471 866,144 776,515 768,759

47,749 44,495 47,397 42,066 40,990 47,392 43,189 42,740 34,314 36,750

16 22 17 17 20 24 19 18 18 19

63,224 62,919 77,994 50,563 69,043 87,956 55,224 77,483 64,667 51,955

34,665,869

1,559,169

693

64,842

Notes: Includes households in PUMAs with identified metropolitan areas; all computations, except sample size and PUMAs, use the PUMS sample weight. Source: 2000 5-Percent Public Use Microdata Sample, U.S. Bureau of the Census (tabulation by the authors).

New York-Northeastern, NJ Los Angeles-Long Beach, CA Chicago-Gary-Lake, IL Philadelphia, PA/NJ Washington, DC/MD/VA Detroit, MI Houston-Brazoria, TX Atlanta, GA Boston, MA Dallas-Fort Worth, TX Phoenix, AZ Minneapolis-St. Paul, MN Riverside-San Bernardino, CA St. Louis, MO/IL Tampa-St. PetersburgClearwater, FL San Diego, CA Baltimore, MD Orange County, CA Pittsburgh-Beaver Valley, PA Seattle-Everett, WA Nassau Co, NY Cleveland, OH Oakland, CA Denver-Boulder-Longmont, CO Miami-Hialeah, FL

Metropolitan Areas

0.462

0.454 0.438 0.446 0.462 0.436 0.435 0.451 0.451 0.432 0.509

0.535 0.496 0.456 0.462 0.428 0.444 0.470 0.440 0.471 0.459 0.439 0.415 0.437 0.440 0.455

Income inequality Gi

0.160

0.135 0.169 0.127 0.117 0.105 0.115 0.166 0.172 0.143 0.148

0.249 0.185 0.173 0.183 0.155 0.167 0.150 0.152 0.152 0.132 0.140 0.113 0.092 0.169 0.095

0.345

0.298 0.386 0.286 0.254 0.241 0.264 0.368 0.382 0.331 0.290

0.465 0.374 0.381 0.395 0.362 0.376 0.318 0.345 0.324 0.288 0.318 0.272 0.211 0.384 0.208

67,051

64,302 59,543 76,229 53,833 67,626 83,193 56,746 74,870 62,140 63,666

79,189 69,626 69,145 63,655 73,974 61,382 65,164 65,606 76,594 69,182 57,938 61,455 51,986 56,637 53,873

19,003

15,840 18,763 17,356 10,777 14,271 19,291 16,175 24,204 17,738 13,943

28,905 20,806 20,972 19,971 22,720 20,299 16,098 18,734 21,070 16,233 15,926 13,579 8,981 17,825 8,459

Neighbor-hood mean income

Individual household income

Neighborhood inequality G j

Segregation Gs

Standard Deviation ($)

Gini Coefficients

0.281

0.246 0.315 0.228 0.200 0.211 0.232 0.285 0.323 0.285 0.219

0.365 0.299 0.303 0.314 0.307 0.331 0.247 0.286 0.275 0.235 0.275 0.221 0.173 0.315 0.157

Neighborhood sorting index (NSI)

Gini Coefficients and Standard Deviations of Individual and Neighborhood Income Distributions, and Economic Segregation Measures, 25 Largest U.S. Metropolitan Areas, 2000.

Notes: Includes households in PUMAs with identified metropolitan areas; all computations, except sample size and PUMAs, use the PUMS sample weight. Source: 2000 5-Percent Public Use Microdata Sample, U.S. Bureau of the Census (tabulation by the authors).

Total

16 17 18 19 20 21 22 23 24 25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Table 2. 66 JEONGDAI KIM AND PAUL A. JARGOWSKY


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Gini Coefficient of Segregation

0.500 r = 0.9771 0.400

0.300

0.200

0.100

0.000 0.000

0.100

0.200

0.300

0.400

Neighborhood Sorting Index

Fig. 3.

Economic Segregation in Metropolitan Areas by PUMAs, 25 Largest Metropolitan Areas, 2000.

of PUMAs, number of sample households, weighted number of total households, and mean household income for the 25 largest metropolitan areas. Table 2 shows the individual, neighborhood, and income segregation Gini coefficients, as well as the standard deviations of the household and neighborhood income distributions and the NSI. Metropolitan areas vary little from each other in terms of income inequality, but have very different levels of sorting across neighborhoods. Gs and NSI rank metropolitan areas in a very similar fashion. Fig. 3 shows the near-perfect correspondence between the NSI and the Gs in the sample of 25 metropolitan areas. The Pearsonian correlation coefficient is 0.9771, and it is highly significant. Clearly, these measures are tapping into a common underlying phenomenon. However, the NSI2, as a function of squared deviations, is known to give greater weight to higher incomes than Gini, which does not overweight any segment of the distribution (James & Taeuber, 1985; White, 1986). This may be considered an advantage in certain applications, particularly those involving smaller, more heterogeneous neighborhood units. However, the optimal weighting depends on the researcher’s implicit social welfare function (Atkinson, 1970; Dalton, 1920).

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CONCLUSION The Gini Index of Neighborhood Inequality, G j, the application of the Gini index to neighborhood-level income, is not a measure of economic segregation. It is nothing more than the between-groups Gini described in the Gini decomposition literature. It reflects a mixture of information about the inequality of the income distribution and the ameliorative effects of neighborhood heterogeneity. Normalizing the neighborhood-level Gini by the individual-level Gini provides a pure measure of segregation and an alternative measure of economic segregation. In our application to household income in U.S. metropolitan areas, the two measures were highly correlated across 25 metropolitan areas. In general, the Gini Index of Segregation may be used with either continuous or dichotomous variables.

NOTE 1. PUMAs are bigger than census tracts, which more frequently served as neighborhood proxies (White, 1987). The 25 largest metropolitan areas contain 693 PUMAs. A few PUMAs on the fringes of metropolitan areas included territory in neighboring metropolitan areas or nonmetropolitan areas, and thus did not have a metropolitan area code. These PUMAs and the households they contain were excluded from the analysis. The average PUMA size was 50,000 households (weighted), based on an average sample size of 2,250.

ACKNOWLEDGMENTS An earlier version of this paper was presented at the conference ‘‘New Frontiers in the Field of Segregation Measurement and Analysis,’’ July 3–8, 2007, Monte Verita, Switzerland. We are grateful to Brian Berry, Marie Chevrier, Yves Flu¨ckiger, Sean Reardon, Jacques Silber, Rebecca Yang, and the conference participants for helpful comments and discussions. We also thank the Centre de Sciences Humaines in New Delhi, India, where Jargowsky was a visiting scholar in 2004 when an earlier version of this paper was written.

REFERENCES Anand, S. (1983). Inequality and poverty in Malaysia: Measurement and decomposition. Oxford: Oxford University Press.


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Atkinson, A. B. (1970). On the measurement of inequality. American Sociological Review, 43, 865–880. Brown, M. C. (1994). Using Gini-style indexes to evaluate the spatial patterns of health practitioners: Theoretical considerations and an application based on Alberta data. Social Science and Medicine, 38, 1243–1256. Dalton, H. (1920). Measurement of the inequality of income. The Economic Journal, 30, 348–361. Deutsch, J., Flu¨ckiger, Y., & Silber, J. (1994). Measuring occupational segregation. Journal of Econometrics, 61, 133–146. Duncan, O. D., & Duncan, B. (1955). A methodological analysis of segregation measures. American Sociological Review, 20, 210–217. Flu¨ckiger, Y., & Silber, J. (1990). Career choice and the measurement of segregation by sex in employment: The case of students in Switzerland. In: Modeling the Labor Market, Proceedings of the Meeting of the Applied Econometrics Association, Strasbourg (pp. 97–108). Gini, C. (1912). Variabilita e mutabilita. Reprinted in E. Pizetti & T. Salvemini (Eds). (1955). Memorie di metodologia statistica. Rome: Libreria Erendi Virgilio Veschi. Gini, C. (1921). Measurement of inequality of incomes. The Economic Journal, 31, 124–126. Hutchens, R. (2004). One measure of segregation. International Economic Review, 45, 555–578. Jahn, J., Schmid, C. F., & Schrag, C. (1947). The measurement of ecological segregation. American Sociological Review, 3, 293–303. James, D. R., & Taeuber, K. E. (1985). Measures of segregation. Sociological Methodology, 15, 1–32. Jargowsky, P. A. (1996). Take the money and run: Economic segregation in U.S. metropolitan areas. American Sociological Review, 61, 984–998. Jargowsky, P. A., & Kim, J. (2009). The information theory of segregation: Uniting segregation and inequality in a common framework. In: Y. Flu¨ckiger, S. Reardon, & J. Silber (Eds), Occupational and residential segregation (Vol. 17). UK: Emerald. Kendall, M. G., & Stuart, A. (1963). The advanced theory of statistics. Vol. 1. Distribution theory. , London: Griffin. Lorenz, M. O. (1905). Methods for measuring the concentration of wealth. American Statistical Association, 9, 209–219. Massey, D. S., & Denton, N. A. (1988). The dimensions of racial segregation. Social Forces, 67, 281–305. Massey, D. S., & Eggers, M. L. (1990). The ecology of inequality: Minorities and the concentration of poverty, 1970–1980. American Journal of Sociology, 95, 1153–1188. Milanovic, B. (2002). True world income distribution, 1988 and 1993: First calculation based on household surveys alone. The Economic Journal, 112, 51–92. Mussard, S. (2004). The bidimensional decomposition of the Gini ratio. A case study: Italy. Applied Economics Letters, 11, 503–505. Pyatt, G. (1976). On the interpretation and disaggregation of Gini coefficients. Economics Journal, 86, 243–255. Reardon, S. F., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32, 33–67. Ruggles, S., Sobek, M., Alexander, T., Fitch, C. A., Goeken, R., Hall, P. K., King, M., & Ronnander, C. (2004). Integrated public use microdata series: Version 3.0 (Machinereadable database). Minneapolis, MN: Minnesota Population Center. Accessed from http://www.ipums.org on November 11, 2004.

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Schwartz, W. B., Newhouse, J. P., Bennett, B. W., & Williams, A. P. (1980). The changing geographic distribution of board-certified physicians. New England Journal of Medicine, 303, 1032–1038. Silber, J. (1989). Factor components, population subgroups and the computation of the Gini index of inequality. The Review of Economics and Statistics, 71, 107–115. Theil, H. (1972). Statistical decomposition analysis. Amsterdam: North Holland. Townshend, I. J., & Walker, R. (2002). The structure of income residential segregation in Canadian metropolitan areas. Canadian Journal of Regional Science, 25, 25–52. White, M. J. (1986). Segregation and diversity measures in population distribution. Population Index, 52, 198–221. White, M. J. (1987). American neighborhoods and residential differentiation. New York: Russell Sage Foundation. Yao, S. (1999). On the decomposition of Gini coefficients by population class and income source: A spreadsheet approach and application. Applied Economics, 31, 1249–1264.

GENERALIZED GINI OCCUPATIONAL SEGREGATION INDICES Satya R. Chakravarty, Conchita D’Ambrosio and Jacques Silber ABSTRACT This article axiomatically derives a class of numerical indices of integration (equality) in the distribution of different types of workers across occupations. The associated segregation (inequality) indices parallel one form of multidimensional generalized Gini inequality indices. A comparison is made with the other Gini-related segregation indices. A numerical illustration of the family of indices is also provided using US occupational data.

1. INTRODUCTION Occupational segregation, that is, inequality in the distribution of types of workers across occupational groups, exists in almost all countries of the world. Examples of different types of workers can be White, Black, Asian, non-Asian, male, female, and so forth. Several reasons may be given for this


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SATYA R. CHAKRAVARTY ET AL.

phenomenon. But one of the most important reasons for this is unequal access to jobs. In contrast to segregation, integration refers to equality in the occupational distribution across groups (see Hutchens, 2004; Chakravarty & Silber, 2007). Complete or perfect integration occurs if employees are allocated to occupations in proportions to their shares in the population. Segregation arises if this condition does not hold. In other words, segregation refers to the comparison of actual distributions of the types of workers among occupations with the distribution that would arise if types had been allocated in proportions to their shares. Equivalently, segregation arises if and only if the ratio between the proportion of persons of any type in any occupation and the proportion of persons of the type considered is different from 1 (Reardon & Firebaugh, 2002). The increasing consciousness about occupational segregation has naturally raised the question of measuring segregation. Since the seminal contribution of Duncan and Duncan (1955), several numerical indices of employment segregation have been suggested in the literature (see Flu¨ckiger & Silber, 1999, for a survey; see also Hutchens, 1991, 2001). Now, segregation indices can give quite different results. Two indices may rank two occupations in different directions. In view of this, it becomes worthwhile to characterize alternative indices axiomatically for understanding which index becomes more appropriate in which situation. In other words, an axiomatic characterization gives us insight into the underlying index in a more elaborate way through the axioms employed in the characterization exercise. Equivalently, a characterization exercise enables us to identify a set of necessary and sufficient conditions for isolating a unique index of segregation in a specific framework. In a recent paper, Hutchens (2004) characterized an index of segregation, the square root index, using subgroup decomposability. According to subgroup decomposability, if we subdivide the occupational structure into, say, two subgroups 1 and 2, then the overall segregation can be decomposed into a component for subgroup 1, a component for subgroup 2, and a between-group component. This enables us to assess how the different components changed through time. Hutchens’s characterization relies on the concepts of aggregation and additive decomposability presented in the analysis of income inequality (Shorrocks, 1980, 1984). Following Duncan and Duncan (1955), Hutchens (1991) defined the segregation curve associated with the occupation-wise distribution of two types of workers as a plot of the cumulative proportions of type 1 people against the cumulative proportions of type 2 people, both proportions being

Generalized Gini Occupational Segregation Indices

73

ranked in increasing order of ratios between numbers of type 1 and type 2 employees in different occupations. The Gini segregation index, which is based on comparisons of gender ratios between various occupations, equals twice the area enclosed between the segregation curve and the diagonal line of equality (see Silber, 1989a; Hutchens, 1991; Flu¨ckiger & Silber, 1999). Following Donaldson and Weymark (1980), Deutsch and Silber (2004) developed a parametric generalization of the Gini segregation index. In a recent paper, Chakravarty and Silber (2007) provided a characterization of a parametric family of integration indices using a separability assumption, which shows how we can calculate overall integration when we subdivide the occupations into two subgroups, say, with respect to regions. The corresponding segregation indices parallel the multidimensional Atkinson (1970) indices of inequality considered by Tsui (1995). Two members of the family of segregation indices are monotonically related to the square root index and the Theil–Finizza (1967) logarithmic index (Theil, 1967).1 However, the Hutchens (2004) and Chakravarty–Silber (2007) families do not contain the Gini index of segregation or its parametric generalized form because the latter do not satisfy subgroup decomposability. Further, all these contributions concentrate only on two types of workers, for example, male and female. Segregation, however, should not be so narrowly focused. It might, for instance, be interesting to know whether in a given country segregation by race is more important than segregation by gender. There are, in fact, no normative or ethical reasons that justify that the measurement of segregation be limited only to two types. In most cases, particularly in the case of the Gini index or its parametric generalization, there has been little discussion on the extension of the index if there are more than two types of workers (see, however, Boisso, Hayes, Hirschberg, & Silber, 1994). Some of these contributions even make the restricted assumption that the number of workers of each type in each occupation is positive. Although the Gini index of segregation has been considered by several authors, no characterization of this index has been developed. Therefore, a separate characterization of the Gini index and/or its generalized form will be worthwhile. The purpose of this paper is to characterize a Gini kind of index of integration under the realistic assumptions that the number of types of workers is arbitrary and numbers of workers of one or more types can be zero in some occupations.2 We transport and adapt the method used for constructing multidimensional generalized Gini measure of welfare and inequality to the appraisal of integration index. The corresponding segregation index parallels one form of the multidimensional generalized

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Gini indices considered by Gajdos and Weymark (2005). However, the crucial difference between multidimensional welfare measurement and assessment of integration is that while in the former situation the variables are ratio scale or ordinally measurable, in the latter case the variables – namely, worker type and occupation – are purely categorical. This means that no particular meaning is assigned to the way of arrangement of occupations and types. The next section of the paper presents the formal framework and the characterization theorem. In Section 3 a comparison is made with other Gini-related segregation indices. In Section 4 we give a numerical illustration of several indices of segregation using US occupational data. Finally, Section 5 concludes.

2. FORMAL FRAMEWORK We consider a society with qZ2 types of people distributed over T>1 groups or occupations. If q ¼ 2, for concreteness, we may suppose that the two types of people are male and female. Let xij be the number of persons of type i in occupation j (i ¼ 1,2, y, q; j ¼ 1,2, y, T ). We may also regard xij as the number of hours spent by workers of type i in occupation j. Each xij is a nonnegative real number. Noninteger values of xij are possible because parttime workers and child labors can be regarded pffiffiffiffiffiffiffi as fractional workers. For example, a child labor can be counted as 0:5. Let X be the q T matrix of these xij values. We refer to X as an occupational distribution matrix. For any X, the number of type i people is denoted by ni (X ) (or simply by ni), that is, ni ¼ STj¼1 xij 40; i ¼ 1; 2; . . . ; q. For any X, let S(X), or simply S, be the matrix associated with X showing shares of employees of different types in different occupations. That is, sij, the (i, j)th entry of S, will be xij/ni (i ¼ 1,2, y, q; j ¼ 1,2, y, T ). Therefore, for all i ¼ 1,2, y, q, STj¼1 sij ¼ 1. We denote the set of all q T share matrices with nonnegative entries by M. Given any SAM, the jth column of S, s.j, represents the distribution of the shares of q types of persons in occupation j. For any q-coordinated vector u, let u be that permutation of u for which u1 u2 ::: uq . An index of employment segregation examines the inequality in the distribution of people across occupations. A dissegregation or integration index is concerned with the opposite feature; that is, it can be regarded as a measure of equality in the occupation-wise distribution of the population that exists in S between shares of different types of employees in different occupations. Under ceteris paribus assumptions, an increase in the value of a segregation index is equivalent to a reduction in integration and vice versa.


75

This correspondence is similar in nature to a relationship found in the literature on income inequality between an index of inequality and its associated equality or welfare index. An integration index I is a real valued function defined on M, that is, I:M-R1, where R1 is the real line. For any SAM, I(S) indicates the extent of integration that exists in S between q types of employees in different occupations. We now choose the following weighted sum form of integration index: IðSÞ ¼

q T X X

aij ðSÞsij

(1)

j¼1 i¼1

where aij (S)W0 is the weight on sij. It may be noted that at this stage there is no loss of generality since the weights are assumed to be a function of the matrix S, and no assumption of additive separability is implied by Eq. (1). Since the integration index given by Eq. (1) looks quite similar to one form of the multidimensional generalized Gini welfare function suggested by Gajdos and Weymark (2005) (their Eq. (30)), we can call it a generalized Gini integration index. A set of four axioms constraining the form I(S) in Eq. (1) further will now be proposed. These axioms will identify a particular form of the integration index in Eq. (1) uniquely. Equality of limits (EQL): For any S, SuAM, for any jA{1,2, y, T}, maxi aij (S) ¼ maxi aij (Su) and mini aij (S ) ¼ mini aij (Su ). Equality of weight gaps (EWG): For any S 2 M, for any jA{1,2, y, T}, aiþ1; j ðSÞ ¼ akj ðSÞ akþ1; j ðSÞ. aij ðSÞ Occupation-wise symmetry in types (OST): For any S 2 M; IðS 0 Þ ¼ IðSÞ, where S 0 ¼ ðG1 s:1 ; G2 s:2 ; :::; GT s:T Þ, each Gj being a q q permutation matrix.3 Before we proceed to state the fourth axiom, let us explain EQL, EWG, and OST. Weighted functions of the form (1) are used in different branches of social sciences. For instance, in welfare economics we often use the generalized utilitarian rule, which is a weighted sum of individual utility functions, as a social welfare function. EQL, which parallels an axiom of Sen (1974), demands that, given any occupation j, all q T share matrices have weights lying within the same range. Since each sij lies in the interval [0,1], the weights, which depend on the matrix S, are also assumed to belong to a finite interval. This assumption does not make the weights unnecessarily large. However, it does not say anything about the maximal and minimal values the weights should take. The axiom does not say that the same weight is assigned

76


to different occupations. It only makes the weights bounded within the same interval. This is a quite general and sensible assumption. EWG, which is a variant of Sen’s (1974) ordinal information axiom, makes the weighting system independent of the magnitude of the gaps between shares of employees of types i and i þ 1 in occupation j, given that in the occupation we have arranged the shares of different types of employees in nonincreasing order. That is, given that an occupation has been arranged in nonincreasing order of shares of types of employees, for any two consecutive types i and i þ 1, the excess of the weight on the share of type i over that on the share of type i þ 1 must be the same as the excess of the weight on type k’s share over that of type k þ 1. This implies linearity of the weights. This is an advantage from computational perspective. Apart from this, this axiom does not make any assumption about numerical values of weights. It thus makes the weights ‘‘equi-spaced’’ (see Sen, 1976). One may note the similarity of this axiom to the Borda rank-order voting scheme, as well as Borda’s argument for ‘‘choosing equal distances in the absence of a convincing case for any alternative assumption’’ (Sen, 1981, p. 36). It may be noted here that the weights employed in the well-known Gini index of equality satisfy this axiom. In fact, all segregation/integration indices, which use constant weights in the aggregation, for instance, the Hutchens square root index and Chakravarty– Silber generalized integration index, fulfill this axiom. OST makes the types anonymous in a particular way. This means that the integration index should remain invariant under reordering of the proportions of numbers of employees of different types in the occupations. For instance, if we have only two types, men and women, we want to make the index insensitive to whether women or men are labeled type 1 or 2 in any occupation. A variant of this postulate was introduced by Chakravarty and Silber (1994, 2007) (see also Kakwani, 1994; Hutchens, 1991, 2004). An advantage of OST is that it enables us to rank the occupations in an unambiguous way. The fourth postulate, which was considered by Hutchens (1991, 2004), is concerned with movement between occupations. To understand this postulate, let us consider the following example:

Women Men

Occupation 1 2 3 3 2 6 8 4

(2)

4

Since occupation 2 contains two women and four men, and occupation 3 contains six women and four men, we have s22 ¼ s23 ¼ 1=4;


77

s12 =s22 ¼ 8=11o24=11 ¼ s13 =s23 . Now, if one woman moves from occupation 2 to occupation 3 (so that occupation 2 contains one woman and occupation 3 has seven women), then the occupation-wise distribution of women and men becomes more concentrated. This is because the movement increases the percentage of females in female-dominated occupation 3 and the percentage of males in the male-dominated occupation 2. Hence, segregation should not decrease and integration should not increase. Likewise, integration should not decrease under an equalizing movement. More formally, for any SAM, we say that SuAM is obtained from S by a disequalizing movement of type i people if, for l and j, (i) skl ¼ skj ¼ s0kl ¼ s0kj 40, (ii) ðsil =skl Þoðsij =skj Þ, (iii) s0il ¼ sil d and sij ¼ s0ij þ d for 0od sil , and (iv) shr ¼ s0hr ; h ¼ 1; 2; . . . ; q; kal; j. This then leads to our next axiom: Movement between occupations (MBO): For any SAM if SuAM is obtained by a disequalizing movement from S, then I(Su)rI(S). We can state the following theorem: Theorem 1. The only integration index of the form (1) satisfying axioms EQL, EWG, OST, and MBO is given by IðSÞ ¼

q T X X

ðaj þ ði 1Þtj Þsij

(3)

j¼1 i¼1

ai;j ðSÞ. where SAM is arbitrary, aj ¼ mini aij(S), and tj ¼ aiþ1; j ðSÞ Proof. Since I satisfies OST, we can rank the rows of the matrix S in nonincreasing order of proportions of different types of employees in all occupations j. More precisely, for any j, we can restrict the attention on the permutation s:j of s.j, that is, s1j s2j sqj . We can therefore sij . For any SAM and rewrite I(S) in Eq. (1) as IðSÞ ¼ STj¼1 Sqi¼1 aij ðSÞ given type j, we have by axiom EQL, mini aij ðSÞ ¼ aj and let a2j ðSÞ maxi aij ðSÞ ¼ bj . MBO implies that for any type j, a1j ðSÞ aqj ðSÞ (see Sen, 1974). By axiom EWG, for all types j and occupations i, i þ 1, and i þ 2, ¼ aiþ1; j ðSÞ ai; j ðSÞ ¼ aiþ2; j ðSÞ aiþ1; j ðSÞ tj ðSÞ

(4)

¼ aj and where 1 i q 2. By Eq. (4) and axiom EQL, we get a1j ðSÞ aqj ðSÞ ¼ aj þ ðq 1Þtj ðSÞ ¼ bj . Hence, tj ðSÞ ¼ ððbj aj Þ=ðq 1ÞÞ. Since aj is also independent of S. This in turn and bj are independent of S, tj ðSÞ

78


on S. Therefore, for any given occupation shows independence of aij ðSÞ ¼ aj þ ði 1Þtj . share matrix S, type j and for all occupations i, aij ðSÞ Substituting this specification of aij in Eq. (1), we get the desired form of the integration index. This establishes the necessity part of the theorem. The sufficiency is easy to check. Theorem 1 identifies a specific case of the general class (1) by imposing restrictions on the weight sequence, which follow from the axioms. Essentially in Eq. (3), for each occupation, we employ a Gini kind of aggregation across types of proportions of workers in that occupation and then these occupation-wise Gini values are aggregated to arrive at an index of integration. We now show that the four axioms we have considered in Theorem 1 are independent. Independence means that none of the postulates implies or is implied by another; that is, none of them is redundant. It is thus a minimal condition. Therefore, if one of the postulates is dropped, there will be indices that satisfy the remaining postulates but not the dropped one. Theorem 2. Axioms EQL, EWG, OST, and MBO are independent. Proof. (i) The index I 1 ðSÞ ¼ STj¼1 Sqi¼1 ðq STj¼1 Sqi¼1 s2ij Þsij satisfies all the axioms except EQL because of explicit dependence of the weights on the sij values. (ii) The index I 2 ðSÞ ¼ STj¼1 Sqi¼1 ði2 Þsij is a violator of EWG since for a given j, the weight appears as a nonlinear function of i. However, the remaining axioms are satisfied. (iii) Since the index I 3 ðSÞ ¼ STj¼1 Sqi¼1 ðn þ 1 iÞsij has decreasing weight sequence for a given j, it does not satisfy MBO, but verifies the other axioms. pffiffiffiffiffi pffiffiffiffiffi (v) I 4 ðSÞ ¼ d1 STj¼1 Sqi¼1 sij þ d2 STj¼1 Sqi¼1 sij , where d1 and d2 are two different positive numbers, violates OST since d1 6¼ d2. But it fulfills all the other three axioms. In addition to satisfying the above axioms, the generalized Gini integration index of type (3) satisfies the following axioms: Symmetry in occupations (SYO): For any SAM, if Su is obtained from S by a permutation of the columns of S, then I(S) ¼ I(Su). Continuity (CON): I is continuous on M.


79

OST makes the types anonymous in a particular way. Likewise, the postulate SYO makes the occupations anonymous. This means that the integration index should remain invariant under reordering of the columns of the occupation matrix. This postulate was introduced by Hutchens (1991, 2004). The two notions of symmetry demonstrate the difference between the measurement of welfare and segregation. While in the former, there is anonymity among individuals only, in this case we have anonymity in both occupations and types. Since no particular meaning is attached to the ways types and occupations are arranged, the variables considered in the case of segregation are purely categorical in contrast to the ratio scale or ordinally measurable variables employed in the case of welfare comparisons. CON ensures that minor observational errors on the number of workers will not make the index oversensitive. To derive the segregation index corresponding to the integration index given by Eq. (3), let Se be the matrix associated with S whose (i,j)th entry is given by 1/T. That is, in Se for each type i (i ¼ 1,2), the employees are equally distributed across occupations and occupations are of equal size. Further, complete integration (or no segregation) occurs in Se because the ratio of women to men is the same in all occupations (see Hutchens, 2004; Chakravarty & Silber, 2007). We define the scale factor y(S) implicitly as follows: IðSÞ ¼ IðyðSÞSe Þ

(5)

The factor y(S) is formally similar to Debreu’s (1951) coefficient of resource utilization. It has the same flavor as the equally distributed equivalent income, which is used as an index of equality in the income distribution literature (see also Kolm, 1977). Then as an index of employment segregation, we suggest the use of KðSÞ ¼ 1

yðSÞ T

(6)

The segregation index in Eq. (6) can be interpreted as the proportion of workers of each type that could be taken out of the labor force (occupations) if employees were equally distributed across occupations when they are of equal size and the resulting occupational distribution of workers of different types is indifferent to the original distribution according to the integration index I.

80


For the form of I given by Eq. (3), the segregation index defined by Eq. (6) turns out to be T X X 1 ðaj þ ði 1Þtj Þsij STj¼1 Sqi¼1 ðaj þ ði 1Þtj Þ j¼1 i¼1 q

KðSÞ ¼ 1

(7)

Each of the postulates considered above for an index of integration will have its segregation index counterpart. We also assume that a segregation index should be scale invariant, where scale invariance demands that if one or more of nius are multiplied by a positive scalar and the shares of different types of people in all the T occupations remain the same, then segregation does not change. For example, suppose that there be only two types, female and male. Suppose also that the number of women in the labor force triples because of a three-fold occupation-by-occupation replication of the female labor force. Likewise, assume that the number of men doubles in the labor force under an analogous replication. Then according to scale invariance, such changes in the numbers of female and male in the labor force do not change the level of segregation. Thus, if the types have different ratio scales, rescaling them should not alter the level of segregation. Since we have defined the integration index directly on share matrices, this property is fulfilled. The index K given by Eq. (7) is continuous, bounded between 0 and 1, where the lower bound is achieved if there is no segregation at all. Further, it is symmetric in occupations and types and nondecreasing under a disequalizing movement. Reardon and Firebaugh (2002) defined the Gini index of multigroup segregation as the average of the difference between the ratios that reflect the extent to which different types are disproportionately represented in different occupations. Since our index is based on a weight sequence, it has a more flexible formulation than the Reardon–Firebaugh (2002) index. We now give an example to illustrate the formula in Eq. (7). Let for any given j, aij ¼ (2i 1). For each j, this sequence is nondecreasing and it satisfies EWG. The corresponding segregation index will be T X X 1 ð2i 1Þsij STj¼1 Sqi¼1 ð2i 1Þ j¼1 i¼1 q

I CDS ¼ 1

(8)

Note that in the aggregation in Eq. (8), although in each occupation, higher weights are assigned to types of workers with lower proportions, equal weight ( ¼ 1) is assigned to each occupation. However, one may


81

attach different weights to different occupations depending on their importance. Recall that for any income distribution z over a population of size n, the Gini index is defined as n 1X ð2i 1Þei (9) GðzÞ ¼ 1 n i¼1 where e ¼ ðe1 ; e2 ; . . . ; en Þ is the income share vector corresponding to z. Therefore, the segregation index in Eq. (7) can be regarded as a fairly natural translation of the multidimensional generalized Gini index to the measurement of segregation. Clearly, given a sequence of weights of the type specified in Eq. (3), we have a corresponding index of segregation. These indices will differ only in the way we define the weight sequence.

3. A COMPARISON WITH THE OTHER GINI-RELATED SEGREGATION INDICES Several other forms of Gini-type segregation indices have been suggested in the literature. It may be worthwhile to make a comparison of these indices with our index. We concentrate our analysis on the multigroup (e.g., ethnic groups), multicategories (e.g., occupations) case.

3.1. The Generalized Gini Index As indicated in Deutsch, Flu¨ckiger, and Silber (2009), this index may be expressed as I GG ¼

q X T h n m i h x i X i j ij ...; G ...; ;... n n n i¼1 j¼1

(10)

where mj is the total number of workers in occupation j. Clearly, n ¼ Sqi¼1 ni ¼ STj¼1 mj . The expression ½. . . ; ðni =nÞðmj =nÞ; . . . is then a row vector of (q T ) ‘‘a priori’’ shares, while the expression ½. . . ; ðxij =nÞ; . . . is a (q T ) column vector of ‘‘a posteriori’’ shares, where ‘‘share’’ refers to the probability to belong to category i and to work in occupation j. Finally, G is a (q T ) by (q T ) square matrix called G-matrix whose typical element gij is equal to 0 if i ¼ j, 1 if jWi, and þ1 if iWj (see Silber, 1989b, for more

82


details on an algorithm for computing the Gini index that is based on the concept of G-matrix). Note that the elements in both the vectors have to be ranked by decreasing ratios ðxij Þ=ðni mj Þ. This index was first proposed in Boisso et al. (1994).

3.2. Reardon and Firebaugh’s (2002) Formulation for the Multigroup Gini Index Reardon and Firebaugh’s formulation, using our notation and ignoring their normalization factor (so that we limit ourselves to the numerator of their expression), is as follows: I RF ¼

q T X T X ni X mh mk i¼1

where

n

h¼1 k¼1

n n

fg

ðxih =mh Þ ðxik =mk Þ ðxih =ni Þ ðxik =ni Þ ¼ fg ¼ ðni =nÞ ðni =nÞ ðmh =nÞ ðmk =nÞ

But IRF can be rewritten as q X ni h mj i0 xij I RF ¼ G n n ni i¼1

(11)

(12)

(13)

where G, as before, is a G-matrix and the elements of the vectors on both sides of the matrix are ranked by decreasing values of the ratios ðxij =ni Þ=ðmj =nÞ. Graph 1 is a representation of both the Generalized Gini index and the Gini index proposed by Reardon and Firebaugh (2002). The area between the curve OBA and the diagonal is equal to half the value of the Generalized Gini index IGG, while the sum of the areas lying between the sequence of curves OMNA and the diagonal is equal to half the value of the Gini index IRF proposed by Reardon and Firebaugh (2002).4 Note that one may think of another Gini index I SRF , which would be a ‘‘symmetric variant’’ of the index proposed by Reardon and Firebaugh (2002). This Gini segregation index I SRF would be defined as T h n i0 x X mj i ij S (14) I RF ¼ G n n m j j¼1

83

Generalized Gini Occupational Segregation Indices Graph 1: The Generalized Gini Approach and the Reardon-Firebaugh Approach

Cumulative values of (xij /n)

A

1

N (n1 /n) + (n2 /n) M

B

(n1/n) 1 O

Graph 1.

(n1/n)

(n1/n) + (n2/n)

Cumulative values of (ni /n)(mj /n)

The Generalized Gini Approach and the Reardon–Firebaugh Approach.

where the elements of the vectors on both sides of the G-matrix in Eq. (14) are ranked by decreasing values of the ratios ðxij =mj Þ=ðni =nÞ. Graph 2 gives a representation of such an index.

3.3. The Formulation Proposed in the Present Paper We can rewrite ICDS alternatively as I CDS

0 T X 1 1 xij ¼ G mj T q j¼1

(15)

In Eq. (15) the elements of the vector ½ ðxij =mj Þ on the RHS are ranked by decreasing values of the ratios ðxij =mj Þ. Graph 3 gives a representation of this index. Note that here points like Mu, Nu, Pu, and Qu, which define the broken curve OMuNuPuQuC, do not lie on the diagonal because the proportions ðxij =mj Þ are compared not with ðni =nÞ but with ð1=qÞ.

84

SATYA R. CHAKRAVARTY ET AL. Graph 2: The Symmetric Approach to the ReardonFirebaugh Approach

Cumulative values of (xij /n)

A 1

N (m1/n) + (m2/n) B M (m1/n)

1 O

(m1/n)

(m1/n) + (m2/n) Cumulative values of (ni /n)(mj/n)

Graph 2.

The Symmetric Approach to the Reardon–Firebaugh Approach.

Graph 3: The Chakravarty - D′Ambrosio - Silber Approach C

Cumulative 1 values of xij /n

Q′

P′

B

N′ M′

A O

1/qT

Graph 3.

2/T q/qT = 1/T

3/T

1

Cumulative values of 1/qT

The Chakravarty–D’Ambrosio–Silber Approach.

85


In Graph 3, OA and OB are defined as Sqi¼1 ðxi1 =ni Þ Sqi¼1 ðxi1 =ni Þ ¼ q STj¼1 Sqi¼1 ðxij =ni Þ

(16)

Sqi¼1 ððxi1 =ni Þ þ ðxi2 =ni ÞÞ ½Sqi¼1 ððxi1 =ni Þ þ ðxi2 =ni ÞÞ ¼ q STj¼1 Sqi¼1 ðxij =ni Þ

(17)

OA ¼

OB ¼

3.4. The Weighted Version of the Chakravarty–D’Ambrosio–Silber Index The unweighted Gini index in Eq. (8) may be expressed as I uCDS ¼ 1

ðSTj¼1 Sqi¼1 ð2i 1Þ=Sqi¼1 ð2i 1ÞÞsij T P ð1Þ

(18)

j¼1

which we rewrite in a more compact form as I uCDS ¼ 1

STj¼1 sej T

(19)

Therefore, the weighted version, I wCDS , is defined as I wCDS ¼ 1

q T X mj X j¼1

n

ð2i 1Þ sij 1Þ

Sq ð2i i¼1 i¼1

2I wCDS ¼ 1

T X mj j¼1

n

sej

(20)

(21)

where

q X ½ð2i 1Þsij sej ¼ ½Sqi¼1 ð2i 1Þ i¼1

(22)

Note the parallelism between the index I uCDS and Hutchens’s square root index H where the index H is defined as: H ¼1

T T 1X 1X s~ej ðs1j s2j Þ1=2 ¼ 1 T j¼1 T j¼1

(23)

86


Observe that in Eq. (23) s~ej is defined using the geometric averaging principle. In contrast, in Eq. (22) we use the Gini-type averaging to define sej . One major difference between the generalized Gini index suggested in this paper and other forms of Gini indices mentioned previously is that the former corresponds to an axiomatically derived class of integration indices. Under ceteris paribus assumptions, an increase in integration is regarded as a reduction in segregation and vice versa.

4. AN EMPIRICAL ILLUSTRATION The data sources cover 8 occupations, 5 ethnic groups, and the 50 states of the USA plus the District of Columbia (see Appendix C for the list of states and their symbols). The data set used is the 5-percent IPUMS from the 1990 Census. A simple classification into eight categories was used (see Appendix D). Five ethnic groups were distinguished: (i) White; (ii) Black; (iii) American Indian, Eskimo, or Aleutian; (iv) Asian or Pacific Islander; and (v) other. The last category includes any other race except the four mentioned. Note that the 1990 Census does not identify Hispanic as a separate racial category. The following segregation indices were computed: the unweighted Gini index introduced in the present paper (I U CDS ); the weighted Gini index ); the Reardon and Firebaugh (2002) introduced in this paper (I W CDS Gini index of multigroup segregation(IRF); the generalization (see Silber, 1992, and Appendix B) of the index proposed by Karmel and MacLachlan (1988), which turns out to be a generalized Duncan index (henceforth, IDG); and finally, the generalized Gini index (IGG) proposed originally by Boisso et al. (1994). The value of these five segregation indices for each of the states in the USA is given in Table 1 for the year 1990. In Table 2 we report their ranks. The coefficients of correlation among the indices are given in Table 3. The correlation values are very high, ranging from 0.885 (between the indices IU CDS and IDG) to 0.999 (between the indices IGG and IRF). According to all the indices, but the index I W CDS (which puts Hawaii in the first position and the District of Columbia in the second position of the ranking), the most segregated state is the District of Columbia, followed by Mississippi. At the opposite end we have states like Vermont,

87


Table 1.

Segregation by Race in the United States in 1990.

State

Unweighted Gini Index (I U CDS )

Weighted Gini Index (I W CDS )

Generalized Duncan Index (IDG)

Generalized Gini Index (IGG)

Reardon and Firebaugh Index (IRF)

AK AL AR AZ CA CO CT DC DE FL GA HI IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD

0.83113 0.83119 0.82185 0.82974 0.8298 0.81278 0.81548 0.84128 0.82142 0.81986 0.82935 0.83702 0.80572 0.8075 0.83202 0.81959 0.8193 0.81497 0.83651 0.81351 0.82612 0.8064 0.82628 0.80992 0.81988 0.83894 0.8114 0.82701 0.81015 0.8162 0.80308 0.82107 0.82065 0.82166 0.82742 0.81966 0.82253 0.80838 0.81871 0.81395 0.83458 0.81806

0.4547 0.46426 0.33033 0.40735 0.54315 0.2488 0.26615 0.58819 0.37764 0.35891 0.49544 0.61272 0.074965 0.11548 0.43192 0.23942 0.24637 0.20384 0.53329 0.2195 0.49226 0.062717 0.3498 0.12647 0.27621 0.55059 0.15597 0.4487 0.11682 0.17981 0.043708 0.40442 0.41956 0.34425 0.46715 0.27095 0.36643 0.15205 0.25984 0.2006 0.51958 0.18224

0.15042 0.18021 0.10549 0.1188 0.15412 0.070421 0.080464 0.34208 0.1428 0.12499 0.1827 0.13903 0.016381 0.048824 0.10643 0.044551 0.057146 0.049875 0.21253 0.050202 0.1276 0.0061961 0.080081 0.023203 0.067074 0.26231 0.044409 0.16893 0.029439 0.03276 0.0074084 0.11944 0.11774 0.12172 0.11139 0.05861 0.091299 0.041401 0.051242 0.064697 0.23203 0.044464

0.10944 0.12289 0.077123 0.088877 0.11683 0.055286 0.061521 0.22794 0.10278 0.088938 0.12196 0.10072 0.014289 0.038286 0.081806 0.03665 0.045898 0.039343 0.13967 0.038661 0.096802 0.0055206 0.061082 0.018489 0.053559 0.16587 0.033883 0.11532 0.0238 0.027852 0.0063287 0.088707 0.085609 0.089296 0.086289 0.04724 0.068242 0.032331 0.041909 0.051151 0.15169 0.035135

0.093059 0.10473 0.061832 0.074714 0.10037 0.045115 0.05129 0.2065 0.085731 0.071671 0.10577 0.093721 0.011256 0.030328 0.07026 0.030887 0.037545 0.031682 0.1218 0.031327 0.086696 0.0044085 0.050904 0.015024 0.045899 0.14601 0.025776 0.096853 0.018772 0.022418 0.0050127 0.073856 0.074024 0.072511 0.077777 0.040033 0.056198 0.026183 0.036135 0.040735 0.13171 0.027672

88


Table 1. (Continued ) State






TN TX UT VA VT WA WI WV WY

0.82216 0.82473 0.80925 0.82451 0.80219 0.81575 0.81781 0.80644 0.80717

0.34201 0.46289 0.14501 0.43405 0.035608 0.25768 0.19069 0.095537 0.11345

0.099163 0.15523 0.043608 0.14477 0.0071787 0.06012 0.046471 0.018602 0.035513

0.072935 0.11343 0.033461 0.10479 0.006436 0.046384 0.036455 0.01592 0.029146

0.06047 0.096593 0.026237 0.089941 0.0048622 0.038516 0.028897 0.012325 0.023452

Note: see Appendix C for the list of states and the symbols used to represent them.

New Hampshire, Iowa, Maine, West Virginia, and Wyoming. Here, among the least segregated states, the differences in the ranks between the indices are not so pronounced as those we observe at the opposite end of the scale, where higher disagreement is registered. There are nevertheless some discrepancies. For example, the state with the third highest segregation rank according to the index I U CDS , Hawaii, has a much lower ranking according to the other indices (10th position according to the index IRF and 13th according to the other indices).

5. CONCLUSIONS Inequality in the distribution of employees in occupations by sex is an important issue in labor market analysis. An indicator of this type of inequality is called a segregation index. In this paper we have characterized a family of integration (equality) indices of the distribution of different types of workers across occupations using a set of intuitive axioms, assuming that the number of types of workers is arbitrary and the numbers of one or more types of workers may be zero in some occupations. The corresponding family of segregation indices parallels one form of the multidimensional generalized Gini inequality indices. The paper therefore provides a simple translation of the multidimensional generalized Gini inequality indices into the measurement of segregation.

89


Table 2.

Ranks of the Segregation Indices.

Rank

State






1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

DC MS HI LA SC IL AL AK CA AZ GA NY NC MI MD TX VA OK TN AR NV DE NJ NM MO FL OH IN KS PA SD WI NE WA CT KY RI MA CO MT ND MN

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

2 3 1 5 6 15 10 12 4 17 7 9 13 22 8 11 14 20 24 25 23 19 18 16 26 21 27 33 32 29 38 37 39 30 28 35 36 34 31 40 44 43

1 2 13 4 3 21 6 10 9 18 5 20 7 26 14 8 11 24 23 22 16 12 17 19 28 15 31 38 32 33 39 37 44 30 25 35 29 34 27 40 45 46

1 2 13 4 3 21 5 10 7 17 6 19 8 26 14 9 11 24 23 22 15 12 18 20 28 16 30 37 32 33 39 38 44 31 25 34 29 35 27 40 45 46

1 2 10 4 3 21 6 11 7 16 5 15 8 26 13 9 12 24 23 22 19 14 18 17 27 20 30 36 32 33 39 38 44 31 25 34 29 35 28 42 45 46

90


Table 2. (Continued ) Rank

State






43 44 45 46 47 48 49 50 51

UT OR ID WY WV ME IA NH VT

43 44 45 46 47 48 49 50 51

42 41 45 46 47 49 48 50 51

41 42 36 43 47 51 48 49 50

41 42 36 43 47 51 48 50 49

40 41 37 43 47 51 48 49 50

Table 3.

Correlation Matrix between the Various Segregation Indices. Weighted Generalized Generalized Gini Index Gini Index Duncan Index (IDG) (IGG) (I W CDS )

Unweighted Gini index ðI U CDS Þ Weighted Gini index (I W CDS ) Generalized Duncan index (IDG) Generalized Gini index (IGG)

0.956

0.885 0.895

0.900 0.918 0.996

Reardon and Firebaugh Index (IRF) 0.902 0.918 0.996 0.999

NOTES 1. Mora and Ruiz-Castillo (2003) and Frankel and Volij (2006) provide further discussion on the Theil–Finizza index. 2. Note that the focus of this paper is on occupational segregation and not on residential segregation. For a general derivation and evaluation of measures of multigroup segregation, see Reardon and Firebaugh (2002). 3. A q q matrix with entries 0 and 1 is called a permutation matrix if each of its rows and columns sums to 1. 4. The denominator of the formulation of the Gini index proposed by Firebaugh and Reardon (2002) corresponds to the maximal value of the sum of the areas lying between the sequence of curves OMNA and the diagonal. We believe, however, that using the non-normalized version of the formulation of the Gini index proposed by Reardon and Firebaugh (2002) makes easier the comparison with other formulations of the Gini index of segregation.


91

REFERENCES Atkinson, A. B. (1970). On the measurement of inequality. Journal of Economic Theory, 2(3), 244–263. Boisso, D., Hayes, K., Hirschberg, J., & Silber, J. (1994). Occupational segregation in the multidimensional case: Decomposition and tests of statistical significance. Journal of Econometrics, 61(1), 161–171. Chakravarty, S. R., & Silber, J. (1994). Employment segregation indices: An axiomatic characterization. In: W. Eichhorn (Ed.), Models and measurement of welfare and inequality (pp. 912–920). New York: Springer-Verlag. Chakravarty, S. R., & Silber, J. (2007). A generalized index of employment segregation. Mathematical Social Sciences, 53(2), 185–195. Debreu, G. (1951). The coefficient of resource utilization. Econometrica, 19(3), 273–292. Deutsch, J., Flu¨ckiger, Y., & Silber, J. (2009). Analyzing changes in occupational segregation: The case of Switzerland (1970–2000). In: Y. Flu¨ckiger, S. F. Reardon & J. Silber (Eds), Occupational and residential segregation (Vol. 17). UK: Emerald. Deutsch, J., & Silber, J. (2004). Comparing segregation by gender in the labor force across ten European countries in the 1990s: An analysis based on the use of normative segregation indices. International Journal of Manpower, 26(3), 237–264. Donaldson, D., & Weymark, J. (1980). A single parameter generalization of the Gini indices of inequality. Journal of Economic Theory, 22(1), 67–80. Duncan, O. D., & Duncan, B. (1955). A methodological analysis of segregation indices. American Sociological Review, 20(2), 210–217. Frankel, D., & Volij, O. (2006). Measuring segregation. Mimeographed, Iowa State University, USA, and Ben-Gurion University, Israel. Flu¨ckiger, Y., & Silber, J. (1999). The measurement of segregation in the labor force. New York: Physica-Verlag. Gajdos, T., & Weymark, J. (2005). Multidimensional generalized Gini indices. Economic Theory, 26(3), 471–496. Hutchens, R. M. (1991). Segregation curves, Lorenz curves and inequality in the distribution of people across occupations. Mathematical Social Sciences, 21(1), 31–51. Hutchens, R. M. (2001). Numerical measures of segregation: Desirable properties and their implications. Mathematical Social Sciences, 42(1), 13–29. Hutchens, R. M. (2004). One measure of segregation. International Economic Review, 45(2), 555–578. Karmel, T., & MacLachlan, M. (1988). Occupational sex segregation — increasing or decreasing? Economic Record, 64(186), 187–195. Kakwani, N. C. (1994). Segregation by sex: Measurement and hypothesis testing. In: S. Neuman & J. Silber (Eds), Research on economic inequality (pp. 1–26). Greenwich: JAI Press. Kolm, S. C. (1977). Multidimensional egalitarianisms. Quarterly Journal of Economics, 91(1), 1–13. Mora, R., & Ruiz-Castillo, J. (2003). Additively decomposable segregation indexes: The case of gender segregation by occupations and human capital levels in Spain. Journal of Economic Inequality, 1(2), 147–179. Reardon, S. F., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32(1), 33–67.

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Sen, A. K. (1974). Information bases of alternative welfare approaches. Journal of Public Economics, 3(4), 387–403. Sen, A. K. (1976). Poverty: An ordinal approach to measurement. Econometrica, 44(2), 219–231. Sen, A. K. (1981). Poverty and famines: An essay on entitlement and deprivation. Oxford: Clarendon. Shorrocks, A. F. (1980). The class of additively decomposable inequality measures. Econometrica, 44(3), 613–625. Shorrocks, A. F. (1984). Inequality decomposition by population subgroups. Econometrica, 48(6), 1369–1386. Silber, J. (1989a). On the measurement of employment segregation. Economics Letters, 30(3), 237–243. Silber, J. (1989b). Factors components, population subgroups and the computation of the Gini index of inequality. Review of Economics and Statistics, 71(2), 107–115. Silber, J. (1992). Occupational segregation indices in the multidimensional case: A note. Economic Record, 68(202), 276–277. Theil, H. (1967). Economics and information theory. Amsterdam: North-Holland. Tsui, K. Y. (1995). Multidimensional generalizations of the relative and absolute inequality indices: The Atkinson–Kolm–Sen approach. Journal of Economic Theory, 67(1), 251–265.

APPENDIX A. COMPARING THE REARDON AND FIREBAUGH (2002) GINI INDEX OF MULTIGROUP SEGREGATION AND THE GINI INDEX DERIVED IN THIS PAPER According to the notion of segregation considered in this paper, perfect integration occurs if and only if xqh x1h x2h ¼ ¼ ¼ n1 n2 nq

(A.1)

for all h ¼ 1,2, y, T. The Reardon and Firebaugh (2002) disproportionality axiom for segregation says that segregation is zero, that is, integration is perfect, when and only when ðxih = ðni =

q P j¼1 q P

xjh Þ ¼ nj Þ

j¼1

for all i ¼ 1,2, y, q and h ¼ 1,2, y, T.

pih ¼1 pi

(A.2)

93


According to Eq. (A.1), perfect integration requires equality of proportions of different types of workers across occupations. The numerator in Eq. (A.2) is the proportion of workers in occupation h who belong to type i. The denominator is the proportion of type i workers. The ratio pih/pi reflects the extent to which type i is disproportionally represented in occupation h. If this ratio is greater than 1, type i is overrepresented, and a value less than 1 indicates underrepresentation of type i. We now establish that Eqs. (A.1) and (A.2) are equivalent. Proposition 1. Conditions (A.1) and (A.2) are equivalent. Proof. We assume for simplicity that q ¼ 2. Then from Eq. (A.1) we have x1h/x2h ¼ n1/n2, which gives x1h =ðx1h þ x2h Þ ¼ n1 =ðn1 þ n2 Þ. This implies that p1h/p1 ¼ 1. We can similarly show that p2h/p2 ¼ 1. Hence, (A.1) implies (A.2). By an analogous argument we can show that (A.2) implies (A.1). This completes the proof of the proposition.

APPENDIX B. DEFINITION OF THE KARMEL AND MACLACHLAN SEGREGATION INDEX (ALSO CALLED ‘‘GENERALIZED DUNCAN INDEX’’) The generalized Duncan index (IDG) can be defined (see Deutsch et al., 2007), using the notation of the present paper, as I DG ¼

q X T n m X xij i j n n n i¼1 j¼1

APPENDIX C. LIST OF THE STATES AND THE SYMBOLS USED TO REPRESENT THEM Symbols used for the states: AK AL AR AZ CA

Alaska Alabama Arkansas Arizona California

(B.1)

94

CO CT DC DE FL GA HI IA ID IL IN KS KY LA MA MD ME MI MN MO MS MT NC ND NE NH NJ NM NV NY OH OK OR PA RI SC SD TN TX UT


Colorado Connecticut District of Columbia Delaware Florida Georgia Hawaii Iowa Idaho Illinois Indiana Kansas Kentucky Louisiana Massachusetts Maryland Maine Michigan Minnesota Missouri Mississippi Montana North Carolina North Dakota Nebraska New Hampshire New Jersey New Mexico Nevada New York Ohio Oklahoma Oregon Pennsylvania Rhode Island South Carolina South Dakota Tennessee Texas Utah


VA VT WA WI WV WY

95

Virginia Vermont Washington Wisconsin West Virginia Wyoming

APPENDIX D. LIST OF OCCUPATIONS Code of the Occupation 1 2 3 4 5 6 7 8

Label of the Occupation

Farming, forestry, and fishing occupations Managerial and professional specialty occupations Military occupations Operators, fabricators, and laborers Precision production, craft, and repair occupations Service occupations Technical, sales, and administrative support occupations Unemployed since 1984

PART III MEASURING SEGREGATION WITH ORDERED CATEGORIES

OCCUPATIONAL SEGREGATION WITH ECONOMIC DISADVANTAGE: AN INVESTIGATION OF DECOMPOSABLE INDEXES Robert Hutchens ABSTRACT Purpose – This paper considers methods for decomposing indexes that incorporate economic disadvantage into a measure of segregation. According to such indexes, segregation in high-economic-status occupations is worse than similar segregation in low-economic-status occupations. The paper presents three decompositions of these indexes. Methodology/Approach – The paper first characterizes a class of segregation indexes that include economic disadvantage in the index. It then develops mathematical methods for decomposing a change in such an index. The change is decomposed into two or more components: components that indicate either the effect of changes in economic disadvantage or the effect of changes in a standard measure of segregation – a measure that essentially ignores economic disadvantage. The paper then implements the decompositions using data on U.S. occupational segregation by gender between 1970 and 2000.


99

100

ROBERT HUTCHENS

Findings – The primary finding is that a segregation index that incorporate economic disadvantage can be decomposed in interesting ways. A secondary finding is that such indexes indicate reduced segregation between 1970 and 2000. The dominant forces associated with the reduction were (a) the convergence of occupational gender ratios and (b) the movement of women out of less advantaged occupations and into the comparatively well-compensated professional and managerial occupations. Research limitations/Implications – The 1970–2000 results are mainly illustrative. They are based on three broad occupational categories for which there were compatible earnings data, and the analysis could quite feasibly be done with more detailed occupational categories.

Measures of segregation usually ignore economic disadvantage. For example, when measuring occupational segregation of men and women, we usually examine variation in gender ratios across occupations without worrying about whether women tend to be concentrated in poorly compensated occupations. Similarly, when measuring residential segregation, we often consider the ethnic or racial mix of a neighborhood without worrying about whether minorities tend to be concentrated in more impoverished neighborhoods.1 In ignoring the disadvantages associated with an occupation or a neighborhood, measures of segregation deviate from both common parlance and common practice. For example, in 1991 the U.S. Congress established the ‘‘Glass Ceiling Commission’’ to study barriers to the advancement of minorities and women within corporate hierarchies (Glass Ceiling Commission, 1995). While women were underrepresented in other occupations (e.g., repair services), Congress indicated a particular concern with their underrepresentation in the prestigious executive occupations. This is presumably because that underrepresentation was perceived to be accompanied by economic disadvantage. The point can be made with the example in Table 1. The example starts with three occupations each containing five men and five women. It then moves women between occupations so as to create two alternative distributions, labeled Case I and Case II. The variation in gender ratios is identical in the two alternative distributions; that is, the unequal representation of men and women across the three occupations is the same for both cases. The two cases differ, however, in that Case I is generated by

101

Occupational Segregation with Economic Disadvantage

Table 1. Occupation

A B C Total

Illustration of Segregation and Economic Disadvantage. Individual Earnings $100,000 $100,000 $20,000

Before

After – Case I

After – Case II

Female

Male

Female

Male

Female

Male

5 5 5

5 5 5

8 7 0

5 5 5

0 7 8

5 5 5

15

15

15

15

15

15

moving women from low- to high-wage occupations, while Case II is generated by moving women from high to low. The point of the example is that Case II is arguably a worse form of segregation since it involves reduced compensation for some people. Even though the gender ratios are the same in the two cases, there is good reason to say that one is more segregated than the other. To provide a more detailed treatment of this example, note that while occupations A and B in Table 1 pay high wages ($100,000 to each man and woman in the occupation), occupation C pays only $20,000. Now consider a transfer whereby all five women are moved out of the low-wage C and into high-wage A and B; two go to B and three go to A. As displayed in Case I, occupation C now contains only men and is completely segregated, while the other two occupations contain both men and women. Although gender ratios are more dispersed than before, the five women from C are now in higher paid occupations. Next consider a similar transfer of five women, but this time let the women move out of high-wage A and into B and C; two go to high-wage B and three go to low-wage C. As displayed in Case II, occupation A is now completely segregated, and three of the five transferred women have suffered a loss in pay. While both cases depict ‘‘crowding’’ of one type of people (women) into a subset of occupations, in Case II that crowding is tied to economic disadvantage. Thus, despite the fact that both cases depict the same range of gender ratios, because Case II involves a transfer with reduced compensation, it is arguably a worse form of segregation than Case I.2 Aspirations provide another way to argue that Case II is a more egregious form of segregation. Whereas in Case I young men and women can be expected to aspire to the well-compensated occupations A and B, that may not be true in Case II. One could reasonably claim that in Case II, young women will be less likely to prepare for jobs in occupation A. Thus,

102

ROBERT HUTCHENS

segregation like that in Case II could influence investment in human capital and thereby economic efficiency. Such arguments are certainly consistent with the thinking of those who established the Glass Ceiling Commission. In another paper I try to characterize measures whereby segregation in Case II is worse than that in Case I (see Hutchens, 2007). The paper proposes two methods for incorporating economic disadvantage into empirical work. One is to rank groups from highest to lowest economic status and apply a dominance criteria. The second is to invoke a cardinal measure of economic status and compute a numerical index. This paper extends that second line of attack. It derives numerical indexes of segregation that use the average level of compensation in an occupation as a cardinal measure of economic status. Thus, occupation A is ‘‘disadvantaged’’ relative to B if A has lower average compensation than B. The paper goes on to develop methods for decomposing changes in these indexes into analytically meaningful components. It concludes with an application: analysis of occupational segregation of men and women in the United States between 1970 and 2000.

THE PROBLEM AND AN APPROACH TO THE PROBLEM Consider T groups that contain two types of people. Let xij be the number of type i people in group j (i ¼ 1,2; j ¼ 1, y, T ). For example, type 1 could be women, type 2 could be men, and group j could be one of T occupations. The total number of people in group j is then x1j þ x2j. Since this paper is concerned with assessing segregation in situations where one group is disadvantaged relative to another, let x0j be a nonnegative scalar that indicates the economic status of people in group j. If x0jox0k, then group j is disadvantaged relative to group k; if x0j ¼ x0k, then j and k have the same economic status and are thereby equally advantageous. Let x be a matrix of the form 2 3 x0 6x 7 x ¼ 4 15 x2 where xi are row vectors of the measure of economic status and numbers of [xi1, xi2, y, xiT,]; type 1 and 2 people in the T occupations, that is, xi ¼P i ¼ 0,1,2, and let the sum of xij be denoted as N i ¼ Tj¼1 xij ; i ¼ 0; 1; 2.


103

Using the same notation to denote an alternative matrix y, this paper is concerned with measuring whether x is more segregated than y. A numerical measure provides a good way to address that issue. Thus, let OA(x) be a continuous real-valued measure of segregation defined on the domain D, where OA(x)ZOA( y) iff x is at least as segregated as y, for each x,y element of D.3 Given this notation, consider an index of the form T X x1j x2j x1j x2j OAðxÞ ¼ C þ lðx0j Þ (1) ½Vðx0j ÞH ; þ N1 N2 N1 N2 j¼1 where C is a constant; V( ) and l( ) are continuous, nonnegative, increasing, and strictly concave functions; and H( ) is a continuous function that is homogeneous of degree 1, nonnegative, increasing, symmetric, and strictly concave with H(0,0) ¼ 0. The Eq. (1) index – henceforth termed an augmented index – has two interesting properties. The first property arises when groups are equally advantageous and the elements of the x0 vector are equal, that is, x01 ¼ x02 ¼ ¼ x0T ¼ x 0 . In this case, the index in (1) reduces to T X x1j x2j (2) OAðxÞ ¼ C 2lðx 0 Þ Vðx 0 Þ H ; N1 N2 j¼1 where Vðx 0 Þ and lðx 0 Þ are constants. From the perspective of the previous literature on measuring segregation, the Eq. (2) index has the properties of a ‘‘good’’ index in the sense of Hutchens (2001). That paper presents seven desirable properties for an index of segregation and derives the class of measures that satisfies these properties. The index in Eq. (2) belongs to that class. A second property – henceforth termed the MBG (movements between group) property – of the Eq. (1) augmented index arises when groups are not equally advantageous and the x0j are not all the same. A consequence of this property is that the Eq. (1) index will declare the segregation in Case II of Table 1 as worse than that in Case I. To see this, it is useful to distinguish between two kinds of movements of people between groups: ‘‘a disequalizing movement with unchanged economic status’’ and ‘‘a disequalizing movement with reduced economic status.’’ The formal definitions are given below. Note that the definitions are identical except for part (d); for the movement with unchanged economic status, part (d) stipulates that

104

ROBERT HUTCHENS

x0f ¼ x0g, and for the movement with reduced economic status, part (d) stipulates that x0fWx0g. A disequalizing movement with unchanged economic status. The distribution y is obtained from x by a disequalizing movement of type 1 people if, for f and g, (a) x2f ¼ x2g ¼ y2f ¼ y2gW0, (b) x1f/x2f o x1g/x2g, (c) y1f ¼ x1f d and y1g ¼ x1g þ d, for 0odrx1f, (d) x0f ¼ x0g, and (e) xhk ¼ yhk, h ¼ 1,2,3; k 6¼ f,g. A disequalizing movement with reduced economic status. The distribution y is obtained from x by a disequalizing movement of type 1 people if, for f and g, (a) x2f ¼ x2g ¼ y2f ¼ y2gW0, (b) x1f /x2f ox1g/x2g, (c) y1f ¼ x1f d and y1g ¼ x1g þ d, for 0odrx1f, (d) x0f W x0g, and (e) xhk ¼ yhk, h ¼ 1,2,3; k 6¼ f, g. A disequalizing movement with unchanged economic status is similar to a regressive transfer in the income inequality literature. Given two groups, f and g, with x1f /x2f ox1g/x2g, a disequalizing movement reduces x1f /x2f and increases x1g/x2g. To illustrate, let type 1 people be women and type 2 people be men, and suppose that the initial distribution is as in Table 2. Note that since occupation 1 initially contains three women and five men and occupation 2 contains six women and five men, x11/x21 ¼ 3/5o6/ 5 ¼ x12/x22. Now, as depicted in Case I, suppose that ‘‘d ’’ women move from occupation 1 to occupation 2 (so that occupation 1 contains 3–d women and five men and occupation 2 contains 6 þ d women and five men). This movement increases the female-to-male ratio in the more ‘‘female’’ occupation 2 while decreasing that ratio in the more ‘‘male’’ occupation 1. The MBG property below addresses the effect on measured segregation of such a disequalizing movement with unchanged economic status. It essentially says that a segregation index should either increase or remain unchanged as a result of such a movement.

Table 2. Illustrating the Two Disequalizing Movements. Occupation

1 2 3 Total

Individual Earnings $100,000 $100,000 $20,000

Before

After – Case I

After – Case II

Female

Male

Female

Male

Female

Male

3 6 6

5 5 5

3d 6þd 6

5 5 5

3d 6 6þd

5 5 5

15

15

15

15

15

15

105


A disequalizing movement with reduced economic status is the same except that x0f Wx0g. Thus, when the type 1 person moves from group f to group g, the person enters not only an occupation with a higher ratio of type 1 to type 2 people (since x1f /x2f ox1g/x2g), but also an occupation with lower economic status (since x0f Wx0g). Case II in Table 2 illustrates this kind of movement. Since it involves a reduction in x0, a disequalizing movement with reduced economic status arguably increases segregation more than an otherwise identical disequalizing movement with unchanged economic status. In particular, in Table 2, segregation should increase more in Case II than in Case I. We can state this as a property. The Movements between Groups Property A disequalizing movement with unchanged economic status should not decrease and may increase segregation. A disequalizing movement with reduced economic status should have an effect on measured segregation that is greater than or equal to an otherwise identical disequalizing movement with unchanged economic status.

It can be shown that the augmented index in Eq. (1) exhibits this property. A good way to conclude this section is by addressing a potential critique of the augmented index in Eq. (1). One could argue that in mixing together measures of segregation and income inequality, this index creates a hodgepodge of two dissimilar things. It is true that the index increases when either a conventional measure of income inequality increases or a conventional segregation index increases, but what is the gain? We know just as much from looking at the two separate measures. Why combine them into this augmented index? One way to address this is with yet another table. Consider Table 3. Table 3 is quite similar to Table 1 in that the population of 15 men and 15 women is initially evenly distributed over occupations A, B, and C; Table 3 Table 3. Occupation

A B C Total

Another Illustration of Segregation and Economic Disadvantage.

Individual Earnings $100,000 $20,000 $20,000

Before

After – Case I

After – Case II

Female

Male

Female

Male

Female

Male

5 5 5

5 5 5

5 8 2

5 2 8

2 8 5

8 2 5

15

15

15

15

15

15

106

ROBERT HUTCHENS

is, however, different in that the income in occupation B is now $20,000, and both men and women move. Note that a conventional measure of income inequality (e.g., a Gini coefficient) will find no difference between ‘‘After – Case I’’ and ‘‘After – Case II.’’ In both cases 20 people earn $20,000, and 10 people earn $100,000. Note also that a conventional measure of segregation (for example, a dissimilarity index) will find no difference between the two cases. In both cases the ratio of women to men in the three occupations is 2:8, 8:2, and 5:5. Thus, if one were to use two separate conventional measures – a conventional measure of income inequality and a conventional measure of segregation – the appropriate conclusion would be that ‘‘After – Case I’’ and ‘‘After – Case II’’ exhibit identical levels of inequality. Yes, the two cases are more unequal than the ‘‘Before’’ distribution, but in comparison to each other they are ‘‘equally unequal.’’ The augmented index comes to a different conclusion about Table 3, and thereby not only conveys more information than the two separate conventional measures, but also gets closer to what we mean by segregation. The augmented index indicates that Case II is a more egregious form of segregation than Case I because for Case II one type of people (men) are overrepresented in the high-income occupation while another type of people (women) are overrepresented in the two low-income occupations. Thus, the two cases are not ‘‘equally unequal.’’ The augmented index captures this aspect of segregation, and thereby conveys more information than is obtained from either a conventional measure of segregation or a conventional measure of income inequality or both.

THREE DECOMPOSITIONS Analyses of segregation frequently involve decompositions. For example, in order to analyze the change in occupational segregation between two points in time, Blau and Hendricks (1979) decomposed their segregation index into a component attributable to changing occupational structure and a component attributable to changing gender ratios. A similar approach has been used to analyze differences in segregation between countries, regions, or sectors of the economy (e.g., see Charles & Grusky, 1995; Flu¨ckiger & Silber, 1999, and cites therein). The approach used here is similar to that of Blau and Hendricks (1979) in that it focuses on changes in the Eq. (1) augmented index between two points in time.


107

This section introduces three decompositions denoted I, II, and III. These decompositions separate a change in the augmented index into several components. For example, Decomposition I separates a change in the augmented index into two components, one indicating how the index would change if economic status were ignored (as in a conventional segregation index) and the second indicating the effect of changes in economic status. The other two decompositions separate a change in the augmented index into three components. For example, Decomposition II takes the second component in Decomposition I (the effect of changes in economic status) and splits it into two. The resulting new second component indicates how the movement of people between occupations affects economic status and thereby the index, and the new third component indicates the effect of changes in economic status over time. It should perhaps be noted that an alternative way to proceed with decompositions involves an analysis of within-sector and betweensector changes. For example, changes in occupational segregation can be analyzed in terms of changes in segregation within the blue-collar sector, within the white-collar sector, and between the blue- and white-collar sectors. That is the approach taken in Hutchens (2004), and it is quite feasible with the Eq. (1) index. It will, however, have to be a topic for the future. Before introducing the three decompositions, it is useful to write the Eq. (1) index more compactly. Let s1j ¼ x1j/N1, s2j ¼ x2j/N2, and qj ¼ s1j/s2j. In addition, since H( ) is homogeneous of degree 1, we have H

x1j x2j ; N1 N2

^ jÞ ¼ Hðs1j ; s2j Þ ¼ s2j Hðqj ; 1Þ ¼ s2j Hðq

Using this notation the index in (1) becomes

OAðxÞ ¼ C

T X

^ j Þ þ lðx0j Þðs1j þ s2j Þ ½Vðx0j Þs2j Hðq

(3)

j¼1

One way to analyze OA(x) is in terms of one component that indicates segregation when all occupations have the same economic status and a second component that indicates the effect of unequal economic status. Specifically, OAðxÞ ¼ OðxÞ þ yðxÞ

(4)

108

ROBERT HUTCHENS

where OðxÞ ¼ C Vðx 0 Þ

T X

^ jÞ s2j Hðq

j¼1

yðxÞ ¼

T X

^ j Þ þ lðx0j Þðs1j þ s2j Þ ½ðVðx0j Þ Vðx 0 ÞÞs2j Hðq

j¼1

and x 0 is the value of x0j such that all x0j are equal, that is, x01 ¼ x02 ¼ ¼ x0T ¼ x 0 . The O(x) term in Eq. (4) is a measure of occupational segregation that treats occupations as equally advantageous. It is simply a function of the numbers of type 1 and type 2 people (x1j and x2j); x0j do not affect this measure. In contrast, the y(x) term indicates how unequal advantage – as signified by x0j – influences segregation. In particular, y(x) behaves as required by the MBG property; that is, y(x) behaves properly in response to a disequalizing movement with reduced economic status. Note that y(x) can be either positive or negative; its sign depends on the correlation between x0, x1, and x2. For example, if l equals zero and occupations with high levels of economic status tend to contain only one type of person, then y(x) will be positive. As proportions of type 1 and 2 people become more equal in higheconomic-status occupations, y(x) becomes negative. The Eq. (4) restatement of OA(x) is particularly useful when analyzing changes in the augmented index through time. Writing the change in OA(x), O(x), and y(x) between time t and tu (totu) as OAðxt0 Þ OAðxt Þ ¼ DOA Oðxt0 Þ Oðxt Þ ¼ DO yðxt0 Þ yðxt Þ ¼ Dy we obtain what is henceforth termed ‘‘Decomposition I.’’ DOA ¼ DO þ Dy ðDecomposition IÞ

(5)

‘‘Decomposition I’’ separates a change in OA(x) into two components. The first (DO) indicates how segregation would change if economic status were ignored, simply reflecting the effect of changes in the distribution of type 1 and type 2 people across occupations. As explained below, the second component (Dy) indicates the effect of changes in economic status on the index.


109

To clarify P what is meant by Dy, a further decomposition is useful. Let Dy ¼ Tj¼1 Dyj , where ^ jt0 Þ þ lðx0jt0 Þðs1jt0 þ s2jt0 Þ Dyj ¼ ½ðVðx0jt0 Þ Vðx 0 ÞÞs2jt0 Hðq ^ jt Þ þ lðx0jt Þðs1jt þ s2jt Þ ½ðVðx0jt Þ Vðx ÞÞs2jt Hðq 0

Applying the method of Mookherjee and Shorrocks (1982), write Dyj as Dyj ¼ ðV j DPj þ lj Dmj Þ þ ðDV j Pj þ Dlj mj Þ

(6)

where DV j ¼Vðx0jt0 Þ Vðx0jt Þ ðVðx0jt0 Þ þ Vðx0jt Þ 2Vðx 0 ÞÞ Vj ¼ 2 ^ jt0 Þ s2jt Hðq ^ jt ÞÞ DPj ¼ðs2jt0 Hðq ^ jt0 Þ þ s2jt Hðq ^ jt ÞÞ ðs2jt0 Hðq Pj ¼ 2 Dlj ¼lðx0jt0 Þ lðx0jt Þ ðlðx0jt0 Þ þ lðx0jt ÞÞ lj ¼ 2 Dmj ¼s1jt0 þ s2jt0 s1jt s2jt ðs1jt0 þ s2jt0 þ s1jt þ s2jt Þ mj ¼ 2 Thus, one can write Dy as Dy ¼ DISTRIB þ PRIZE

(7)

where DISTRIB ¼

T X ðV j DPj þ lj Dmj Þ j¼1

PRIZE ¼

XT j¼1

ðDV j Pj þ Dlj mj Þ

It follows that the change in OA(x) between time t and tu can be decomposed into three components, henceforth termed Decomposition II: DOA ¼ DO þ DISTRIB þ PRIZE

ðDecomposition IIÞ

(8)

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As in Decomposition I, the DO component in Decomposition II indicates how segregation would change if economic status were ignored. The second component, DISTRIB, indicates how changes in type 1 and type 2 people are distributed across the occupational hierarchy, that is, the extent to which these changes occur in occupations with high- or low-economic status (x0j). For example, consider the case where DPj is positive because of an increase in type 1 people in occupation j. The effect of this positive DPj on DISTRIB depends in part on the magnitude of x0j. If occupation j is high status, then both x0j and the associated V j will be large positive numbers, thereby increasing the effect of DPj on DISTRIB. If, however, occupation j is low status, then that same DPj has a smaller effect on DISTRIB. Finally, the third component, PRIZE, indicates how changes in economic status or the ‘‘prize’’ associated with different occupations (the x0j) alters OA(x) through effects on both V(x0j) and l(x0j). Even if there were no change in the occupational distribution of type 1 and type 2 people, this third component would change as a result of changes in the x0j between t and tu. Yet another way to decompose an augmented index – termed Decomposition III – builds on the earlier literature on decomposing segregation. That literature decomposes a segregation index into two components: (1) the change in the index that is due to changes in the sex (or racial) composition within occupations and (2) the change in the index that is due to changes in the distribution of the population across occupations. Decomposition III takes a similar approach. Looking at Eq. (3), it is clear that OA(x) can change as a result of changes in the ratio of type 1 to type 2 people (the qj), holding constant the other key variables (i.e., holding constant x0j and s2j for j ¼ 1, y, T ). OA(x) can also change as a result of changes in the share of type 2 people (the s2j), ceteris paribus. For example, if type 2 people were men, a movement of men from male-dominated occupations in manufacturing to less-male-dominated occupations in the service sector would change the Eq. (3) index even if the qj did not change. To implement Decomposition III, let (9)

DOAj ¼ DOj Dyj where Dyj is defined above, DOA ¼ Using Eq. (6), rewrite Eq. (9) as

PT

j¼1 DOAj ,

and DO ¼

DOAj ¼ DOj ðV j DPj þ lj Dmj Þ ðDV j Pj þ Dlj mj Þ

PT

j¼1 DOj .

(10)

111


and note that Dmj can be written as Dmj ¼ Ds2j ð1 þ qj Þ þ s2j Dqj where Ds2j ¼ ðs2jt0 s2jt Þ; s2j ¼ ðs2jt0 þ s2jt Þ=2; Dqj ¼ ðqjt0 qjt Þ; qj ¼ ðqjt0 þ qjt Þ=2 Applying the method of Mookherjee and Shorrocks (1982) to DOj, DOj ¼ Vðx 0 Þ½s2j DHðqj Þ þ Ds2j Hðqj Þ ^ jt0 Þ Hðq ^ jt ÞÞ; where Vðx 0 Þ; s2j , and Ds2j are defined above; DHðqj Þ ¼ ðHðq ^ jt0 Þ þ Hðq ^ jt ÞÞ=2. and Hðqj Þ ¼ ðHðq Substituting terms into Eq. (10) and rearranging, DOAj ¼ ðs2j DHðqj Þ þ Ds2j Hðqj ÞÞð1 þ V j Þ lj ðDs2j ð1 þ qj Þ þ s2j Dqj Þ ðDV j Pj þ Dlj mj Þ ¼ s2j ½DHðqj Þð1 þ V j Þ þ Dqj lj Ds2j ½Hðqj Þð1 þ V j Þ þ lj ð1 þ qj Þ ðDV j Pj þ Dlj mj Þ Thus, Decomposition III partitions the change in OA(x) between time t and tu as follows: DOA ¼

T X

DOAj ¼ RATIO þ SHARE þ PRIZE

ðDecomposition IIIÞ

j¼1

(11) where PRIZE is as defined above, and RATIO ¼

T X

s2j ½DHðqj ÞðVðx 0 Þ þ V j Þ þ Dqj lj

j¼1

SHARE ¼

T X

Ds2j ½Hðqj ÞðVðx 0 Þ þ V j Þ þ lj ð1 þ qj Þ

j¼1

The ‘‘RATIO’’ component indicates how a change in the qj – the ratio of type 1 to type 2 people – affects OA(x); for example, how a change in the within-occupation ratio of women to men alters the index. The ‘‘SHARE’’ component indicates how a change in the share of type 2 people affects

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OA(x); for example, how a flow of men out of manufacturing and into other occupations can affect the index even without changes in the qj. In summary, this section has introduced three related decompositions. Decomposition I. separates a change in the augmented index into two components: the first component (DO) indicates how the index would change if economic status were ignored (as in a conventional segregation index), and the second (Dy) indicates the effect of changes in economic status. Decomposition II. separates the change in the augmented index into three components. The first component (DO) is identical to the first component in Decomposition I – it indicates how the index would change if economic status were ignored. The second and third components split up Dy into two parts: DISTRIB and PRIZE. DISTRIB reveals how the movement of people between occupations affects economic status and thereby the index, and PRIZE reveals the effect of changes in economic status over time. For example, DISTRIB could indicate how the index changed as a result of movement of women into higher paid professional and managerial occupations, while PRIZE could indicate how the index changed as a result of increased occupational wage differentials. Decomposition III. is similar to Decomposition II in that it splits the change in the augmented index into three components, one of which is PRIZE. However, the other two components, labeled RATIO and SHARE, differ from those in Decomposition II: RATIO indicates the effect of changes in the sex (or racial composition) within occupations, while SHARE indicates how a change in the distribution of the population across occupations alters the index.

IMPLEMENTATION In order to illustrate the Eq. (1) index and its associated decompositions, this paper undertakes an analysis of changes in occupational segregation by gender between 1970 and 2000. The data come from work by Zvi Eckstein and E´va Nagypa´l who used several years of Current Population Surveys to analyze earnings inequality in the United States (see Eckstein & Nagypa´l, 2004, as well as Nagypa´l’s website, http://www.faculty.econ.northwestern. edu/faculty/nagypal/QRproject/).


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Eckstein and Nagypa´l divide the full-time full-year workforce into three broad occupation groups: professional and managerial, white collar, and blue collar. (A full description of these data is available on Nagypa´l’s abovenoted website.) Although it would be interesting to analyze more detailed occupations, due to changes in the U.S. occupation classification system, it is not a simple matter to assemble a data set that both tracks detailed occupations between 1970 and 2000 and also contains data on compensation in those occupations. The Eckstein and Nagypa´l data are ideal in the sense that they permit a useful first step toward implementing the above indexes and decompositions. The top panel of Table 4 presents occupational shares for men and women in 1970, 1980, 1990, and 2000. These are the key elements of the x1 and x2 vectors. Note that both men and women are similar in that their shares decrease in blue- and white-collar occupations while they increase in professional and managerial occupations. They differ, however, in the pace that change occurs. Whereas 8.3% of employed women were in professional and managerial occupations in 1970, by 2000 this had more than tripled to 28.5%. In contrast, over the same period the male share in professional occupations did not even double, going from 23.7% to 32.5%. With regard to the x0 vector, the ideal would be a measure of the relative income per unit of time for people in the occupation. Income should be measured in a Haig–Simons sense, that is, consumption plus change in savings. Thus, the income associated with a specific occupation would include not only wages and salary, but also fringe benefits like health insurance, vacation time, pension accruals, stock options, and free services (child care, personal trainer, and so forth). Since it involves a form of consumption, the ‘‘cost’’ of working in the occupation (e.g., foregone leisure, required training to obtain or maintain requisite skills over the relevant time period, required uniforms, and expected loss from on-the-job injury) should be subtracted from income. All of this should be measured over a consistent unit of time (e.g., income per year), and would ideally be expressed relative to other occupations. Thus, a period of economic growth that raises occupational incomes without altering the relative standing of the occupations would change neither the x0 vector nor the measured segregation. For present purpose, x0j will be measured as yj x0j ¼ ½Maxfy1 ; y2 ; . . . ; yT g þ Minfy1 ; y2 ; . . . ; yT g where yj is the average annual earnings of full-time full-year workers (males and females combined) in the jth occupation. While this is certainly not the

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Table 4.

Men, Women, and Salaries in Three Occupations: Professional and Managerial, White Collar, and Blue Collar.

Panel 1: Shares of Men and Women in the Three Occupations Year

1970 1980 1990 2000

Men

Women

Professional

White collar

Blue collar

Total

Professional

White collar

Blue collar

Total

0.237 0.284 0.299 0.325

0.197 0.191 0.176 0.171

0.566 0.526 0.525 0.505

1.000 1.000 1.000 1.000

0.083 0.140 0.232 0.285

0.569 0.583 0.523 0.477

0.348 0.277 0.246 0.239

1.000 1.000 1.000 1.000

Panel 2: Average Annual Salary and x0 in the Three Occupations Year

Average Annual Salary (in 2002 Dollar)

Value of x0 Used in the Analysis

Professional ($) White collar ($) Blue collar ($) Professional White collar Blue collar 1970 1980 1990 2000

51,769 53,120 54,882 66,239

30,871 30,943 33,501 38,129

28,833 31,121 29,789 32,136

0.642 0.632 0.648 0.673

0.383 0.368 0.396 0.388

0.358 0.370 0.352 0.327

ideal measure described above, it is both a good place to start and useful for purposes of illustration. Moreover, since yj are divided by the sum of the minimum and maximum value of yj, this measure has the desirable feature of being a relative measure. In particular, if every occupation has the same average earnings, then regardless of the level of those earnings, x01 ¼ x02 ¼ y ¼ x0T ¼ 1/2. The lower panel of Table 4 presents data on the average earnings (in 2002 dollars) of workers (men and women combined) in the three occupations as well as the associated values of x0. Looking at the x0 data, note that consistent with increasing earnings inequality in the United States between 1970 and 2000, the relative income of the professional occupations rose, while that of the white- and blue-collar occupations fell. Implementation of the Eq. (1) class of indexes also requires specification of C, V( ), H( ), and l( ). This paper utilizes two members of the Eq. (1) class: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T pffiffiffiffiffiffiffiffiffi X x1j x2j cx0j Index A : 1 N1 N2 j


115

(sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) T kH ð1 kL Þ X x1j x2j ð1 kH Þ x1j x2j pffiffiffiffiffiffi þ Index B : þ x0j N1 N2 kH kL 2ðkH kL Þ N 1 N 2 j where c ¼ 1 þ Min{y1, y2, y, yT}/Max{y1, y2, y, yT}, and kH (kL) is the square root of the largest (smallest) element in the x0 vector (1WkHWkLW0). Index A sets C ¼ 1, l(x0j) ¼ 0, V(x0j) ¼ Ocx0j, and H(x1j/N1,x2j/ N2) ¼ O[x1j/N1][x2j/N2]. It can be shown that the index not only exhibits all of the properties of the Eq. (1) class, but also lies between 0 and 1. It equals 0 (thereby declaring no segregation) when all men and all women are employed in the occupation (or occupations) with the highest economic status (highest x0j) and the share of all men in an occupation equals the share of all women in that occupation. It equals 1 (thereby declaring complete segregation) when men and women are in separate occupations regardless of economic status; that is, complete segregation in low-status occupations is no worse than complete segregation in high-status occupations. That is a rather strange maximum, and it is happily not shared by index B. It should also be noted that in the special case where every occupation has the same average earnings (i.e., y1 ¼ y2 ¼ y ¼ yT), index A reduces to the square root index of Hutchens (2001, 2004). Index B is similar to index A in that it uses the same specification of H( ), since H(x1j/N1,x2j/N2) ¼ O[x1j/N1][x2j/N2]. It differs, however, in that V(x0j) ¼ Ox0j, C ¼ KH(1 kL)/KH kL and l(x0j) ¼ (1 kH)Ox0j/2(kH kL). Like index A, index B exhibits all of the properties of the Eq. (1) class, lies between 0 and 1, and equals 0 (thereby declaring no segregation) when all men and all women are employed in the occupation (or occupations) with the highest economic status (highest x0j) and the share of all men in an occupation equals the share of all women in that occupation. Unlike index A, index B equals 1 (thereby declaring complete segregation) when all men and women are employed in occupations with the lowest economic status (lowest x0j) and men and women have no occupation in common. Fig. 1 traces a square root index as well as indexes A and B through time. All three are computed on the Eckstein and Nagypa´l data. Note that the square root index declines between 1980 and 2000. That is consistent with Hutchens (2004), which computes square root indexes over 21 occupations in 1980, 1990, and 2000. It is also consistent with a broader literature indicating declining occupational segregation in the United States between 1980 and 2000 (see Weeden, 2004, and cites therein). Thus, despite being

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ROBERT HUTCHENS

Fig. 1.

The Segregation Indexes 1970–2000.

based on only three occupations, the segregation measures presented here yield plausible trends over time. Of course, from the perspective of the present paper, the two new indexes are of particular interest. Both decline between 1970 and 2000. Indeed, both index A and B decline more than the square root index. This can be seen most clearly in Table 5. While the 1970–2000 change in the square root index is 0.0169 (from 0.0807 to 0.0638), the change in index A is 0.0317 (from 0.2696 to 0.2379) and in B is 0.0582 (from 0.3663 to 0.3081). Table 5 also presents the three decompositions. The top panel presents Decomposition I, which separates the 1970–2000 change into two components: DO and Dy. Recall that DO indicates how segregation would change if economic status were ignored, and Dy indicates the change resulting from bringing unequal economic status into the index. Since the square root index effectively treats all occupations as having the same economic status, the only entry for the square root index in the top panel is DO ¼ 0.0169, which indicates that segregation declined between 1970 and 2000; Dy is effectively set to zero. In contrast, the top panel includes entries for both DO and Dy for index A and B, since both indexes allow occupations to have different economic status. And the important result here is that Dy is negative for both indexes. Thus, allowing for changes in economic status between 1970 and 2000 strengthens the conclusion that segregation declined over the period. Decomposition II in the middle panel of Table 5 provides a more complete understanding of the decline in y. Here the change in y is separated into two

117


Table 5. Decomposition I

Square root index Augmented indexes Index A Index B Decomposition II

Square root index Augmented indexes Index A Index B Decomposition III

Square root index Augmented indexes Index A Index B

Sources of Change in Segregation Indexes 1970–2000: Three Decompositions. Index Value in 1970

2000

0.0807

0.0638

0.2696 0.3663

0.2379 0.3081

1970–2000 Change DOA

DO

Dy

0.0169 0.0317 0.0582

Index Value in

0.0169 0.0120

0.0147 0.0462

1970–2000 Change

1970

2000

DOA

0.0807

0.0638

0.0169

0.2696 0.3663

0.2379 0.3081

0.0317 0.0582

Index Value in

AÖ

DISTRIB

PRIZE

0.0169 0.0120

0.0386 0.0551

0.0239 0.0089

1970–2000 Change

1970

2000

DOA

RATIO

SHARE

0.0807

0.0638

0.0169

0.0397

0.0228

0.2696 0.3663

0.2379 0.3081

0.0317 0.0582

0.0552 0.0682

0.0003 0.0011

PRIZE

0.0239 0.0089

components: DISTRIB and PRIZE. The second, PRIZE, is positive for both index A and B. That essentially means that had the occupational distribution of men and women not changed between 1970 and 2000, the changes in x0 would have caused y – and thereby the augmented segregation index – to increase. The increased earnings inequality of the period aggravated segregation by increasing the relative wage of the well-compensated professional and managerial occupations. But this positive PRIZE effect is completely overwhelmed by the large and negative DISTRIB effect. DISTRIB is negative because the occupational distribution of men and women did change between 1970 and 2000, with both genders moving up the occupational hierarchy into the well-compensated professional and managerial occupations. In effect, the change in the occupational distribution was such that the high-economic-status occupations ceased to be disproportionately populated by one type of people – men. And that reduced y.

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Decomposition III in the lower panel of Table 5 helps nail down that last point. Here the change in the indexes is split into three components: RATIO, SHARE, and PRIZE. The last component, PRIZE, is the same here as in Decomposition II. In both cases this component is positive for index A and B because of the 1970–2000 increase in earnings inequality. What is important about the other two components is the much larger magnitude of RATIO. The RATIO component indicates how a change in the qj, the ratio of type 1 to type 2 people, affects the indexes. It is because the qj became less dispersed over time that all three indexes fell between 1970 and 2000. Moreover, it is because of the fall in qj in the well-compensated professional and managerial occupations that index A and B dropped so dramatically.

CONCLUSION This paper examines indexes for which segregation in high-economic-status occupations is worse than similar segregation in low-economic-status occupations. It presents three decompositions that provide a clearer picture of how and why such indexes change. Using these methods, the paper provides an illustrative analysis of changes in occupational segregation in the United States between 1970 and 2000. The principal result in this analysis is that measured segregation declined over the period. This decline was despite the influence of changing earnings inequality, which tended to exacerbate measured segregation by decreasing the relative earnings of less advantaged occupations. The dominant forces associated with the decline were (a) the convergence of occupational gender ratios and (b) the large fraction of women who moved out of less advantaged occupations and into the comparatively well-compensated professional and managerial occupations.

NOTES 1. Of course, ‘‘often’’ does not mean ‘‘always.’’ There are papers in the literature on residential segregation that discuss links between racial segregation and economic status. For example, see Erbe (1975), Farley (1977), and Jargowsky (1996). Similarly, there are papers in the literature on occupational segregation that discuss how gender and racial segregation are related to wages and economic status. For example, see Rubery, Fagan, and Maier (1996), Flu¨ckiger and Silber (1999), and Hutchens (2004).


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2. Some may be concerned that the ‘‘before’’ society has $2.2 million while Case I has $2.6 million and Case II has $1.96 million. Where does the money come from and go to? One way to handle this is to assume a national bank (in the United States, the Federal Reserve) that adds and subtracts money with goods prices adjusting accordingly. 3. Denoting the vector space of all 3 S T real matrices with nonnegative elements 1 3T by R3T : N xi 40; i ¼ þ , the domain of O(x; k) is D ¼ T¼2 DT where DT ¼ fx 2 Rþ 1; 2; 3g. One could expand the problem to allow for more than two groups and within-group variation in economic status. The strategy here is to take one step at a time. Before the problem can be expanded, it is important to fully examine the simpler problem.

ACKNOWLEDGMENTS I am grateful to the participants in the conference on New Frontiers in the Field of Segregation Measurement and Analysis for several helpful suggestions and to two anonymous referees for their insightful detailed comments. All remaining errors are, of course, strictly my own.

REFERENCES Blau, F. D., & Hendricks, W. E. (1979). Occupational segregation by sex: Trends and prospects. Journal of Human Resources, 14, 197–210. Charles, M., & Grusky, D. B. (1995). Models for describing the underlying structure of sex segregation. American Journal of Sociology, 100, 931–971. Eckstein, Z., & Nagypa´l, E´. (2004). The evolution of U.S. earnings inequality: 1961–2002. Federal Reserve Bank of Minneapolis Quarterly Review, 28, 10–29. Erbe, B. M. (1975). Race and socioeconomic segregation. American Sociological Review, 40, 801–812. Farley, R. (1977). Residential segregation in urbanized areas of the United States in 1970: An analysis of social class and racial differences. Demography, 14, 497–518. Flu¨ckiger, Y., & Silber, J. (1999). The measurement of segregation in the labor force. Heidelberg: Physica-Verlag. Glass Ceiling Commission. (1995). Washington, D.C.: U.S. Department of Labor. Hutchens, R. M. (2001). Numerical measures of segregation: Desirable properties and their implications. Mathematical Social Sciences, 42, 13–29. Hutchens, R. M. (2004). One measure of segregation. International Economic Review, 45, 555–577. Hutchens, R. M. (2007). Measuring segregation when hierarchy matters. Ithaca, New York: ILR School. Jargowsky, P. A. (1996). Take the money and run: Economic segregation in U.S. metropolitan areas. American Sociological Review, 61, 984–998.

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Mookherjee, D., & Shorrocks, A. F. (1982). A decomposition analysis of the trend in U.K. income inequality. The Economic Journal, 92, 886–902. Rubery, J., Fagan, C., & Maier, F. (1996). Occupational segregation, discrimination, and equal opportunity. In: G. Schmid, J. O’Reilly & K. Schomann (Eds), International handbook of labour market policy and evaluation. Cheltenham, UK: Edward Elger. Weeden, K. A. (2004). Profiles of change: Sex segregation in the United States, 1910–2000. In: M. Charles & D. B. Grusky (Eds), Occupational ghettos: The worldwide segregation of women and men (pp. 131–178). Palo Alto, CA: Stanford University Press.

COMMENT ON ROBERT HUTCHENS, ‘‘OCCUPATIONAL SEGREGATION WITH ECONOMIC DISADVANTAGE: AN INVESTIGATION OF DECOMPOSABLE INDEXES’’ Paul A. Jargowsky The basic premise of Hutchens’s paper is that there are cases in which measures of segregation need to take account of the relative status of the groups into which members of a population are segregated. Segregation by occupation, Hutchens argues, is in some sense worse for a group if that group is segregated into lower status occupations. Hutchens proposes several measures that incorporate group status information, shows their properties, and works out their decompositions. I argue in this comment that the measures proposed by Hutchens have questionable utility in that they combine two fundamentally dissimilar types of information: a segregation dimension and a disparity dimension. Segregation often has important consequences. Otherwise, there would be no reason to study it. However, the consequences of segregation should not be equated with the phenomenon itself. Exposure to radiation can sometimes lead to cancer, and it is certainly worse for those exposed if Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 121–124 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017009

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they do, in fact, develop cancer. Yet we do not alter the measure of radiation exposure depending on the consequences of that exposure. Similarly, we can show examples where occupational sex segregation leads to unequal pay between men and women and relative impoverishment of one group relative to the other. We could also ask about other consequences of occupational sex segregation, such as exposure to hazardous conditions and chemicals, opportunities for advancement, on-the-job training, and psychic satisfaction. The same sorting pattern may produce very different disparities on these different dimensions; for example, stock brokers may be rich but very unhappy. Differences between men and women on any of these dimensions may be explained in whole or in part by occupational sex segregation, but they do not themselves constitute segregation. Hutchens describes a situation, illustrated in his Table 1, where men and women are initially equally represented in three occupations. Then women are moved up into higher paying occupations (Case I) or down into lower paying occupations (Case II). By construction, the raw gender ratios are the same, but Hutchens argues that ‘‘Case II is arguably a worse form of segregation’’ and ‘‘there is good reason to say that one is more segregated than the other.’’ Segregation is explicitly a measure applied to two or more groups, measuring the degree of separation between them, so it makes sense to examine the measure from the other group’s points of view. In Case I, men are disproportionately located in lower paying occupations, whereas in Case II, men are disproportionately located in higher paying occupations. Hence, by the logic of Hutchens’s argument, men are more segregated from women in Case I, but women are more segregated from men in Case II. This puzzling asymmetry, which also plagues the proposed ‘‘Movement Between Groups (MBG)’’ property, results from combining two dissimilar types of information: sorting by occupation and gender inequality in income. The former is inherently symmetric (men are separated from women, and therefore vice versa), whereas the latter is inherently comparative (one group is doing better than the other). A simpler and more accurate summary of Table 1 is that men and women are equally segregated in Cases I and II, but the relative income of the two groups varies. Hutchens confronts the issue that the augmented index is merely a hodgepodge of two dissimilar things. His counterargument is illustrated in Table 3. He argues that ‘‘a conventional measure of income inequality (e.g., a Gini coefficient) will find no difference’’ between Cases I and II, and they are clearly equally segregated in the conventional sense. Hutchens concludes that the augmented index ‘‘not only conveys more information than the two separate conventional measures, but also gets closer to what we mean by

Comment on Robert Hutchens

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segregation.’’ However, the issue is not the overall inequality in the income distribution, but the inequality between men and women. In Case I, both men and women have average incomes of $46,667, but in Case II the average salary of men is more than twice that of women ($62,667 vs. $30,667). One can decompose the variance in salaries to within- and between-gender components, calculate the R2 and F statistics, and so forth. Conventional measures can easily capture the unequal compensation of men and women in Case II. Hutchens further argues that Case II is ‘‘a more egregious form of segregation,’’ which is captured by the augmented index. But one could also say that Case II is equally segregated but more unequal. One could argue that this is simply a matter of taste: some people like their segregation served straight up, and others prefer a more inclusive measure. But there are substantive implications of the augmented measure that should be considered. Measuring segregation in a way that incorporates income disparity makes an implicit assumption about the causal link between the two dimensions. In Hutchens’s examples, the incomes of the various occupations are stated a priori. However, suppose that rates of pay for different occupations are determined after the sorting into occupations by sexist assumptions about the value of women’s work relative to men’s work. A predominantly female occupation like teaching may have lower compensation than a male occupation like plumbing precisely because of the gender ratios, despite the fact that the former have more education and do arguably more important work. Alternatively, suppose that all women are paid $20,000 and all men are paid $40,000, and they are randomly distributed across occupations. By chance, some occupations will have more women and therefore lower average compensation. In both of these scenarios, there is segregation and women earn less than men, but the disparity is not caused by segregation into lower paying occupations. Further, if income disparity is to be included in a measure of segregation, the proposed measure does not go far enough. In the examples of Table 1, each occupation pays a fixed wage ($100,000 or $20,000). Thus, there is no inequality within occupations. In the development that follows, x0j represents the average economic status of occupation j. But the variance should also matter to the argument that is being made. For example, if all women are in occupation A and all men are in occupation B, and A is greater than B in average economic status, it makes a difference whether the income distributions of A and B are disjoint or substantially overlapping. With sufficient within-occupation variance, assignment to A or B may have little economic consequence.

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PAUL A. JARGOWSKY

In fairness, Hutchens’s ‘‘augmented measure’’ was introduced in a previous paper (2007). This paper is primarily about decompositions in the augmented measure. I have no argument with the mathematical development of the decompositions, other than to note that the goal of decomposition is to unlump that which was previously lumped together. The shares revealed by the decompositions will be sensitive to the specification of C, V( ), H( ), and l( ) in the proposed measures. Thus, the decompositions tell one about the functional form of the measure and the assumptions of the researcher, not just the underlying phenomenon. For the decomposition to be valuable, the composition has to make sense in the first place and not be the result of an arbitrary weighting. In summary, the paper develops a class of indices that combine two dissimilar types of information: segregation (in the conventional sense) by occupation and inequality between men and women in a characteristic of those occupations (income). It may be more properly described as a measure of occupational disadvantage, combining disadvantage from segregation per se (separate is never equal) and inequality on an economic or status dimension. The two types of disadvantage can be weighted differently by specifying the parameters of the function. The decompositions presented allow a researcher to examine the extent to which changes in the combined index result from changes in segregation in the traditional sense versus changes in the economic status of the occupational categories. However, the relative shares obtained in the decomposition will be a function both of changes in the underlying data matrix and of the decisions on weighting and functional form made by the researcher.

A RESPONSE TO PAUL JARGOWSKY’S COMMENT Robert Hutchens I want to thank Paul Jargowsky for his comment. I take it as a serious comment by a thoughtful scholar. I admit, however, that I cannot make sense of much of it. My paper has its origins in the 1980s. I was presenting a seminar on my first paper on measuring segregation (‘‘Segregation Curves, Lorenz Curves, and Inequality in the Distribution of People Across Occupations’’) to a group of labor economists. Of course, segregation curves play a prominent role in the work of Otis and Beverly Duncan, and are related to the kind of gender ratio measures of segregation favored by Jargowsky in his comment. As the seminar came to an end, my labor economist friends told me that I was talking about segregation in a way that failed to capture what they meant by the term. To their minds segregation was associated with hardship. When a society is segregated, a group not only is separated from other groups, but also suffers reduced opportunity and hardship. Racial segregation in the American South and corporate glass ceilings were cited as examples of such segregation. I thought this was an interesting line of criticism, and I filed it away for further thought. Jargowsky would have evidently told those labor economists that they misunderstood what was meant by segregation and were ‘‘combining two fundamentally dissimilar types of information.’’ He makes that point in his comment by focusing on Table 1 and its associated text. He writes, ‘‘Hence, by the logic of Hutchens’ argument, men are more segregated from women Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 125–127 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017010

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in Case I, but women are more segregated from men in Case II.’’ At this point I am lost. I have no idea what is meant by ‘‘men are more segregated from women’’ as distinct from ‘‘women are more segregated from men.’’ I do not use that language in my paper, nor am I familiar with such language. This leads, however, to a modest proposal that may help move this and related discussions forward. Segregation indexes are based on mathematics. In line with the methods of mathematics, it would first be helpful if Jargowsky would provide a mathematical definition for ‘‘men are more segregated from women’’ and ‘‘women are more segregated from men’’ and explain how these statements differ in a world with two genders. Second, when Jargowsky starts the above sentence with, ‘‘by the logic of Hutchens’ argument,’’ it appears that he has in mind a statement of the form, ‘‘If A then B,’’ where ‘‘A’’ is the class of indexes in my paper and ‘‘B’’ is a suitably defined ‘‘men are more segregated from women in Case I, but women are more segregated than men in Case II.’’ If that was Jargowsky’s intent, then there is a centuries-old method for arguing that the statement ‘‘If A then B’’ is true. Present a proof. If Jargowsky had defined terms and presented a proof, I would not only understand what he meant, but by virtue of his logic also quite possibly be compelled to agree with him. In that same paragraph Jargowsky writes about ‘‘this puzzling asymmetry, which also plagues the proposed ‘Movement Between Groups (MBG)’ property.’’ I too am puzzled. The index in my paper satisfies ‘‘symmetry in types’’ whereby if two types of people change labels (type 1 becomes type 2, and type 2 becomes type 1) then measured segregation does not change. Thus, the argument for Table 1 Case I would be the same if the men moved so that occupation A has five women and eight men, occupation B has five women and seven men, and occupation C has five women and zero men. That relabeling of men and women does not change measured segregation. Where is the asymmetry? If Jargowsky is using a different definition of symmetry, then it would be useful to not only acknowledge that but also write down the mathematical definition (or provide an appropriate citation) so that everyone knows what we are talking about. Jargowsky’s next paragraph tackles Table 3 in an interesting way. We apparently agree that Case II is a less desirable (less lovely?) distribution than Case I, but disagree on what language should be used to describe that. I say that Case II is a more egregious form of segregation than Case I, and Jargowsky says that ‘‘Case II is equally segregated but unequal.’’ Consider an index that orders all feasible distributions of men and women across occupations such that not only Case II is less desirable than Case I, but also

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other distributions are similarly ordered. In addition, suppose the values of that index – denoted Z – are identical to those for a suitably specified version of the augmented index in Eq. (1), for example, index A in the implementation section of the paper. Now consider the statement: there exists a function f( ) such that Z ¼ f(X1, X2, X3, X4), where X1 is a standard measure of segregation (i.e., a measure that Jargowsky would find acceptable), X2 is a standard measure of income inequality for the full population, X3 is a standard measure of income inequality for men, and X4 is a standard measure of income inequality for women. Is that statement true? I do not know the answer, but it is an interesting question. Suppose it was true, would it make a difference whether one used Z or computed f( )? I see no reason to prefer one over the other. What is important here is that we agree that some distributions of people and incomes across occupations are less desirable than others. I would like Z to be an index that can be used to order those distributions from most to least desirable. As indicated in my 2007 paper, I am also quite willing to accept a partial ordering whereby some but not all distributions can be ordered as more desirable than others. If there exists an f(X1, X2, X3, X4) that yields a defensible ordering, then that is interesting. But the argument should be made with mathematics. In his remaining paragraphs, Jargowsky argues that ‘‘measuring segregation in a way that incorporates income disparity makes an implicit assumption about the causal link between the two dimensions.’’ Again, I am lost. Does measuring family income inequality imply a causal link between the earnings of husbands and earnings of wives? Does a body mass index imply a causal link between height and weight? Surely an index can combine different kinds of information without implying causality. Happily, there are areas where I not only understand Jargowsky’s argument, but also agree with it. We agree that the proposed index could go further to incorporate inequality within occupations. One step at a time. We also agree that the decompositions will be sensitive to the functional form of C, V( ), H( ), and l( ). This is, however, true for any decomposition of a numerical index. For example, the decompositions of Mookherjee and Shorrocks (1982) are potentially sensitive to the functional form of the chosen income inequality index as well as the implicit weights on incomes at different points in the income distribution (or on income shares versus population shares). To conclude, I thank Jargowsky for his words, but this field would benefit from more mathematics.

MEASURES OF ORDINAL SEGREGATION Sean F. Reardon ABSTRACT Purpose – To develop measures of segregation that are appropriate when either the groups or the organizational units are defined by ordered categories. These methods allow the measurement of segregation among groups defined by ordered educational attainment categories or among ordered occupational categories, for example. Approach – I define a set of desirable properties of such measures, develop a general approach to constructing such measures, derive three such measures, and show that these measures satisfy the required properties. Originality – Traditional methods of measuring segregation focus on the measurement of segregation among groups defined by nominal categorical variables (e.g., race and gender) among organizational units also defined by nominal categorical units (e.g., schools and neighborhoods). Such methods are not appropriate to the measurement of occupational segregation, for example. The methods developed here are widely applicable and appropriate for such cases.


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Traditional measures of segregation are designed to measure segregation between two groups (e.g., between black and white individuals or between males and females) among nominal categorical units (e.g., schools and neighborhoods) (James & Taeuber, 1985; Zoloth, 1976; Taeuber & Taeuber, 1976). More recently, these measures have been extended to allow the measurement of segregation among multiple unordered groups (e.g., multiple racial/ethnic categories) among nominal categorical units (Reardon & Firebaugh, 2002). However, neither the binary nor the multiple group measures are appropriate when either the groups or the units have an inherent ordering. For example, the study of gender segregation among occupations requires the measurement of segregation of two groups among ordered categorical units (occupations). Likewise, the study of residential segregation among groups defined by educational attainment requires the measurement of multiple ordered categorical groups (defined by ordered levels of educational attainment) among nominal categorical groups. In this paper, I develop measures of segregation appropriate to such cases. The paper proceeds as follows. The section ‘‘Motivating Examples’’ provides two motivating examples to illustrate the measurement problem. ‘‘Prior Methods of Measuring Ordinal Segregation’’ briefly discusses existing approaches of measuring segregation among ordered groups, concluding that little useful work exists. The section ‘‘Notation’’ introduces the notation used in the paper. In ‘‘Desirable Properties of an Ordinal Segregation Measure,’’ I describe a set of desirable properties of ordinal segregation measures. ‘‘Measuring Segregation by an Ordinal Category’’ derives a set of such measures and demonstrates their relation to existing binary segregation measures. ‘‘Properties of the Ordinal Segregation Measure’’ shows that three of the proposed measures satisfy the criteria outlined in the fourth section. Finally, ‘‘Discussion and Conclusion’’ discusses the strengths and limitations of the measures.

MOTIVATING EXAMPLES Consider the following patterns of occupational location by gender. Under Scenario A (top panel of Table 1), of the 100 males and females in the labor market, males are disproportionately located in the highest occupational category, while females are disproportionately located in the lowest. In Scenario B (bottom panel), the distributions in occupations 1 and 2 are switched, as are the distributions in occupations 3 and 4. As a result, men

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Measures of Ordinal Segregation

Stylized Occupational Distribution Patterns.

Table 1.

Occupational Level

Total

1

2

3

4

Scenario A Male Female Total

10 40 50

20 30 50

30 20 50

40 10 50

100 100 200

Scenario B Male Female Total

20 30 50

10 40 50

40 10 50

30 20 50

100 100 200

Table 2.

Stylized Educational Neighborhood Distribution Patterns. Neighborhood

Total

1

2

3

4

Scenario A Less than HS HS diploma BA BAþ Total

50 40 10 0 100

40 30 20 10 100

10 20 30 40 100

0 10 40 50 100

100 100 100 100 400

Scenario B Less than HS HS diploma BA BAþ Total

40 50 0 10 100

30 40 10 20 100

20 10 40 30 100

10 0 50 40 100

100 100 100 100 400

and women are somewhat less concentrated at one extreme of the occupational distribution. A traditional (nominal) segregation measure – such as the dissimilarity index – would rank Scenarios A and B as identically segregated. In each case, there is one occupation with a 40:10 male/female ratio, one with a 30:20 ratio, one with a 20:30 ratio, and one with a 10:40 ratio. A good measure of occupational segregation, however, should describe A as more segregated than B, given the more dramatic segregation between males and females in A.

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Likewise, consider the patterns of educational attainment by neighborhood described in Table 2. In Scenario A (top panel), those with the lowest level of educational attainment are concentrated in neighborhoods 1 and 2, where there are few individuals with the highest levels of education. In Scenario B, the distributions of the lowest two educational categories are switched with one another, as are the distributions of the highest two categories. Intuitively, educational attainment segregation is lower in B than in A, because those with the lowest and highest educational levels are more likely to occupy the same neighborhoods. As in Table 1, standard (nominal) segregation measures will rank Scenarios A and B identically segregated, because they will treat the ordered educational attainment categories as unordered and interchangeable. In both Tables 1 and 2, then, the need for measures of segregation that account for the ordered nature of one of the two categories involved is apparent.

PRIOR METHODS OF MEASURING ORDINAL SEGREGATION There is little prior methodological work developing measures of ordinal segregation. Most research on occupational segregation treats occupational categories as unordered (Charles & Grusky, 1995; Watts, 1992, 1997, 1998; Grusky & Pager, 1998; Hutchens, 1991; Silber, 1992; Siltanen, 1990). Some research on economic segregation relies on ordered income categories, but again, much of this work treats income as either binary or unordered (Telles, 1995; Fong & Shibuya, 2000). One exception is a recent paper that modifies traditional exposure-type segregation indices by introducing a ‘‘social difference coefficient’’ – a factor that accounts for the ordered nature of income categories (Meng, Hall, & Roberts, 2006). This paper, however, relies on somewhat arbitrary definitions of social differences (it assigns interval values to an ordinal scale) and derives exposure measures but not evenness measures of segregation. A second type of prior work comes from measures of nominal–ordinal association. Reardon and Firebaugh argue that one way to conceptualize segregation measures is to view them as measures of association between categorical variables (Reardon & Firebaugh, 2002). While they were concerned with segregation as the association between two nominal categorical variables (e.g., racial groups and schools), measures of association between a nominal and ordinal category might be thought of as segregation

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measures. The generalized delta and alpha measures introduced by Agresti, for example, could perhaps be seen as measures of ordinal segregation (Agresti, 1981), though I am aware of no work that has made this link. Moreover, the properties of such measures are not clear. The third strand of related work is the development of methods for measuring segregation along a continuously measured variable, such as income. A considerable body of such work exists in the literature on residential household income segregation. For example, some work relies on ordered income categories and computes two-group segregation measures for many or all possible pairs of income categories, and then constructs some average or summary measure of these multiple pairwise indices (Massey & Eggers, 1990; Farley, 1977; Telles, 1995). This approach is rather ad hoc in nature, and depends heavily on the number of income categories used. Other measures compute a ratio of the between-neighborhood variation to overall variation, constructing measures of the proportion of variation in incomes that lies between neighborhoods (Davidoff, 2005; Wheeler, 2006; Wheeler & La Jeunesse, 2006; Jargowsky, 1996, 1997; Ioannides, 2004; Ioannides & Seslen, 2002; Hardman & Ioannides, 2004; Watson, 2006). While there are many more examples of such measures, it is not clear how they inform the measurement of segregation along an ordinal variable, since they depend on the interval scale of income to compute between- and within-neighborhood variations. In this paper, I develop a new approach to the measurement of ordinal segregation that is derived from the variation ratio approach described in Reardon and Firebaugh (2002) and similar to that used in many of the income segregation papers referred to above. This approach is based on the idea that segregation can be thought of as a measure of the extent to which variation within organizational units (unordered categories) is less than the total variation in the population. By using well-defined measures of ordinal variation, the Reardon and Firebaugh approach is readily adapted to the case of ordinal segregation.

NOTATION Let k be a variable denoting ordered categories 1,2, y, K, and let m index unordered categories 1,2, y, M. For example, k might index K ordered educational attainment categories (less than high school, high school diploma, bachelor’s degree, or graduate degree) and m might index neighborhoods. Conversely, m might index (unordered) racial groups and

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k might index (ordered) occupational status categories (manual labor, technical, or professional). These two cases may seem conceptually quite different – in the first case, the ordered educational attainment category is a fixed characteristic of individuals (for the purposes of this example) and the unordered neighborhood category is changeable; while in the second, the opposite is true – but we shall see that they are mathematically identical. Moreover, suppose we have a population of size T where each individual is cross-classified by m and k. Let tm, tk, and tmk indicate the total population in category m, category k, and their intersection, respectively. Within each unordered category m, denote the cumulative proportion of the population in m who are in ordered category k or below as cmk; that is, cmk ¼

k X tmj j¼1

tm

(1)

Now we can write the distribution of individuals in category m as the K-tuple cm ¼ (cm1, cm2, y, cmK). (Note that cmK ¼ 1 for all m, so we need only the [K1]-tuple cm ¼ (cm1, cm2, y, cm[K1]) to fully characterize the distribution in m). Likewise, we let ck indicate the cumulative proportion of the population who are in category k or below, and write the distribution of individuals in the population as the [K-1]-tuple c ¼ (c1, c2, y, c[K1]). One additional piece of terminology will be useful in the following sections. We say that the distribution in m dominates the distribution in n over categories h to j (where 1rhojrK) if cmkocnk for all kA(h, y, j). In other words, m dominates n over h to j if there is a greater proportion of the population of n than m at or below each category from h to j.

DESIRABLE PROPERTIES OF AN ORDINAL SEGREGATION MEASURE The goal of this chapter is to define a class of ordinal segregation indices that measure what is generally termed the ‘‘evenness’’ dimension of segregation. That is, we wish to measure the extent to which ordered groups are evenly distributed across unordered categories. In the case of occupational segregation, this corresponds to measuring the extent to which ordered occupational categories are proportionately represented among unordered groups (typically gender or racial/ethnic groups). In the case of educational residential segregation, this corresponds to measuring the extent


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to which different educational attainment groups are evenly distributed among residential locations (e.g., neighborhoods). In order to define a set of properties that a measure of ordinal segregation should satisfy, it is useful to consider several simple examples. Below we describe a set of desirable properties of an ordinal segregation measure, using examples of residential educational segregation or of racial occupational segregation in order to make the properties concrete. Scale Interpretability A segregation index is maximized iff within each unordered category m, all individuals occupy a single ordered category (if cmkA{0, 1} for all m and k). A segregation index is minimized iff within each unordered category m, the distribution of individuals is identical to that in the population (if cmk ¼ ck, for all m and k).1 The intuition here is that maximum segregation occurs when there is no variation in ordered category k within each unordered unit, and minimum segregation occurs when the distribution of k within each unit m mirrors that of the population as a whole, as illustrated in Scenarios A and B of Table 3. It is important to note that this definition of maximum segregation does not depend on the overall amount of ordinal variation in the population, but only on the relative distribution among categories. This is a necessary requirement for defining evenness measures of segregation. In Scenario C of Table 3, for example, there is less variation in educational attainment than in Scenario A (everyone in C has either an HS diploma or a BA; no one has less than an HS diploma or more than a BA), yet by the definition here, Scenarios A and C are equally segregated, because in each case the unordered groups are as unevenly distributed among neighborhoods as possible (there is no neighborhood with any diversity of educational attainment). Conversely, to claim that Scenario A is more segregated than Scenario C would be to confound the extent of variation in educational attainment in the population with the unevenness of their distribution among neighborhoods. Given our interest in measuring the evenness dimension of segregation, this would be undesirable. Unordered Category Equivalence If the populations in two unordered categories m and n have the same distribution over k, and m and n are combined, segregation is unchanged.

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Table 3.

Maximum and Minimum Segregation Patterns. Neighborhood

Total

1

2

3

4

Scenario A Less than HS HS diploma BA BAþ Total

0 100 0 0 100

100 0 0 0 100

0 0 0 100 100

0 0 100 0 100

100 100 100 100 400

Scenario B Less than HS HS diploma BA BAþ Total

25 25 25 25 100

25 25 25 25 100

25 25 25 25 100

25 25 25 25 100

100 100 100 100 400

Scenario C Less than HS HS diploma BA BAþ Total

0 100 0 0 100

0 100 0 0 100

0 0 100 0 100

0 0 100 0 100

0 200 200 0 400

This is analogous to the standard organizational equivalence condition used in nominal measures of segregation (James & Taeuber, 1985; Reardon & Firebaugh, 2002). The intuition here is that two identically distributed unordered categories are not segregated from one another, so combining them does not alter overall segregation patterns. In Table 4, for example, Black and Hispanic individuals have the same distribution among occupations, so combining them into a ‘‘minority’’ group should not alter conclusions about the levels of segregation.

Size Invariance If the number of members of each category k in each category m is multiplied by a positive constant, segregation is unchanged. The intuition here is simply that the total population counts do not matter, but only the distributions among categories.

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Table 4.

Unordered Category Equivalence. Occupational Level

Total

1

2

3

4

Scenario A White Black Hispanic Total

10 20 10 40

20 20 10 50

30 10 5 45

40 0 0 40

100 50 25 175

Scenario B White Minority Total

10 30 40

20 30 50

30 15 45

40 0 40

100 75 175

Null Group Effects If an ordered category j with no population (so that tmj ¼ tj ¼ 0 for all m) is added to the vector of possible ordered categories k ¼ 1,2, y, K, the following conditions should yield unambiguous results. First, if j ¼ 0 (i.e., category j is lower than each of the K ordered categories) or j ¼ K þ 1 (i.e., category j is higher than each of the K ordered categories), then segregation is unchanged. Second, if 1ojoK and cmjA{0,1} for all m (in each unordered category m, all the population has koj or kWj) and there exist some unordered categories m and n where cmj ¼ 0 and cnj ¼ 1, then segregation will increase, unless segregation is already at its maximum possible value. The intuition here is that the ordinal categories have some inherent meaning. Adding a null category at the extremes doesn’t change the separation of groups, since it introduces no additional distance between any two individuals. But adding a null category that unambiguously increases the distance between higher and lower ordered categories (because for each unordered category m, all individuals are above or below the new ordered category) should increase segregation (unless it is already at its maximum).

Exchanges If the distribution in category m dominates that in category n over categories j to k, and if an individual of category j is moved from n to m while an

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individual of category k is moved from m to n, then segregation is reduced. Likewise, if an individual in m is moved from category k to j, while an individual in n is moved from j to k, then segregation is reduced.2 The intuition here is that if we exchange individuals of different ordered categories in a way that makes the distributions of individuals in two unordered categories more similar to one another, segregation should be reduced. For example, if neighborhood m has a lower proportion of its population at or below each education level than neighborhood n, and if a highly educated individual in neighborhood m swaps houses with a less educated individual in n, then segregation should be reduced – because we have raised the education levels in n and lowered them in m. Likewise, if there are more females than males at or below each occupation status level, and if a man in a high-status occupation changes jobs with a woman in a low-status occupation, segregation should be reduced. The latter case is illustrated in Table 4. In Scenario A, the male occupational distribution dominates the female distribution over categories 1–3 (there is a greater proportion of females than males at or below each occupational level from 1 to 3). If a male moves from occupation 3 to 2, while a female moves from 2 to 3, we expect segregation to be reduced, because it has reduced the disparity in cumulative proportions at or below income categories 2 and 3, while leaving the others unchanged (Table 5).

Table 5.

Exchange Properties of Segregation. Occupational Level

Total

1

2

3

4


10 40 50

20 30 50

30 20 50

40 10 50

100 100 200


10 40 50

21 29 50

29 21 50

40 10 50

100 100 200

Scenario C Male Female Total

11 39 50

20 30 50

29 21 50

40 10 50

100 100 200


139

Ordered Exchanges If the distribution in unordered category m dominates that in category n over categories j to l (with jokol), and if an individual of category j is moved from n to m while an individual of category l is moved from m to n, then the resulting reduction in segregation will be greater than that resulting if an individual of category j is moved from n to m while an individual of category k is moved from m to n. Likewise, if an individual in m is moved from category l to j, while an individual in n is moved from j to l, then the resulting reduction in segregation will be greater than that resulting if an individual in m is moved from category k to j, while an individual in n is moved from j to k. The intuition here is that the principle of exchanges ought to be sensitive to the ordering of the categories involved. Exchanges of individuals who are farther apart (in their ordered category levels) should change segregation more than exchanges of individuals who are closer. If a male doctor switches jobs with a female janitor, segregation should be reduced more than if a male doctor switches jobs with a female nurse. This is illustrated in the bottom panel of Table 4. Scenario C is obtained from A by having a category 3 male switch occupations with a category 1 female. By the principle of exchanges, segregation C is less than B, and B is less than A, so segregation in C should be less than in A.

Additive Unordered Group Decomposability If M unordered categories are clustered into N clusters, then segregation is decomposable into a sum of independent within- and between-cluster components. The intuition here is that we would like to be able to decompose segregation into between- and within-category components, as described by Reardon and Firebaugh (2002).

Additive Ordered Group Decomposability If K ordered categories are clustered into J supercategories through the combination of adjacent categories, then segregation is decomposable into a sum of independent within- and between-supercategory components. For example, we might like to be able to describe segregation among occupations as a sum of a component of segregation between professional

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and nonprofessional occupations, a component of segregation among nonprofessional occupations, and a component of segregation among professional occupations.

MEASURING SEGREGATION BY AN ORDINAL CATEGORY Reardon and Firebaugh (2002), Watson (2006), and Jargowsky and Kim (2009, this volume) suggest that one way of constructing a segregation measure is to think of it as a measure of the ratio of between-category inequality/variation to the total inequality/variation. In this framework, segregation is the proportion of the total variation of some characteristic in a population that is due to differences in population composition of different categorical units (e.g., schools or census tracts). For an unordered categorical variable (such as race), population variation may be measured by diversity or entropy, but for an ordered (ordinal or continuous) variable, variation is typically measured using some index of the spread of the distribution (e.g., the variance or standard deviation, in the case of an interval-scaled variable). Following Reardon and Firebaugh (2002, Eq. (9), p. 45), if we have a suitable measure of ordinal variation n (a measure of the variation in ordinal variable k) we can define L(n) – an ordinal segregation measure based on the ordinal variation measure n – as follows:

LðnÞ ¼

M X tm ðn nm Þ Tn m¼1

(2)

where m indexes unordered categories, tm and nm are the population count and ordered variation of k within category m, respectively, and T is the total population. Note that if n is the variance function (treating k as an interval-scale variable), then L(n) is equivalent to Z2 (or the R2 from a regression of k on a set dummy variables for categories m ¼ 1,2, y, M). One way to construct a measure of ordinal segregation, then, is to define a suitable measure of ordinal variation (I define ‘suitable’ below), and then use it to construct a segregation measure as above.

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Ordinal Variation Measuring the variation in a population of a quantity measured with an ordinal variable requires us to define what we mean by variation. For an ordinal variable k that can take on any of K ordered categories 1,2, y, K, we define variation as having a maximum (which we can normalize to equal 1) when half the population has k ¼ 1 and half has k ¼ K. Variation is at a minimum (normalized to equal 0) when all observations have k ¼ c for some cA(1,2, y, K). Measuring ordinal variation then amounts to measuring how close the distribution of x is to these minimum and maximum variation states. As described above, we can express the distribution of values of x in a sample as a [K-1]-tuple of cumulative proportions, c ¼ (c1,c2, y, c[K1]), where cj is the cumulative proportion of the sample with values of k in category j or below (note that cK ¼ 1 by definition, so is not needed to characterize the distribution of k). A distribution of k has maximum variation when c ¼ c0 ¼ (1/2, 1/2, 1/2, y, 1/2), corresponding to the case where half the population has the lowest possible value and half has the highest possible value of k. There are K possible distributions of k such that there is no variation in k, corresponding to the K cases of the pattern c ¼ (0,0, y 0,0,1,1, y 1,1) (where all observations have the same value of k ¼ c, so that cj ¼ 0 for jok and cj ¼ 1 for all jZk). Blair and Lacy (1996) suggest that it is helpful to think of c as a point in [K1] space, which leads to the insight that variation can be measured as an inverse function of the distance from c to c0, the point of maximum variation (it is easier to define variation in terms of the distance from the single point of maximum variation rather than from one of the K points of zero variation). Alternately, we can think of c as describing a cumulative density function of an ordinal variable, where variation is measured as a function of the closeness of c to the line ck ¼ 1/2. This suggests a general form of a variation measure as:

n¼

1 1 KX f ðcj Þ K 1 j¼1

(3)

where f(c) is a continuous function defined on the interval [0,1] such that f(c) is increasing for cA(0,1/2) and decreasing for cA(1/2,1), maximized on the interval [0,1] at f(1/2) ¼ 1, and minimized on the interval [0,1] at

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SEAN F. REARDON

f(0) ¼ f(1) ¼ 0. Four such possible functions f are defined below:3 f 1 ðcÞ ¼ ½clog2 c þ ð1 cÞlog2 ð1 cÞ f 2 ðcÞ ¼ 4cð1 cÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 3 ðcÞ ¼ 2 cð1 cÞ

(4)

f 4 ðcÞ ¼ 1 j2c 1j Substituting these functions into Eq. (3) above yields four potential measures of ordinal variation, which are denoted n1, n2, n3, and n4.4 Each of these ordinal variation measures equals its maximum value of 1 if and only if the distribution of k is such that k ¼ 1 for half the observations and k ¼ K for the other half (corresponding to the cumulative proportion vector c ¼ (1/2, 1/2, 1/2, y, 1/2, 1)), and each equals its minimum value of 0 if and only if all observations have the same value of k (corresponding to a cumulative proportion vector c ¼ (0,0, y 0,1, y, 1,1)).

Ordinal Segregation We can substitute n1, n2, n3, and n4 into Eq. (2) above to produce measures of ordinal segregation L1, L2, L3, and L4: L1 ¼

M P m¼1

L2 ¼

M P m1

L3 ¼

M P m1

L4 ¼

M P m1

tm Tn1

ðn1 n1m Þ

tm Tn2

ðn2 n2m Þ

tm Tn3

ðn3 n3m Þ

tm Tn4

ðn4 n4m Þ

(5)

The index L1 is an ordinal generalization of the information theory index H, so we will denote L1 as HO, and call it the ordinal information theory index. In the case where K ¼ 2, HO is identical to H, the conventional two-group information theory index (Theil, 1972; Theil & Finezza, 1971; Zoloth, 1976). Similarly, L2 is an ordinal generalization of the variation ratio index R (which goes by many names in the literature – see, e.g., James & Taeuber, 1985; Reardon & Firebaugh, 2002), and is identical to that index when

143


K ¼ 2, so we will denote L2 as RO, and call it the ordinal variation ratio index. In addition, L3 is an ordinal generalization of Hutchens’s square root index (Hutchens, 2001, 2004), and is identical to that index when K ¼ 2, so we will denote L3 as SO, and call it the ordinal square root index. Finally, L4 might be termed the ordinal absolute difference index, denoted DO. Unlike the other three indices, however, DO does not correspond to any standard binary index when K ¼ 2; in particular, it is not equivalent to the dissimilarity index (James & Taeuber, 1985; Taeuber & Taeuber, 1976), despite its apparently similar form. Each of these four indices is interpreted as the average difference between overall and within-unit ordinal variation, expressed as a ratio of the overall ordinal variation of the population. They differ only in the functional forms of the ordinal variation measures each employs. Alternate Formulations of the Ordinal Segregation Measures The ordinal segregation measures can be written in another form that illustrates their relation to existing measures. First, we require some additional notation. Let nj ¼ f(cj) and nmj ¼ f(cmj); these can be thought of as measures of the (ordinal) variation in the population and in category m, respectively, in the case where the population is divided into only two categories – those with krj and those with kWj. Note that Eq. (3) implies P K1 j i¼1 vj ¼ vðK 1Þ. In addition, let L indicate the segregation (as defined in Eq. (2)) when the population is similarly divided into two categories at j. With some substitution and algebraic manipulation, Eq. (2) can be written as L¼

M X tm m¼1

¼

¼

ðv vm Þ

K 1 X tm ½ f ðcj Þ f ðcmj Þ TvðK 1Þ j¼1 m¼1 M X

K1 X j¼1

¼

Tv

K 1 X

M X vj tm ðvj vmj Þ vðK 1Þ m¼1 Tvj

(6)

wj Lj

j¼1

PK1

PK1 where wj ¼ vj = i¼1 vi and j¼1 wj ¼ 1. The key insight to be gained from Eq. (6) is that a measure of ordinal segregation defined by Eq. (2)

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SEAN F. REARDON

(as a function of the difference between the ordinal variation of the population and the average ordinal variation within each category m) can be rewritten as the weighted average of K – 1 binary (non-ordinal) segregation indices. The weights are proportional to vj ¼ f(cj), the variation in k when collapsed into two categories corresponding to those above and below category j. Because f(cj) is defined to have its maximum at cj ¼ 1/2, Eq. (6) weights segregation most heavily between categories that most evenly split the population. For three of the four ordinal measures defined in Eq. (4), the pairwise indices Lj in Eq. (6) are familiar indices: Lj1 ¼ H j (the two-group Theil information theory index), Lj2 ¼ V j (the two-group variance ratio index), and Lj3 ¼ Oj (the two-group Hutchens square root index). This means that the properties of L in these cases can be derived easily from known properties of the corresponding Ljs. The index defined by Lj4 , however, corresponds to no existing pairwise index. Moreover, as we shall see, it fails to satisfy the scale interpretability and principle of exchanges properties desired in a segregation index.

PROPERTIES OF THE ORDINAL SEGREGATION MEASURES Scale Interpretation Each of the four indices I describe is bounded between 0 and 1, obtaining their maximum when segregation is complete and their minimum when each unordered category has an identical distribution of members of the ordered categories. For three of the indices (except L4), this follows most obviously from Eq. (6) and the fact that each of their corresponding binary indices is bounded between 0 and 1 (Hutchens, 2001, 2004; James & Taeuber, 1985; Reardon & Firebaugh, 2002). For L4, the minimum value of segregation can be obtained even when segregation is not complete (i.e., when members are not evenly distributed among categories). To see this, consider the example in Table 6. The value of L4 in this example is 0, despite the fact that the distribution of males and females among occupations is not equal. Thus, L4 does not meet the scale interpretability criterion. Unordered Category Equivalence Each index of the form defined in Eq. (2) satisfies this property. This follows readily from the definition in (2) and the fact that v depends only on

145


Table 6.

Scale Interpretability Example. Occupational Level


Total

1

2

3

4

10 40 50

40 10 50

25 25 50

25 25 50

100 100 200

cumulative proportions and not on cell totals. Let r be the unordered category formed by combining unordered categories p and q, where cpk ¼ cqk for all kA{1,2, y, K}. From Eq. (1), we then have crk ¼ cpk ¼ cqk for all kA{1,2, y, K}, and thus vr ¼ vp ¼ vq. Now, X tm tp tq ðv vm Þ þ ðv vp Þ þ ðv vq Þ L¼ Tv Tv Tv map;q X tm tp þ tq ðv vm Þ þ ðv vr Þ ¼ (7) Tv Tv map;q

¼

X tm tr ðv vm Þ þ ðv vr Þ Tv Tv map;q

Size Invariance Each index of the form defined in Eq. (2) satisfies this property. This follows readily from the fact that each ck and cmk is unchanged if the number of members of each category k in each category m is multiplied by a constant. As a result, v and each vm are unchanged as well. It follows that Eq. (2) is unchanged under scalar multiplication of all cell sizes. Null Group Effects Recall from (3) that ordinal variation is the average v¼

1 1 KX f ðcj Þ K 1 j¼1

If we add a null category i below category 1 (i ¼ 0) or above K (i ¼ K þ 1), the cumulative proportion in category i will be 0 (if i ¼ 0) or 1 (if i ¼ K þ 1)

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SEAN F. REARDON

for the population as a whole and for each group m. As a result, both v and vm are multiplied by a factor of (K 1)/K (because f(ci) ¼ f(cmi) ¼ 0); these factors will cancel in Eq. (2), leaving L unchanged. If we add a null category i such that 1oioK, 0ocio1, and cmiA{0,1} for all m, each vm will be multiplied by a factor of (K 1)/K as well. The overall ordinal variation in the population, v, will be multiplied by a factor greater than (K 1)/K, because f(ci)W0. As a result, L will increase. If, however, 0ocmjo1 for some m, then, in general, L may increase, decrease, or remain unchanged.

Exchanges To investigate the exchange property, it is simplest to use the form of L in Eq. (6). L¼

K 1 X

wj Lj

j¼1

Under an exchange, the marginal distributions of individuals among the categories do not change; that is, the cks and the tms do not change (and because the cks are unchanged, so are the vjs and v). As a result, L can change only if the Ljs change. Recall that Lj measures the segregation of those with xWj from those with xrj. In an exchange involving individuals of category j and k (where jok), there will be no change in the cumulative proportions cmi, where ioj or iWk. As a result, there will be no change in Li. Now consider categories i such that jrirk. If the distribution in category m dominates that in category n from j to k, then cmiocni for all these i. Exchanging an individual of category j from n with an individual of category k from m will increase cmi and decrease cni for jrirk. If Li is a binary segregation index satisfying the principle of exchanges, then Li will decrease for each jrirk in such a case. As a result, L will decrease as well. Three of the Ljs described correspond to binary segregation indices satisfying the principle of exchanges. The information theory index, variance ratio index, and square root index each satisfy the principle of exchanges (James & Taeuber, 1985; Hutchens, 2001, 2004; Reardon & Firebaugh, 2002). The fourth index does not satisfy the principle of exchanges except when the exchange involves transfers from categories whose cumulative proportion is above 0.5 to those whose cumulative

147


Table 7.

Principle of Exchanges Example. Occupational Level

Total

1

2

3

4


10 40 50

40 10 50

25 25 50

25 25 50

100 100 200


9 41 50

41 9 50

25 25 50

25 25 50

100 100 200

proportion is less than 0.5, or vice versa. In other cases, exchanges will leave the index unchanged. To see this, consider the example in Table 7. In this example, Scenario B is obtained from Scenario A by the exchange of a male in category 1 with a female in category 2 (a male moves from occupation category 1 to category 2, while a female moves from category 2 to category 1). Intuitively, this should increase segregation, because this exchange exacerbates the disproportionate representation of females in the lowest category. However, in both cases, L4 ¼ 0.5 Thus, L4 does not meet the exchange criterion.

Ordered Exchanges Following the logic above, an exchange between ordered categories j and l, where jokol, will reduce each Li for iA{ j, y, l}. An exchange between categories j and k will reduce only the Lis where iA{ j, y, k}. As a result, an index that can be written as in Eq. (6) and that satisfies the principle of exchanges will also satisfy the principle of ordered exchanges.

Additive Unordered Group Decomposability If the binary index Lj is additively decomposable in the sense described by Reardon and Firebaugh (2002), then so is L. Let the M unordered

148

SEAN F. REARDON

categories be combined into some smaller number of supercategories, indexed by g. Then for an additively decomposable index, we can write: L¼

K 1 X

wj Lj

j¼1

¼

K 1 X

" wj LjG

þ

X tg vg

¼

K 1 X

Tv

g

j¼1

wj LjG þ

1 X KX g

j¼1

¼LG þ

g

¼LG þ

X t g vg g

Tv

Ljg

wj

j¼1

1 X tg vg KX

Tv

#

t g vg j L Tv g

(8)

wj Ljg

j¼1

Lg

Thus, L can be written as a sum of a component due to segregation among the supercategories and a weighted sum of the segregation within each supercategory.

Additive Ordered Group Decomposability None of the four indices described here satisfy the ordered group decomposability criterion. While this is a desirable analytic property, it is not clear whether any index can satisfy it.

DISCUSSION AND CONCLUSION The approach I have described above yields four potential measures of segregation along an ordered dimension. Of these, three of the measures satisfy the set of criteria described at the outset. The characteristics of each measure are described in Table 8. Each of the measures can be interpreted in two ways: (1) as a measure of relative ordinal variation (a measure of the difference in the ordinal variation of the population and the average ordinal variation within each unordered category) and (2) as a weighted average of the binary segregation between those above and below each threshold of the ordered variable. The first interpretation links the measures to the analysis

RO

SO

DO

Ordinal variation ratio index

Ordinal square root index

Ordinal absolute difference index

cj ð1 cj Þ

j¼1

j¼1

K1 P

j2cj 1j

K1 P pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

j¼1

cj ð1 cj Þ

½cj log2 cj þ ð1 cj Þlog2 ð1 cj Þ

K1 P

j¼1

K1 P

1 v4 ¼ 1 K1

2 v3 ¼ K1

4 v2 ¼ K1

1 v1 ¼ K1

Variation Formula

None

S

R

H

Two-Group Analog

Measures of Ordinal Segregation.

n/a

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi vj ¼ 2 cj ð1 cj Þ

vj ¼ 4cj(1 cj)

vj ¼ [cjlog2cj þ (1 cj)log2(1cj)]

Weight

Notes: H is the Theil information theory index (Theil, 1972; Theil & Finezza, 1971); R is the relative variation index (which goes by many names in the literature – see, e.g., James & Taeuber, 1985; Reardon & Firebaugh, 2002); S is Hutchens’ square root index (Hutchens, 2001, 2004).

H

O

Symbol

Ordinal information theory index

Ordinal Index

Table 8.

Measures of Ordinal Segregation 149

150

SEAN F. REARDON

of within- and between-group variation; the measures are analogous to Z2-type statistics, denoting the proportion of variation in a population that lies between, rather than within, organizational units. The second interpretation links the measures to traditional binary segregation measures. It shows that we can compute ordinal segregation by computing binary segregation measures between the groups of individuals above and below each threshold of an ordered categorical variable, and then computing a weighted average of these binary measures over all thresholds. In each case, the weights are greatest for thresholds that divide the ordered variable such that half the population is above and half below the threshold. The weights wj in Eq. (6) have intuitive interpretations in some cases. In the case of the ordinal information theory index, the weight wj assigned to Hj in (6) can be interpreted as the expected information content contained in Hj about a randomly chosen individual from the population, where information is defined as log2cj, where cj is the proportion of the population in or below the jth ordered category (Theil, 1972; Pielou, 1977). If cj is near 0 or 1, then Hj contains little information about the segregation experienced by an individual, since it distinguishes among individuals only at one extreme of the income distribution. Conversely, if cj is near 0.5, then Hj contains maximal information, since it distinguishes at the median of the population. In the case of the ordinal relative variation index, the weight wj likewise has an appealing interpretation. For a given threshold j, the probability that two randomly selected individuals from the population will have incomes on opposite sides of threshold j is 2cj(1 cj), which is proportional to f(cj). Since the segregation level Rj describes the extent of segregation between individuals on either side of the jth ordinal threshold, we can interpret Eq. (6) in this case as a weighted average of the segregation between individuals on either side of the jth ordinal threshold, where the segregation level at each threshold is weighted by how informative segregation measured at that threshold is for a randomly chosen pair of individuals in the population. In the case of the ordinal square root index, the weights have no clear intuitive interpretation (they are the square root of the weights in the ordinal relative variation index). Nonetheless, because each of the variation functions in Eq. (4) has their maximum at cj ¼ 1/2, and their minima at cj ¼ 0 and cj ¼ 1, each of the ordinal segregation measures weights binary segregation between groups defined by the median of the ordinal distribution most heavily, and segregation between the extreme groups and the remainder least heavily. Intuitively, this makes sense, since a segregation

151


level computed between those in a small extreme ordinal category and all others, for example, tells us very little about the segregation between two randomly chosen individuals, while segregation between those above and below the median income tells us more about overall income segregation. Although the weights in some cases have intuitive interpretations, it is important to note that the class of ordinal segregation measures described here were not derived originally as a weighted sum of K 1 pairwise segregation indices and thus do not rely on the interpretation of the weights for their derivation. Rather, the weights derived in Eq. (6) are simply a consequence of the definition of ordinal segregation as a function of the ratio of within-unit to overall ordinal variation. If the weights are less intuitive than one might like, they can nonetheless be understood as providing a means of computing the ordinal segregation measures from easily computed and familiar pairwise indices. One could perhaps imagine an alternate class of segregation measures defined explicitly as a weighted average of pairwise measures, where the weights were based on some preferred value function rather than derived from the approach here. Weighting each of the pairwise measures equally, for example, might be appealing under some circumstances. In some cases, one might prefer a segregation measure that is not normalized by the variance in the population. Some recent papers have argued that measures of absolute, rather than relative, deviation, such as the mutual information index (Theil, 1971), are preferable to normalized indices because of their superior decomposition properties (Frankel & Volij, 2008; Mora & Ruiz-Castillo, 2003).6 In fact, Eq. (6) can be easily rewritten to suggest that the ordinal segregation measures here can be understood as normalized unweighted averages of pairwise segregation measures that are not normalized by population variation. Rearranging the terms in the third line of Eq. (6) yields L¼

1 1 KX v j Lj v j¼1 ðK 1Þ

1 1 KX L j ¼ v j¼1 ðK 1Þ

(9)

where Lj ¼ v jLj is a measure of pairwise segregation that is not normalized by dividing it by v (see Eq. (2)). The mutual information theory index M is such an index. Because Mj ¼ v1Hj by definition (see Theil, 1971), Eq. (9), in fact, can be readily used to show that the ordinal information theory index

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SEAN F. REARDON

HO is easily transformed into an alternate index one might call the ordinal mutual information theory index, MO: HO ¼

1 1 KX Mj v1 j¼1 ðK 1Þ

(10)

1 ¼ MO v1 where Mj is the mutual information theory index when the population is divided into two categories at j, and MO is the (unweighted) average of Mj over the K1 ordered thresholds. Thus, the approach described in this paper easily produces a set of measures that are not based on normalized measures. These measures would not satisfy the scale interpretability and null group effects properties described above (though advocates of the mutual information theory index typically discard scale interpretability in any case). A full investigation of the properties of this set of measures, however, is beyond the scope of the present paper. In this paper, I have described an approach to measuring segregation among groups defined by ordered categories. The approach described here combines the appealing features of defining segregation as a ratio of variation within categories to variation in a population and those of averaging binary segregation measures among pairs of categories defined by an ordered variable. Three measures of ordinal segregation derived from this approach – the ordinal information theory index (HO), the ordinal variation ratio index (RO), and the ordinal square root index (SO) – all satisfy the set of desired characteristics for such measures defined here. In addition to the characteristics described above, each of the indices described here can easily be adapted to incorporate spatial proximity, using the methods described by Reardon and O’Sullivan (2004). This may be useful, for example, if one wished to measure the spatial residential segregation among groups defined by ordered categories of educational attainment.

NOTES 1. Note that in the case where the number of non-empty ordinal categories is greater than the number of groups, maximum segregation is not possible by this definition. 2. Note that we could specify transfer properties, but the exchange properties have the advantage of leaving the margins constant. 3. Note that we define 0log2 ð0Þ ¼ lim xlog2 x ¼ 0. x!0


153

4. The second of these, v2, is the index of ordinal variation (IOV), which has been derived in a number of alternate, but equivalent, forms (Berry & Mielke, 1992a, 1992b; Blair & Lacy, 1996; Kva˚lseth, 1995a, 1995b). The other three indices of ordinal variation are new, to my knowledge. Note that, in the case where K ¼ 2, several of these variation measures collapse to familiar measures of binary variation. When K ¼ 2, v1 ¼ [clog2c þ (1 c)log2(1 c)], which is the entropy (Theil, 1972) of a two-group population when one group makes up c% of the population. Likewise, when K ¼ 2, v2 ¼ 4c(1 c), which is twice the Simpson interaction index diversity measure for a two-group population when one group makes up c% of the population. 5. The failure of L4 to meet the scale interpretability and exchange criteria is a direct result of the fact that the variation function v4 is not a concave down function of cj. 6. The choice between measures like the mutual information theory index M (which is not normalized by the population variation) and the information theory index H (which is) lies more in how one believes indices should compare segregation between populations that differ in their overall variation. The decomposition differences are, in my opinion, a minor issue compared to this issue. A fuller discussion of the comparison is beyond the scope of this paper.

ACKNOWLEDGMENTS This paper benefited from helpful conversation with Paul Jargowsky, Robert Hutchens, and Glenn Firebaugh, as well as the thoughtful comments of an anonymous and insightful reviewer. Support was provided by the William T. Grant Foundation and the National Science Foundation (SES-0520400). I take responsibility for all errors.

REFERENCES Agresti, A. (1981). Measures of nominal-ordinal association. Journal of the American Statistical Association, 76, 524–529. Berry, K. J., & Mielke, P. W., Jr. (1992a). Assessment of variation in ordinal data. Perceptual and Motor Skills, 74, 63–66. Berry, K. J., & Mielke, P. W., Jr. (1992b). Indices of ordinal variation. Perceptual and Motor Skills, 74, 576–578. Blair, J., & Lacy, M. G. (1996). Measures of variation for ordinal data as functions of the cumulative distribution. Perceptual and Motor Skills, 82, 411–418. Charles, M., & Grusky, D. B. (1995). Models for describing the underlying structure of sex segregation. American Journal of Sociology, 100, 931–971. Davidoff, T. (2005). Income sorting: Measurement and decomposition. Berkeley, CA: UC-Berkeley.

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Farley, R. (1977). Residential segregation in urbanized areas of the United States in 1970: An analysis of social class and racial differences. Demography, 14, 497–518. Fong, E., & Shibuya, K. (2000). The spatial separation of the poor in Canadian cities. Demography, 37, 449–459. Frankel, D. M., & Volij, O. (2008). Measuring school segregation. Mimeo. Grusky, D. B., & Pager, D. (1998). Models for analyzing occupational segregation by race. Unpublished manuscript. Hardman, A., & Ioannides, Y. M. (2004). Neighbors’ income distribution: Economic segregation and mixing in U.S. urban neighborhoods. Journal of Housing Economics, 13, 368–382. Hutchens, R. M. (1991). Segregation curves, Lorenz curves, and inequality in the distribution of people across occupations. Mathematical Social Sciences, 21, 31–51. Hutchens, R. M. (2001). Numerical measures of segregation: Desirable properties and their implications. Mathematical Social Sciences, 42, 13–29. Hutchens, R. M. (2004). One measure of segregation. International Economic Review, 45, 555–578. Ioannides, Y. M. (2004). Neighborhood income distributions. Journal of Urban Economics, 56, 435–457. Ioannides, Y. M., & Seslen, T. N. (2002). Neighborhood wealth distributions. Economics Letters, 76, 357–367. James, D. R., & Taeuber, K. E. (1985). Measures of segregation. Sociological Methodology, 14, 1–32. Jargowsky, P. A. (1996). Take the money and run: Economic segregation in U.S. metropolitan areas. American Sociological Review, 61, 984–998. Jargowsky, P. A. (1997). Poverty and place: Ghettos, barrios, and the American city. New York: Russell Sage Foundation. Jargowsky, P. A., & Kim, J. (2009). The information theory of segregation: Uniting segregation and inequality in a common framework. In: Y. Flu¨ckiger, S. Reardon & J. Silber (Eds), Occupational and residential segregation (Vol. 17). UK: Emerald. Kva˚lseth, T. O. (1995a). Coefficients of variation for nominal and ordinal categorical data. Perceptual and Motor Skills, 80, 843–847. Kva˚lseth, T. O. (1995b). Comment on the coefficient of ordinal variation. Perceptual and Motor Skills, 81, 621–622. Massey, D. S., & Eggers, M. L. (1990). The ecology of inequality: Minorities and the concentration of poverty, 1970–1980. American Journal of Sociology, 95, 1153–1188. Meng, G., Hall, G. B., & Roberts, S. (2006). Multi-group segregation indices for measuring ordinal classes. Computers, Environment and Urban Systems, 30, 275–299. Mora, R., & Ruiz-Castillo, J. (2003). Additively decomposable segregation indexes. The case of gender segregation by occupations and human capital levels in Spain. Journal of Economic Inequality, 1, 147–179. Pielou, E. C. (1977). Mathematical ecology. New York: Wiley. Reardon, S. F., & Firebaugh, G. (2002). Measures of multi-group segregation. Sociological Methodology, 32, 33–67. Reardon, S. F., & O’sullivan, D. (2004). Measures of spatial segregation. Sociological Methodology, 34, 121–162. Silber, J. (1992). Occupational segregation indices in the multidimensional case: A note. The Economic Record, 68, 276–277.


155

Siltanen, J. (1990). Social change and the measurement of occupational segregation by sex: An assessment of the sex ratio index. Work, Employment and Society, 4, 1–29. Taeuber, K. E., & Taeuber, A. F. (1976). A practitioner’s perspective on the index of dissimilarity. American Sociological Review, 41, 884–889. Telles, E. E. (1995). Structural sources of socioeconomic segregation in Brazilian metropolitan areas. American Journal of Sociology, 100, 1199–1223. Theil, H. (1971). Principles of Econometrics. New York: Wiley. Theil, H. (1972). Statistical decomposition analysis. Amsterdam: North-Holland Publishing Company. Theil, H., & Finezza, A. J. (1971). A note on the measurement of racial integration of schools by means of informational concepts. Journal of Mathematical Sociology, 1, 187–194. Watson, T. (2006). Metropolitan growth, inequality, and neighborhood segregation by income. Williamstown, MA: Williams College. Watts, M. (1992). How should occupational sex segregation be measured? Work, Employment and Society, 6, 475–487. Watts, M. (1997). Multidimensional indexes of occupational segregation: A critical assessment. Evaluation Review, 21, 461–482. Watts, M. (1998). Occupational gender segregation: Index measurement and econometric modeling. Demography, 35, 489–496. Wheeler, C. H. (2006). Urban decentralization and income inequality: Is sprawl associated with rising income segregation across neighborhoods? St. Louis, MO: Federal Reserve Bank of St. Louis. Wheeler, C. H., & La Jeunesse, E. A. (2006). Neighborhood income inequality. St. Louis, MO: Federal Reserve Bank of St. Louis. Zoloth, B. S. (1976). Alternative measures of school segregation. Land Economics, 52, 278–298.

PART IV EXPLORING CHANGES IN SEGREGATION

EXPLORING CHANGES IN THE SPATIAL PATTERN OF INCOME SEGREGATION DURING THE 1990s Casey J. Dawkins ABSTRACT Purpose – Evidence suggests that during the 1990s, many US metropolitan areas saw fundamental changes in the spatial distribution of household income. Following two decades of increasing economic segregation, many metropolitan neighborhoods saw declines in economic segregation, particularly those neighborhoods located within central cities and rural areas. This paper adapts the Spatial Ordering Index proposed by Dawkins (2007b) to explore these trends. Methodology/Approach – Using US Census data, I calculate economic segregation indices for a sample of 205 US metropolitan areas in 1990 and 2000 and decompose changes in the indices into portions attributable to changes in the spatial distribution of households and portions capturing changes in the spatial distribution of aggregate income. I also examine regional variations in the decompositions. Findings – The results suggest that changes in the spatial distribution of households and of income each influenced metropolitan economic segregation in different ways during the 1990s. Furthermore, the spatial dynamics of income segregation exhibited significant regional heterogeneity. Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 159–170 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017012

159

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Originality/Value of paper – This paper presents a new approach to measuring the dynamics of economic segregation.

I. INTRODUCTION When analyzing the dynamics of income segregation, researchers face two challenges. First, existing measures of income segregation tend to emphasize the degree of income sorting within neighborhoods, ignoring larger spatial patterns in the neighborhood income distribution. Second, given that changes in a given pattern of income segregation may result from changes in the spatial distribution either of households or of income, it is difficult to isolate the factors contributing to an observed pattern of income sorting. These two challenges limit the researcher’s ability to identify whether observed patterns of income sorting reflect substantive changes in the degree of interaction among different income groups. This paper addresses these two shortcomings. Using a result from Flu¨ckiger and Silber (1999), I demonstrate how to quantify the relative influence of income changes versus household changes toward a variety of spatial patterns of segregation by income, including unevenness, centralization, and clustering-three of the spatial dimensions of segregation identified by Massey and Denton (1988). The proposed approach is applied toward a descriptive analysis of changes in metropolitan income segregation for a sample of 205 US metropolitan areas over the 1990–2000 period. The decomposition suggests that declines in unevenness and clustering observed during the 1990s were driven primarily by changes in the spatial pattern of income, while changes in the degree of income centralization were influenced by changes in the spatial distribution of households.

1.1. Decomposing Changes in the Spatial Distribution of Household Income Economic segregation is often understood in terms of the relationship between overall income inequality and between-neighborhood income inequality. For a given personal or household income distribution, economic segregation is thought to increase as the variability in income within neighborhoods decreases relative to variability in per household incomes across neighborhoods. Massey and Denton (1988) define this

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spatial dimension of residential segregation as ‘‘unevenness.’’ Several authors have proposed measures of economic segregation that are consistent with this definition (see, e.g., Jahn, Schmid, & Schrag, 1947; Jargowsky, 1996; Kim & Jargowsky, 2005). It is well known that measures of this type generally ignore information about the spatial configuration of neighborhoods. Since such measures are insensitive to patterns of spatial autocorrelation in household income, a problem known as the ‘‘checkerboard problem’’ arises, where a given pattern of variability in household income across neighborhoods may take on a variety of spatial patterns (Dawkins, 2004; Morrill, 1991; White, 1983). Dawkins (2007b) proposes a new spatial ordering index that can be used to overcome the checkerboard problem. The index has several desirable properties. First, it can be used to quantify the relative importance of a variety of spatial patterns of economic segregation compared to an overall degree of economic segregation or household income inequality. Second, unlike measures of spatial autocorrelation based on the variance, the spatial ordering index is less sensitive to the presence of outliers in the underlying distribution of neighborhood per capita incomes. Third, unlike many measures of spatial segregation proposed in the literature, the index can be easily calculated using a simple ranking algorithm. These and other properties of the spatial ordering index are discussed in Dawkins (2007b). If we assume that a given metropolitan area with a total population of H is divided into j neighborhoods, the spatial ordering index (Sr) can be calculated as follows (Dawkins, 2007a): PJ PJ GS 2 j¼1 ðRjðnÞ ððH þ 1Þ=2ÞÞyj =HY j¼1 ðRjðnÞ ððH þ 1Þ=2ÞÞyj ¼ PJ Sr ¼ ¼ PJ GN 2 j¼1 ðRj ððH þ 1Þ=2ÞÞyj =HY j¼1 ðRj ððH þ 1Þ=2ÞÞyj (1) where GS is a spatial Gini (1912) index (Dawkins, 2004) that captures the spatial reordering of the neighborhood income distribution, GN is a Gini index of between-neighborhood income inequality that captures the spatial dimension of unevenness discussed by Massey and Denton (1988), yj is the aggregate household income earned by residents of the jth neighborhood, Y is the aggregate household income earned by the residents of the region, Rj is the average household income rank of neighborhood j residents, and RjðnÞ is the average spatial household income rank of neighborhood j residents. To calculate Rj , one must first construct a neighborhood-level average household income ‘‘parade’’ organized such that the neighborhood earning

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the lowest average household income is ranked 1 in the parade and that earning the highest average household income is ranked J in the parade, or y1ry2r y ryjr y yJ. Then, calculate the cumulative number of households within each successive position within the neighborhood income parade and define this cumulative total as Nj. The cumulative total of households one rank lower within the income parade can similarly be defined as Nj1. The jth neighborhood’s average rank is then equal to: (Nj þ Nj1 þ 1)/2. To calculate RjðnÞ , one must first define a spatial ordering that reflects the jth neighborhood’s spatial position. This paper follows Dawkins (2007a) and examines two spatial orderings that are commonly discussed in the literature: income clustering and income centralization. Income clustering refers to the similarity in average household incomes for adjacent neighborhoods. This is the dimension of segregation that is captured by the Moran-I statistic of spatial autocorrelation. Income centralization refers to the extent to which neighborhood average household income rises or declines with increasing distance from the region’s urban core. The variability in household income with distance from the region’s core has important implications for estimates of the income elasticity of demand for housing in the monocentric urban model (Muth, 1969; Mills, 1972; Alonso, 1964). The spatial centralization of income also has important implications for city-suburb tax base inequities and intra-urban access to economic opportunities. A spatial Gini index of income clustering (GCL) can be calculated by pairing each neighborhood with its most spatially proximate neighbor j(n) and assigning the jth neighborhood a rank equal to the rank of j(n). The cumulative number of households within each successive position within the spatially ordered income parade can now be defined as Nj(n), and the cumulative total of households one rank lower within the spatially ordered income parade can similarly be defined as Nj1(n). The jth neighborhood’s average rank is then equal to: (Nj(n) þ Nj1(n) þ 1)/2. The spatial Gini index constructed from this type of spatial ordering can be interpreted as a measure of the degree of clustering in aggregate household incomes (or weighted average household income) among adjacent neighborhoods. As neighborhoods take on spatial rankings that correspond closely to their ranking in the original income parade, GCL approaches GN. Similarly, as neighborhoods take on rankings that differ from their rank in the original income parade, GCL approaches –GN. Finally, in the case where each neighborhood’s spatial ranking is unrelated to their ranking in the original income parade, GCL approaches zero. Although the index calculated using this approach ignores higher-order


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patterns of clustering, it is possible to construct a spatial ordering that is based on higher-order nearest neighbor pairs (Dawkins, 2004). One can also construct a spatial Gini index of income centralization (GCE) that is defined by ranking neighborhoods by distance from a given point within the region. In the analysis that follows, the GCE index is constructed by ranking each neighborhood by distance from the region’s central business district (CBD). The GCE index approaches GN as each neighborhood’s rank in the original income parade (from lowest to highest household income) becomes identical to the neighborhood’s relative position with respect to the CBD (from greatest to smallest distance from the CBD). A value of GCE equal to GN indicates extreme centralization of household incomes, while a value of –GN indicates extreme decentralization of household incomes with respect to the CBD. Sr can also be calculated using a version of the matrix algebra formulation of the Gini index of segregation proposed by Flu¨ckiger and Silber (1999): Sr ¼

½hJðnÞ =H; hJ1ðnÞ =H; . . . ; hjðnÞ =HG½ yJðnÞ =Y; yJ1ðnÞ =Y; . . . ; yjðnÞ =Y0 ½hJ =H; hJ1 =H; . . . ; hj =HG½ yJ =Y; yJ1 =Y; . . . ; yj =Y0 (2)

where G ¼ Silber’s (1989) G-matrix, a J J matrix with elements equal to zero along the diagonal, 1 above the diagonal, and 1 below the diagonal; hj ¼ total households residing in neighborhood j; H ¼ total households residing in the entire metropolitan area; yj ¼ aggregate income earned by households residing in neighborhood j; Y ¼ aggregate income earned by all households in the metropolitan area; yj(n) ¼ yj, ranked according to a given spatial ordering; and hj(n) ¼ hj, ranked according to a given spatial ordering. The vectors [hJ/H, hJ1/H, y, hj/H ] and [ yJ/Y, yJ1/Y, y, yj/Y ] in Eq. (2) are each ranked in descending order by the ratio [(yj/Y)/(hj/H)], while the vectors [hJ(n)/H, hJ1(n)/H, y, hj(n)/H] and [ yJ(n)/Y, yJ1(n)/Y, y, yj(n)/ Y ]u are ranked in descending order by the ratio [( yj(n)/Y)/(hj(n)/H )]. An advantage of this formula is its computational simplicity. All that is required to calculate Sr is a sorting vector that gives the ranking of either the nearest neighborhoods’ positions in the income parade or the distances from a given point within the region. If we calculate Sr for two different time periods and examine the difference between the two, it is apparent from Eq. (2) that changes in the value of Sr may result from changes in either the relative distribution of households across neighborhoods (represented by [hJ/H, hJ1/H, y, hj/H ] and [hJ(n)/H, hJ1(n)/H, y, hj(n)/H ]) or in the relative distribution of

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neighborhood income (represented by [ yJ/Y, yJ1/Y, y, yj/Y ] and [yJ(n)/Y, yJ1(n)/Y, y, yj(n)/Y ]u). This issue arises because income data are typically not available at the household level. If it were possible to calculate Sr from household-level microdata, all elements of the vectors [hJ/H, hJ1/H, y, hj/ H ] and [hJ(n)/H, hJ1(n)/H, y, hj(n)/H ]) would be equal to 1/H, and changes in Sr would be solely determined by changes in household income. In the case of aggregate data, it is necessary to decompose changes in the aggregate distribution of households to determine whether changes in Sr result primarily from income or household changes. Following an approach suggested by Flu¨ckiger and Silber (1999), it is possible to decompose the decadal change in any given spatial pattern of economic segregation into two types of changes: (1) changes in the relative share of households residing in each neighborhood, and (2) changes in the share of aggregate income earned by households residing in each neighborhood. A decomposition of changes in the index Sr is not very useful, because it would not distinguish between changes influencing spatial pattern, as represented by GS, and changes in the variability of household income, as represented by GN. To overcome this limitation, I propose decomposing changes in the numerator and denominator separately. Using the formulation of GN shown in the denominator of Eq. (2), the total change in GN, or DGN, can be written as follows: 1 DGN ¼ ð½H 01 GY 0 H 00 GY 0 þ ½H 01 GY 1 H 00 GY 1 Þ 2 1 þ ð½H 01 GY 1 H 01 GY 0 þ ½H 00 GY 1 H 00 GY 0 Þ ð3Þ 2 where H1 is the vector [hJ/H, hJ1/H, y, hj/H ] for time period 1, H0 is the same vector for time period 0, Y1 is the vector [ yJ/Y, yJ1/Y, y, yj/Y ] for time period 1, and Y0 is the same vector for time period 0. Eq. (3) can be written more compactly as: 1 1 (4) DGN ¼ ðDGH Þ þ ðDGY Þ 2 2 where (DGH/2) is the change in GN attributable to changes in the spatial distribution of households (holding incomes constant), and (DGY/2) is the change in GN attributable to changes in the distribution of aggregate income (holding the distribution of households constant). The total change in each spatial Gini index (DGCL and DGCE) can be decomposed in a similar manner.


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II. DESCRIBING THE DYNAMICS OF INCOME SEGREGATION DURING THE 1990S Following two decades of increasing economic segregation, many metropolitan neighborhoods saw declines in economic segregation during the 1990s, particularly those neighborhoods located within central cities and rural areas (Jargowsky, 2003; Fischer, 2003). What is not known is whether these trends reflect changes in the spatial distribution of households or changes in the spatial distribution of income. If households became more evenly distributed across neighborhoods, segregation would have declined even if the spatial distribution of household income remained unchanged. Similarly, if all households remained in the same place but earned higher incomes, or if the same number of households of different incomes switched places, income segregation would likewise change. This section differentiates which of these two factors was more important in shaping the spatial patterns of segregation observed during the 1990s. The data used in the analysis comes from the Geolytics, Inc., Neighborhood Change Database (NCDB). This database is one of the few sources of longitudinal Census data made available at the Census tract level. One unique advantage of these data for the current exercise is that all 1990 variables are provided for normalized 2000 Census tract definitions. The normalization procedure involves overlaying 2000 tract boundaries onto those of an earlier year and estimating the proportion of persons in earlier tracts that make up the new 2000 tract. Population weights reflecting tract changes are then applied to 1990 tract-level variables to convert the data to 2000 boundary definitions (see Tatian, 2002, for more detail on this normalization procedure). This approach allows one to examine total changes in economic segregation over time for a consistent boundary definition. Normalization also makes it possible to decompose total changes into portions attributable to spatial changes in households and spatial changes in aggregate income across tracts. The sample of metropolitan areas examined in the analysis consists of 205 metropolitan areas defined using June 30, 1999, metropolitan statistical area (MSA) definitions. When MSAs are combined into consolidated metropolitan statistical areas (CMSAs), the constituent primary metropolitan statistical areas (PMSAs) are utilized as the unit of analysis. The sample is somewhat smaller than the total number of 1999 metropolitan areas, because the calculation of the spatial Gini index of income centralization requires identification of a CBD for each MSA. I rely on the only known

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Table 1.

Decomposition of Changes in Gini Indices of Income Segregation, 1990–2000. 1990 Value

2000 Value

Change, 1990–2000

GN DGN, household component DGN, income component

0.1764

0.1743

0.0022 0.0325 0.0347

DGCL DGCL, household component DGCL, income component

0.1046

0.1005

0.0041 0.0342 0.0384

DGCE DGCE, household component DGCE, income component

0.0428

0.0477

0.0049 0.0531 0.0482

source of such data: the 1982 Economic Census. Geographic Reference Manual (US Bureau of the Census, 1993), which is based on 1980 (S)MSA definitions. Since fewer metropolitan areas were identified in 1980, the sample is restricted somewhat by this data requirement. Table 1 displays the results of this decomposition for GN, GCL, and GCE. The decomposition is carried out for averages of all 205 metropolitan areas in the sample. In the table, GN represents the Gini index of betweenneighborhood household income inequality (the denominator in Eq. (2)), GCL represents the spatial Gini index of income clustering, and GCE represents the spatial Gini index of income centralization. Both spatial Gini indices are calculated using the generalized formula shown in the numerator of Eq. (2). The most interesting trend revealed by this decomposition is the counteracting impact of changes in the distribution of households compared to changes in the distribution of aggregate income. For all indices, changes in the spatial distribution of income were in the opposite direction of changes in the spatial distribution of households. Income changes have a similar influence on both the between-neighborhood Gini index (GN) and the spatial Gini index of clustering (GCL). In both cases, the impact of income changes outweighs the influence of household changes. Such is not the case with the spatial Gini index of income centralization (GCE). This index is influenced primarily by changes in the spatial distribution of households, which serve to further reduce the value of GCE in 2000. Table 2 decomposes the changes displayed in Table 1 by US region. The average trends displayed in Table 1 are quite dissimilar across region,

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Table 2. Regional Decomposition of Changes in Gini Indices of Income Segregation, 1990–2000. Northeast

Midwest

South

West

DGN DGN, household component DGN, income component

0.0042 0.0322 0.0280

0.0070 0.0322 0.0392

0.0041 0.0327 0.0367

0.0053 0.0330 0.0276

DGCL DGCL, household component DGCL, income component

0.0003 0.0290 0.0287

0.0059 0.0355 0.0414

0.0074 0.0333 0.0407

0.0026 0.0371 0.0345

DGCE DGCE, household component DGCE, income component

0.0032 0.0278 0.0246

0.0062 0.0496 0.0434

0.0035 0.0616 0.0581

0.0063 0.0548 0.0485

Table 3.

Direction of Changes in Household Components versus Income Component.

Household Component

DGN DGCL DGCE

Direction of Change (# of MSAs) Negative Positive Negative Positive Negative Positive

Income Component Negative

Positive

8 194 17 182 6 5

2 1 3 3 188 6

particularly for the between-neighborhood Gini index (GN) and the spatial Gini index of clustering (GCL). For each of these indices, changes in the South and Midwest largely mirrored changes for the average metropolitan area in the sample. The Northeast and West displayed the opposite pattern, however, with increasing between-neighborhood income segregation and increased clustering, both of which were driven primarily by household changes. Changes in the degree of income centralization, on the other hand, did not display significant regional patterns. Table 3 examines a cross tabulation of metropolitan areas to determine the number of metropolitan areas exhibiting positive and negative changes in each component of income segregation during the 1990s. As this table suggests, most metropolitan areas fall into the quadrant typical of the

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average metropolitan area. For the between-neighborhood Gini index (GN), 95% of metropolitan areas exhibited positive changes in the distribution of households and negative changes in the distribution of income. For the spatial Gini index of income clustering (GCL), 89% of metropolitan areas exhibited the same pattern of change. For the spatial Gini index of income centralization (GCE), 92% of metropolitan areas exhibited negative changes in the spatial distribution of households and positive changes in the spatial distribution of income. The regularity of these patterns, combined with the small magnitude of changes in segregation suggested by Table 1, suggests that spatial patterns of income segregation established as of 1990 remained persistent throughout the decade. Amidst these overall trends, a few outliers exhibit interesting patterns of changes. The New York, NY, PMSA, for example, exhibits positive changes in both the distribution of households and the income for all three indices, suggesting that both income and households became more unevenly distributed across neighborhoods, in addition to becoming more highly clustered and more highly centralized. The San Francisco, CA, PMSA is another interesting outlier. This metropolitan area exhibits a pattern of between-neighborhood segregation that is opposite of the average MSA, with negative changes in the distribution of households and positive changes in the distribution of income. The change in the spatial Gini index of income centralization (positive change in the centralization of households and negative change in the centralization of income) is also in the opposite direction of the average MSA. For the spatial Gini index of income clustering, San Francisco exhibits positive changes in the distribution of both households and income. The increased centralization of households within each of these cities is consistent with anecdotal evidence pointing to a significant ‘‘back to the city’’ gentrification trend. Further research is needed to determine why these cities exhibit atypical changes in the spatial pattern of income segregation compared to the majority of US metropolitan areas.

III. CONCLUSION This paper has demonstrated an application of Flu¨ckiger and Silber’s (1999) decomposition of the Gini index of segregation. I have shown how the decomposition can be useful for understanding the role of household changes versus income changes in shaping spatial patterns of residential segregation by income. The technique is implemented using data from a sample of 205 US metropolitan areas. The differences in the influence of


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household mobility versus income changes, in addition to the regional variations in these two factors, underscore the utility of a decomposable index for describing patterns of spatial income segregation. Metropolitan areas exhibiting atypical income segregation trends, particularly New York and San Francisco, deserve additional study. Further research is also needed to understand the policy implications of different evolving patterns of spatial income segregation.

ACKNOWLEDGMENT An earlier version of this paper was presented at the 2006 Association of American Geographers Conference in Chicago, Illinois. The author wishes to thank William A.V. Clark, David Wong, and Michael Reibel for helpful comments that greatly improved the paper.

REFERENCES Alonso, W. (1964). Location and land use. Cambridge, MA: Harvard University Press. Dawkins, C. J. (2004). Measuring the spatial pattern of residential segregation. Urban Studies, 41(4), 833–851. Dawkins, C. J. (2007a). Exploring changes in income clustering and centralization during the 1990s. Journal of Planning Education and Research, 26(4), 404–414. Dawkins, C. J. (2007b). Space and the measurement of income segregation. Journal of Regional Science, 47(2), 255–272. Fischer, M. J. (2003). The relative importance of income and race in determining residential outcomes in US urban areas, 1970–2000. Urban Affairs Review, 38, 669–696. Flu¨ckiger, Y., & Silber, J. (1999). The measurement of segregation in the labor force. Heidelberg, Germany: Physica-Verlag. Gini, G. (1912). Variabilita e mutabilita. Studio economico-giuridice dell’Univ di Cagliari, 3, 1–158. Jahn, J., Schmid, C. F., & Schrag, C. (1947). The measurement of ecological segregation. American Sociological Review, 12, 293–303. Jargowsky, P. A. (1996). Take the money and run: Economic segregation in US metropolitan areas. American Sociological Review, 61, 948–998. Jargowsky, P. A. (2003). Stunning progress, hidden problems: The dramatic decline of concentrated poverty in the 1990s. Working Paper, The Brookings Institution. Kim, J., & Jargowsky, P. A. (2005). The GINI coefficient and segregation on a continuous variable. National Poverty Center Working Paper Series #05-2. Massey, D. S., & Denton, N. A. (1988). The dimensions of residential segregation. Social Forces, 67, 281–315. Mills, E. S. (1972). Studies in the structure of the urban economy. Baltimore: Johns Hopkins. Morrill, R. L. (1991). On the measure of segregation. Geography Research Forum, 11, 25–36.

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Muth, R. (1969). Cities and housing. Chicago: Chicago University Press. Silber, J. (1989). Factor components, population subgroups, and the computation of the Gini index of inequality. Review of Economics and Statistics, LXXI, 107–115. Tatian, P. A. (2002). Neighborhood change data base: Data user’s guide. Washington, DC: Urban Institute and Geolytics, Inc. US Bureau of the Census. (1993). 1982 Economic census. Geographic reference manual. Washington, DC: US Bureau of the Census. White, M. J. (1983). The measurement of spatial segregation. American Journal of Sociology, 88, 1008–1018.

ANALYZING CHANGES IN OCCUPATIONAL SEGREGATION: THE CASE OF SWITZERLAND (1970–2000) Joseph Deutsch, Yves Flu¨ckiger and Jacques Silber ABSTRACT Purpose – The aim of this paper is to analyze the changes that took place in occupational segregation by gender, nationality, and age in Switzerland during the period 1970–2000. Methodology – The paper starts by using correspondence analysis to detect changes in occupational segregation by gender and nationality. It then generalizes a decomposition procedure originally proposed by Karmel and McLachlan by combining their approach with what is now known as the Shapley decomposition. Such a generalization offers a clear breakdown of the variation over time in occupational segregation into a component measuring changes in net segregation and another one corresponding to changes in the margins, the latter itself including variations in the occupational structure and in the shares of the subpopulations (e.g., the genders) in the labor force.


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Findings – Between 1970 and 2000 there was a slight increase in gross segregation by gender but a decrease in net segregation. The change in gross segregation is because the change in the margins more than compensated that in the internal structure. But even the change in the margins is the consequence of opposite forces since variations in the occupational structure would have per se led to a decrease in gross segregation. Originality – The results of the empirical illustration based on Swiss data for 1970 and 2000 prove the usefulness of the approach. They stress in particular that in several instances, variations in gross and net segregation worked in opposite directions.

1. INTRODUCTION The study of occupational segregation by gender has mainly been the work of sociologists (see James & Taeuber, 1985; Reardon & Firebaugh, 2002, for clear surveys of this topic). In recent years some economists have, however, shown interest in the analysis of gender differences in the occupational structure (see, e.g., Anker, 1998). Moreover, several economists (see Butler, 1987; Silber, 1989) have stressed that the study of occupational segregation could greatly benefit from the ‘‘import’’ of techniques used when measuring income inequality (see Flu¨ckiger & Silber, 1999, for a survey on the measurement of segregation in the labor force). Silber (1992) also showed how a generalization of the famous Duncan dissimilarity index allowed one to study multidimensional segregation, which refers either to the case when more than two categories are distinguished (e.g., more than two ethnic groups rather than only two genders) or to that where segregation is analyzed along more than two dimensions (e.g., segregation by occupation, industry, and gender). More recently Deutsch and Silber (2005) took an additional step in adapting the approach used in income inequality measurement to the study of occupational segregation, by proposing a ‘‘normative approach’’ to the analysis of segregation. Such a normative view of segregation allows one to determine whether occupational segregation by gender is mainly the consequence of the presence of ‘‘male-intensive’’ occupations or whether it is rather due to the existence of ‘‘female-intensive’’ occupations. Moreover, it gives policy makers the possibility to select the weight they want to give to very ‘‘male- or female-intensive’’ occupations.

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The main purpose of this paper is to analyze changes that occurred in Switzerland, during the period 1970–2000, in occupational segregation by gender, nationality, or age. We first apply correspondence analysis (CA), a multivariate analysis technique that originated in France and does not seem to have been applied previously, at least in the English-speaking literature, to the study of occupational segregation. The graphs that are the output of such an approach allow us to detect some important changes that occurred in Switzerland, between 1970 and 2000, in occupational segregation by gender and nationality. CA, however, does not indicate whether the changes in occupational segregation that may have occurred were mainly the consequence of variations in the margins (i.e., in the relative importance of the various occupations or in the share of the two genders or of the categories of nationality) or of a change in the degree of dependence between the lines (the occupations) and the columns (the genders and/or the nationality groups). In order to disentangle these different potential effects, we show, in the second part of the paper, how an additional tool recently introduced in income inequality analysis, the game theory concept of Shapley value, could be combined with a technique originally suggested by Deming and Stephan (1940) and applied to the field of segregation by gender by Karmel and MacLachlan (1988) and Watts (1998), to make a very clear distinction between changes in ‘‘gross segregation’’ and changes in ‘‘net segregation,’’ the term ‘‘net’’ referring here to ‘‘net of changes in occupational weights and in the labor force participation rates of both genders.’’1 The empirical section of the paper will indeed show that such a distinction is useful and that one can obtain quite different results, depending on whether one refers to changes in ‘‘gross’’ or ‘‘net’’ segregation. The paper is organized as follows. Section 2 gives the main elements and steps of CA and applies it to the study of occupation segregation by gender and nationality in Switzerland in 1970 and 2000. Section 3 then explains how it is possible to generalize the traditional Duncan index of occupational segregation to take into account more than two dimensions. It also recalls how Karmel and MacLachlan (1988) suggested to decompose variations over time in the generalized Duncan index and shows how this decomposition procedure may be made more systematic, once the concept of Shapley decomposition is applied. Section 4 then finally applies this technique to Swiss Census data for the years 1970 and 2000 and studies variations in occupational segregation by gender, nationality, and age as well as changes in occupational segregation by gender, separately for Swiss and foreign workers. Concluding comments are given in Section 5.

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2. USING CORRESPONDENCE ANALYSIS TO OBSERVE THE CHANGES THAT TOOK PLACE IN SWITZERLAND BETWEEN 1970 AND 2000 IN THE OCCUPATIONAL STRUCTURE BY GENDER AND NATIONALITY To analyze the changes that took place between 1970 and 2000 in the occupational structure by gender and nationality of the labor force in Switzerland, we have applied what is known as CA.

2.1. On Correspondence Analysis CA was introduced by Benzećri (1973) and his French school. It is an exploratory data analytic technique aiming at analyzing simple two-way (or multi-way) tables where some measure of correspondence is assumed to exist between the rows and the columns. CA is extremely useful to transform a set of complex data into quite a simple description of almost all the implicit information provided by the data. A very useful characteristic of CA is that it allows one to obtain a graphical display of row and column points in biplots, which helps discovering some structural relationships that may exist between the variables and the observations.2 Although CA may be defined as a special case of principal components analysis (PCA) of the rows and columns of a table, one should stress that CA and PCA each has specific uses. PCA is a useful tool when one has tables consisting of continuous measurement, whereas CA is typically applied to the case of contingency tables. While the Chi-square test is the usual procedure adopted for analyzing, in a cross-tabulation, the degree of association between rows and columns, this test does not allow us to find out which are the important individual associations between a specific pair of row and column. CA, on the contrary, indicates how the variables are related and not simply whether there is such a link. Assume a contingency table that has I rows and J columns. The plot given by a CA gives then a set of (I þ J) points, I points corresponding to the rows and J points to the columns. If two row points are close, one can then conclude that their conditional distributions across the columns are similar. Given the symmetry of the role played by lines and columns in CA we can


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also conclude that if two column points are close on the biplot provided by the CA, it implies that their conditional distributions across the rows are similar. Like PCA, CA provides the researcher with principal components that are orthogonal. More specifically, each component is a linear combination of the variables, on the one hand, and the observations, on the other. The coefficients of these variables (observations) for the two first components give us, in fact, the coordinates that allow us to plot these variables (observations) in the graph previously mentioned (for more details, see Appendix A and Johnson & Wichern, 1999, Chapter 12).

2.2. Applying Correspondence Analysis to Occupational Data Provided by the 1970 and 2000 Swiss Censuses Table 1 gives the coefficients of the variables (the four ‘‘gender by nationality’’ categories) as well as of the observations (the various two-digit occupations) for the first two components (dimensions) of the CA.3 The results of Table 1 were derived on the basis of the matrices given in Appendix B. Using the sets of coefficients for the two components (dimensions), we were able to plot both the occupations and the ‘‘gender by nationality’’ categories, for both Censuses (those of the years 1970 and 2000). These geometric representations are given in Figs. 1 and 2. We first observe that in both years (both figures), the horizontal axis stresses the gender, making a separation between the males (SM or FM) and the females (SF or FF) while the vertical axis emphasizes the nationality, separating the Swiss (SM or SF) from the foreigners (FM or FF). The analysis also indicates4 that in 1970 the contribution of the first dimension (axis) is 65.5% and that of the second dimension is 25.9%. In 2000 the contribution of the first dimension is 79% and that of the second is 16%. Then we may note that in 1970 the variable FF (foreign females) was very close to the observations 22 and 61, where code 22 refers to textile- or leather-related professions while code 61 refers to professions related to work in hotels or restaurants.5 If we now look at Fig. 2, which gives the graphical representation for the year 2000, we observe that a third occupation, code 62, is now close to the variable SF. Code 62 refers to various professions related to cleanliness or hygiene, such as laundresses and also hairdressers. It hence appears that these professions that in 1970 were more common among Swiss women were in 2000 relatively more common among foreign women. To give another illustration, in 1970, the variable ‘‘foreign men’’ (FM) was close to two occupations whose codes are 41

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Table 1.


Row and Column Coefficients for the First Two Components of the Correspondence Analysis.

Rows (Occupational Code)

11 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 41 42 51 52 53 54 61 62 71 72 73 74 75 81 82 83 84 85 86 91

1970

2000

First Dimension (Component)

Second Dimension (Component)



0.1644 0.4108 0.7548 0.3338 0.5421 0.1543 0.5584 0.2335 0.0339 0.0928 0.6723 0.6817 0.3484 0.5776 0.6736 0.4248 0.8516 0.7819 0.4945 0.0704 0.5505 0.2288 0.8294 0.6549 0.5334 0.5270 0.5829 0.5100 0.5091 0.1178 0.1652 0.2112 0.3675 0.5132 0.7201 0.2460

0.4434 0.0338 0.7466 0.3866 0.2852 0.0257 0.0660 0.0499 0.0946 0.0630 0.0360 0.1319 0.2650 0.3265 0.0552 0.1380 0.5847 0.6703 0.2733 0.2653 0.2796 0.4534 0.4766 0.1397 0.3915 0.2071 0.5045 0.5738 0.4898 0.0395 0.0192 0.1817 0.2564 0.2358 0.0682 0.0104

0.3710 0.3965 0.4907 0.2381 0.7267 0.6541 0.7605 0.2910 0.0187 0.3829 0.7034 0.7582 0.3048 0.6794 0.7508 0.5797 0.8335 0.7666 0.3348 0.1260 0.5491 0.2539 0.4129 0.5959 0.3456 0.6079 0.0380 0.5406 0.1651 0.0836 0.0237 0.5138 0.4078 0.1221 0.7376 0.3126

0.2927 0.0260 0.6702 0.4919 0.2069 0.0278 0.1442 0.0564 0.0222 0.3172 0.1043 0.1853 0.1950 0.1429 0.5610 0.0464 0.2936 0.6320 0.0423 0.1124 0.0446 0.2133 0.5857 0.4933 0.1239 0.1568 0.1748 0.3504 0.3043 0.1060 0.0026 0.1450 0.2330 0.0905 0.0369 0.0221

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Table 1. (Continued ) 1970

Columns (Gender and Nationality)

SM (Swiss males) SF (Swiss females) FM (foreign males) FF (foreign females)

2000





0.3242 0.7461 0.6299 0.8373

0.2199 0.1061 0.6337 0.7230

0.4378 0.6323 0.5184 0.6237

0.1213 0.0999 0.4653 0.5927

Correspondence analysis biplot 1 rows

columns

Dimension 2 (25.9%)

.5

54 51 84 72

52 83

SF 0

11

91 86 62

33 SM 36 29 27 81 21 25 82

28

74 7573 71 34 53 32 31 26 35 85 24

23 61

-.5

41 FM 42

FF22 -1 -1

-.5

0 Dimension 1 (65.5%)

.5

coordinates in principal normalization

Fig. 1.

Geometrical Representation for the Year 1970.

1

178

JOSEPH DEUTSCH ET AL. Correspondence analysis biplot .8 22

Dimension 2 (16.0%)

.6

FF

42 35

61

62

23

FM

.4 rows

columns

29

41 24

.2 28 82

86

0

51 SF 72 83

-.2

81 84

85 52

73

54

75

25 21 91 53 36 27 31 71 SM 34 26 32 33 11 74

-.4 -1

-.5

0

.5

1

Dimension 1 (79.0%) coordinates in principal normalization

Fig. 2.

Geometrical Representation for the Year 2000.

and 42. Code 41 refers to various professions in the building industry, while code 42 refers to profession related to mining or quarrying. In 2000 the three occupations that are now the closest to FM are those with the codes 23, 29, and 35, although codes 41 and 42 are not much further away. Code 23 refers to professions related to work with glass, code 29 to professions related to wrapping activities, and code 35 to ‘‘machinists.’’ Although in 1970 code 23 was also closer to FM than to SM, codes 29 and 35 were clearly closer to SM than to FM. It should now be clear that CA allows us to quickly find out which changes in the occupational structure by gender and nationality took place between the two Censuses.6 What such an analysis does not tell us is whether these changes were a consequence of variations in the margin (in the shares of the genders or nationalities, or in the weights of the professions) of the original data matrix (such as that which appears in Appendix B) or in the degree of dependence between the rows (the occupations) and the columns (the genders or the two categories of nationality). In the next section we present a methodology that allows one to disentangle these three effects.


179

3. SEPARATING THE IMPACT OF CHANGES IN THE MARGINS FROM CHANGES IN THE DEGREE OF DEPENDENCE BETWEEN THE ROWS AND THE COLUMNS Before describing the methodology used to disentangle the three effects previously mentioned, we describe the segregation index that we use and that is a generalization of the famous Duncan index. 3.1. On a Generalization of the Duncan Index The so-called Duncan index (Duncan & Duncan, 1955) remains certainly the most popular measure of occupational segregation by gender (see James & Taeuber, 1985, for a clear presentation of various segregation indices and a thorough discussion of the desirable properties of such indices; see also Reardon & Firebaugh, 2002, for an extension of this discussion to the case of multigroup segregation). The Duncan or dissimilarity index belongs to the family of relative mean deviations with respect to the mean. Despite its extreme popularity, the Duncan index has quite a few shortcomings (see, e.g., Flu¨ckiger & Silber, 1999), and this is why Silber (1989) recommended applying the Gini index to the measurement of occupational segregation. What characterizes the Gini index is that instead of comparing the gender ratios in the various occupations with the overall gender ratio in the labor force (what the Duncan index does), one makes all binary comparisons between the gender ratios in the different occupations. It can be shown (see Flu¨ckiger & Silber, 1999, for more details) that the Gini index of occupational segregation by gender amounts to comparing a set of ‘‘prior shares’’ (giving the distribution of males among the various occupations) with a set of ‘‘a posteriori shares’’ (giving the distribution of females among the various occupations). Such an interpretation of the Ginisegregation index is also at the basis of the well-known segregation curve (see Duncan & Duncan, 1955). The traditional dissimilarity (Duncan) and Gini indices that have just been mentioned are, however, limited to the case where only two categories (e.g., men and women) are compared. One can, however, imagine cases where one is interested in comparing the occupational distribution of more than two categories (e.g., men, single women and married women, or four ethnic groups). To solve this problem, Silber (1992), following a suggestion of Karmel and MacLachlan (1988), proposed a generalization of the

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Duncan index. The basic idea of this generalization is as follows. Assume we know the occupational distribution of the two genders. We have hence a data matrix where the lines i correspond to the various occupations and the columns j to the two genders. The typical element Tij of this matrix will then tell usPhowPmany individuals of gender j are employed in occupation i. Let T ¼ ni¼1 2j¼1 T ij be the total number of individuals in the labor force. The ratio (Tij/T) will then give us the proportion of individuals employed in occupation i and of gender j. This ratio may be also interpreted as the ‘‘posterior probability’’ that an individual of gender j is employed in occupation i. It should, however, be clear that if there is independence between the genders and the occupations, the probability that an individual of gender j will be employed in occupation i will be equal to the product ðT id =TÞðT dj =TÞ, where Tid refers to the total number of individuals employed in occupation i, whatever their gender, and Tdj the total number of individuals of gender j. If there is at least one element (i,j) of the matrix for which there is no identity between the posterior probability (Tij/T) and the prior probability ðT id =TÞðT dj =TÞ, one will conclude that there is occupational segregation by gender. Such an approach may clearly be generalized to the case where there are more than two categories. Such a generalization of the Duncan index will be expressed as I Dg ¼

n X m X

½jðT ij =TÞ ððT id =TÞðT dj =TÞÞj

(1)

i¼1 j¼1

Although the generalization of the Duncan index7 given in Eq. (1) stresses the link between the concept of occupational segregation and that of dependence between the occupations and the categories examined (e.g., the genders), this dependence may be checked on the basis of other indices. Boisso et al. (1994) have thus used a generalization of the Gini index to measure multigroup segregation. Entropy indices like those proposed by Theil (1967) or Mora and Ruiz-Castillo (2003) may also be used. The present study will, however, limit itself to the use of the generalized Duncan index, but this does not imply that this index should be preferred to the generalized Gini or Theil indices.

3.2. The Decomposition of Changes over Time in Occupational Segregation Assume we know the value of one of the segregation indices at times t and tu and that the index decreased over time. It may then be interesting to find out


181

why it decreased. One may think of two sets of causes: either the relative weight of the various occupations varied over time (e.g., the share of occupations where the ratio (Fi/Mi) was very high or very low decreased) or the occupations that were essentially ‘‘female occupations’’ became less ‘‘female’’ ((Fi/Mi) decreased) and/or those that were essentially ‘‘male’’ became less ‘‘male’’ ((Fi/Mi) increased). Such a decomposition stressing the respective role of the occupational weights and the occupation-specific ‘‘gender ratios’’ (Fi/Mi) has been proposed by Blau, Hendricks, and Wallace (1979), who used the Duncan index, and by Boisso, Hayes, Hirschberg, and Silber (1994), who used a similar generalization of the G-segregation index, while Flu¨ckiger and Silber (1999) used both generalizations. Karmel and MacLachlan (1988), however, proposed a different approach based on the idea of a ‘‘marginal free’’ decomposition of the variation in segregation. A segregation index is said to be ‘‘marginal free’’ if it is not affected by changes in the overall gender and occupational composition of the labor force (see Charles, 1992; Blackburn, Jarman, & Siltanen, 1993; Watts, 1998; Flu¨ckiger & Silber, 1999, for more details on this concept). In other words, Karmel and MacLachlan (1988) made a distinction between a variation in occupational segregation by gender that is due to a variation in the relative weights of the occupations or to a change in the overall proportions of men and women in the labor force and a ‘‘net change in segregation,’’ a change that has nothing to do with changes in the relative weights of the occupations or of the genders. In addition to these three changes, there is usually also an interaction term. Flu¨ckiger and Silber (1999), as well as Watts (1998), also adopted Karmel and MacLachlan’s approach. In Appendix D we show that it is possible to combine the Karmel and MacLachlan approach with an income inequality decomposition technique based on the concept of Shapley value, which is used in cooperative game theory (see Chantreuil & Trannoy, 1999; Shorrocks, 1999; Sastre & Trannoy, 2002, for a presentation of this application of the concept of Shapley value). Such decomposition has no interaction term and makes a distinction between three sources of variation. A first impact is the consequence of variations over time in the relative weights of the different occupations. The second effect is the consequence of variations over time in the relative weights of the genders in the total labor force. Finally, a third element measures the ‘‘net change’’ in segregation (net of the two first impacts) and refers to changes in the ‘‘internal structure’’ of the matrix because it is assumed in this case that no change in the margins of the matrix took place. The sum of these three components will then be called ‘‘gross variation’’ in occupational segregation by gender. The approach taken by Karmel and

182


MacLachlan (1988) is based on a technique originally introduced by Deming and Stephan (1940). As shown in Appendix D it is possible to derive a decomposition that is more general than the one introduced by Karmel and MacLachlan (1988). The decomposition that is proposed in Appendix D has the advantage of being systematic and general since it takes into account the possibility that the variation in segregation will be measured by going from period 1 to period 0. Moreover, it is easy to generalize this analysis to the case where the matrix has more than two dimensions (e.g., segregation by occupation, industry, and gender).

4. AN EMPIRICAL ILLUSTRATION: DECOMPOSING CHANGES IN OCCUPATIONAL SEGREGATION BY GENDER, NATIONALITY, AND AGE IN SWITZERLAND BETWEEN 1970 AND 2000 The data sources we used were the Swiss Censuses for the years 1970 and 2000. We first computed the value of the Duncan generalized index of occupational segregation by gender for both years. Then we similarly computed the value of this index for occupational segregation by nationality (Swiss versus foreigners, making no distinction between the genders of the workers). Finally, we repeated the exercise to compute the value of the Duncan generalized index to measure occupational segregation by age, making no distinction between the genders or the nationalities of the workers. Then in each of these three cases, we computed the change observed, over this 30-year period, in the value of the generalized Duncan index, and using the methodology previously described, we decomposed this change into two components, one measuring the changes in the margins, the other the change in the ‘‘internal structure.’’ Finally, the change in the margins was itself broken down into a change in a component measuring the change in the occupational structure and another one measuring the variations in the shares of the subpopulations analyzed. The results of this decomposition are given in Table 2. We also computed bootstrap intervals for the overall change in occupational segregations as well as for the components of these changes. Let us first take a look at occupational segregation by gender. Remembering that the overall variation in the value of the generalized Duncan index will be called ‘‘gross variation in segregation’’ while the change in the internal structure is labeled ‘‘net variation in occupational

183


Table 2. Decomposition of the Change in Switzerland between 1970 and 2000 in the Generalized Duncan Index (Occupational Segregation by Gender, Nationality, or Age).

A. Occupational Segregation by Gender Value of the index in 1970 Value of the index in 2000 Change observed between 1970 and 2000 Component of the change due to variations in the ‘‘internal structure’’ Component of the change due to variations in the ‘‘margins’’ Component due to variation in the occupational structure Component due to variations in the shares of the subpopulations B. Occupational Segregation by Nationality Value of the index in 1970 Value of the index in 2000 Change observed between 1970 and 2000 Component of the change due to variations in the ‘‘internal structure’’ Component of the change due to variations in the ‘‘margins’’ Component due to variation in the occupational structure Component due to variations in the shares of the subpopulations C. Occupational Segregation by Age Value of the index in 1970 Value of the index in 2000 Change observed between 1970 and 2000 Component of the change due to variations in the ‘‘internal structure’’ Component of the change due to variations in the ‘‘margins’’ Component due to variation in the occupational structure Component due to variations in the shares of the subpopulations

Actual Value

Lower Bound (2.5%)

Upper Bound (97.5%)

0.4787 0.4875 0.0088 0.0216

0.4738 0.4836 0.0024 0.0303

0.4845 0.4947 0.0173 0.0163

0.0304

0.0284

0.0369

0.0237

0.0257

0.0189

0.0542

0.0519

0.0581

0.2449 0.1446 0.1003 0.0524

0.2408 0.1412 0.1071 0.0586

0.2509 0.1506 0.0935 0.0456

0.0479

0.0519

0.0434

0.0224

0.0258

0.0184

0.0255

0.0282

0.0232

0.1325 0.0651 0.0673 0.0691

0.1277 0.0610 0.0743 0.0770

0.1382 0.0716 0.0595 0.0608

0.0017

0.0015

0.0058

0.0036

0.0005

0.0075

0.0019

0.0029

0.0009

Note: The bounds are derived from a bootstrap analysis with a 95% confidence interval based on 1,000 random samples.

184


segregation,’’ we observe opposite trends between these gross and net variations. Table 2, part A, indicates a slight increase in gross segregation by gender but a decrease in net segregation. The overall increase (change in gross segregation) occurred because the change in the margins more than compensated that in the internal structure. But even the change in the margins is the consequence of opposite forces. Variations in the occupational structure would have per se led to a decrease in gross segregation. This impact of the occupational structure is probably due to the fact that the share of professions belonging to the ‘‘third sector’’ increased, and in this sector the share of women is relatively high. The change in the relative shares of the genders in the labor force, which clearly is a consequence of an increase in the labor force participation of women, worked in the opposite direction, was stronger in absolute value, and would per se have led to an increase in gross segregation. This is probably a consequence of the fact that the women who entered the labor force between 1970 and 2000 worked in occupations traditionally considered as feminine. Note also that the bootstrap procedure indicates that all the variations and all the components of these variations are statistically significant (different from zero). This first illustration thus clearly shows how important it is to make a distinction between changes in gross and net segregation. The second illustration is given in Table 2, part B. It refers to changes in occupational segregation by nationality. Here we observe first, not surprisingly, that this type of segregation is much smaller than occupational segregation by gender, and second that the variations in gross and net segregation are in the same direction (of an important decrease in segregation). The respective impacts of variations in the ‘‘internal structure’’ and in the margins are quite similar both in magnitude and in sign. Even the respective impacts of variations in the occupational structure and in the shares of the Swiss and foreigners in the labor force are similar in magnitude and sign. Here again all the changes and components of these changes are significantly different from zero. The third illustration concerns occupational segregation by age, a distinction being made between workers, whatever their gender or nationality, younger than 50 and those who are 50 years old or more. Table 2, part C, indicates that there was more than a halving of this type of occupational segregation (which in any case is much smaller than the two other ones) and that the whole change was a consequence of a change in the ‘‘internal structure.’’ In other words, in this third illustration the picture given by changes in gross segregation is identical to that given by changes in net segregation. Here again all the results are significant.

185


In Table 3 we look only at occupational segregation by gender, but we make a distinction between Swiss and foreign workers. It then appears that among Swiss workers there was an increase in the gross variation in occupational segregation by gender between 1970 and 2000 but a decrease in the net variation in this occupational segregation. So here is another illustration of the usefulness of making a distinction between gross and net variations. The data thus show that among Swiss workers, between 1970 and 2000, ‘‘pure’’ occupational segregation by gender decreased. This

Table 3. Decomposition of the Variation in Switzerland between 1970 and 2000 of the Degree Occupational Segregation by Gender, Separately for Swiss and Foreign Workers (Based on the Use of the Generalized Duncan Index).

A. Swiss Workers Value of the index in 1970 Value of the index in 2000 Change observed between 1970 and 2000 Component of the change due to variations in the ‘‘internal structure’’ Component of the change due to variations in the ‘‘margins’’ Component due to variation in the occupational structure Component due to variations in the shares of the subpopulations B. Foreign Workers Value of the index in 1970 Value of the index in 2000 Change observed between 1970 and 2000 Component of the change due to variations in the ‘‘internal structure’’ Component of the change due to variations in the ‘‘margins’’ Component due to variation in the occupational structure Component due to variations in the shares of the subpopulations

Actual Value

Lower Bound (2.5%)

Upper Bound (97.5%)

0.4683 0.4905 0.0223 0.0224

0.4634 0.4857 0.0151 0.0299

0.4738 0.4965 0.0297 0.0159

0.0447

0.0413

0.0496

0.0114

0.0146

0.0073

0.0561

0.0535

0.0593

0.5210 0.4705 0.0505 0.0269

0.5167 0.4659 0.0577 0.0344

0.5268 0.4762 0.0429 0.0204

0.0235

0.0286

0.0181

0.0617

0.0658

0.0572

0.0382

0.0345

0.0419

Note: The bounds are derived from bootstrap a 95% confidence interval based on 1,000 random samples.

186


decrease, however, was more than compensated by variations in the ‘‘margins,’’ that is, by changes in the occupational structure or in the shares of the genders in the overall Swiss labor force. Note, however, that changes in the occupational structure per se would have led to a decrease in the overall segregation by gender, but this variation was more than compensated by changes in the shares of men and women in the labor force (of workers having the Swiss nationality) so that the total impact of changes in the ‘‘margins’’ is an increase in segregation. Among foreign workers the results are simpler in the sense that variations both in the ‘‘internal structure’’ and in the ‘‘margins’’ were of almost equal magnitude and of equal sign and both led to a decrease in segregation. Note, however, that here also variations in the occupational structure would have per se led to a decrease in segregation, while changes in the shares of the genders would have per se led to an increase in segregation, the magnitude (in absolute value) of this latter change being higher than that of the former. Note that here again all the changes and components of the changes are statistically different from zero. The illustrations given in Tables 2 and 3 indicate, therefore, quite clearly how useful the methodology, originally proposed by Karmel and MacLachlan and extended in the present study, is and how important it is to make a distinction between gross and net variations in occupational segregation.

5. CONCLUDING COMMENTS This paper attempted to analyze changes in occupational segregation by gender, nationality, and age that took place in Switzerland between 1970 and 2000. It first used CA to detect graphically some important variations in the degree of proximity between specific occupations and each of the four categories distinguished (male and female Swiss workers and male and female foreign workers). Then in the second stage, the paper showed how the decomposition procedure originally proposed by Karmel and MacLachlan (1988) could be extended and applied, for example, to what may be called the ‘‘generalized Duncan index.’’ The idea is to combine their approach with what is now known as the Shapley decomposition. Such a generalization offers a clear breakdown of the variation over time in occupational segregation (change in gross segregation) into a component measuring changes in net segregation and another one corresponding to


187

changes in the margin, the latter itself including variations in the occupational structure and in the shares of the subpopulations (e.g., the genders) in the labor force. At the difference of the approach of Karmel and MacLachlan (1988), such decomposition includes no interaction term.8 This decomposition may easily be extended to the cases where more than two categories are distinguished (e.g., several ethnic groups) or when there are more than three dimensions (e.g., segregation by gender, occupation, and economic branch). The results of the empirical illustration, which looked at Swiss data for the years 1970 and 2000, proved the usefulness of such an approach. They showed in particular that in several instances variations in gross and net segregation worked in opposite directions, the same being sometimes true of changes in the two margins, that referring to variations in the occupational structure, and that measuring variations in the relative shares of the subpopulations. Finally, it should be stressed that although this study selected as segregation measure the generalization of the Duncan index, other indices could have been used, such as the Gini multigroup segregation index or information theory–based entropy indices. It is certainly true that the approach presented here advocates the use of indices to compare the actual distribution among occupations of individuals belonging to various categories with what this distribution would have been in the case of independence between the occupations and the categories distinguished. Log linear or multiplicative models take an alternative strategy, one that attempts to develop a model that tries to fit the data. We certainly hope, in future work, to compare, if not combine, these alternative strategies. Most sociologists seem to have already made a choice that favors the use of the log linear or multiplicative models. The latter approach is, for the time being at least, less popular among economists although it seems clear that in the near future, attempts will be made to combine the more descriptive approach based on the use of indices with one that is based on statistical inference.

NOTES 1. It seems that in recent years the sociological literature on the analysis of occupational segregation has given the preference to the use of log linear or log multiplicative models derived from contingency tables analysis. Given that there is already quite a vast literature applying such a technique (e.g., Charles, 1992; Charles & Grusky, 1995; Weeden, 1998, 2004; Weeden & Sorensen, 2004), we preferred using an alternative and probably less known technique, which is also derived from contingency tables, namely correspondence analysis. In addition, the emphasis

188


in this paper is still on the use of segregation indices. Given the lasting popularity of the Duncan index, we stress the possibility of extending the use of such an index to cases where more than two categories are used and show how the so-called Shapley decomposition, when mixed with the Deming and Stephan algorithm, may give some very useful insight into the sources of the variation in occupational segregation. 2. See Appendix A for more details on this technique. 3. As mentioned previously, it does not really matter whether we call observations the occupations and variables the gender by nationality categories, or the reverse. 4. These results were not included in Table 1 but may be obtained on request from the authors. 5. The Swiss codification of professions is in fact a five-digit classification. In the present paper, we used only the two first numbers of this code, that is, a two-digit classification. Unfortunately there is no available translation in English of the detailed five-digit classification. Moreover, there is no official two-digit classification, whether in German, French, or Italian (the three main languages spoken in Switzerland). We have, however, tried to give names to these various two-digit occupations. This list of occupations is given in Appendix C. The more detailed list of five-digit occupations (in German, French, or Italian) may be obtained from the authors on request. 6. It should be stressed that correspondence analysis treats ordinal categories as unordered. In other words, the ordering of the rows/columns does not matter when constructing the biplot. As a consequence, correspondence analysis cannot, for example, take into account directly the prestige of occupations. However, if, say, the two-digit codes we used were ordered according to prestige scores, and if, as is the case in Fig. 1, the horizontal axis stresses the gender and the vertical axis the nationality, we could draw the following type of conclusions. First, if more prestigious occupations were located more on the right of the graph, we would know that male workers are more likely to belong to prestigious occupations. Similarly, if more prestigious occupations were located more often in the upper part of the graph (in Fig. 1), we could conclude that it is more common to find Swiss than foreigners in prestigious occupations. 7. Note that IDg is not standardized. It is, however, possible to divide it by its maximal value (which occurs when there is perfect segregation; that is, when in any occupation there is only one category). Thus, in the case of gender (two categories), its maximal value will be equal to 4MF/(M þ F)2. 8. Some may view the absence of interaction term as a shortcoming rather than an advantage, but we believe that one of the most attractive features of the Shapley decomposition procedure is that it takes into account all possible orders of eliminations of variables and should thus be considered superior to alternative breakdown techniques.

REFERENCES Anker, R. (1998). Gender and jobs: Sex segregation of occupations in the world. Geneva: International Labor Organization.


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Benzećri, J.-P. (1973). L’analyse des donneés. Tome 2. L’analyse des correspondances. Paris: Dunod. Blackburn, R. M., Jarman, J., & Siltanen, J. (1993). The analysis of occupational gender segregation over time and place: Considerations of measurement and some new evidence. Work, Employment and Society, 7. Blau, F., Hendricks, D., & Wallace, E. (1979). Occupational segregation by sex: Trends and prospects. Journal of Human Resources, 14(2), 197–210. Boisso, D., Hayes, K., Hirschberg, J., & Silber, J. (1994). Occupational segregation in the multidimensional case: Decomposition and tests of statistical significance. Journal of Econometrics, 61, 161–171. Butler, R. J. (1987). New indices of segregation. Economics Letters, 24, 359–362. Chantreuil, F., & Trannoy, A. (1999). Inequality decomposition values: The trade-off between marginality and consistency. THEMA Discussion Paper, Universite´ de Cergy-Pontoise. Charles, M. (1992). Cross-national variation in occupational sex segregation. American Sociological Review, 57(4), 482–503. Charles, M., & Grusky, D. B. (1995). Models for describing the underlying structure of sex segregation. American Journal of Sociology, 100(4), 931–971. Deming, W. E., & Stephan, F. F. (1940). On a least squares adjustment of a sampled frequency table when the expected marginals are known. Annals of Mathematical Statistics, 11, 427–444. Deutsch, J., & Silber, J. (2005). Comparing segregation by gender in the labour force across ten European countries in the late 1990s: An analysis based on the use of normative segregation indices. International Journal of Manpower, 26(3), 237–264. Duncan, O. D., & Duncan, B. (1955). A methodological analysis of segregation indices. American Sociological Review, 20, 210–217. Flu¨ckiger, Y., & Silber, J. (1999). The measurement of segregation in the labor force. Heidelberg: Physica Verlag. James, D. R., & Taeuber, K. E. (1985). Measures of segregation. Sociological Methodology, 15, 1–32. Johnson, R. A., & Wichern, D. W. (1999). Applied multivariate statistical analysis. Upper Saddle River, NJ: Prentice Hall. Karmel, T., & MacLachlan, M. (1988). Occupational sex segregation – increasing or decreasing. Economic Record, 64, 187–195. Mora, R., & Ruiz-Castillo, J. (2003). Additively decomposable segregation indexes. The case of gender segregation by occupations and human capital levels in Spain. Journal of Economic Inequality, 1, 147–179. Reardon, S. F., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32, 33–67. Sastre, M., & Trannoy, A. (2002). Shapley inequality decomposition by factor components: Some methodological issues, Journal of Economics, Suppl. 9, 51–89. Shorrocks, A. F. (1999). Decomposition procedures for distributional analysis: A unified framework based on the Shapley value. Mimeo, University of Essex. Silber, J. (1989). On the measurement of employment segregation. Economic Letters, 30, 237–243.

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Silber, J. (1992). Occupational segregation indices in the multidimensional case: A note. Economic Record, 68, 276–277. Theil, H. (1967). Economics and information theory. Amserdam: North-Holland. Watts, M. (1998). Occupational gender segregation: Index measurement and econometric modeling. Demography, 35(4), 489–496. Weeden, K. A. (1998). Revisiting occupational segregation in the United States, 1910–1990: Results from a log-linear approach. Demography, 35(4), 475–487. Weeden, K. A. (2004). Profiles of changes: Sex segregation in the United States, 1910–2000. In: M. Charles & D. B. Grusky (Eds), Occupational ghettos: The worldwide segregation of men and women (pp. 131–178). Palo Alto, CA: Stanford University Press. Weeden, K. A., & Sorensen, J. B. (2004). A framework for analyzing industrial and occupational sex segregation in the United States. In: M. Charles & D. B. Grusky (Eds), Occupational ghettos: The worldwide segregation of men and women (pp. 245–296). Palo Alto, CA: Stanford University Press.

APPENDIX A. A QUICK SUMMARY OF THE MAIN FEATURES OF CORRESPONDENCE ANALYSIS Let X be an I J two-way frequency table whose typical element xij gives the number of observations that belong to the row category i as well as to the column category j. The rows and columns of such a table correspond, therefore, to two types of characteristics, in our illustration to occupations and ‘‘gender by nationality’’ categories. Let n refer to the total number of frequencies in matrix X. Let us now build a matrix of proportions P whose P typical element pP ij is defined as pij ¼ ðxij =nÞ. Let us also define ri as ri ¼ Jj¼1 pij and cj ¼ Ii¼1 pij . Let us also define a centered matrix P~ whose typical element p~ij is defined as p~ij ¼ pij ðri cj Þ. We now define two diagonal matrices Dr and Dc as Dr ¼ diag ðr1 ; . . . ; rI Þ and Dc ¼ diag ðc1 ; . . . ; cJ Þ. In the next step we construct the scaled matrix P where P ¼ ð1=2Þ ~ ð1=2Þ PDc Dr element p ij is defined as whose pffiffiffiffiffiffiffiffiffiffi typical

pij ¼ ðpij ðri cj ÞÞ= ðri cj Þ . We will now explain the various steps leading to the geometric representation given by CA.

191


First one has to find the singular value decomposition of the matrix P; that is, we write that P ¼ ULV 0

(A.1)

with UU 0 ¼ V 0 V ¼ I, I being the identity matrix, while the diagonal matrix L ¼ diag ðl1 ; . . . ; lJ1 Þ contains the singular values, ordered from the largest to the smallest along the main diagonal of this matrix. U and V~ ¼ Dð1=2Þ V. Second we define the matrices U~ and V~ as U~ ¼ Dð1=2Þ r c One can then show that ~ V~ 0 ¼ P~ ¼ P rc0 ¼ UL

J 1 X

lj u~j v~0j

(A.2)

j¼1

~ where u~j is the jth column vector of U~ and vj the jth column vector of V. One can also prove that 0 ~ ~ 10 ~ U~ D1 r U ¼ VDc V ¼ I

(A.3)

The columns of U~ give the coordinate axes for the points representing the column profiles of P. Similarly the columns of V~ define the coordinate axes for the points representing the row profiles of P. Third we can now compute the coordinates of the row profiles ~ Y ¼ D1 r UL

(A.4)

and the coordinates of the column profiles ~ Z ¼ D1 c VL

(A.5)

The first two columns of Y contain the pairs of coordinates of the row points in the best two-dimensional representation of the data. Similarly the first two columns of Z contain the pair of coordinates of the column points in the best two-dimensional representation of the data. The points corresponding to these two sets of coordinates can then be plotted on the same graph and give the kind of graph that appears in Figs. 1 and 2. Source: We borrowed the main elements of this summary from Johnson & Wichern (1999, pp. 773–774).

192


APPENDIX B. ORIGINAL DATA ON THE OCCUPATIONAL STRUCTURE BY GENDER AND NATIONALITY IN SWITZERLAND IN 1970 AND 2000 The Data for 1970 Occupation Swiss Males Swiss Females Foreign Males Foreign Females

Total

11 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 41 42 51 52 53 54 61 62 71 72 73 74 75 81 82 83 84 85 86 91 Total

169859 35848 17828 1299 134733 79884 43022 21131 18472 43683 19711 32586 18176 53410 21169 7334 107311 4726 84201 10492 107838 24016 27847 25427 50624 138769 7096 16825 6028 7333 15367 7895 33948 7132 18081 595

52189 3683 37019 300 12275 27238 2816 5733 7945 15565 216 71 2621 1722 302 1000 160 186 111469 4317 6207 17568 104043 43216 2522 186452 93 697 319 2672 5411 5857 34945 960 39594 433

9546 8594 13574 785 71521 20832 15266 5310 5612 9102 6058 8185 2262 7064 7774 1618 106153 5275 7076 1380 17156 329 24246 6015 4953 12033 363 257 312 1787 4303 1118 3470 3676 4612 113

1007 3073 37116 243 11931 8343 2142 1793 3493 5344 260 132 861 407 464 276 163 220 7885 474 800 321 50085 14860 345 22903 7 42 22 815 1450 894 3049 488 9644 139

232601 51198 105537 2627 230460 136297 63246 33967 35522 73694 26245 40974 23920 62603 29709 10228 213787 10407 210631 16663 132001 42234 206221 89518 58444 360157 7559 17821 6681 12607 26531 15764 75412 12256 71931 1280

1419696

737816

397730

191491

2746733

193


The Data for 2000 Occupation Swiss Males Swiss Females Foreign Males Foreign Females

Total

11 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 41 42 51 52 53 54 61 62 71 72 73 74 75 81 82 83 84 85 86 91 Total

67268 17232 2578 394 63792 42694 24502 10394 8783 24420 42939 30074 10000 35112 12090 51803 92686 965 77139 34800 71221 16003 26417 18996 139305 84813 32158 27318 10613 14919 16704 14961 52017 9077 33214 10693

20631 4872 6295 213 3571 4007 936 4328 6731 7441 2719 1085 3766 2842 807 7503 1872 87 108468 18799 13714 19311 61967 42115 45528 220993 21199 4427 5115 12750 13118 32172 86731 5386 125758 3856

6129 4268 2425 280 24235 10738 4079 2452 2139 11104 7615 4256 1429 5652 9094 10924 45375 874 14337 5448 17181 1872 26841 7245 23226 11938 4180 1517 662 2460 3977 2524 5889 2770 8473 2377

1096 1289 3807 93 2560 1649 223 525 1425 3541 963 301 327 651 640 1925 436 29 19042 3165 1515 1685 29230 23814 7092 23465 2594 350 430 1889 2469 3373 6322 1273 24226 812

95124 27661 15105 980 94158 59088 29740 17699 19078 46506 54236 35716 15522 44257 22631 72155 140369 1955 218986 62212 103631 38871 144455 92170 215151 341209 60131 33612 16820 32018 36268 53030 150959 18506 191671 17738

1228094

921113

295985

174226

2619418

194


APPENDIX C. LIST OF TWO-DIGIT OCCUPATIONS Occupational Code 11 21 22 23 24 25 26 27 28 29 31 32 33 34 35 36 41 42 51 52 53 54 61 62 71 72 73 74 75 81 82 83 84 85 86 91

Description of Occupation

Professions related to the agricultural sector Food-related professions (e.g., baker) Textile- and leather-related professions Professions related to work with glass Professions related to work with metal Professions dealing with electricity, electronics, watches, and cars Professions related to work with wood or cardboard Printing-related professions Work with chemicals or plastic Professions related to packing Architects and engineers Technicians in various branches (e.g., electronics, construction, and textile,) Drawers and draftsmen Foremen Professions related to work with machines Computer scientists and programmers Professions in building and civil engineering Professions related to mining and quarrying Salesmen and related professions Professions in marketing, public relations, tourism, and accounting Profession in sector of transportation Profession related to post office Professions in sector of hotels and restaurants Professions related to cleaning and hygiene Entrepreneurs, senior and middle managers, civil servants of high level Specialists of import–export, real estate, and related professions Banking- and insurance-related professions Soldiers, policemen, firemen, and other security-related professions Judges, lawyers, and notaries Journalists, translators, librarians, and photographers Artists, dancers, sculptors, and similar professions Social workers and priests University professors and teachers Historians, biologists, physicists, mathematicians, psychologists, geographers, etc. Physicians, pharmacists, and other medical and paramedical professions Professions related to sport and entertainment

As mentioned previously, there is no official list of these two-digit occupations. We derived such a list on the basis of the list of five-digit occupations currently used in Switzerland (list that exists only in German, French, or Italian). The title we gave for each occupation should hence be considered with caution and was not approved by any official Swiss institution.

195


APPENDIX D. THE DECOMPOSITION OF CHANGES OVER TIME IN THE SEGREGATION INDEX To simplify, let pij ; pid , and pdj refer, respectively, to the ratios ðT ij =TÞ; ðT id =TÞ, and ðT dj =TÞ defined previously. Since the product ðpid pdj Þ is in fact equal to the product of the margins i and j of the matrix {pij} whose typical element is pij, we will call qij the product ðpid pdj Þ. The generalized Duncan index may then be expressed as I Dg ¼ hðpij ; qij Þ ¼

m X 2 X

jpij qij j

(D.1)

i¼1 j¼1

To compare occupation segregation by gender at two periods 0 and 1, Karmel and MacLachlan (1988), following Deming and Stephan (1940), proceeded as follows. The idea, when comparing two matrices of proportions {pij} and {vij}, is to build a third matrix {sij} that will have the ‘‘internal structure’’ of the matrix {pij} but the margins of the matrix {vij}. To derive {sij} one has to multiply first all the elements (pij) of the matrix {pij} by the ratios ðvid =pid Þ, where vid and pid refer to the horizontal margins of the matrices {pij} and {vij}. Call {xij} the matrix you get after such a multiplication. Then multiply all the elements (xij) of this matrix {xij} by the ratios ðvdj =xdj Þ, where vdj and xdj refer to the vertical margins of the matrices {vij} and {xij}. Call {yij} the matrix you get after this second multiplication. If we continue this procedure, multiplying now the elements (yij) of the matrix {yij} by the ratios ðvid =yid Þ where ðvid Þ and ðyid Þ are the horizontal margins of the matrices {vij} and {yij}, and so on, the matrices one successively derives will converge, as proven by Deming and Stephan (1940), toward a matrix {sij} that will have the margins of the matrix {vij} but the internal structure of the matrix {pij}. We could evidently have started with the matrix {vij} and ended up with a matrix {wij} that would have the margins of the matrix {pij} but the internal structure of the matrix {vij}. As was previously explained, this transition from a matrix {pij} to a matrix {vij} includes in fact two stages: a first one where only the margins of the matrix {pij} vary and a second one where the internal structure of this matrix is modified. Call DI ¼ IðvÞ IðpÞ the overall change between two periods in the degree of occupational segregation by gender, I being the generalized Duncan index IDg. DI may also be expressed as DI ¼ f ðDm; DsÞ, where Dm and Dis measure, respectively, the change in the margins and the variation in the internal structure of the original matrix.

196


We can now borrow a decomposition technique based on the concept of Shapley value that was originally suggested by Chantreuil and Trannoy (1999), extended by Shorrocks (1999), and applied by Sastre and Trannoy (2002).

A Short Summary of the Simplest Application of the Concept of Shapley Decomposition Let F(a,b) be a function depending on two variables a and b. Such a function need not be linear. Although Chantreuil and Trannoy (1999) and Sastre and Trannoy (2002) limited their application of the Shapley value to the decomposition of income inequality, Shorrocks (1999) has shown that such a decomposition could be applied to any function. The idea of the Shapley value is to consider all the possible sequences allowing us to eliminate the variables a and b. Let us start with the elimination of the variable a. This variable may be the first one or the second one to be eliminated. If it is eliminated first, the function F(a,b) will become equal to F½ða ¼ 0Þ; ðba0Þ since the variable a has been eliminated, so that in this case the contribution of a to the function Fða; bÞ is equal to F½ðaa0Þ; ðba0Þ F½ða ¼ 0Þ; ðba0Þ. If the variable a is the second one to be eliminated, the function F will then be equal to F½ðaa0Þ; ðb ¼ 0Þ. Since both elimination sequences are possible and assuming the probability of these two sequences is the same, we may conclude that the contribution C(a) of the variable a to the function F(a,b) is equal to 1 F½ðaa0Þ; ðba0Þ F½ða ¼ 0Þ; ðba0Þ CðaÞ ¼ 2 1 þ F½ðaa0Þ; ðb ¼ 0Þ F½ða ¼ 0Þ; ðb ¼ 0Þ 2 (D.2) 1 ¼ F½ðaa0Þ; ðba0Þ F½ða ¼ 0Þ; ðba0Þ 2 1 þ F½ðaa0Þ; ðb ¼ 0Þ 2 since we assume that F½ða ¼ 0Þ; ðb ¼ 0Þ ¼ 0.


197

Similarly, one can prove that the contribution C(b) of the variable b to the function Fða; bÞ is 1 F½ðaa0Þ; ðba0Þ F½ðaa0Þ; ðb ¼ 0Þ CðbÞ ¼ 2 1 þ F½ða ¼ 0Þ; ðba0Þ F½ða ¼ 0Þ; ðb ¼ 0Þ 2 (D.3) 1 ¼ F½ðaa0Þ; ðba0Þ F½ðaa0Þ; ðb ¼ 0Þ 2 1 þ F½ða ¼ 0Þ; ðba0Þ 2 Combining (D.2) and (D.3) we observe that CðaÞ þ CðbÞ ¼ F½ðaa0Þ; ðba0Þ

(D.4)

Applying Shapley’s Decomposition to the Analysis of Variations over Time in the Value of the Generalized Duncan Index Using expressions (D.2)–(D.4), we may express the contribution CDm of the variations in the margins to the overall change DI in occupational segregation by gender as 1 f ½ðDma0Þ; ðDis ¼ 0Þ C Dm ¼ 2 1 þ f½f ½ðDma0Þ; ðDisa0Þ f ½ðDm ¼ 0Þ; ðDisa0Þ ðD:5Þ 2 where Dm and Dis refer, respectively, to the change in the margins and to that in the internal structure of the original matrix. Similarly, the contribution CDis of the variation in the internal structure of the matrix to the overall change DI in occupational segregation by gender will be 1 f ½ðDm ¼ 0Þ; ðDisa0Þ C Dis ¼ 2 1 þ f½f ½ðDma0Þ; ðDisa0Þ f ½ðDma0Þ; ðDis ¼ 0Þ ðD:6Þ 2 It is easy to observe that C Dm þ C Dis ¼ DI.

198


Using the various matrices that were defined previously, one may prove that the contributions CDm and CDis may also be expressed as 1 C Dm ¼ f½IðsÞ Ið pÞ þ ½IðvÞ IðwÞg (D.7) 2 C Dis ¼

1 f½IðwÞ Ið pÞ þ ½IðvÞ IðsÞg 2

(D.8)

so that, as expected, C Dm þ C Dis ¼ IðvÞ IðpÞ

(D.9)

If we now apply the concept of ‘‘Nested Shapley Decomposition,’’ as suggested by Sastre and Trannoy (2002), we can also decompose the contribution CDm into two components corresponding, respectively, to the contributions of the horizontal and vertical margins. The idea is to derive first a matrix l that would have the internal structure of the matrix p, the vertical margins of this same matrix p, but the horizontal margins of the matrix v. We therefore need to build a matrix that will have the vertical margins of the matrix p and the horizontal margins of the matrix v. There are many such matrices, including a matrix n where each element nij is equal to the product of the margins vdj and pid . If we now apply the technique proposed by Deming and Stephan (1940) to the case where the original matrix is p and the final matrix is n, the matrix p will converge toward a matrix l that will have the internal structure of the matrix p, the vertical margins of this same matrix p, but the horizontal margins of the matrix v since the matrix n has the horizontal margins of the matrix v and the vertical margins of the matrix p. We can use the same procedure to define: a matrix k that will have the internal structure of the matrix p, the vertical margins of the matrix v, and the horizontal margins of the matrix p; a matrix c that will have the internal structure of the matrix v, the vertical margins of the matrix v, and the horizontal margins of the matrix p; a matrix f that will have the internal structure of the matrix v, the vertical margins of the matrix p, and the horizontal margins of the matrix v. Let Dm1 be the change defined as Dm1 ¼ ½IðsÞ Ið pÞ

(D.10)


199

Since the matrices s and p that were previously defined have the same internal structure (is), we may write that Dm1 ¼ g½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ

(D.11)

where Dh and Dt correspond to horizontal and vertical variations of the margins. The contributions CDh1 and CDf1 of changes in the horizontal and vertical margins to the difference Dm1 may therefore be expressed as C Dh1

1 g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ ¼ 2 1 þ f g½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ 2

(D.12)

g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ and C Dt1

1 ¼ g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ 2 1 fg½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ g½ðDha0Þ; þ 2 ðDt ¼ 0Þ; ðDis ¼ 0Þ

ðD:13Þ

We must now derive the expressions corresponding to g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ and g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ. Using the definitions of the matrices l and p that were given previously we conclude that g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ ¼ IðlÞ Ið pÞ

(D.14)

Similarly, we derive that g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ ¼ IðkÞ Ið pÞ

(D.15)

200


Combining expressions (D.12)–(D.14), we derive 1 g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ C Dh1 ¼ 2 1 þ g½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ 2 1 1 ¼ ½IðlÞ IðpÞ þ ½IðsÞ IðpÞ ½IðkÞ IðpÞ 2 2 1 1 ¼ ½IðlÞ IðpÞ þ ½IðsÞ IðkÞ 2 2 and C Dt1

1 ¼ g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ 2 1 f g½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ þ 2 1 1 ¼ ½IðkÞ IðpÞ þ f½IðsÞ Ið pÞ ½IðlÞ Ið pÞg 2 2 1 1 ½IðkÞ IðpÞ þ ½IðsÞ IðlÞ ðD:17Þ ¼ 2 2

It is easy to observe that C Dh1 þ CDt1 ¼ IðsÞ Ið pÞ

(D.18)

Let us now similarly decompose the difference Dm2 defined previously as being equal to Dm2 ¼ ½IðvÞ IðwÞ

(D.19)

Given the definitions of the matrices c and f, we define the contributions C Dh2 and C Dt2 of the horizontal and vertical margins to the difference Dm2


201

as being equal to 1 C Dh2 ¼ g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ 2 1 fg½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ þ 2 1 1 ½IðvÞ IðcÞ þ f½IðvÞ IðwÞ ½IðvÞ Iðf Þg ¼ 2 2 1 1 ¼ ½IðvÞ IðcÞ þ ½Iðf Þ IðwÞ ðD:20Þ 2 2 and C Dt2

1 ¼ g½ðDh ¼ 0Þ; ðDta0Þ; ðDis ¼ 0Þ 2 1 fg½ðDha0Þ; ðDta0Þ; ðDis ¼ 0Þ g½ðDha0Þ; ðDt ¼ 0Þ; ðDis ¼ 0Þ þ 2 1 1 ½IðvÞ Iðf Þ þ f½IðvÞ IðwÞ ½IðvÞ IðcÞg ¼ 2 2 1 1 ¼ ½IðvÞ Iðf Þ þ ½IðcÞ IðwÞ ðD:21Þ 2 2

It easy to observe that C Dh2 þ C Dt2 ¼ IðvÞ IðwÞ

(D.22)

Combining now equations (D.5)–(D.22), we conclude that the contribution CDm may be defined as being equal to C Dm ¼ Ch þ Ct

(D.23)

where 1 ½C Dh1 þ C Dh2 2 1 1 ½f½IðlÞ IðpÞ þ ½IðsÞ IðkÞg ¼ 2 2

Ch ¼

þ f½IðvÞ IðcÞ þ ½Ið f Þ IðwÞg

ðD:24Þ

202

and


1 ½C Dt1 þ C Dt2 2 1 1 ½f½IðkÞ Ið pÞ þ ½IðsÞ IðlÞg ¼ 2 2

Ct ¼

þ f½IðvÞ Ið f Þ þ ½IðcÞ IðwÞg

ðD:25Þ

Combining (D.23), (D.24), and (D.25) we finally derive that, as expected, 1 f½IðsÞ Ið pÞ þ ½IðvÞ IðwÞg (D.26) C Dm ¼ 2

PART V WAGE INEQUALITY AND SEGREGATION

WAGE INEQUALITY AND SEGREGATION BETWEEN NATIVE AND IMMIGRANT WORKERS IN SWITZERLAND: EVIDENCE USING MATCHED EMPLOYEE–EMPLOYER DATA Tobias Mu¨ller and Jose´ Ramirez ABSTRACT Purpose – We analyze segregation between immigrants and natives at the firm level and explore the connection between segregation and wage inequality in Switzerland. Methodology/Approach – Our approach accounts for the interaction between skill level and immigration status (work permit). First, we calculate exposure rates in order to analyze segregation at the firm level along these two dimensions. Second, we examine the role of segregation in the explanation of wage inequality between different skill–nationality groups. We use data from the Swiss Wage Structure Survey 2002, an employer–employee database that records individual wages among a very large sample of establishments in all industries, covering approximately 42,000 firms and 1 million workers. Occupational and Residential Segregation Research on Economic Inequality, Volume 17, 205–243 Copyright r 2009 by Emerald Group Publishing Limited All rights of reproduction in any form reserved ISSN: 1049-2585/doi:10.1108/S1049-2585(2009)0000017014

205

206

TOBIAS MU¨LLER AND JOSE´ RAMIREZ

Findings – Our results show that interfirm segregation is particularly pronounced for unskilled foreign workers and for recently arrived, highly skilled foreigners. The former earn lower wages than equally skilled Swiss workers, and the latter are paid higher wages than highly skilled Swiss workers. In both cases, interfirm segregation accounts for almost the entire wage differential. Originality/Value of paper – This paper presents a generalization of the approach used by Groshen (1991) to the multigroup case by defining segregation with respect to the two dimensions of nationality and skill. The use of multigroup exposure rates is common in studies of neighborhood segregation (e.g., Bayer et al., 2004), but our paper shows that they can also be fruitfully applied in the analysis of interfirm segregation and wage inequality.

1. INTRODUCTION In many European countries, immigration policy has undergone major changes in recent years. Even traditional guest-worker countries, such as Germany or Switzerland, try to attract more highly skilled immigrants and to limit the arrival of unskilled workers. While this policy change has been relatively successful in Switzerland, an important number of unskilled workers continue to arrive because of the inertia produced by migration laws (e.g., family reunion). As a result, the group of recently arrived immigrants is highly heterogeneous with respect to skill. What is the situation of this new generation of immigrants in the labor market? In the past, several empirical studies documented the fact that immigrant workers are, on average, paid lower wages than natives. Moreover, it has often been observed that unskilled immigrants hold jobs that native workers would not accept. Does this still hold true with the recent changes in immigration policy? This paper seeks to answer these questions by analyzing wage inequality and segregation between natives and immigrants. We focus on the role of segregation at the firm level between natives and immigrants and explore the connection between segregation and wage inequality. In order to take the skill heterogeneity of the immigrant population into account, we analyze segregation as a two-dimensional phenomenon, considering the dimensions of skill and nationality (or type of work permit). The interaction between these two dimensions turns out to be crucial for the decomposition of wage inequality.

Wage Inequality and Segregation

207

To our knowledge, there has been little research on the link between segregation and the native/immigrant wage gap. Liu, Zhang, and Chong (2004) analyze the link between occupational segregation and wage inequality between immigrants and natives in Hong Kong, using Brown, Moon and Zoloth’s (1980) methodology. Several other papers focus on the related issue of segregation between ethnic groups. Neuman and Silber (1996) explore the role of occupational segregation in the wage differential between the two main ethnic groups in Israel. They expand Oaxaca’s (1973) decomposition method in order to account also for occupational segregation. Hirsch and Schumacher (1992) and Hirsch and Macpherson (2004) introduce a racial density variable in Mincer-type wage equations and evaluate its contribution to the racial wage gap. It should be noted that these papers define segregation at the occupational level (Hirsch and Schumacher define segregation within industry–occupation–region cells). Our approach to segregation is closer to that of Carrington and Troske (1998), who discuss segregation between black and white workers at the firm level. The concept of interfirm segregation is interesting as it captures the actual interactions between different population groups in the workplace. When decomposing the black/white wage gap, these authors find that interfirm segregation accounts only for a small share of the wage gap. Our paper draws also on the approach adopted by Groshen (1991) and Bayard, Hellerstein, Neumark, and Trotske (2003) in order to determine the contribution of segregation to the explanation of the male/female wage gap. In her decomposition procedure, Groshen uses a segregation index that turns out to be based on male/female exposure rates. Our paper generalizes this approach to the multigroup case by defining segregation with respect to the two dimensions of nationality and skill. Exposure rates are particularly well suited for such a generalization since they have a natural interpretation in the multigroup case. The use of multigroup exposure rates is common in studies of neighborhood segregation (e.g., Bayer, McMillan, & Rueben, 2004), but, to our knowledge, they have never been used in the context of interfirm segregation. Although exposure rates can be aggregated into a single-valued segregation index (James, 1986), the link between segregation and wage inequality can be modeled in a more satisfactory way by using the full information on exposure rates by skill/nationality groups.1 This is crucial if one wants to model the link between segregation and wage inequality at both ends of the skill distribution. We use data from the Swiss Wage Structure Survey (SWSS) 2002, which records individual wages among a very large sample of establishments in all industries, covering approximately 42,000 firms and 1 million workers.

208


Our results show that there is segregation along both the skill and nationality dimensions and point to the existence of two types of firms. Some firms follow the traditional guest-worker strategy and hire mostly unskilled foreign workers. Other firms benefit from the recent bilateral agreements with the European Union (EU) by employing a great number of highly skilled foreign specialists. In a further step, we relate wage inequality between natives and immigrants to firm-level segregation and find that the skill composition of firms does not account for a significant share of wage differences between individuals of different skill. However, there are important interaction effects at the firm level between skill and nationality. Firms with a great share of unskilled foreign workers pay lower wages to all their employees (whether they are Swiss or immigrants), while firms that employ a large share of highly skilled foreign workers pay higher wages to all their workers. The structure of our data does not enable us to establish a causal link between segregation and wage inequality, but our descriptive evidence helps to shed some light on the relevance of different theories of discrimination and segregation. Our results are difficult to reconcile with an explanation of taste-based discrimination (Becker, 1957), but they could be consistent either with a ‘‘quality sorting’’ explanation (Hirsch & Macpherson, 2004) or with the existence of rationing of high-wage jobs. From a theoretical viewpoint, sorting of workers across firms according to their skill level can be explained by technological complementarity, as in the competitive labormarket model of Kremer and Maskin (1996). If skill levels are (partially) unobserved, the variables capturing a firm’s workforce composition could be ‘‘proxies’’ of the workers’ unobserved characteristics and there would be no causal link between segregation and wage inequality. Although we find some evidence for this view at the low end of the skill spectrum, our results could also be consistent with another view that sees the labor market as being imperfectly competitive, where the most productive firms pay higher wages to all their employees because of rent-sharing motives. In this case, workforce composition variables could be proxies of a firm’s productivity. The remainder of the paper is structured as follows. The next section summarizes the main features of Swiss migration policy and presents the database. Section 3 gives a brief account of the approach we use in a simplified framework where the skill and nationality dimensions are characterized by two alternatives each (skilled/unskilled and Swiss/foreign). Some interesting results can already be derived in this framework. The two following sections generalize the setting to four skill levels and five categories of nationality/work permit. The patterns of segmentation along


209

these two dimensions are discussed in Section 4, and the link between segregation and wage inequality is analyzed in Section 5. The last section concludes.

2. INSTITUTIONAL CONTEXT AND DATA Under the traditional guest-worker policy applied by the Swiss government after World War II, unskilled workers were actively recruited in South European countries. Although active recruitment came to a halt in the 1970s, a temporary migration regime subsisted until the 1990s. This policy has had important consequences for the skill composition of the permanent population in Switzerland until today. Many immigrants were first granted a seasonal work permit, allowing them to work in a limited number of sectors. Because of these severe restrictions, this policy attracted mainly unskilled workers. After several years of uninterrupted seasonal work, these workers were entitled to a more permanent, annually renewable work permit without sectoral restrictions. More recently, the Swiss government has reoriented its policy by negotiating bilateral agreements on the free movement of persons between the EU and Switzerland. The main features of the Swiss migration policy are the following. Foreigners are classified into four main categories that entail different economic rights: 1. 2. 3. 4.

seasonal workers and short-term workers: ‘‘permit A’’; annual workers: ‘‘permit B’’; settled workers: ‘‘permit C’’; cross-border workers: ‘‘permit G.’’

The seasonal work permit existed until June 2002 and allowed workers to stay in Switzerland at most for nine months during a calendar year. Seasonal workers were constrained to work in industries characterized by seasonal activities (i.e., mainly construction, tourism, and agriculture) and were not free to move to another canton. It is important to note that after four consecutive years of seasonal work in Switzerland, seasonal workers could apply for an annual permit. The seasonal permit was abolished in June 2002 with the entry into force of the bilateral agreements with the EU. Nowadays, Swiss authorities issue more often short-term work permits (duration less than 12 months), particularly for skilled workers. The quotas for short-term workers (as was the case for seasonal workers) are set at both the cantonal and federal levels, depending on the size and economic situation of the canton.

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Annual permits have to be renewed every year but, since 1993, the holders of such permits are no longer limited in their mobility between cantons. The population possessing an annual permit is very heterogeneous. Many of these workers are highly skilled, but a significant number of mostly unskilled workers have obtained their annual permit by ‘‘upgrading’’ a seasonal permit. Finally, settlement permits are delivered to workers who have stayed in Switzerland for several years (citizens of EU, EFTA, USA, and Canada: 5 years; other countries: 10 years). This category of workers can therefore be characterized as ‘‘old’’ immigrants. Settled workers have the same economic rights as the Swiss and are free to move from one canton to another. Finally, cross-border workers must live close to the Swiss border and have to return home every day. Obviously, this population of workers has significantly grown since the signing of the bilateral agreements with the EU. The number of cross-border work permits is not limited by the federal legislation. Each canton bordering a frontier can issue such authorizations. Turn now to the database we use in the analysis of segregation and wage inequality. The SWSS was launched in 1994 by the Federal Office of Statistics (OFS). The survey is repeated every two years. The SWSS records individual wages within a sample of firms of all industries (including the public sector at the federal level exclusively). The sampling has two levels: at the first level, firms are sampled, and at the second level, individuals employed at these sampled units are also sampled. Concerning firms, the universe to be sampled includes all firms with at least two employees. In 2002, almost 1.1 million of employees from 42,000 firms were included, representing almost one-third of the workforce in Switzerland. For each worker we know whether he or she is Swiss or not and, if not, the type of work permit he or she holds; his or her age, level of seniority in the firm, educational level and marital status; and the number of ‘‘normal’’ hours he or she normally puts into the job.2 In addition, the level of skill required by the job (four positions), hereafter the ‘‘skill,’’ the hierarchical level, as well as a 24-position variable defining the ‘‘activity domain’’ of the job are given. Finally, the information about the firm comprises two- and four-digit industry codes, total employment, post code, and the existence or not of a branch agreement (referred to below as a collective agreement) or of a firmlevel agreement. Our final sample was selected as follows. We use the SWSS data for the year 2002 and exclude public sector firms, the agricultural sector, and firms employing less than five workers. We also exclude workers under the age of 20 or over the legal retirement age for men (i.e., 65 years). In the end, once the


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observations for which information was missing on the variables of interest were excluded, our sample contained 24,512 firms and 829,452 workers. Descriptive statistics of main variables are shown in Table A1 in the appendix. The wage variable we use in the regressions is a full-time equivalent wage rate: the gross monthly earnings linearly standardized at 40 h per week. For both descriptive statistics and estimations, we use individual (probability) weights and the firm’s number of identification as cluster.

3. OVERVIEW OF THE DECOMPOSITION METHOD In order to analyze the link between segregation at the firm level and wage inequality, we distinguish population groups along two dimensions: skill and nationality. The two-dimensional nature of the analysis enables us to account for the heterogeneity characterizing the group of foreign workers in Switzerland. In this section, we introduce our decomposition method by focusing on the simplified ‘‘two-by-two’’ case where each dimension is characterized by only two alternatives (skilled–unskilled; Swiss–foreign). In the following sections, the method will be expanded to four skill groups and five nationality groups.3 In contrast to other studies (Groshen, 1991; Bayard et al., 2003) we are considering segregation as a multigroup phenomenon. In the ‘‘two-by-two’’ case presented in this section, population can be divided into four groups: Swiss skilled, Swiss unskilled, foreign skilled, and foreign unskilled workers. Several indices of multigroup segregation have been proposed in the literature (see an overview of these measures in Reardon & Firebaugh, 2002). As we focus on the link between segregation and wage inequality, the use of a singlevalued index would, however, be too reductionist in our context. Instead, we prefer to measure segregation by using a set of ‘‘exposure rates’’ at the firm level. Exposure rates give an intuitive and simple description of the work environment of each individual worker: they measure, from the viewpoint of a representative member of a group, the average composition of his or her firm’s workforce. By using a set of exposure rates rather than a single-valued index, the link between segregation and wage inequality can be specified in a much more flexible way. Consider first the measurement of segregation. The skill dimension is captured by a dummy variable sij, which takes the value 1 if individual i in firm j is skilled, and 0 otherwise. As to nationality, fij is a dummy variable indicating whether individual i in firm j is a foreigner. The composition of


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firm j’s workforce in terms of skill and nationality can be described by the following variables: P y fj ¼ ð1=nj Þ P i f ij ¼ f j =nj , the fraction of foreigners in firm j’s workforce; y sj ¼ ð1=nj Þ i sij ¼ sj =nj , the fraction of skilled workers in firm j’s workforce; P yxj ¼ ð1=nj Þ i f ij sij ¼ xj =nj , the fraction of skilled foreigners in firm j’s workforce; P P P where f j ¼ i f ij , sj ¼ i sij , xj ¼ i f ij sij , and nj denotes firm j’s total workforce. For later use, we also P total number Pof foreign (skilled, P define the foreign skilled) workers: f ¼ j f j , s ¼ j sj , and x ¼ j xj . Exposure rates can now be defined on the basis of these compositional variables. The exposure rate em,n measures, in the context of firms, the average exposure of a worker of group m to workers belonging to group n. Exposure rates are usually defined along a single dimension (e.g., sex, race, nationality, and skill). As an illustrative example, we first calculate exposure rates with respect to nationality (Swiss or foreign): X 2 X X f ij yjf fj 1 ¼ eforeign;foreign ¼ f nj f j j i eforeign;Swiss ¼

X X f ij ð1 yjf Þ j

eSwiss;foreign ¼

f

i

XX j

X 1 ¼ f j ðnj f j Þ=nj ¼ 1 eforeign;foreign f j

ð1 f ij Þyjf =ðn f Þ ¼ ð1=ðn f ÞÞ

X

f j ðnj f j Þ=nj

j

i

eSwiss;Swiss ¼ 1 eSwiss;foreign A segregation index can be defined by normalizing the exposure rate of one group to the other (James, 1986): eSwiss;foreign eforeign;Swiss ¼1 (1) S nationality ¼ 1 f =n 1 ð f =nÞ where the share of foreign workers in population f/n can be interpreted as the expected exposure of Swiss workers to foreigners if there were perfect integration. In the case of two population groups, an equivalent definition of the segregation index is based on the ‘‘overexposure’’ to the own group: S nationality ¼

eforeign;foreign ð f =nÞ eSwiss;Swiss ð1 f =nÞ ¼ 1 ð f =nÞ f =n

(2)

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Table 1.

Exposure Rates by Nationality (Switzerland, 2002). Average Composition of Firms

Swiss Foreigner Overall composition

Overexposure to own group James’s index

Share Swiss

Share foreigners

0.788 (0.006) 0.499 (0.007)

0.212 (0.006) 0.501 (0.007)

0.702

0.298

Swiss

Foreigners

0.086 0.289

0.203 0.289

Note: Standard errors (in parentheses) are adjusted for clustering at the firm level.

Table 1 illustrates the extent of firm-level segregation in Switzerland with respect to nationality. The work environment of Swiss workers is composed, on average, by 78.8% Swiss workers; the remaining 21.2% are foreign workers. As the share of Swiss workers in total working population is 70.2%, the ‘‘overexposure’’ of Swiss workers to workers of their own nationality is 8.6% and the value of the segregation index is 0.289. It is instructive to compare these values with segregation measured along the skill dimension. The segregation index calculated from exposure rates in Table 2 indicates that segregation is more pronounced along the skill dimension (0.341) than according to the criterion of nationality (0.289). This analysis can be refined by interacting the skill dimension with the nationality dimension. Table 3 presents exposure rates for the resulting four population groups. James (1986) proposed different generalizations of the segregation index defined in Eqs. (1) and (2) to the multigroup case. In Table 3, we report the index that measures the segregation of one group from all others; it is a straightforward generalization of (2) to the case of several groups. According to this index, segregation is particularly pronounced for the group of unskilled foreigners, as well as for skilled Swiss workers. Moreover, it seems that unskilled workers tend to be located in the same firms even if they have different nationalities. Indeed, the exposure rate of unskilled Swiss workers to unskilled foreign workers (14.5%) is greater than the latter’s share in total population (11.8%), and the reciprocal exposure


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Table 2.

Exposure Rates by Skill (Switzerland, 2002). Average Composition of Firms

Skilled Unskilled

Share skilled

Share unskilled

0.843 (0.007) 0.501 (0.009)

0.157 (0.007) 0.499 (0.009)

0.761

0.239

Skilled

Unskilled

0.082 0.341

0.260 0.341

Overall composition



Table 3.

Exposure Rates by Skill and by Nationality (Switzerland, 2002). Average Composition of Firms

Swiss skilled Swiss unskilled Foreigner skilled Foreigner unskilled Overall composition

Overexposure to own group Segregation index (James) b^p

Share Swiss skilled

Share Swiss unskilled

Share foreigners skilled

Share foreigners unskilled

0.705 (0.009) 0.427 (0.011) 0.457 (0.008) 0.312 (0.007)

0.089 (0.005) 0.329 (0.009) 0.066 (0.005) 0.150 (0.008)

0.142 (0.004) 0.099 (0.004) 0.369 (0.008) 0.165 (0.004)

0.063 (0.003) 0.145 (0.005) 0.107 (0.003) 0.374 (0.012)

0.581

0.121

0.180

0.118

Swiss skilled

Swiss unskilled

Foreigners skilled

Foreigners unskilled

0.125

0.208

0.188

0.257

0.298

0.237

0.230

0.291

0.059

0.012

0.013

0.257


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rate (15.0%) is also greater than the population share of unskilled Swiss workers (12.1%). Turn now to the determination of wages and their link with segregation. We use a wage equation that includes the above-defined dummy variables at the individual level and the compositional variables at the firm level: log wij ¼ a0 þ a1 f ij þ a2 sij þ a3 f ij sij þ b1 yfj þ b2 ysj þ b3 yxj þ g0 xi þ d0 zj þ ij

ð3Þ

where xi is a vector of individual characteristics (age, age squared, dummy female, dummy never married, and interaction terms) and zj is a vector of other firm characteristics (share of female workers in firm and log size of the firm). It should be emphasized that this equation allows for the possibility that Swiss and foreign workers face different returns to skill since the individual dummy variables include an interaction term, f ij sij . By contrast, the parameters applying to individual demographic characteristics and firm composition variables are constrained to be identical for all groups (this assumption is relaxed in the extensions of the model). Together with zj, the variables yjf , ysj , and yxj give a detailed description of the composition of firm j’s workforce. In order to illustrate our decomposition method in the simple two-by-two case, we estimate Eq. (3) by ordinary least squares (OLS). One might be worried by the fact that omitted firm-level variables could induce biased estimates of the individual coefficients ai. It can be shown that by including the means by firm of all three individual dummy variables, the ai are estimated without bias (see Skrondal & Rabe-Hesketh, 2004, p. 52). The difference between the average log wages of two groups m and n can be decomposed as follows: X X b^p ðem;p en;p Þ a^ k ðdm;k dn;k Þ þ log w m log w n ¼ p

k 0

þ g^0 ðx m x n Þ þ d^ ðzm zn Þ

ð4Þ

where x m and zm are the averages of variables x and z in group m, dm;k is a generic notation for the group averages of individual dummy variables ( fij, sij, and the interaction term fijsij), em;p denotes the exposure rate of group m to group p, and a hat indicates estimates by OLS.4 Eq. (4) can be interpreted as an extended Oaxaca (1973) decomposition, with the coefficients on demographic variables (but not on skill) constrained to be identical for all groups.


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The first term on the right hand side of (4) includes the individual effects of nationality and skill. In general, this term includes two different effects that are familiar from the traditional Oaxaca decomposition:5 (i) the estimated effect of the difference in average skill between the two groups and (ii) the unexplained part of the wage difference that reflects differences in returns to skill between Swiss and foreign workers and that is possibly due to discrimination or group differences in unobserved characteristics.6 One or the other of these effects might be zero, depending on which groups are compared. For example, if unskilled Swiss workers are compared with unskilled foreigners, the first term on the right hand side of (4) reduces to â1 , which can be interpreted as the unexplained part of the wage difference. Table 4 (second panel, column I) shows that this term represents more than half of the mean wage differential (which is equal to 8.4 log points). Table 4. Decomposition of Wage Differentials by Nationality and by Skill. Wage Differential

(I) Swiss–Foreigners

(II) Skilled–Unskilled

Skilled

Swiss

Delta log wage [% sex, age, civil status] [% nationality/unexplained] [% skill] [% firm composition] [% firm size]

0.0786 [22.37] [87.20] – [33.48] [1.68]

0.3711 [16.13] – [73.47] [12.14] [1.74]

Subsample:

Unskilled

Foreigners

Delta log wage [% sex, age, civil status] [% nationality/unexplained] [% skill] [% firm composition] [% firm size]

0.0835 [36.83] [53.18] – [73.12] [10.54]

0.3759 [12.42] – [72.65] [21.10] [0.27]

All

All

0.1628 [5.84] [37.19] [36.26] [30.66] [1.74]

0.3935 [12.42] [69.40] [1.42] [17.39] [0.62]

Subsample:

Sample: Delta log wage [% sex, age, civil status] [% skill] [% nationality/unexplained] [% firm composition] [% firm size]


217

If the wage comparison is carried out for skilled workers, the first term on the right hand side of (4) is equal to â1 a^ 3 . In this case, almost the entire wage difference between Swiss and foreign skilled workers (equal to 7.9 log points) remains unexplained. The interpretation of the first term on the right hand side of (4) is different if two groups of the same nationality are compared. For example, if mean wages of skilled and unskilled workers of Swiss nationality are compared, the first term on the right hand side of (4) is equal to a^ 2 , capturing the contribution of skill differences (more than 70%) to the mean log-wage differential of 0.371. The preceding examples make clear that the first term on the right hand side of (4) captures both the ‘‘endowment’’ and the ‘‘discrimination’’ components of the traditional Oaxaca decomposition if the mean wages of two groups of different skill and nationality composition are compared. As Table 4 shows, the endowment effect accounts for 37% and the discrimination (or ‘‘unexplained’’) component for 36% of the mean log-wage differential between Swiss and foreign workers. The impact of firm-level segregation on wage differences is captured by the second term on the right hand side of (4). This term is a weighted sum of the differences in exposure rates of the two groups being compared (i.e., the rows in Table 3). The estimated weights b^p are linear combinations of the coefficients b^ k in the wage regression and are reported in the last row of Table 3.7 According to these estimates, workers in firms with a high share of foreign unskilled workers are highly disadvantaged in terms of wages, especially in comparison to individuals working in firms with a high share of skilled foreign workers. The exposure rates given in the last column of Table 3 indicate that foreign unskilled workers are likely to experience the most important wage losses due to their work environment. Indeed, the exposure rate of unskilled foreign workers to workers of their own group is 25.7 percentage points higher than their share in population. For this group of workers, segregation at the firm level translates (with an estimated bûnskilled;foreign equal to 0.257) into a wage loss of 6.6 log points. This simple calculation can be refined by calculating the decomposition given in Table 4. Of the total wage differential between Swiss and foreign unskilled workers, amounting to 8.4 log points, almost three quarters can be explained by firmlevel segregation. The contribution of segregation to the wage differential between skilled Swiss and skilled foreign workers turns out to be much smaller: only about one-fifth of the wage differential is explained by segregation. This result should, however, be taken with a grain of salt because the category of ‘‘skilled’’ workers is an aggregate of three different skill levels. As the more


218

detailed analysis carried out in the next section will make clear, segregation accounts for an important fraction of wage differentials between equally skilled Swiss and foreign workers. When evaluating wage differentials by skill (see column II of Table 4), the contribution of segregation seems to be rather modest in relative terms, especially for Swiss workers.

4. SEGREGATION BY SKILL AND BY WORK PERMIT In this section we apply the approach outlined in Section 3 to a more detailed breakdown of skill levels and of the immigration status (work permits). For each employee, employers were asked to define the skill level required for the job. The four skill levels are defined as follows:8

skill skill skill skill

1: 2: 3: 4:

job job job job

involving the most demanding and most difficult tasks; requiring independent and qualified work; requiring specialized professional knowledge; involving simple and repetitive tasks.

Our database does not provide any information on the workers’ nationality. However, we are able to distinguish foreign workers by work permit. This distinction enables us to analyze the impact of different types of migration. Foreigners with seasonal or short-term permits (permit A) can be considered as temporary migrants. In contrast, holders of settlement permits (permit C) can be seen as permanent immigrants because these permits are granted only after 5 or 10 years of uninterrupted stay in Switzerland. Annual permits (permit B) are held by recently arrived immigrants; this category might include both temporary and permanent immigrants. The category of cross-border workers (permit G) differs from the others by the fact that these workers are hired only in regions close to the border. These permits are handled by authorities in a quite flexible way. Firm-level segregation is particularly strong for cross-border workers and seasonal/short-term permits (see segregation indicators in Table 5). Although the skill profile of cross-border workers is similar to the skill structure of Swiss workers (see Table A1), there is a large fraction of firms (75% in our sample) who do not hire any of these workers, obviously for geographical reasons. Seasonal and short-term workers are hired predominantly in certain sectors, implying that more than 87% of firms in our sample do not employ any of these, largely unskilled, workers. Settled workers have a similar skill profile, but they are distributed much more evenly among firms: only 34% of

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Table 5.

Exposure Rates by Nationality and Work Permit. Average Work Permit Composition of Firms

Swiss Permit A Permit B Permit C Permit G Overall composition


Share Swiss

Share Permit A

Share Permit B

Share Permit C

Share Permit G

0.788 (0.006) 0.424 (0.027) 0.500 (0.009) 0.537 (0.008) 0.422 (0.010)

0.014 (0.001) 0.300 (0.035) 0.040 (0.003) 0.021 (0.001) 0.017 (0.002)

0.034 (0.001) 0.081 (0.005) 0.185 (0.006) 0.062 (0.002) 0.044 (0.002)

0.128 (0.004) 0.150 (0.011) 0.220 (0.006) 0.324 (0.006) 0.155 (0.004)

0.036 (0.002) 0.044 (0.004) 0.055 (0.003) 0.056 (0.002) 0.362 (0.011)

0.702

0.023

0.047

0.167

0.060

Swiss

Permit A

Permit B

Permit C

Permit G

0.086 0.289

0.277 0.284

0.137 0.144

0.157 0.189

0.302 0.321

Notes: Standard errors (in parentheses) are adjusted for clustering at the firm level. Permit A: seasonal and short-term workers. Permit B: annual work permit. Permit C: settlement permit. Permit G: cross-border workers.

firms do not employ any of them. The James (1986) index of segregation is even lower for annual permits than for settlement permits. This is probably because both ends of the skill distribution are well represented in this category of recently arrived immigrants who occupy jobs in very different types of firms. The heterogeneity of the category of annual permits – including temporary and permanent immigrants – is also reflected by the exposure rates to other foreign worker categories. In particular, the exposure rate of annual workers to seasonal and short-term workers (4%), on the one hand, and to settled workers (22%), on the other hand, is higher than the average population share of the two latter groups. Turning to segregation by skill (see Table 6), a similar close link exists between the two most highly skilled population groups. More generally, the degree of segregation, as measured by James’s (1986) index, seems to diminish with skill. Before turning to the analysis of the impact of firm-level segregation on wage inequality, it is useful to take a closer look at the interaction between segregation by skill and by work permit. Here the approach proposed by


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Table 6.

Exposure Rates by Skill Levels. Average Skill Composition of Firms

Skill 1 Skill 2 Skill 3 Skill 4 Overall composition


Percent skill 1

Percent skill 2

Percent skill 3

Percent skill 4

0.235 (0.008) 0.084 (0.003) 0.058 (0.003) 0.042 (0.002)

0.289 (0.008) 0.458 (0.006) 0.185 (0.007) 0.153 (0.006)

0.340 (0.010) 0.314 (0.007) 0.588 (0.007) 0.306 (0.008)

0.136 (0.007) 0.144 (0.005) 0.169 (0.009) 0.499 (0.009)

0.074

0.254

0.432

0.239

Skill 1

Skill 2

Skill 3

Skill 4

0.161 0.174

0.204 0.273

0.156 0.274

0.260 0.341

Note: Standard errors (in parentheses) are adjusted for clustering at the firm level. Skill levels range from highest (Skill 1) to unskilled (Skill 4).

Bayer et al. (2004) turns out to be useful. They notice that exposure rates can be calculated by regressing the fraction ypj of workers of type p in a worker’s firm on individual dummy variables characterizing the worker’s type. For example, the first column of Table 5 can be obtained by regressing the fraction of Swiss workers in an individual’s firm on a set of dummy variables characterizing nationality and work permits (omitting the constant term). Besides being a convenient way of calculating standard errors, this method can be usefully extended by including other explanatory variables in the regression.9 We use this approach to explore first the influence of skill on segregation by nationality or work permit. Table 7 shows the marginal impact of skill on the exposure of workers of a certain nationality (work permit) to other workers of the same nationality (work permit).10 The first column of Table 7 indicates that unskilled Swiss individuals work in firms with 3.7 percentage points less Swiss workers than Swiss workers with specialized professional knowledge (‘‘skill 3,’’ the reference in the regression). For Swiss workers, the other skill categories do not have a marked influence on the exposure rates to other Swiss workers. For foreign workers, the effect of skill on segregation is much more pronounced. In the case of settled workers and cross-border workers, a clear pattern emerges: the exposure of these workers

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Table 7. Dependent Variable

The Influence of Skill on Exposure Rates by Nationality/Work Permit.

Share of Swiss in Firm

Skill level (reference ¼ skill 3) Skill 1 0.013 (0.006) Skill 2 0.001 (0.004) Skill 4 0.037 (0.006)

Share of Short-Term Permits in Firm

Share of Annual Permits in Firm

Share of Settlement Permits in Firm

Share of Cross-Border Permits in Firm

0.094 (0.050) 0.048 (0.033) 0.088 (0.040)

0.041 (0.020) 0.010 (0.008) 0.022 (0.008)

0.070 (0.007) 0.036 (0.005) 0.055 (0.009)

0.045 (0.013) 0.036 (0.010) 0.075 (0.015)

Subsample

Swiss

Short-term

Annual

Settlement

Cross-border

Sample size R2

591,513 0.059

16,725 0.154

37,583 0.213

133,169 0.134

50,462 0.130

Notes: The table shows the results of regressing the share of a work permit or nationality in a firm’s total workforce on dummy variables indicating skill levels. Other explanatory variables in the regression are: female, age, age2, never married, femaleage, femaleage2, female(never married), ln (firm size), and ln2 (firm size). Sample weights are used in regressions, and standard errors (in parentheses) are adjusted for clustering at the firm level.

to other workers of their group is strongly decreasing with skill. For highly skilled foreigners with a settlement permit, the exposure to other settled foreign workers is 12.5 percentage points lower than for unskilled foreign workers with a settlement permit. Recently arrived immigrants (with annual permits) show a rather different pattern: the highest exposure rates to workers of their own type are measured at both extremes of the skill distribution. In other words, not only unskilled foreigners tend to work predominantly with other foreigners, but also highly skilled immigrants (with annual permits) tend to work with other foreigners. The analysis can be refined by taking the opposite perspective, analyzing the influence of work permits on segregation by skill (see Table 8). Segregation by skill seems to be more pronounced for foreign workers, especially for recently arrived immigrants (holding annual or short-term permits). This holds true for both highly skilled and unskilled workers, but not for jobs requiring specialized professional knowledge (skill 3) where most Swiss workers are located. For example, unskilled foreign workers holding an annual permit work in firms with 7.5 percentage points more unskilled workers than unskilled Swiss workers (the reference category). Conversely,


222

Table 8.

The Influence of Work Permits on Exposure Rates by Skill.

Dependent Variable

Share of Skill 1 in Firm

Work permit (reference ¼ Swiss) Short-term 0.070 (0.027) Annual 0.090 (0.021) Settlement 0.039 (0.008) Cross-border 0.039 (0.014)




0.022 (0.025) 0.027 (0.010) 0.009 (0.006) 0.003 (0.009)

0.072 (0.026) 0.003 (0.009) 0.013 (0.006) 0.002 (0.010)

0.047 (0.018) 0.075 (0.015) 0.040 (0.012) 0.083 (0.013)

Subsample

Skill 1

Skill 2

Skill 3

Skill 4

Sample size Adjusted R2

58,113 0.130

202,021 0.122

366,097 0.019

203,221 0.028

Notes: The table shows the results of regressing the share of a skill level in a firm’s total workforce on dummy variables indicating nationality/work permit. Other explanatory variables in the regression are: female, age, age2, never married, femaleage, femaleage2, female(never married), ln (firm size), and ln2 (firm size). Sample weights are used in regressions and standard errors (in parentheses) are adjusted for clustering at the firm level.

the work environment of highly skilled foreign workers with an annual permit is characterized by a 9.0 percentage points greater share of highly skilled workers than the work environment of highly skilled Swiss workers. These results point to the existence of two very different types of firms employing mostly foreign workers. On the one hand, some firms hire predominantly unskilled foreign workers; on the other hand, there are firms that employ a great share of highly skilled foreign workers who have been recruited recently, possibly as a consequence of bilateral agreements with the EU. Whether these firms pay different wages than other firms is the question we turn to now.

5. WAGE INEQUALITY AND SEGREGATION In this section, we explore the link between segregation and wage inequality in more detail. By taking all four skill levels into account, we are able to analyze the roles of the unskilled and the truly high-skilled individuals. The disaggregation by work permit enables us to distinguish in particular between ‘‘old’’ immigrants (with settlement permits) and ‘‘new’’ immigrants (with


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annual permits) and to analyze the specific role of cross-border workers and workers with short-term/seasonal permits. We modify Eq. (3) by replacing the individual dummy variables (nationality, skill) by a set of dummy variables describing the four levels of skill and the five categories of nationality/work permit. Possible links between skill and nationality are captured by a full set of interaction dummies. Similarly, the three variables describing the composition of the workforce are replaced by share variables capturing the structure of each firm’s workforce in terms of all skills and work permits (and their interactions). Moreover, we add sectoral dummy variables in order to account for unobserved factors at the sector level.11 Together with the same set of demographic variables as in the ‘‘two-by-two’’ case, this forms our basic wage equation. In the estimation of the wage equation, we have to account for the possibility that our firm-level variables do not include all relevant information that helps to explain wage differences between firms. In the presence of unobserved factors at the firm level, the parameters associated with individual characteristics can be estimated consistently if firm fixed effects are included in the regression. Unfortunately, it is impossible to include in the same equation firm fixed effects and variables that are defined at the firm level. As our main focus is on estimating the impact of workforce composition variables on wage differences, we use two approaches, each one having its advantages and disadvantages.12 One possibility (labeled Model I) is to adopt a two-step estimation procedure, as suggested by Hsiao (2003, p. 52).13 First, we regress the log wage on individual variables and firm dummies. In a second step, we regress the firm fixed effects on the firm-level variables and on dummy variables capturing sectors at the two-digit level. As mentioned above, the advantage of this estimation procedure is that coefficients of individual characteristics are consistently estimated. The disadvantage is that standard errors of the second-step estimates should be corrected because the dependent variable in the second-step regression is generated.14 Another possibility (Model II), following Wooldridge (2002, p. 331), is to estimate the complete wage equation by pooled OLS and to adjust standard errors for clustering at the firm level. This procedure also allows consistent estimation of the effects of both individual characteristics and workforce composition variables. As it turns out, the two approaches yield very similar results for the entire sample (see Table A2 in the appendix). For practical purposes, the estimates of the effects of skill and nationality/work permit are undistinguishable in the two approaches, both at the individual and at the firm level. Moreover, the estimated standard errors of the coefficients of

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firm-level variables are almost identical.15 We are therefore confident that our decomposition of wage differences along the dimensions of skill and nationality/work permit does not depend on the estimation method.

5.1. Wage Effects of Individual Characteristics We discuss first the wage effects of individual characteristics. A first glance at the coefficients of the individual interaction terms (Table A2, upper panel) reveals that the return to skill is slightly higher for foreigners with annual and settlement permits than for Swiss workers. A similar pattern seems to hold for the composition of the workforce (Table A2, lower panel): whereas a large fraction of highly skilled foreign workers in a firm is beneficial for the wages of all their colleagues, a large share of unskilled foreign workers in a firm exercises downward pressure on all wages paid by the firm. It is instructive to compare these results with those of a simple model without firm-level variables (Model III). The model’s adjustment to the data deteriorates markedly since it explains only 48.8% of the total variance (compared to 75.0% in the first stage of Model I and 59.5% in Model II). The omission of firm-level variables yields biased estimates of the contribution of individual characteristics to wage inequality. Fig. A1 in the appendix uses the estimated coefficients of individual characteristics in Models I and III in order to compare the wage effects of skill for Swiss workers and the two main categories of foreign workers (with annual and settlement permits). According to Model III, the individual return to skill is more pronounced for foreign workers with settlement permits than for Swiss workers, and even greater for the more recently arrived workers with annual permits. Model III estimates that the log-wage differential between the lowest and the highest skill is equal to 0.743 for Swiss workers, 0.882 for settled workers, and 1.047 for workers with annual permits. By contrast, the differences in individual returns to skill are greatly attenuated in Model I, which takes firm-level variables into account: the log-wage differentials between the lowest and the highest skill are 0.732 for Swiss workers, 0.774 for settled workers, and 0.804 for workers with annual permits. These results suggest that firm-level segregation plays an important role in the explanation of these differences.

5.2. Wage Effects of Segregation and Workforce Composition In order to get a clearer idea of the impact of segregation on wages, we calculate the ‘‘marginal’’ effect of changing the work environment of an


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individual. The underlying idea is the following. The typical work environment of, say, a foreign unskilled worker with an annual permit is characterized by a large share of unskilled foreign coworkers and a relatively small share of skilled Swiss coworkers. Individuals (of any skill and any nationality) who work in such an environment receive lower wages than people who work in an ‘‘average’’ work environment. To evaluate precisely this effect, we calculate, for each skill–nationality group, the ‘‘marginal’’ effect on log wages of moving from the work environment of the average worker in the economy to the work environment of an average member of that specific group.16 In this calculation, the ‘‘overexposure’’ of workers to other members of their group quantitatively plays an important role. An example will illustrate the procedure. Unskilled annual workers represent 2.0% of the total workforce in our sample. In the typical work environment of these workers, the share of members of their own group is 16.2% and overexposure is therefore equal to 14.2 percentage points. According to the estimates in Table A2, this overexposure reduces wages by 6.9 log points. The change in exposure rates of unskilled annual workers to other groups moderates this effect slightly, leading to a total wage reduction of 6.6 log points. Fig. 1 illustrates these effects for the three main population groups and all skill levels, using the estimates of Model I. A clear pattern emerges. On the one hand, individuals who work in the typical work environment of an unskilled foreign worker receive lower wages than those who work in the average environment.17 On the other hand, workers who hold jobs in a work environment that is similar to the environment of a highly skilled foreigner are paid higher wages. These effects are stronger for recently arrived migrants (holding annual permits) than for settled foreign workers and are quantitatively important. The typical work environment of a highly skilled annual worker offers a 12.7 log points higher wage than the average work environment, whereas the work environment of an unskilled annual worker is characterized by a negative wage differential of more than 10 log points. Interestingly, the structure of the Swiss workforce in a firm has relatively little influence on wages. Before we turn to a more systematic decomposition of wage differentials, it is useful to check whether the results obtained so far depend on the assumption that the wage impact of segregation on natives and immigrants is identical. This assumption can be relaxed by estimating the wage equation separately for Swiss and for foreign workers, allowing for a differential impact of the demographic and the workforce composition variables. Fig. 2 shows the marginal impact of segregation on wages of Swiss workers; Fig. 3 does the same for foreign workers. Interestingly, Swiss workers benefit more


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log wage differential

0.140

Swiss

0.100

settlement permit

0.060

annual permit

0.020 -0.020

skill 1

skill 2

skill 3

skill 4

-0.060 -0.100 -0.140

Fig. 1. Wage Effects of Firm-Level Segregation (by Work Permit and by Skill). Note: This figure depicts the ‘‘marginal’’ effect on log wages of moving from the work environment of the average worker in the economy to the work environment of an average member of a specific skill–nationality [work permit] group. Marginal effects are calculated using exposure rates by skill–nationality group and estimates of the wage equation [second stage of Model I] for all workers, Swiss and foreign.

than foreigners from working in a professional environment that is typical for highly skilled foreigners. This result suggests that a ‘‘quality sorting’’ argument is probably not the only explanation of this phenomenon. By contrast, the typical work environment of an unskilled foreigner is associated with similar reductions in wages for Swiss and foreign workers. 5.3. Decomposition of Wage Differentials between Skill–Nationality Groups A more complete account of the relative importance of individual characteristics and segregation at the firm level can be obtained by decomposing the average log-wage differentials between skill–nationality groups. The differences in average wages of Swiss and foreign workers, differentiated by work permit, are given in the lower part of Table 9. Total wage differentials between Swiss workers and different categories of foreign workers vary between 9.0 log points (cross-border workers) and 30.5 log points (seasonal/short-term permits). Observed skill differences explain the

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0.140

Swiss

0.100

settlement permit

0.060

annual permit

0.020 -0.020

skill 1

skill 2

skill 3

skill 4

-0.060 -0.100 -0.140


Fig. 2. Wage Effects of Firm-Level Segregation for Swiss Workers (by Work Permit and by Skill). Note: See Fig. 1 for details. Marginal effects are calculated using exposure rates by skill–nationality group and estimates of the wage equation [second stage of Model I] for Swiss workers only.

0.140

Swiss

0.100

settlement permit

0.060

annual permit

0.020 -0.020

skill 1

skill 2

skill 3

skill 4

-0.060 -0.100 -0.140

Fig. 3. Wage Effects of Firm-Level Segregation for Foreign Workers (by Work Permit and by Skill). Note: See Fig. 1 for details. Marginal effects are calculated using exposure rates by skill–nationality group and estimates of the wage equation [second stage of Model I] for foreign workers only.


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Table 9. Decomposition of Wage Differentials by Nationality and Work Permit. Swiss-Permit A

Swiss-Permit B

Swiss-Permit C

Swiss-Permit G

Skill 1: Delta log wage [% sex, age, civil status] [% nationality/unexplained] [% firm size] [% total firm composition]

0.687 [18.20] [86.92] [2.68] [2.44]

0.1268 [29.84] [29.95] [16.17] [83.72]

0.0494 [2.07] [10.66] [7.47] [98.88]

0.0331 [11.21] [259.97] [34.13] [114.63]


0.1749 [15.48] [29.94] [0.26] [54.84]

0.0581 [48.97] [10.26] [9.80] [128.91]

0.0590 [15.21] [68.09] [9.09] [38.03]

0.0370 [46.73] [143.99] [7.69] [10.43]


0.1748 [4.27] [38.66] [0.43] [56.64]

0.0714 [5.12] [42.22] [13.96] [38.70]

0.0574 [52.11] [77.11] [7.30] [67.70]

0.0259 [170.48] [138.50] [19.26] [112.72]


0.1779 [3.77] [25.10] [10.15] [60.99]

0.1407 [1.64] [24.25] [8.13] [65.98]

0.0571 [48.77] [63.72] [13.78] [71.28]

0.0714 [46.69] [39.57] [17.55] [89.57]

Mean: Delta log wage [% sex, age, civil status] [% skill] [% nationality/unexplained] [% firm size] [% total firm composition]

0.3050 [8.25] [36.48] [18.17] [1.60] [35.50]

0.1319 [13.76] [46.47] [14.06] [2.40] [23.31]

0.1780 [6.04] [58.88] [21.55] [1.74] [23.88]

0.0900 [29.96] [48.41] [45.03] [3.13] [33.38]

[2.08] [37.58]

[0.07] [23.38]

[1.44] [25.32]

[6.71] [40.09]

of which: [% women] [% skill-work permit]

Notes: Permit A: seasonal and short-term workers. Permit B: annual work permit. Permit C: settlement permit. Permit G: cross-border workers.

greatest share (36–59%) of the wage differentials. Firm-level segregation, as measured by the composition of the workforce, accounts for 23–36% of these differences in average wages. Except for border workers, segregation represents a more important share of wage differentials than the Oaxaca/Blinder term (which accounts for 14–22% of wage differentials for permits A to C, and 45% for border workers). Although this


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aggregate decomposition analysis is useful as a benchmark, only a more disaggregated analysis is able to reveal interesting structural features of the data. As a first step toward a more disaggregated analysis, we calculate Swiss– foreign wage differentials at different skill levels (see upper part of Table 9). These numbers reveal that highly skilled foreigners with annual or settlement permits earn on average higher wages than comparatively skilled Swiss workers. Interestingly, the composition of the firms’ workforce explains 84– 99% of the wage differential between highly skilled workers of different nationality. If foreign workers had the same demographic characteristics as Swiss workers, the estimated wage equation would predict their wages to be lower: this explains the negative contribution of demographic variables to the wage differential. The unexplained ‘‘Oaxaca–Blinder’’ part of the wage differential accounts for 11% (settled workers) and 30% (annual workers) of the wage differential. This latter term most likely reflects unobserved skill differences between Swiss and foreign workers. At the other end of the skill distribution, the wage differential – which is always in favor of Swiss workers – is also explained primarily by segregation. Indeed, unskilled Swiss workers earn 6–18 log points higher wages than foreigners, and at least 60% of this differential can be attributed to the composition of the firm’s workforce. Moreover, differences in firm size explain another 8–18% of the wage difference, which indicates that foreign unskilled workers are in smaller firms compared to Swiss unskilled workers. The unexplained part of the wage differential, reflecting unobserved differences in skill or discrimination, accounts for 24–64% of the wage differential (corresponding to 2.8–4.5 log points of the wage differential). Consider now wage differences between different skill levels. Our analysis of segregation revealed that segregation is stronger along the dimension of skill than along the dimension of nationality. For Swiss workers, firm-level segregation by skill does not seem to have important consequences in terms of wages. The first part of Table 10 shows that the composition of a firm’s workforce does not contribute to the explanation of the differences in wages between the three highest skill categories. Only unskilled Swiss workers receive slightly lower wages because of firm-level segregation. Closer inspection of exposure rates and of the estimates of the wage equation reveals that this effect is mainly due to the presence of a high fraction of foreign (rather than Swiss) unskilled workers in their firms. This latter interpretation is confirmed by the fact that the wage gap between skilled and unskilled workers is greater for foreign workers than for


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Table 10.

Decomposition of Wage Differentials by Skill. Skill 1–Skill 3

Skill 2–Skill 3

Skill 3–Skill 4

Swiss: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% firm composition]

0.6511 [13.94] [86.23] [1.55] [1.38]

0.2445 [15.68] [91.82] [5.76] [1.74]

0.2179 [7.16] [78.45] [0.60] [14.99]

Permit A: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% firm composition]

0.7572 [11.32] [75.19] [0.99] [14.48]

0.2444 [7.66] [98.07] [5.27] [0.47]

0.2210 [6.71] [66.97] [7.24] [19.07]

Permit B: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% firm composition]

0.8494 [6.66] [74.13] [2.40] [16.81]

0.3741 [3.62] [69.68] [0.43] [26.28]

0.2871 [4.96] [60.91] [0.05] [34.07]

Permit C: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% firm composition]

0.7580 [7.89] [80.61] [1.27] [12.76]

0.2429 [7.17] [94.11] [6.28] [5.01]

0.2176 [8.11] [74.93] [1.05] [15.86]

Permit G: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% firm composition]

0.64394 [7.82] [79.40] [0.96] [11.82]

0.2337 [4.87] [88.80] [2.69] [9.02]

0.2634 [10.01] [62.01] [2.37] [25.61]

Mean: Delta log wage [% sex, age, civil status] [% skill] [% firm size] [% total firm composition]

0.6768 [12.04] [84.42] [1.11] [4.65]

0.2530 [12.51] [90.39] [5.12] [2.23]

0.2428 [5.21] [72.99] [0.95] [20.85]

[0.57] [4.08]

[1.01] [1.23]

[2.76] [18.09]

of which: [% women] [% skill-work permit]

Notes: Permit A: seasonal and short-term workers. Permit B: annual work permit. Permit C: settlement permit. Permit G: cross-border workers.


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Swiss workers (see Table 10). Interestingly, the fact that the skill wage gap is greater for foreigners can be almost entirely attributed to firm-level variables. For example, the log-wage difference between skill levels 1 and 3 (the reference category), on the one hand, and skill levels 3 and 4, on the other hand, is greater for annual workers (0.85 and 0.29, respectively) than for Swiss workers (0.65 and 0.22). Segregation, as measured by the composition of a firm’s workforce, explains almost the entire difference between Swiss and foreign skill wage gaps. From these decompositions, a more complete picture emerges of the role of firm-level segregation in the explanation of wage inequality between Swiss and foreign workers. Firms with a large share of foreign skilled workers tend to pay higher wages to all their employees, whereas the opposite holds for firms with a large proportion of foreign unskilled workers. As a consequence, segregation at the firm level accounts for almost the entire wage differential between identically skilled Swiss and foreign workers. Moreover, the difference between foreign and Swiss skill wage gaps can also be explained by firm-level segregation.

5.4. Sectoral Wage Differentials and Segregation To explore further the role of firm-level segregation, we turn now to the analysis of sectoral wage differentials. The literature has documented large and persistent wage differentials between economic sectors. In our context, it is interesting to see to what extent these wage differentials can be explained by the composition of a firm’s workforce. This analysis also sheds some light on the sectoral identity of low-wage and high-wage firms. Before we turn to these questions, it is useful to describe our treatment of sectoral variables. As every firm belongs to one sector, there is a problem of collinearity between our firm composition variables and sectoral dummies. We want to avoid that part of the firm composition effects are captured by the sectoral dummy variables in our wage regression. Therefore, we decompose the sectoral dummies into two components: a first component that can be explained by firm composition variables and a second component that is orthogonal to the firm composition variables.18 It is the latter component that is added to our (second-step) wage equation instead of the original dummy variables. This procedure ensures that the transformed variables capture only sectoral effects that are not correlated with the composition of a firm’s workforce.


16

65

64

.2

67

0

80 72 91 70 22 51 62 85 33 21 92 26 10 4525 15 27 36 50 90 52 20 60

17 -.2

93 19

2366 73 40

29

30

63

61

55 18

-.4

firm-level fixed effects (sector averages)

.4

232

-.15

-.1

-.05

0

.05

.1

predicted firm-level fixed effects (sector averages)

Fig. 4. Sectoral Wage Differences and Composition of Firms’ Workforce. Note: Labels in the graph refer to the two-digit NOGA classification. On the y-axis, we plot sectoral averages of firm-level fixed effects [as estimated in the first stage of Model I in Table A2]. On the x-axis, we plot sectoral averages of firm-level effects that are ‘‘predicted’’ by variables capturing the composition of the firms’ workforce [predictions are obtained from the estimation of the second stage of Model I in Table A2].

Sectoral wage differentials at the two-digit level and the role of firm composition variables are depicted in Fig. 4. The vertical axis shows the estimated sectoral wage differentials, and the horizontal axis depicts the wage differentials that are predicted by firm-level variables. More precisely, sectoral log-wage differentials are calculated by averaging over sectors the firm fixed effects obtained in the first step of Model I (they are shown on the y-axis of Fig. 4). To capture the role of firm composition, we run a regression of firm fixed effects on all firm-level variables (firm composition, size of the firm, without sectoral dummies). The sectoral averages of the predicted log wages are depicted on the horizontal axis of Fig. 4. At first glance, firm-level variables seem to predict sectoral wage differentials quite well.19 Low-wage firms can be found in services (55: Hotels and restaurants) and in industry (17: Textile, 18: Wearing apparel,


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19: Leather). Hotels and restaurants cumulate several unfavorable characteristics: the share of unskilled seasonal and annual workers is high and firms are small. In the Wearing apparel industry, there is not only a high share of annual and cross-border workers but also a high share of female employees. At the other extreme, there are high-wage firms in sectors such as Post and telecommunications (64), Financial intermediation (65) and their auxiliaries (67), and Manufacturing of tobacco products (16). Whereas the firm-level variables predict high wages for the two former sectors (because of the structure of the workforce and the predominance of large firms), this is not the case for the two latter sectors, mainly because of the much smaller average firm size (67) and the mainly unskilled workforce (16). A plausible explanation of the high wages paid in Manufacturing of tobacco products might be compensatory differences.

5.5. Robustness Tests The observed pattern of segregation and wage inequality could be consistent with two types of explanations. On the one hand, it is possible that firms choose their employees on the basis of unobserved (by the analyst) productivity characteristics. If this is the case, the variables capturing a firm’s workforce composition are ‘‘proxies’’ of the workers’ unobserved characteristics. Our results could also be consistent with another view that sees the labor market as being segmented, with rationing of high-wage jobs because of frictions in the labor market. The data at our disposal do not enable us to discriminate clearly between these two views. However, it is possible to test the validity of the first view by following the procedure proposed by Hirsch and Macpherson (2004). The ‘‘quality sorting’’ explanation given by Hirsch and Macpherson (2004) in their analysis of the black–white wage gap in the USA relies on the assumption that the proportion of black workers in a job is correlated with observed and unobserved productivity of workers.20 In our context, the ‘‘quality sorting’’ hypothesis could be interpreted as follows. Highly skilled foreign and Swiss workers are sorted into firms that employ a large proportion of highly skilled foreigners, but the correlation between this proportion and wages does not reflect a causal relationship. According to the quality sorting hypothesis, the share of highly skilled foreigners is simply an indicator of the unobserved skills of the firm’s employees. If this hypothesis is correct, the introduction of job characteristics in the wage equation should diminish the effect of firm-level variables since job

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characteristics are presumably correlated with unobserved characteristics of workers. In order to check this hypothesis, we add three sets of dummy variables to the wage equation: (i) variables describing the occupation of the worker (our database includes a two-digit classification of occupations), (ii) variables describing the hierarchical position of the employee (five positions), and (iii) variables describing the type of collective agreement. Fig. A2 in the appendix summarizes the results for the two main groups of foreign workers. Both for annual and settled workers, the introduction of job characteristics lowers the effect of segregation on wage inequality, but not drastically. At the lower end of the skill distribution, the effect of segregation is reduced by respectively one-half (settled workers) and onethird (annual workers). This result suggests that the low wages paid by firms with a large share of foreign unskilled workers can in part be explained by the fact that they hire mostly workers with a low level of unobserved skills. At the other end of the skill spectrum, the introduction of job characteristics in the wage equation hardly reduces the effect of segregation. Therefore, the quality sorting explanation does not find much support for highly skilled foreigners.

6. SUMMARY AND CONCLUSIONS In this paper, we analyze segregation at the firm level between native and immigrant workers in Switzerland and explore the link between segregation and wage inequality. We argue that in the Swiss context it is crucial to measure segregation along the two dimensions of skill and nationality. To take the multidimensional aspect of segregation into account we use exposure rates – a concept borrowed from the literature on residential segregation – in order to analyze segregation between Swiss and foreign workers of different skill levels. Exposure rates are an intuitively appealing concept and can be easily incorporated in the analysis of wage inequality. We propose a simple decomposition method that allows to evaluate the role of segregation in the explanation of wage inequality between Swiss and foreign workers. In a first step, we calculate exposure rates by nationality and by skill and we analyze interaction effects between these two dimensions, using a regression method proposed by Bayer et al. (2004). Our results show that there is segregation along both the skill and the nationality dimensions and that the most segregated groups can be found among recently arrived foreign workers at both extremes of the skill distribution. This result points


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to the existence of two types of firms that reflect the recent evolution of Swiss immigration policy. On the one hand, some firms follow the traditional guest-worker strategy and hire mostly unskilled foreign workers. On the other hand, there are firms that benefit from the recent bilateral agreements with the EU by employing a great number of highly skilled foreign specialists. The second step of our analysis relates firm-level segregation to wage inequality between natives and immigrants. We estimate wage equations that account for the composition (in terms of skill and nationality) of the firm’s workforce to which an individual belongs. On the basis of these wage equations, we propose a simple decomposition method that relates differences in average wages between skill–nationality groups to the exposure rates of these groups to other groups in the workforce. It turns out that the skill composition of firms does not explain a significant share of wage differences between individuals of different skill. However, there are important interaction effects at the firm level between skill and nationality. On the one hand, firms with a great share of unskilled foreign workers pay lower wages to all their employees (whether they are Swiss or immigrants). On the other hand, those firms that employ a large share of highly skilled foreign workers with an annual or settlement permit pay higher wages to all their workers. These effects are significant from both a statistical and an economic viewpoint. Highly skilled foreign workers with an annual permit earn on average 13% higher wages than highly skilled Swiss workers, and this difference is almost entirely explained by the difference in work environments of these workers (i.e., firm-level segregation). By contrast, unskilled foreign workers get paid lower wages, on average, than unskilled Swiss workers (15% for recently arrived workers with annual permits, 6% for settled workers). Two-thirds of these differences in mean wages are explained by segregation at the firm level. From a theoretical viewpoint, our descriptive evidence is difficult to reconcile with an explanation of taste-based discrimination (Becker, 1957) since both foreign and Swiss workers receive lower wages if they work in a firm with a high share of low-skill foreign workers. Moreover, the fact that highly skilled foreign workers are paid higher wages than highly skilled Swiss workers is also at odds with this theory. Our results could be consistent either with a ‘‘quality sorting’’ explanation or with the existence of rationing of high-wage jobs in an imperfectly competitive labor market. In our data, we find some support for the first view, but only at the lower end of the (measured) skill spectrum. One interpretation of our results is that unskilled workers with low unmeasured skills tend to be hired in

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specific firms and that unskilled foreigners have lower unmeasured skills, on average, than unskilled Swiss workers, perhaps because of lacking language proficiency. For highly skilled employees, we do not find evidence for such a ‘‘quality sorting’’ effect. Due to the limitations of our data, we would like to emphasize that these conclusions can only be tentative and that further research is warranted in this area.

NOTES 1. In fact, our database constrains us to a breakdown by work permit rather than by nationality. As the description of the Swiss immigration policy will make clear, this distinction has its own merits. In particular, recently immigrated workers can be distinguished from settled immigrants on the basis of their work permits. 2. There is, however, no direct information on overtime worked in October, or on night- and shift-work time. 3. More precisely, in our database, foreign workers are distinguished by type of work permit. 4. Note that the average residual is zero for all four population groups as the estimation method is OLS and the model contains dummy variables for all population groups. 5. In general, the distinction between the two first effects can only be made explicit if a reference group is chosen. For an extensive discussion of this issue, see Oaxaca and Ransom (1994). Here we choose the majority group (Swiss workers) as the reference unless otherwise indicated. 6. In the context of immigration, this second term might in particular reflect unobserved differences in education quality. In our setting, this bias is likely to be rather small since skill is measured by a variable describing the required skill level for the job. 7. Because of the linear dependence of exposure rates, these coefficients must be normalized. Setting b^Swiss;unskilled ¼ c yields b^Swiss;skilled ¼ b^ 2 þ c, b^foreign;skilled ¼ b^ 1 þ b^ 2 þ b^ 3 þ c, and b^foreign;unskilled ¼ b^ 1 þ c. In Table 3, c is chosen in such a way that the population-weighted sum of the b coefficients is zero. 8. Note that in the aggregated version of the model presented above, ‘‘unskilled’’ corresponds to ‘‘skill 4’’ and the ‘‘skilled’’ category is an aggregation of skill levels 1 to 3. 9. Bayer et al. (2004) develop this approach in order to analyze neighborhood segregation patterns in the San Francisco Bay Area. 10. These ‘‘own-group’’ exposure rates can be estimated by running separate regressions for each group. This is the approach taken in Tables 7 and 8. 11. As every firm belongs to one sector, there is a problem of collinearity between sectoral dummies and variables capturing the composition of a firm’s workforce. For a detailed discussion of how we address this issue, see Section 5.4. 12. Because of data limitations, we are not able to account for the possibility of endogenous sorting. Therefore, our empirical results should be interpreted in a descriptive manner; they do not necessarily reflect a causal link between segregation


237

and wage inequality. See the Conclusion section for some more discussion on this issue. 13. For applications of this approach, see Kramarz, Lollivier, & Pele (1996) and Ramirez (2000). 14. Because of the large size of our sample, we are unable to carry out the correction suggested by Dumont, Rayp, Thas, and Willeme´ (2005). However, the comparison of our two estimation methods suggests that this correction would not matter much in our context (see following paragraph), presumably because of the large sample size. 15. The only noticeable difference between the two approaches concerns the effects of gender and civil status. In this context, our first approach yields better results since the use of firm fixed effects can be interpreted as a nonparametric specification of firm-level segregation. 16. To carry out this calculation, we compute exposure rates of all skill– nationality/work permit groups to each other (20 20 matrix of exposure rates). 17. As the work environment of an individual is characterized mainly by ‘‘overexposure’’ to members of his or her own skill–nationality group, this implies, more simply stated, that firms with a high fraction of unskilled foreign workers pay lower wages than the average firm. 18. Technically speaking, the second component is obtained as the residuals of a linear regression of each sectoral dummy variable on all firm composition variables. 19. In a weighted regression of firm fixed effects (sectoral averages) on the predicted firm-level fixed effects (sectoral averages), the R2 is equal to 0.50. 20. In contrast to our analysis of segregation at the level of the firm, Hirsch and Macpherson (2004) consider segregation by industry–occupation–region groups (‘‘jobs’’).

ACKNOWLEDGMENTS We thank participants at the first ECINEQ conference 2005 in Palma de Mallorca and at the ESPE conference 2006 in Verona for helpful comments on an earlier draft. We also thank Jacques Silber and an anonymous referee for their useful remarks and suggestions. We are grateful to David Zavaleta for excellent research assistance. Financial support by the Swiss National Science Foundation (Grant 612-066094) is gratefully acknowledged.

REFERENCES Bayard, K., Hellerstein, J., Neumark, D., & Trotske, K. (2003). New evidence on sex segregation and sex differences in wages from matched employee–employer data. Journal of Labor Economics, 21(4), 887–922. Bayer, P., McMillan, R., & Rueben, K. (2004). What drives racial segregation? New evidence using census microdata. Journal of Urban Economics, 56(3), 514–535.

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Becker, G. S. (1957). The economics of discrimination. Chicago: University of Chicago Press. Brown, R. S., Moon, M., & Zoloth, S. (1980). Incorporating occupational attainment in studies of male–female differentials. Journal of Human Resources, 15(1), 3–28. Carrington, W. J., & Troske, K. R. (1998). Interfirm segregation and the black/white wage gap. Journal of Labor Economics, 16(2), 231–260. Dumont, M., Rayp, G., Thas, O., & Willeme´, P. (2005). Correcting standard errors in two-stage estimation procedures with generated regressands. Oxford Bulletin of Economics and Statistics, 67(3), 421–433. Groshen, E. (1991). The structure of the female/male wage differential: Is it who you are, what you do, or where you work? Journal of Human Resources, 26(3), 457–472. Hirsch, B. T., & Macpherson, D. A. (2004). Wages, sorting on skill, and the racial composition of jobs. Journal of Labor Economics, 22(1), 189–210. Hirsch, B. T., & Schumacher, E. J. (1992). Labor earnings, discrimination and the racial composition of jobs. Journal of Human Resources, 27(4), 602–628. Hsiao, C. (2003). Analysis of panel data (2nd ed.). Cambridge, UK: Cambridge University Press. James, F. J. (1986). A new generalized ‘‘exposure-based’’ segregation index: Demonstration in Denver and Houston. Sociological Methods and Research, 14(3), 301–316. Kramarz, F., Lollivier, S., & Pele, L. (1996). Wage inequalities and firm-specific compensation policies in France. Annales d’ećonomie et de statistique, 41/42, 369–386. Kremer, M., & Maskin, E. (1996). Wage inequality and segregation by skill. NBER Working Paper 5718. Liu, P. W., Zhang, J., & Chong, S.-C. (2004). Occupational segregation and wage differentials between natives and immigrants: Evidence from Hong Kong. Journal of Development Economics, 73, 395–413. Neuman, S., & Silber, J. (1996). Wage discrimination across ethnic groups: Evidence from Israel. Economic Inquiry, 34, 648–661. Oaxaca, R. L. (1973). Male-female wage differentials in urban labor markets. International Economic Review, 14(3), 693–709. Oaxaca, R. L., & Ransom, M. R. (1994). On discrimination and the decomposition of wage differentials. Journal of Econometrics, 61(1), 5–21. Ramirez, J. (2000). Inter-industry and inter-firm wage and hours differentials in Switzerland. Swiss Journal of Economics and Statistics, 136(3), 371–395. Reardon, S. F., & Firebaugh, G. (2002). Measures of multigroup segregation. Sociological Methodology, 32(1), 33–67. Skrondal, A., & Rabe-Hesketh, S. (2004). Generalized latent variable modeling: Multilevel, longitudinal and structural equation models. Boca Raton, FL: Chapman & Hall/CRC. Wooldridge, J. M. (2002). Econometric analysis of cross section and panel data. Cambridge, MA: MIT Press.

40.95% 31.29% 7.43% 25.45% 43.22% 23.90% 40.08 5763.49

Female Never married Skill 1 (higher) Skill 2 Skill 3 (lower) Skill 4 (unskilled) Age Wage (SFR 2002)

42.96% 34.36% 8.56% 29.02% 45.13% 17.29% 40.61 6050.52

591,513 70.20%

Swiss

30.91% 41.00% 2.67% 13.81% 38.69% 44.83% 34.73 4459.98

16,725 2.33%

Permit A

38.44% 25.12% 8.97% 19.92% 28.60% 42.51% 39.87 5302.85

37,583 4.75%

Permit B

Descriptive Statistics (Switzerland, 2002).

38.01% 19.98% 3.77% 14.61% 37.68% 43.94% 39.87 5063.43

133,169 16.71%

Permit C

32.37% 28.40% 5.06% 22.07% 49.59% 23.28% 40.46 5529.76

50,462 6.01%

Permit G

Notes: Mean values for entire sample and subsamples by nationality/work permit. Permit A: seasonal and short-term workers. Permit B: annual work permit. Permit C: settlement permit. Permit G: cross-border workers.

829,452 100%

All

N observations % subpopulation

Variable

Table A1.

APPENDIX

Wage Inequality and Segregation 239


240

Table A2. Variable

Estimated Log-Wage Equation. Model I

Model II

Coefficient

Standard error

Coefficient

Standard error

–0.0606 0.0311 –0.2962 –0.0537 –0.0023 –0.0026 0.0692

0.0106 0.0004 0.0042 0.0011 0.0005 0.0064 0.0017

–0.0453 0.0354 –0.3429 –0.0531 –0.0034 0.0093 0.0908

0.0200 0.0009 0.0109 0.0023 0.0010 0.0116 0.0033

–0.0676 –0.0302 –0.0443 –0.0358

0.0048 0.0033 0.0014 0.0020

–0.0664 –0.0286 –0.0441 –0.0375

0.0125 0.0075 0.0030 0.0045

0.5615 0.2245 –0.1709

0.0024 0.0011 0.0012

0.5580 0.2226 –0.1700

0.0177 0.0065 0.0054

0.0079 0.0152 0.0229 0.0681 0.0361 –0.0040 0.0496 0.0041 0.0079 –0.0502 –0.0170 0.0076

0.0243 0.0098 0.0061 0.0092 0.0053 0.0041 0.0078 0.0030 0.0021 0.0076 0.0034 0.0032

0.0070 0.0149 0.0243 0.0673 0.0353 –0.0026 0.0494 0.0042 0.0083 –0.0489 –0.0161 0.0075

0.0328 0.0205 0.0114 0.0367 0.0087 0.0076 0.0111 0.0045 0.0043 0.0194 0.0082 0.0065

Firm-level variables % of permit A in firm % of permit B in firm % of permit C in firm % of permit G in firm

–0.3316 –0.0241 –0.2364 –0.1097

0.0375 0.0622 0.0314 0.0348

–0.3307 –0.0234 –0.2360 –0.1088

0.0379 0.0602 0.0315 0.0349

% of skill 1 in firm % of skill 2 in firm % of skill 4 in firm

–0.0475 –0.0681 –0.0488

0.0348 0.0209 0.0254

–0.0460 –0.0670 –0.0487

0.0369 0.0222 0.0260

Individual variables Female Age Age2 (/1,000) Never married Femaleage Femaleage2 FemaleNever married Permit Permit Permit Permit

A B C G

Skill 1 (highly skilled) Skill 2 Skill 4 (unskilled) Permit Permit Permit Permit Permit Permit Permit Permit Permit Permit Permit Permit

Askill 1 Askill 2 Askill 4 Bskill 1 Bskill 2 Bskill 4 Cskill 1 Cskill 2 Cskill 4 Gskill 1 Gskill 2 Gskill 4

241


(Continued ).

Table A2. Variable

Model I

Model II

Coefficient

Standard error

Coefficient

Standard error

% of permit A skill 1 in firm % of permit A skill 2 in firm % of permit A skill 4 in firm % of permit B skill 1 in firm % of permit B skill 2 in firm % of permit B skill 4 in firm % of permit C skill 1 in firm % of permit C skill 2 in firm % of permit C skill 4 in firm % of permit G skill 1 in firm % of permit G skill 2 in firm % of permit G skill 4 in firm

0.5063 –0.1183 –0.0512 0.6636 0.7692 –0.4831 0.7431 0.0576 0.1129 0.4363 0.0961 –0.1175

0.2804 0.0845 0.0678 0.2987 0.1169 0.0965 0.1267 0.0665 0.0579 0.1848 0.0587 0.0552

0.5035 –0.1171 –0.0537 0.6642 0.7653 –0.4815 0.7398 0.0581 0.1122 0.4344 0.0968 –0.1192

0.2698 0.0869 0.0675 0.2666 0.1146 0.0949 0.1242 0.0661 0.0575 0.1792 0.0576 0.0548

% of women in firm Log firm size

–0.0959 0.0251

0.0091 0.0020

–0.0942 0.0250

0.0095 0.0020

Adjusted R squared

1st step: 0.750; 2nd step: 0.420

0.595

Notes: N ¼ 829,452. Model I is estimated in two steps. First, the log wage is regressed on individual variables and on fixed effects at the firm level. Second, fixed effects are regressed on firm-level variables. Model II is estimated in one step (pooled OLS). Sample weights are used in all estimations, and standard errors are adjusted for clustering within firms. Permit A: seasonal and short-term workers. Permit B: annual work permit. Permit C: settlement permit. Permit G: cross-border workers. Other firm-level variables used in the regressions: sectoral variables (i.e., residuals obtained from regressions of two-digit NOGA dummies on all firm-level variables).


242

(a) Model I: individual characteristics and fixed firm effects 0.8

0.6

Swiss


settlement permit 0.4

annual permit

0.2

0.0 skill 1

skill 2

skill 3

skill 4

-0.2

-0.4 (b) Model III: individual characteristics only 0.8

0.6 Swiss log wage differential

settlement permit 0.4

annual permit

0.2

0.0 skill 1

skill 2

skill 3

skill 4

-0.2

-0.4

Fig. A1. Wage Effects of Individual Skill by Nationality.

243


Foreign workers with annual permit 0.150 difference original/extended model


0.100

extended model 0.050 0.000 skill 1

skill 2

skill 3

skill 4

-0.050 -0.100 -0.150 Foreign workers with settlement permit


0.080 0.060

difference original/extended model

0.040

extended model

0.020 0.000 skill 1

skill 2

skill 3

skill 4

-0.020 -0.040 -0.060

Fig. A2. Wage Effects of Firm-Level Segregation – Introducing Job Characteristics. Note: This figure depicts the ‘‘marginal’’ effect on log wages of moving from the work environment of the average worker in the economy to the work environment of an average member of a specific skill–nationality (work permit) group. The bars show the effect of segregation in the original model (Model II), and the dark grey part depicts the effect in the extended model (Model II augmented by job characteristic variables, see text.

GENDER SEGREGATION IN THE WORKPLACE AND WAGE GAPS: EVIDENCE FROM URBAN MEXICO 1994–2004 Sebastiań Calońico and Hugo Nõpo ABSTRACT This paper analyzes the evolution of gender segregation in the workplace in Mexico between 1994 and 2004, using a matching comparisons technique to explore the role of individual and family characteristics in determining gender segregation and wage gaps. The results suggest that the complete elimination of vertical segregation would reduce the observed gender wage gaps by 5 percentage points, while the elimination of occupational segregation would have increased gender wage gaps by approximately 6 percentage points. The results also indicate that the role of occupational segregation in wage gaps has been increasing in magnitude during the period of analysis, while the role of vertical segregation on the determination of wage gaps has been decreasing.


245

246

SEBASTIAŃ CALOŃICO AND HUGO NÕPO

1. INTRODUCTION Occupational segregation by sex, the clustering of the labor force in male-dominated and female-dominated occupations, is particularly high in Latin America (Blau, Ferber, & Winkler, 2006). Similarly, vertical segregation, the fact that managers tend to be males and subordinates females, is commonly accepted as the norm in the region’s labor markets. To an important extent, there are reasons to believe that these phenomena correspond to the existence of traditional gender roles and not necessarily to an efficient allocation of the human resources. A reduction in genderbased segregation in the workplace therefore represents an area in which policy interventions can have an interesting role in improving the overall efficiency of the labor markets. To what extent is this the case? Is it most effective to design interventions aimed at the reduction of occupational segregation rather than vertical segregation, or the other way around? This paper attempts to contribute to the discussion of those issues. The particularly low female labor force participation rates seen in Mexico make this an interesting country to analyze from a gender perspective. Mexico is the country with the lowest female participation rate in the Latin American region, at 36 percent, only after Chile and Paraguay, at 39 percent. At the other extreme, Uruguay (46 percent), Colombia, and Peru (both 43 percent) exhibit the highest female participation rates.1 Although the literature on gender and labor markets in Mexico has included many interesting pieces, most of them have focused on wage differentials. We outline here some of those that have attempted a segregation perspective. Brown, Pagan, and Rodriguez-Oreggia (1999) study the effect of occupational attainment on the increase in gender wage differentials observed between 1987 and 1993. Using data from the National Survey of Urban Employment (ENEU for its acronym in Spanish [Encuesta Nacional de Empleo Urbano]), they found that the decline in gender differences in occupational attainment somewhat alleviated the increase in male–female earnings differential. They also found important roles for labor supply decisions (hours of work per week) and changes in the regional structure of wages. This somehow contrasts with the results of Parker (1999), who examines the gender wage gap in rural areas of Mexico between 1986 and 1992, considering different skill levels within groups of occupations. She finds that wage differentials among wage earners are low, and they remained roughly constant throughout the period, although they vary greatly among occupations. She finds the largest wage gaps in managerial positions (in both

Gender Segregation in the Workplace and Wage Gaps

247

the private and public sectors) and the smallest among public service workers and administrative positions. Rendoń (2003) analyzes the gender differences in employment, segregation, and wages. She documents that, in spite of the large increase in the female labor force participation in recent decades, there is still a large concentration of women in certain activities. As a matter of fact, she documents an increase in segregation by production sectors from 1990 to 2000. However, she suggests that there are reasons to believe that such segregation should decrease in the future, as it has been seen that first women enter activities more populated by other women but then start to enter other activities with more gender neutrality. She also provides estimators for the high degree of vertical segregation, where men hold higher-ranking positions. When analyzing the evolution of the gender wage gap, she argues that the observed reduction in that gap can be explained by an increase in women’s working hours. In a related study, Rendon & Maldonado (2004) study the relationship among domestic work, occupational segregation, and the gender wage gap in Mexico. Their motivation is the large increase observed in female labor force participation, in part due to cultural factors, but also because of changes in the country’s occupational and productive structure (there was an increase in the relative importance of occupations such as professionals, office workers, and salespersons). However, this increase in participation did not imply that conditions faced by men and women became more equal. The occupational segregation and wage gaps are still notable, partly due to the number of hours worked, and they vary substantially among different sectors and occupations. Colmenares (2006) also analyzes occupational segregation by sex and its relation with the wage difference in the industry sector. By comparing different Mexican cities, she finds an interesting variability across regions in gender occupational segregation. This paper complements the literature previously outlined by exploring the linkages between gender differences in human capital observable characteristics, (occupational and vertical) segregation and wages. The rest of the paper proceeds as follows. In Section 2 we present the patterns of occupational and vertical segregation in urban Mexico in the period 1994–2004. In Section 3 we present a simple model exploring the relationship between (vertical or occupational) segregation and wage gaps. In Section 4 we introduce the matching comparisons technique to explore the impact of gender differences in human capital characteristics on segregation and wages, respectively. In Section 5 we present an empirical exploration of the linkages between segregation and wages, and in Section 6 we conclude.

248


2. ASSESSING OCCUPATIONAL AND VERTICAL SEGREGATION IN MEXICO 2.1. The Data The data for this study are drawn from the ENEU, Mexico’s national urban employment survey. The data are collected quarterly, using dwellings or housing structures as sampling units. Each individual aged 12 and above is asked about occupation, labor hours, labor earnings, and employment conditions. The survey is designed as a rotating panel, with households interviewed for five consecutive quarters, but we are not exploiting the panel structure for this work. The analysis of this paper covers the period from the third quarter of 1994 to the fourth quarter of 2004. At the beginning of the period under consideration, the cities covered in our sample represented about 40 percent of the population of Mexican working individuals, and during the 10-year span used in this study, that coverage has increased to 60 percent (48 cities). It is also important to note that by the end of 2001 the Encuesta Nacional de Empleo (ENE) replaced the ENEU, extending the urban coverage of the latter to national coverage. For the later years of this study we restrict our attention to the urban subsample of the ENE, which is comparable to the ENEU sample. Summary statistics of the sample under analysis are presented in Tables 1a and 1b. The working population has aged slightly during the 10-year span considered. On average, the working individuals were 34 years old in 1994 and 36 years old in 2004. Also, average schooling has increased from 9 years in 1994 to 10 years in 2004. The number of children per household has remained roughly constant at 2.6. Another trend to highlight relates to marital status. In 1994, 34 percent of the sampled population was single, a figure that declined to 30 percent by 2004. In a parallel trend, the percentage of individuals in formal and informal unions increased from 58 percent to 61 percent during the same time span. To complete the figure, the prevalence of divorced individuals increased from 7 percent to 9 percent. These trends are not related to the expansion of the sampling framework of the survey, as restricting the attention to the cities that were part of the sampling framework of 1994 delivers the same tendencies. 2.1. Measuring Occupational Segregation In this paper we measure occupational and vertical segregation by gender using the Duncan index (Duncan & Duncan, 1955). Introducing some

249


Table 1.

Descriptive Statistics. (a)

Period

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Gender (Males)

Age

Years of Schooling

Number of Kids

Total

Total

Male

Female

Total

Male

Female

Total

Male

Female

0.65 0.65 0.64 0.64 0.64 0.65 0.64 0.64 0.64 0.63 0.62

34.03 34.39 34.51 34.54 34.49 34.73 35.03 35.42 35.92 36.20 36.22

34.75 35.03 35.14 35.12 35.00 35.21 35.50 35.83 36.31 36.50 36.44

32.65 33.22 33.38 33.50 33.56 33.85 34.18 34.70 35.24 35.67 35.85

9.12 9.24 9.32 9.42 9.40 9.45 9.63 9.72 9.83 9.89 9.92

9.00 9.15 9.23 9.33 9.29 9.31 9.48 9.55 9.66 9.73 9.76

9.34 9.41 9.48 9.59 9.61 9.70 9.89 10.00 10.13 10.18 10.16

2.45 2.45 2.45 2.43 2.40 2.41 2.41 1.90 1.91 1.94 1.96

2.47 2.46 2.46 2.44 2.42 2.43 2.42 – – – –

2.43 2.42 2.42 2.40 2.38 2.39 2.38 1.90 1.91 1.94 1.96

(b) Period

1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004

Single

Formal or Informal Union

Divorced/Separated/ Widowed

Total

Male

Female

Total

Male

Female

Total

Male

Female

0.35 0.33 0.33 0.33 0.32 0.32 0.32 0.31 0.30 0.31 0.31

0.29 0.28 0.28 0.28 0.28 0.28 0.27 0.27 0.26 0.26 0.27

0.46 0.43 0.43 0.42 0.40 0.40 0.40 0.39 0.38 0.38 0.37

0.59 0.60 0.59 0.59 0.61 0.61 0.61 0.62 0.62 0.62 0.61

0.68 0.69 0.69 0.69 0.69 0.69 0.69 0.70 0.70 0.70 0.70

0.40 0.42 0.42 0.43 0.45 0.45 0.45 0.46 0.46 0.47 0.48

0.07 0.07 0.08 0.07 0.07 0.07 0.08 0.08 0.08 0.08 0.08

0.03 0.03 0.03 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04

0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.16 0.16 0.16 0.16

notation will make clear how the indicator works. Let us begin by focusing on segregation by occupation. Let K denote the number of occupations in the economy, and fi the fraction of females working in occupation i ( f1 þ f2 þ y þ fK ¼ 1). Analogously, let mi be the fraction of males working in occupation i (m1 þ m2 þ y þ mK ¼ 1). Then the Duncan index, D, is computed as one-half of the sum of the absolute values of the differences between Pmales’ and females’ fractions in each occupation (D ¼ 1=2 K i¼1 j f i mi j). The index ranges from 0 to 1, a higher index


250

Fig. 1.

Occupational Segregation by Region.

representing higher segregation. It can be interpreted as the percentage of the working male (female) population that would need to switch jobs, from those that are male-dominated (female-dominated) to other femaledominated (male-dominated) jobs in order to achieve a labor force with no segregation. To put magnitudes for the Duncan index in context, in Fig. 1 we present measures of the index for different regions of the world (extracted from Deutsch, Morrison, Piras, & Nõpo, 2002). In this paper we computed the Duncan index using disaggregated information on seven occupations at the one-digit level (professionals and technicians, managers, administrative personnel, salespersons, workers in the service sector, workers in agricultural activities, and workers in industrial activities).2 The computation of the Duncan index of vertical segregation follows the same approach but uses hierarchical categories instead of occupations; the ENEU survey provided five categories (managers, independent workers, piece-rate or commission workers, fixedsalary workers and members of a cooperative). Table 2 reports average measures of occupational and vertical segregation, for different segments of the market, for the whole period under analysis. Occupational segregation by gender in Mexico, as in most of the labor markets, is less pronounced among those with more years of schooling. But interestingly, vertical segregation is notoriously more pronounced among those with more years of schooling. While the younger workers display

251


Table 2.

Mexico 1994–2004. Average Duncan Indexes of Segregation for Different Segments of the Labor Markets. Segregation Occupational

Vertical

Schooling 0 years 1–6 years 7–12 years 13 years and more

0.405 0.393 0.330 0.238

0.107 0.098 0.097 0.143

Age group 15–24 25–49 50–64

0.305 0.338 0.335

0.095 0.098 0.094

Marital status Single (never married) Married Separated

0.295 0.343 0.304

0.105 0.092 0.116

Firm size [1–5] [5–50] (50 þ )

0.493 0.297 0.279

0.192 0.086 0.009

Management Nonpublic Public

0.336 0.329

0.101 0.003

Firm activity Agricultural Extractive and electricity Manufacture Construction Commerce Communications and transports Services Public administration and defense

0.311 0.572 0.049 0.808 0.184 0.703 0.310 0.402

0.251 0.025 0.071 0.289 0.107 0.502 0.194 0.003

All In 1994: III In 2004: IV

0.326 0.349 0.325

0.095 0.107 0.080

lower levels of occupational segregation, vertical segregation appears to remain constant over the life cycle. Occupational segregation is lower among single individuals (including both never-married and separated persons) than among married individuals, and the opposite is true for

252


vertical segregation. Both types of segregation are significantly more pronounced in smaller firms. Although the Mexican public sector exhibits almost no vertical segregation, it displays levels of occupational segregation similar to those in the private sector. The ENEU records eight firm activities (agricultural, extractive and electricity, manufacture, construction, commerce, communications and transport, services, and public administration and defense), and the overall rankings of sectors according to occupational and vertical segregation show some differences. The highest occupational segregation by gender is found in construction firms, followed by communications and transport; the lowest segregation is found in manufacture. On the other hand, the highest vertical segregation by gender is among those who work in communications and transport; the lowest is in public administration and defense. Overall, occupational segregation is substantially greater than vertical segregation, and both have been decreasing, albeit slightly, in the Mexican labor markets. During the 10-year span that we analyze, occupational segregation dropped 2 percentage points (from 0.35 to 0.33), while vertical segregation has dropped 3 points (from 0.11 to 0.08). All in all, the patterns observed for these two measures of segregation within different segments of the labor markets show interesting differences. Nonetheless, at this point the reader should be cautioned that a direct comparison of these two types of segregations is far from straightforward. Not only are the two measures computed on the basis of a different number of categories (seven vs. five), but these categories are also not meant to be comparable. What we attempt next in the paper is a parallel exploration of both occupational and vertical segregation, especially with regard to their link to wage gaps.

3. EXPLORING THE LINK BETWEEN SEGREGATION AND WAGE GAPS Next, we provide a simple but useful theoretical framework to understand the linkages between segregation and wage gaps (and below, in Section 5, we will estimate the implications of this model). Assume there are two occupations in the economy, one of them being male dominated and the other female dominated. The fraction of females working in the economy is denoted by a. The fraction of males in the male-dominated occupation is m, and the fraction of females in the female-dominated occupation is f (the assumptions about the occupations converts into mW1 f ). Wages in the

253


male-dominated occupation are denoted by WMM for males and WFM for females. Analogously, in the female-dominated occupation, wages are denoted by WMF and WFF for males and females, respectively. The table below summarizes the notation. Males ( ¼ 1 a) Occupation 1 (Male dominated) Occupation 2 (Female dominated)

Wage ¼ WMM Males ¼ m Wage ¼ WMF Males ¼ 1 m Average male wages ¼ m W MM þ ð1 mÞ W MF

Females ( ¼ a) Wage ¼ WFM Females ¼ 1 f Wage ¼ WFF Females ¼ f Average female wages ¼ ð1 f Þ W FM þ f W FF

The wage gap is the difference between the average wages for males and females: G ¼ m W MM þ ð1 mÞ W MF ð1 f Þ W FM f W FF

(1)

The Duncan index is: 1 1 D ¼ ½j1 f mj þ j f 1 þ mj ¼ ½jm 2 2 þ f 1j þ jm þ f 1j ¼ m þ f 1

ð2Þ

Note that both expressions are independent of the fraction of females in the economy (a). From (1), it can be obtained that: @G @G ¼ W MM W MF ¼ W FM W FF @m @f The element of interest is @G=@D, the rate at which the wage gap (G) varies for infinitesimal changes in occupational segregation (D): @G @G @m @G @f ¼ þ ¼ ðW MM W MF Þ þ ðW FM W FF Þ @D @m @D @f @D

(3)

The result stated in (3) reveals that the way in which the wage gap would change for a change in occupational segregation depends on two gaps: one for males (average wages for males in male-dominated occupations minus average wages for males in female-dominated occupations) and another for females (average wages for females in male-dominated occupations minus average wages for females in female-dominated occupations). There are no theoretical

254


reasons to expect these two gaps to be positive or negative. Then, within this context, the question about the sign of the relationship between segregation and wage gaps is an empirical one. Next, in Section 5 we turn to this question. Before that we turn to explore the role of observable individuals’ human capital characteristics on the determination of segregation.

4. HOW CAN WE EXPLAIN GENDER SEGREGATION IN MEXICO? THE ROLE OF INDIVIDUAL CHARACTERISTICS To some extent, the existence of gender differences in occupations and hierarchies could be explained by the gender differences in individual characteristics that are important for the labor markets. It could be the case that gender disparities in education, which are still prevalent in Mexican labor markets, somehow determine the occupational and vertical sorting by gender, as previously shown. In this section we address questions under the heading of ‘‘How would (occupational or vertical) gender segregation in Mexico change if individual characteristics of females and males became identical?’’ Additionally, to explore the linkages between the two types of segregation, we provide answers to the counterfactual question ‘‘How would occupational (vertical) segregation change if vertical (occupational) segregation were eliminated?’’3 Our approach to answering these questions consists of generating counterfactual situations that can be estimated with the available data. Such situations can be generated with a range of different techniques. In this case we follow a nonparametric matching-on-characteristics approach that consists of creating synthetic samples of males of females, extracted from the original ENEU samples such that the resulting distribution of observable characteristics of females is exactly the same as the distribution for males. These synthetic samples are created through resampling the individual observations of males with replacement and of females without replacement and matching them. In such resampling, females and males are matched such that their observable characteristics (age, schooling, and marital status) are exactly the same. As a result, the created dataset of females and males reflects a synthetic labor market in which males and females do not differ in the characteristics that were used for the matching. Then, the segregation observed in the matched data can be considered as the segregation that remains after controlling for individual observable characteristics.


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This way of generating counterfactual situations has strong linkages with the one implicit in the Blinder–Oaxaca wage gap decomposition (Blinder, 1973; Oaxaca, 1973). In that regard, it is important to note that with this matching algorithm, it is not only the case that the average observable characteristics of males and females get equalized, but also that the distributions of those characteristics are the same for females and males. As a result, what we employ here is an extension of the Blinder–Oaxaca decomposition in two dimensions. On the one hand, as already stated, the counterfactual situation is generated on the distribution of individuals’ characteristics (hence the counterfactual on the average characteristics is granted). On the other hand, as this is a nonlinear and nonparametric matching, it can be applied beyond wage gap decompositions – in this case, especially for the analysis of nonlinear indicators such as the Duncan index of segregation.4 The first three counterfactual exercises correspond to situations in which (i) there are no gender differences in observable characteristics, namely age, schooling, and marital status (Counterfactual Situation 1), (ii) there are no gender differences in hierarchies (Counterfactual Situation 2a), and (iii) the combination of the previous two; that is, there are no gender differences in age, schooling, marital status, and hierarchies (Counterfactual Situation 3a). For the three counterfactual situations, we explore the evolution of the occupational segregation, comparing the original Duncan index with the one that would prevail in each hypothetical counterfactual situation (Fig. 2). The results suggest that a complete elimination of the gender differences in age, schooling, and marital status in the labor markets would have an impact of between 2 and 3 percentage points on the occupational segregation for the period 1994–2004. The elimination of gender differences in hierarchies would have a smaller impact than the other characteristics, about 1 percentage point in the period 1994–2004. The combination of both counterfactuals yields a situation for which the Duncan index would have been around 4 percentage points less during 1994–2004. The differences between the counterfactual Duncan index and the actual index are roughly constant over the period. The next three counterfactual exercises are analogous to those from the previous set. The only differences are that now the Counterfactual Situation 2b (Fig. 3) corresponds to one in which the gender differences in occupations are eliminated. Then, correspondingly, the Counterfactual Situation 3b combines Counterfactuals 1 and 2b. For these three counterfactuals we explore the evolution of the vertical segregation, comparing the original Duncan index with the corresponding counterfactuals (Fig. 3).


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Fig. 2.

Counterfactuals for the Analysis of Occupational Segregation.


Fig. 3.

Counterfactuals for the Analysis of Vertical Segregation.

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Overall, the results suggest impacts of the observable characteristics on the reduction of vertical segregation that are greater than those on occupational segregation (especially when taking into account that the original levels of vertical segregation are smaller than those of occupational segregation). There are also interesting differences across time, as the role of occupations decreases in importance during the later portion of the period under analysis. The hypothetical situation in which working males and females have the same age, schooling, and education leads us to a vertical segregation that is smaller than the one actually observed by 1 percentage point between 1994 and 2004. The elimination of occupational segregation in Mexico would yield a situation in which vertical segregation is reduced by as much as 6 to 7 percentage points in the mid-1990s and around 3 percent for the year 2004, with a tendency toward the reduction in such impact. The combined effect of eliminating occupational segregation and gender differences in individual characteristics (age, schooling, and marital status) would have an impact of 7–8 percentage points for the mid-1990s and 4 percentage points around 2004. So far, the results show that individuals’ characteristics in the Mexican labor markets play a (somewhat small) role in the determination of gender segregation. Also, the two types of segregation explored here are linked in the sense that a reduction in one would lead to a reduction in the other. Next, we turn to exploring the linkages between these two types of segregation and wage gaps.

5. WHAT ROLE DOES SEGREGATION PLAY IN THE DETERMINATION OF GENDER WAGE GAPS? In this section we explore the role of individual characteristics and (occupational and vertical) segregation on the determination of gender wage gaps in Mexico. We start with the same methodological approach as in the previous section, using matching comparisons to craft counterfactual situations. Then, we move to a theoretical exploration of the linkages between segregation and wage gaps, estimating the predicted linkages with the data. Regarding the counterfactual analysis, the inquiries are: ‘‘By how much would the gender wage gap change if the (occupational or vertical) segregation were reduced to zero?’’ or ‘‘By how much would the gender wage gap change if the gender differences in observable characteristics were


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reduced to zero?’’ For that purpose, we match females and males based on the same combinations of observable characteristics that we used in the previous section. In particular, we report the counterfactual situations that were previously labeled as (1), (2a), and (2b) in Fig. 4. That is, we match on age, schooling, and marital status in the first counterfactual, we match on hierarchies in the second, and we match on occupation in the third. The gender wage gap shows a decreasing trend during most of the period under analysis, only interrupted by two years of increase between 1999 and 2001. By the mid-1990s, on average, males earned around 18 percent more than females per hour worked, and this gap declined to almost 12 percent by 2004. The role of age, schooling, and marital status in explaining the gender differences in wages has changed as well. During the mid-1990s these characteristics explained almost half of the wage gap, but after 2002 these characteristics seem to play almost no role in determining gender differences in pay. A hypothetical world in which there is no vertical segregation, but everything else remains equal, would show gender wage gaps similar to those in a hypothetical world in which there are no gender differences in age, schooling, and marital status inside the labor markets during the late 1990s. Later, the hypothetical gender wage gap without vertical segregation becomes somewhat smaller than the hypothetical gender wage without age, gender, and marital status differences. For the period between 2000 and 2004, the average gender wage gap has been around 14 percent in Mexico, but in the hypothetical world with no vertical segregation, that gap would have reached only 10. The hypothetical world for which there is no occupational segregation shows results that are somewhat surprising. The wage gap that would be observed in the hypothetical world exceeds the one that is actually observed in the Mexican labor markets. Even more, the difference between the hypothetical (counterfactual) and the actual gap increases during the 1990s and remains nearly constant after that. Why is it that a reduction in vertical segregation would lead to a reduction in wage gaps but a reduction in occupational segregation would not? What forces have been behind this development, and how have they evolved during the period of analysis? Next, we turn to explore these questions under the light of the theoretical model presented in Section 2, estimating these implications with the available data. Empirically, the definition of male and female dominance was implemented on the basis of the gender composition in each occupation


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Fig. 4.

Counterfactuals for the Analysis of Wage Gaps.


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over the period under analysis. Three out of seven occupations at the one-digit level were considered male dominated (managers, workers in agricultural activities, and workers in industrial activities), and the other four were labeled as female dominated (professionals and technicians, administrative personnel, salespersons, and workers in the service sector). The upper panel of Fig. 5 shows the estimation of @G=@D and its constitutive components ðW MM W MF Þ and ðW FM W FF Þ.5 Both components, and hence @G=@D, are negative for the whole period under analysis.

Fig. 5.

Estimation of @G=@D for Occupations and Hierarchical Categories.

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That is, average male and female wages in female-dominated occupations are higher than those in male-dominated occupations. Hence, it is not surprising to observe that a reduction in gender occupational segregation would lead to an increase in gender wage gaps in Mexico. As noted above, the difference between the actual wage gap and the hypothetical wage gap without occupational segregation increased during the period of analysis. This result is observed in the estimation of @G=@D, which has been more negative in the later years of our analysis. A more-than-proportional decrease in wages in male-dominated occupations guides such increasing difference.6 The analogous exercise performed with hierarchies instead of occupations follows. For that purpose we labeled three of the five vertical categories as male dominated (managers, piece-rate or by commission workers, and members of a cooperative) and two of them as female dominated (independent workers and fixed-salaried workers). The results, as opposed to those found with occupations, indicate that @G=@D and its two components have positive signs. Reductions in vertical segregation are expected to be linked to reductions in the wage gap, in line with the results reported earlier in this section.

6. CONCLUSIONS In this paper we have shown that vertical segregation by gender (the fact that managers tend to be males and subordinates females) has been substantially smaller than occupational segregation (the fact that males and females tend to cluster in different occupations) during the last two decades in Mexico. However, these two types of segregation have had highly dissimilar impacts on wages. While a complete reduction in the vertical segregation would reduce the observed gender wage gaps by approximately 5 percentage points, a complete reduction in occupational segregation would actually push the gender wage gap toward an increase of around 6 percentage points. Also, a reduction in gender differences in personal or human capital characteristics (age, schooling, and marital status) would have a greater impact on the reduction in observed occupational segregation as opposed to vertical segregation. To complete the picture, a reduction in occupational segregation would have a significant impact on the reduction in vertical segregation, although the reverse may not necessarily be true. These results call for an interesting set of policy recommendations. The first of them, on a very general level, concerns the importance of fighting against vertical rather than occupational segregation. Providing more


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opportunities for women to climb into managerial positions seems to be more promising than letting them work on traditionally male tasks. Linked to this issue of opportunities, another set of policy interventions that may help to reduce both types of segregation involves attitudes, stereotypes, and social norms. Change in these areas is neither an easy nor an immediate task. The necessary changes may take more than one generation to materialize, and they require active participation by different type of agents: current and future employers and job seekers, both females and males. From this perspective, there is a great deal of scope for interventions in the schools and, also, within households (changing the within-households incentive structures to work, stay at home, and perform certain activities or assume certain roles). The results also suggest that a gender equalization of human capital characteristics would help toward the reduction of not only gender wage gaps but also both vertical and occupational segregation. Along those lines, it is important to remind that Mexico is one of the few Latin American countries that still needs to close gender disparities in schooling for its population (see Duryea, Galiani, Nõpo, & Piras, 2007). Also, training programs, both at school and at the job, promise to be fruitful, as well as policies intended to facilitate the functioning of labor markets, reducing frictions and search costs for women. It is important to note at this point that the results discussed here do not take into account the fact that females and males are allocated in the labor force in a nonrandom way. In that sense, all the estimators in this paper assume that there was no change in labor force participation. It is likely that once occupational and vertical segregation by gender become less important and gender wage differences smaller, there will be additional incentives for females to enter the labor force. The general equilibrium impacts of such changes are, however, beyond the scope of this paper.

NOTES 1. These statistics were extracted from the corresponding national household surveys circa 2002. 2. Computations at the two-digit level, comprising 18 occupations, deliver results that are qualitatively similar (although somewhat higher). These computations are presented in the first appendix. See Anker (1998) for a discussion of the influence of the number of categories on the computation of the Duncan index. 3. However, it should be noted that a complete elimination of segregation is an extreme counterfactual situation. As matter of fact, even a random assignment of

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workers to occupations may lead to nonzero segregation (Carrington & Troske, 1997, 1998). 4. See Nõpo (2008) for additional details on the matching algorithm and a proof of consistency of the estimators obtained from this method. 5. These elements are computed as percentages of average female wages for each year. 6. The estimations related to this last claim are available on request.

REFERENCES Anker, R. (1998). Gender and jobs: Sex segregation of occupations in the world. Geneva: International Labor Organization. Blau, F., Ferber, M., & Winkler, A. (2006). The economics of women, men, and work (5th ed.). Upper Saddle River, NJ: Prentice-Hall. Blinder, A. (1973). Wage discrimination: Reduced form and structural estimates. Journal of Human Resources, 7(4), 436–455. Brown, C., Pagan, J., & Rodriguez-Oreggia, E. (1999). Occupational attainment and gender earnings differentials in Mexico. Industrial and Labor Relations Review, 53(1), 123–135. Carrington, W., & Troske, K. (1997). On measuring segregation in samples with small units. Journal of Business and Economic Statistics, 15, 402–409. Carrington, W., & Troske, K. (1998). Sex segregation in U.S. manufacturing. Industrial and Labor Relations Review, 51(3), 445–464. Colmenares, G. (2006). Segregacioń en el empleo por sexo. Salario y ocupacioń en los modelos de industrializacioń de las regiones centro-occidente y fronteriza. Frontera Norte, 18(35), 87–110. Deutsch, R., Morrison, A., Piras, C., & Nõpo, H. (2002). Working within confines: Occupational segregation by gender in three Latin American countries. Technical Papers Series, Sustainable Development Department, Inter-American Development Bank. Duncan, O. D., & Duncan, B. (1955). A methodological analysis of segregation indexes. American Sociological Review, 20, 210–217. Duryea, S., Galiani, S., Nõpo, H., & Piras, C. (2007). The educational gender gap in Latin America and the Caribbean. RES Working Papers 4510, Inter-American Development Bank, Research Department. Nõpo, H. (2008). Matching as a tool to decompose wage gaps. Review of Economics and Statistics, 90(2), 290–299. Oaxaca, R. (1973). Male-female wage differentials in urban labor markets. International Economic Review, 14(3), 693–709. Parker, S. (1999). Niveles salariales de hombres y mujeres: Diferencias por ocupacioń en las a´reas urbanas de Me´xico. In: Beatriz Figueroa Campos (coordinator). Me´xico diverso y desigual: Enfoques sociodemogra´ficos. Serie Investigacioń Demogra´fica en Me´xico 4. Mexico City, Mexico: Colegio de Me´xico/Sociedad Mexicana de Demografı´ a Rendoń, T. (2003). Empleo, segregacioń y salarios por geńero. In: E. de la Garza, & C. Salas (Eds.), La situacioń del trabajo en Me´xico. Mexico City, Mexico: Plaza y Valde´s-IET. Rendon, T., & Maldonado, V. (2004). Vıńculos entre trabajo dome´stico, segregacioń ocupacional y diferencias de ingreso por sexo, en el Me´xico actual. Mexico City, Mexico: Instituto de Estudios del Trabajo.


APPENDIX A. DATA USING TWO-DIGIT LEVELS OCCUPATIONS

Fig. A1.

Counterfactuals for the Analysis of Occupational Segregation.

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Fig. A2.

Counterfactuals for the Analysis of Vertical Segregation.


Fig. A3.

Counterfactuals for the Analysis of Wage Gaps.

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Fig. A4. Estimation of @G=@D for Occupations and Hierarchical Categories.

APPENDIX B. NONPARAMETRIC MATCHING-ON-CHARACTERISTICS TECHNIQUE In this paper, we follow the nonparametric matching-on-characteristics technique from Nõpo (2008) in order to obtain our main estimates. This method emphasizes gender differences in the support of the distributions of observable characteristics and provides insights into the distribution of unexplained gender differences. The traditional Blinder–Oaxaca (BO) approach based on linear regressions suffers from a potential problem of misspecification due to differences in the supports of the empirical distributions of individual characteristics for females and males (gender differences in the supports). This is because there are combinations of individual characteristics for which it is possible to find males in the labor force, but not females – for example, males who are in their early 30s, married, and hold at least a college degree. There are also combinations of characteristics for which it is possible to find females, but not males – for example, single females that are migrants, in their late 40s, and have less than an elementary school education. With such combinations of characteristics, one cannot compare outcomes across genders. By not considering this restriction, the BO decomposition is implicitly based on an


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‘‘out-of-support assumption’’: it becomes necessary to assume that the linear estimators are also valid out of the supports of individual characteristics for which they were estimated. Nõpo (2008) then proposes a nonparametric alternative to the BO decomposition that divides the gender gap (of any outcome of interest, such as earnings or occupations) into four additive elements: D ¼ ðD_x þ D_f þ D_mÞ þ D_0 where D_x: part accounted by differences between the distributions of males’ and females’ individual characteristics over their common support. D_f: due to the existence of some combinations of females’ characteristics that are not comparable to those of males. D_m: due to the existence of some combinations of males’ characteristics that are not comparable to those of females. D_0: part that cannot be explained by differences in observable individual characteristics. The first three components can be attributed to the existence of differences in individual characteristics that the labor market rewards, while the last one is due to the existence of a combination of both unobservable (by the econometrician) differences in characteristics that the labor market rewards and discrimination. Along with the misspecification problem associated with gender differences in the supports, the BO decomposition is only informative about the average unexplained difference in wages. It is therefore not capable of addressing the distribution of these unexplained differences. The matching technique enables us to highlight the problem of gender differences in the supports and also to provide information about the distribution of the unexplained pay differences. It estimates the four components by resampling all females without replacement and matches each observation to one synthetic male, obtained by averaging the characteristics of all males with exactly the same characteristics. The matching algorithm in its basic form can be summarized as follows: Step 1: Select one female from the sample (without replacement). Step 2: Select all the males that have the same characteristics as the female previously selected.

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Step 3: With all the individuals selected in Step 2, construct a synthetic individual whose characteristics are equal to the average of all of them and ‘‘match’’ him to the original female. Step 4: Put the observations of both individuals (the synthetic male and the female) in their respective new samples of matched individuals. Repeat Steps 1 through 4 until it exhausts the original female sample. As a result of the application of this one-to-many-with-zero-discrepancies matching, the dataset is partitioned. The new dataset contains observations of ‘‘matched females,’’ ‘‘matched males,’’ ‘‘unmatched females,’’ and ‘‘unmatched males,’’ the case being that the sets of matched males and females have the same empirical distributions of probabilities for the selected characteristics. The purpose of resampling without replacement from the sample of females and with replacement from the sample of males is to preserve the empirical distribution of characteristics for females (the case being that the support for that distribution is finite). This allows us to generate the appropriate counterfactual and interpret the four components as we have done in the paper. This generation of a counterfactual can also be done to explore outcomes other than wages. As it is the case in this paper, with segregation indices.

Occupational and Residential Segregation, Volume 17

Grantville Gazette-Volume 17

Current Ornithology Volume 17

Equator & Segregation

Advances in Pharmacology and Chemotherapy Volume 17

Provability, Computability and Reflection, Volume 17

Environmental and Occupational Medicine

Advances in Pharmacology and Chemotherapy Volume 17

Reliability and validity assessment, Volume 17

Nanoparticles and Occupational Health

Advances in Geophysics, Volume 17

Advances in Immunology Volume 17

Poetry for Students Volume 17

Advances in Genetics Volume 17

Plant Breeding Reviews, Volume 17

Advances in Computers, Volume 17

Occupational Hazards

Nanoparticles and Occupational Health

Occupational Therapy and Stroke

Advances in Taxation, Volume 17

Topics in Stereochemistry, Volume 17

Advances in Geophysics, Volume 17

Business Plans Handbook, Volume 17

Advances in Agronomy, Volume 17

Occupational and Residential Exposure Assessment for Pesticides (Wiley Series in Agrochemicals & Plant Protection)

Electrical Wiring Residential

Enhancing Occupational Safety and Health

Palestinians in Israel: Segregation, Discrimination and Democracy

Occupational Asthma

Proceedings of the Aristotelian Society, Volume 17, 1916-17

Occupational and Residential Segregation, Volume 17

Grantville Gazette-Volume 17

Current Ornithology Volume 17

Equator & Segregation

Advances in Pharmacology and Chemotherapy Volume 17

Provability, Computability and Reflection, Volume 17

Environmental and Occupational Medicine

Advances in Pharmacology and Chemotherapy Volume 17

Reliability and validity assessment, Volume 17

Nanoparticles and Occupational Health

Advances in Geophysics, Volume 17

Advances in Immunology Volume 17

Poetry for Students Volume 17

Advances in Genetics Volume 17

Plant Breeding Reviews, Volume 17

Advances in Computers, Volume 17

Occupational Hazards

Nanoparticles and Occupational Health

Occupational Therapy and Stroke

Advances in Taxation, Volume 17

Topics in Stereochemistry, Volume 17

Advances in Geophysics, Volume 17

Business Plans Handbook, Volume 17

Advances in Agronomy, Volume 17

Occupational and Residential Exposure Assessment for Pesticides (Wiley Series in Agrochemicals & Plant Protection)

Electrical Wiring Residential

Enhancing Occupational Safety and Health

Palestinians in Israel: Segregation, Discrimination and Democracy

Occupational Asthma

Proceedings of the Aristotelian Society, Volume 17, 1916-17

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